Low Temperature Physics I / Kältephysik I [1st ed.] 978-3-662-38851-8;978-3-662-39773-2

Table of contents :
Front Matter ....Pages III-XIII
The Production of Low Temperatures Down to Hydrogen Temperature (J. G. Daunt)....Pages 1-111
Helium Liquefiers and Carriers (S. C. Collins)....Pages 112-136
Electrical Conductivity of Metals and Alloys at Low Temperatures (D. K. C. MacDonald)....Pages 137-197
Thermal Conductivity of Solids at Low Temperatures (P. G. Klemens)....Pages 198-281
Low Temperature Heat Capacity of Solids (P. H. Keesom, N. Pearlman)....Pages 282-337
Low Temperature Magnetism (J. van den Handel)....Pages 339-375
Adiabatic Demagnetization (D. de Klerk)....Pages 376-547
Superconductivity. Experimental Part (B. Serin)....Pages 548-611
Theory of Superconductivity (J. Bardeen)....Pages 612-707
Liquid Helium (K. Mendelssohn)....Pages 708-799
Back Matter ....Pages 800-827

Citation preview

ENCYCLOPEDIA OF PHYSICS EDITED BY

S. FLOGGE

VOLUME XIV

LOW TEMPERATURE PHYSICS I WITH 215 FIGURES

Springer-Verlag Berlin Heidelberg GmbH 1956

HANDBUCH DER PHYSIK HERAUSGEGEBEN VON

S. FLOGGE

BAND XIV

K.ALTEPHYSIK I MIT 215 FJGUREN

Springer-Verlag Berlin Heidelberg GmbH 1956

ALLE RECHTE, INSBESONDERE DAS DER ÜBERSETZUNG IN FREMDE SPRACHEN, VORBEHALTEN OHNE AUSDRÜCKLICHE GENEHMIGUNG DES VERLAGES IST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS AUF PHOTOMECHANISCHEMWEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFÄLTIGEN ISBN 978-3-662-38851-8 ISBN 978-3-662-39773-2 (eBook) DOI 10.1007/978-3-662-39773-2

© BY SPRINGER-VERLAG BERLINHEIDELBERG 1956 ORIGINALLY PUBLISHED BY SPRINGER-VERLAG OHG. BERLIN . GÖTTINGEN . HEIDELBERG IN 1956

Softcover reprint of the hardcover 1st edition 1956

Contents. The Production of Low Temperatures Down to Hydrogen Temperature. By JoHN GILBERT DAmn, Professor of Physics, Ohio State University, Columbus, Ohio (USA). (With 101 Figures)

Page

Preface . . . . . . . . . A. B. C. D. E. F. G. H.

Refrigeration by gas (air) machines Compressed vapor refrigerating machines. Cooling of gases by JOULE-THOMSON effect Theory of gas liquefaction using JOULE-THOMSON expansion Practical aspects of gas liquefaction using only JouLE-THoMSON expansion Theory of gas liquefaction using isentropic expansion Practical aspects of gas liquefaction using expansion engines Liquefaction by isentropic expansion.

I. Heat interchangers and regenerators. . . . . . . . . . .

2 16 33 40 57 69 76 87 89

Helium Liquefiers and Carriers. By SAMUEL CoRNETTE CoLLINS, Professor of Mechanical Engineering, The Massachusetts Institute of Technology, Cambridge, Mass. 112 (USA). (With 24 Figures) . Introduction . . . . . . . A. B. C. D.

Principles of refrigeration at low temperatures Components of a liquefying system Representative helium liquefiers. . . Liquid storage and transport vessels .

112 112 120 127 135

Electrical Conductivity of Metals and Alloys at Low Temperatures. By Dr. D . K. C. 137 MAcDONALD, National Research Council, Ottawa (Canada). (With 43 Figures) A. B. C. D.

Historical summary and general concepts. . . Notes on experimental methods and techniques Comparison of experimental data with theory. Concluding remarks

Bibliography . . .

137 154 168 193 197

Thermal Conductivity of Solids at Low Temperatures. By PAuL GusTAv KLEMENs, Senior Research Officer, Division of Physics, Commonwealth Scientific and Indu198 strial Research Organization, Sydney (Australia). (With 18 Figures) I. II . III. IV. V.

Introduction . . . . Thermal conductivity of dielectric solids . Thermal conductivity of metals and alloys: Electronic component . Thermal conductivity of metals and alloys: Lattice component Thermal conductivity of superconductors.

References . . . . . . . . . . . . . . . . .

198 201 228 251 266 276

VI

Contents. Page

Low Temperature Heat Capacity of Solids. By PJETER HENDRIK KEESOM, Asso-

ciate Professor and NoRMAN PEARLMAN, Assistant Professor, Department of Physics, Purdue University, Lafayette, Ind. (USA). (With 29 Figures) . 282 Introduction . A. Theory. .

282 282

I. Lattice modes II. Electronic modes

284 288

B. Experimental techniques . I. Calorimeter . . . . . II. Measuring procedures

294 294 298

C. Experimental results. . . I. Lattice and electronic atomic heats of the elements II. Other sources of heat capacity a) Superconductivity . . . b) Excitation modes. . . . c) Cooperative phenomena . d) Size effect . . . . . . . III. Miscellaneous measurements

301

Appendix . . . . . .

303 325 325 330 332 332 333 335

General References

336

Sachverzeichnis (Deutsch-Englisch) Subject Index (English-German) . .

338 344

ENCYCLOPEDIA OF PHYSICS EDITED BY

S. FLUGGE

VOLUME XV

LOW TEMPERATURE PHYSICS II WITH 318 FIGURES

Springer-Verlag Berlin Heidelberg GmbH 1956

HANDBUCH DER PHYSIK HERAUSGEGEBEN VON

S. FLOGGE

BAND XV

KALTEPHYSIK II MIT 318 FIGUREN

Springer-Verlag Berlin Heidelberg GmbH 1956

Contents. Page

Low Temperature Magnetism. By Dr. JoosT VAN DEN HANDEL, Adjunct-Director of the Kamerlingh Onnes Laboratory of the Leiden University, Leiden (Netherlands). (With 32 Figures) I. Introduction II. Effects of magnetic and of electric fields on the energy levels of the magnetic ions . . . . . . . . . . . . . . . . . . . . . . III. Older research methods . . . . . . . a) Measurements of the paramagnetic susceptibilities b) The influence of magnetic and electric fields on the spectra. c) The FARADAY effect

4 11 11 14 15

IV. Paramagnetic relaxation V . Paramagnetic resonance VI. Antiferromagnetism

19 24 28

Bibliography Adiabatic Demagnetization. By DIRK DE KLERK, Docent of Physics, Leiden University, Scientific head-official at the Kamerlingh Onnes Laboratory, Leiden (Netherlands). (With 122 Figures) . . . . . .

34

38

A. Fundamental considerations I. Introduction . . . . II. Thermodynamics of the demagnetization process III. Absolute temperature determination

38 38 47 54

B. Experimental methods . . . . .

60 60 61 68 71

I. II. III. IV.

Introduction . Demagnetization Cryostats Magnets . . . . . . . . Bridge methods . . . . .

C. Magnetic investigations at relatively high temperatures I. Theoretical considerations . . . . . II. Results obtained with individual salts III. The influence of magnetic fields . . .

76 76 85 120

D. Magnetic investigations at the lowest temperatures I. Cooperative effects . . . . . . . II. Results obtained with invidual salts III. The influence of magnetic fields . .

126 126 133 151

E. Other I. II. III. IV.

165 165 173 197 203

investigations below 1° K Heat transfer and thermal equilibrium Experimental results . . . . . . . . The thermal valve and its applications Nuclear demagnetization and nuclear orientation

Gene ral referen ces

209

VI

Contents. Page

Superconductivity. Experimental Part. By BERNARD SERIN, Associate Professor of Physics, Rutgers University, New Brunswick/New Jersey (United States of America). (With 43 Figures) . . . . . 210 I. Introductory survey . . . . . . . . . . . . . . . . II. Electrical and magnetic properties of macroscopic superconductors . III. Thermodynamic properties of the normal and superconductive phases . IV. Penetration of a magnetic field into a superconductor . . . . . . . .

210 214 230 241

V. Phenomena associated with the surface energy between the superconductive and normal phases . . . . . . . . . 249 VI. Thermal effects . . . . . . . . . . . . . . . . . . . . . 261 VII. Superconductive alloys and compounds 268 VIII. Diverse properties unchanged in the superconductive transition 270 Bibliography . . . .

272

References Appended in Proof, March 1956

272

Theory of Superconductivity. By joHN BARDEEN, Professor of Electrical Engineering and Physics, University of Illinois, Urbana/Illinois (United States). (With 20 Figures) 274 I. Introduction . . . . . . . . . . . . . . . . II. Thermodynamic properties and two-fluid models a) Thermodynamic relations b) Two-fluid models . . . . . . . III. LoNDON theory and generalization . . a) b) c) d) e) f)

LONDON theory . . . . . . . . Solutions of the LoNDON equations The LONDON approach to superconductivity PIPPARD's non-local modification of the LoNDON equation Derivation of diamagnetic properties from energy gap model Non-local theories. . . . . . . . .

274 277 277 280 284 284 290 295 299 303 312

IV. Boundary effects; the intermediate state

321

a) Theory of boundary energies . . b) Applications to specific problems c) The intermediate state

321 330 336

V. Electron-phonon interactions a) Introduction . . , . . . b) Formulation of the electron-phonon interaction problem c) Calculation of interaction energy Genera l referen ces

343 343 347 359 368

Liquid Helium. By KuRT MENDELSSOHN, Reader in Physics, Oxford University, Clarendon Laboratory, Oxford (England). (With 101 Figures) 370 Introduktion

. . .

370

A. Historical survey

3 71

B. The diagram of state

402

C. Entropy . .

405

D. Superfluidity

410

E. Viscosity . .

419

Contens.

VII Page

F. Heat conduction G. Wave propagation.

K. Theoretical Appendix

423 431 437 450 454

Literature references

458

H. The saturated film

J.

The unsaturated film

Sachverzeichnis (Deutsch-Englisch)

462

Subject Index (English-German)

470

The Production of Low Temperatures Down to Hydrogen Temperature1 • By

J. G.

DAUNT.

With 101 Figures.

Preface. The purpose of this article is to give the fundamental physical principles involved in the many techniques for the production of low temperatures down to temperatures attainable with liquid hydrogen. In carrying through this aim, emphasis is laid on the evolution and establishment of new ideas and methods. However, it is not my purpose to detail the technological and mechanical developments attendant on each process of refrigeration, which can be found more suitably in engineering publications. In other words, each process of refrigeration is treated at the stage in which it was or is a problem in physics laboratories; but those aspects of the techniques which are concerned with their engineering or commercial development are omitted. The article is subdivided into nine chapters which are concerned respectively with : A. Gas refrigerating machines; B. Compressed vapor refrigerating machines ; C. D. and E. Cooling by JOULE-THOMSON expansion and the liquefaction of air and hydrogen by the LINDE method; F. and G. Cooling by isentropic expansion and the liquefaction of air, etc., by the CLAUDE method; H. Single adiabatic expansion for hydrogen liquefaction; I. Heat exchangers and regenerators. Although the refrigerating processes used in gas refrigerating machines and in compressed vapor machines are capable of producing deep refrigeration in, for example, the liquefaction of air, the detailed discussion of the physical principles involved in them which is given in chapters A and B is included to provide a thermodynamic background for the rest of the work as well as to recognize their own intrinsic interest. No attempt has been made to recognize all the many significant centers of low temperature research and technology, be they academic or commercial. Instead reference is given only to those which by their detailed publications in readily accessible learned Journals have notably added t o the general knowledge of low t emperature production. I am indebted to many previous authors from whose works on low temperature production I have gained aJJ.d relayed valuable information. Since the reader may wish to consult these works also, I append my list herewith together with brief comments concerning each.

J. A. EWING: The Mechanical Production of Cold. Ca mbridge Univ. P ress 1908. A classic giving detailed r eferences to 19th cent ury work. [2] M. a nd B. RuHEMA NN : Low T emperature P h ysics. Cambridge Univ. P ress 193 7. Chap . I , P art I, gives an h ist orical picture of the d evelopment of low t emperature physics.

[1]

1 A condensed bibliography of the most essential books on the subject is given at the end of the Preface.

Handbuch der Physik, Bd. XIV.

2

J. G. DAUNT: The Production of Low Temperatures.

Sect. 1.

[3] H . LENZ: Handbuch der Experimentalphysik, vol. IX/I, p. 47. 1929. Liquefaction of gases and its thermodynamical foundation. Good account of early work on JOULETHOMSON expansion. [4] W. MEISSNER: Handbuch der Physik, vol. XI, p. 272. 1926. Production of low temperatures and the liquefaction of gases. Excellent survey of the field to 1926. [5] M. RuHEMANN: The separation of gases. Oxford Press 1940. Chap. V gives good outline of refrigeration to low temperatures. [6] J. A. VAN LAMMEREN: Technique of Low Temperatures. Springer 1941. Besides detailing methods of production of low temperatures, this gives a useful chapter on cryostats and a full bibliography. [7] M. DAVIES: The physical principles of gas liquefaction and low temperature rectification. Longmans Green & Co. 1949. A short but authoritative book from a modern standpoint. [8] H. HAUSEN: \Varmelibertragung in Gegenstrom, Gleichstrom and Kreuzstrom. A detailed account of interchangers, regenerators, etc. [9] R. PLANK: Handbuch der Kaltetechnik, vol. I. Springer 19 54. Excellent historical article. [10] H. J . MACINTIRE and F. W . HUTCHINSON: Refrigeration Engineering. Wiley & Sons Inc. 1937. Good text from engineering point of view of refrigeration to - 50° C.

Of course the above list makes no pretense at completeness regarding publications on low temperature production, no more than does the total of references given in the text of the article. It represents only a personal view of the major contributions which present aspects of the field of fundamental physical significance.

A. Refrigeration by gas (air) machines. Refrigeration by gas engines, in which the gas has generally been air, has been carried out by two distinct classes of machine (a) the "open-cycle" machine in which atmospheric air circulates in the low pressure circuit of the machine at atmospheric Compressor pressure and (b) the "closed-cycle" machine a in which the same mass of gas is circulated in li a completely closed circuit, in general everywhere at pressures above atmospheric. Coolon! out

1. The open-cycle gas refrigerating machines.

The pioneering work of practical development of open-cycle machines was due largely to GIFFORD (1873) and to ].COLEMAN and J. and H. BELL (1877)1. A diagrammatic sketch of such a machine is given in Fig. 1. The gas (air) is adiabatically compressed in the compressor from pressure Pr to P2 • It is subsequently cooled to temperature Tc, ideally at the compressor presAdiobolio t:xponsion sure, p2 , in the cooler which may have water engine as the coolant fluid. Thence the gas passes to a the expansion engine in which it is expanded tlo!d tlliomber adiabatically, doing external work. The mechanical work obtained from the expansion engine Fig. 1 . Flow diagram for open·cycle gas (air) is coupled back to the compressor, often by refrigerating machine. having compressor and expander on the same shaft. The cool gas at low pressure, P1 , and temperature, Td, passes from the expansion engine to the cold chamber which it maintains refrigerated. Gas from the cold chamber returns to the low pressure side of the compressor at a temperature Ta, which is approximately that at which the cold chamber is maintained. in

1

See R. PLANK: Handbuch der Kaltetechnik, vol. I., Berlin: Springer 19 54.

Sect. 1.

3

The open-cycle gas refrigerating machines.

Diagrammatically the· operation of the machine may be followed in the indicator diagram of Fig. 2. Here the adiabatic path a-+b represents the adiabatic compression from P1 to P2 (the letters a, b, etc. are also marked at the corresponding points of Fig. 1). The path b-+c is the isobaric cooling from temperature Tb toT,, during which heat Q2 is given off. The path c-+d is the adiabatic expansion down to pressure p1 and path d-+a represents the passage through the cold chamber in which the gas ex- p tracts heat, Q1 , from the chamber. For each mole of gas circulated:

Q2 = Cp2 (Tb- Tel } Ql = Cp 1 (Ta- Td).

(1.1) a

Since in both adiabatic processes the ratio of the pressures, P2 /P1 , is the same, then: (1.2)

v

Fig. 2. Indicator diagram of idealized operation of an open-c ycle gas refrigerating machine.

Combining this with eq. (1.1) and assuming, as for a perfect gas, that CP, = C p2 we get: J?1 !_a_ Td (1 .3) Q2

and the coefficient of performance,

Tb

~'

T,

is given by:

This coefficient of performance is less than the thermodynamical coefficient, which would be given by a reversible (ARNOT engine transferring its heat from the cold chamber at temperature Ta to the cooling fluid in the cooler at T, . ~ideal is :

~ideal,

(1. 5)

and the relative thermodynamic efficiency is therefore:

'Y/rel

for the gas refrigerating machine (1.6)

In an actual air open-cycle machine the temperature of the cold chamber, T. might be, say, 0° C and the temperature, T,, of the cooling fluid in the cooler, say, 24° C. This would give ~ideal= 11.4. For actual running conditions with p1 = 1 atmosphere and p2 = 4 atmosphere Td, the temperature of the air after expansion, would be about -65 ° C, which from eq. (1.4) yields ~ = 2.3. The relative efficiency, 'Y/rel, is therefore ~ 0. 2. In facti the coefficients of performance, ~' are about 0.5 to 0.7. The discrepancies here are due to mechanical inefficiencies in the machine, etc. More important, however, is the fact that the absorption of heat, Q1 , in the cold chamber takes place over a wide range of temperature, the cold gas entering the chamber from the expansion engine at temperatures much less than that of the chamber. The isobaric expansion thereby 1

See J. A. EwiNG: The Mech anical Production of Cold. Cambridge Univ. Press 1908. t*

4

J. G. D.AUNT : The Production of Low T emperatures.

Sect. 2.

taking place in the cold chamber cannot do useful external work, which thereby introduces considerable irreversibility. This irreversibility introduced by the isobaric expansion in the cold chamber can be reduced by reducing the value of T.t, the temperature of the gas issuing from the expansion engine. To effect this, smaller compression ratios, r=P 2/P1 , should be employed. The effect of lowering r on the value of ; can be seen in the following way: In an adiabatic compression from pressure p1 to p2 the work done per mole of perfect gas is : (1.7) Where T1 is the temperature at initial pressure p1 and where y is the ratio Cp/Cv. For the engine of Figs. 1 and 2 the net work, W, is the difference in the work of adiabatic compression and expansion and is given by: (1.8} An alternative expression for the coefficient of performance, ;, therefore is obtained from eqs. (1.1} and (1.8}: (1.9}

From this latter expression for; it will be seen that by decreasing the compression ratio, r = p2 jp1 , the value of; can be increased. Lower values of r must be counterbalanced by increased compressor and expansion engine speeds in order to obtain the same net refrigeration. This is mechanically disadvantageous since greater frictional losses are thereby introduced. The low coefficients of performance associated with air refrigerating machines as described above have resulted in their being largely replaced by compressed vapor refrigerating machines, which, as is shown in chapter B, have much higher efficiencies. The air refrigerating machine in general is only retained where the convenience of having air as the refrigerating fluid is paramount, as is the case in some shipboard applications and, as has been recently introduced, in "comfort cooling" in aircraft. In the latter the same rotary compressor may be used for the cooling systems as is used at higher altitudes for a heating cycle. In order to improve the coefficient of performance of the gas refrigerating machines it is clearly necessary to avoid the loss of useful work in isobaric expansion in the cold chamber and to make the process of compression more economical of energy by carrying it out quasi-isothermally instead of adiabatically. A close approach to such desired isothermal processes of heat evolution and heat absorption has been made recently by KoHLER and JONKERS 1 in a closed-cycle gas refrigerating machine to be described in Sect. 5. 2. The vortex tube. The vortex tube was originally devised by RAN QUE 2 and the first publication of a systematic experimental study of it was due to HILSCH 3. J. W . L. KoHLER and C. 0. JoNKERS: Philips Techn. Rev. 16, 69, 105 (1954). G. J. RANQUE: Bull. bi-mensuel Soc. Fran9aise de Phys. June 2, 1933. p. 112. Publication bound with J . Phys. Radium (7) 4 (1933). See also, for example, U .S. Patent No. 1952281. Dec. 6, 1932. 3 R. HrLSCH: Z. Naturforsch. 1, 203 (1946). English translation in Rev. Sci. Inst. 18, 108 (1947). 1

2

5

The vortex tube.

Sect. 2.

Further experimental studies have been made by joHNSON 1 , by ELSER and HocH 2 , by MAcGEE 3 and by others 4 • A typical arrangement of a vortex tube is given in Fig. 3· Essentially the vortex tube consists of a cylindrical tube into which a high velocity gas jet is introduced from a tangential nozzle placed approximately in the center of the cylinder. A circular iris, I, is placed in the tube just to one side of the plane of the jet nozzle A B (in Fig. 3 this is to the right hand side and close to the nozzle, N) so that gas passing from the region of high pressure through the iris comes from the central region of the tube. A turbulent screwlike flow of gas then takes place in a direction away from the iris (to the left in Fig. 3) and its exit from the tube is partially restricted by the A valve, V. As a result of this restriction

Section A-A Fig. 3. Construction of Vortex tube. T, and T, are the side tubes mounted in the body B. N is the nozzle, tangential to the tube and I the iris in tube T,.

r

'"""""'""""~"""""-""'-'Z\'-"-'"~"""""""''«\\.~ ~

Fig. 4. Flow diagram (neglecting rotation) of the air in the warm tube (Tube T, of Fig. 3) of a Vortex tube. [After ] . ]. VAN DEEMTER: Appl. Sci. Res. A 3, 174 (1952). ]

in the valve, a fraction, f1, of the gas is forced to flow back through the iris and this is found to be cooled. The fraction, (1-f1), passing through the valve is heated. A diagram showing the gas flow paths, but omitting the rotational motions, is shown in Fig. 4. In its practical construction, as is shown in Fig. 3, the gas entering the tube at N is guided into a spiral flow by the eccentric ring, E, which is shown at x and y in the s.ection of Fig. 3· The tul!Je itself is made by attaching side tubes, T1 and T 2 , to the central block, B, with threaded fittings. These side tubes are made from thin walled poor thermally conducting material to minimize heat · transfer between the hot and cold ends of the whole system 5 • In all HILSCH's experiments the gas used was air introduced to the nozzle which was approximately ambient. at a temperature, He studied three tubes of different sizes. Noting the nozzle diameter by dN, the sizes and other data are given in Table 1. The choice of d1 , the iris diameter, was made to secure approximately the maximum cooling effect. In all tubes the lengths of the side tubes were about 50 times the tube diameter, dr.

rv'

A. F . JOHNSON : Canad. J. R es. F 25, 299 ( 194 7). K. ELSER and M. Hoc H: Z. Na turforsch. 6a, 25 (1951) . 3 R. MAcGEE jr. : Refrig. Engng. 58, 975 (1950). 4 For a full bibliography see W. CuRLEY and R. MACGEE jr.: R efrig. Engng. 59, 166 (1951). s See M. P. BLAHER : J . Sci . Instrum. 27, 168 (1950) for a neat construction using plastics. 1

2

Handbuch der Physik, Bd. XIV.

1a

6

J. G. DAUNT: The Production of Low Temperatures.

Sect. 2.

Table 1. Dimensions of HILSCH's 1 Vortex Tubes. T ube No.

1 2

3

dN

dr

dJ

rum.

rum.

rum.

1.1 2.3 4.1

4.6 96 17.6

2.6 4.2 6.5

Gas flow for input pressure

=

11 atm. m.' fh.

7.0 30-5 97-0

Fig. 5 shows his results for tube No.2 for the temperature of the emergent cold gas as a function of the value of fi, which is the fraction equal to the mass of cold gas divided by the total IJ?.ass flow to the nozzle. The results shown in Fig. 5 are for four different input .;I>.:ressures, PN, from 1. 5 to 10 atm. It will be seen that at the higher pres____________ _ sures considerable coolings up to 60° C could ..:: •c be obtained for suitable choice of fi· The value of fi experimentally is varied by varying \; -zo the setting of the value V on the side tube T1 . ~ If T, and Tu are the temperatures of the ~-w ~~~~~---+----r-~ emergent cold and hot streams then the ~ o,z 0,6' 0 0,8 fl !,0 energy balance must give: ~

~zo

p-

/1Qss- of !:old (iQs /1QSS of fo/Q/ f!QS

Fig. 5. P lot of HILSCH's measurements of exit temperature of cold tube of Vortex tube (HILSCH' s Tube No. 2) as function of the fractional mass flow, p, through the cold tube. The four curves are for four different pressures, as marked, of the gas entering the nozzle. [After R. HILSCH: Z. Naturforsch. 1, 203 (1946).]

(2.1)

where TN is the initial temperature of the gas entering the nozzle. The original paper should be consulted for further experimental detail. JOHNSON 2 quoted that his results were in general agreement with those of HrLSCH. ELSER and HocH 3 made measurements with C0 2 , CH 4 , A and He gas well as air and found qualitatively similar results. Their results showed that the total temperature difference, TH - Tc, produced for fi = 0. 5 decreased with increasing atomicity of the molecule, the value for TH- T, for C0 2 and CH 4 being essentially the same. Theoretical studies of the cooling produced in the vortex tube are numerous, having been made for example by TER HAAR and WERGELAND 4 , BURKHARDT 5 , PRINS 6 , WEBSTER 7 , FULTON 8 , SHEPER 9 , VAN DEEMTER 10 •11 , A completely rigorous theoretical treatment of the complex turbulent motion of the gas in the vortex tube is clearly a formidable problem, particularly since the velocity profile within the tube is as yet experimentally undetermined. Qualitatively however, the cooling can be understood as follows. The rotating air stream within the tube produces a radial pressure gradient increasing as one moves from 1

2 3 4

5 6 7

8 9 10 11

HILSCH: Z. ~aturforsch. 1, 203 ( 1946). F. JOHNSON: Canad. J. Res. F 25, 299 ( 194 7). ELSER and M. HocH: Z. Naturforsch. 6a, 25 (1951). TERHAAR and H . vVERGELAND: Forh. Kong. Norske. Vid. Selskat. 20, 55 (1947). BuRKHARDT: Z. Naturforsch. 3a, 46 (1948). J. A. PRI NS: Nederl. Tijdschr. Natuurk. 14, 241 ( 1948). D . S. WEBSTER: Refrig. Engng. 58, 163 (1950). C. D. FuLTON: Refrig. Engng. 58, 473 (1950). G. W. SHEPER: Refrig. Engng. 59, 985 (1951). J. J. VAN DEEMTER: Appl. Sci. Res. A 3, 174 (1952). See W. CURLEY and R. MAc GEE jr. : Refrig. Engng. 59, 166 (19 51) for a fuller bibliography. R. A. K. D. G.

The coefficient of performance of the vortex tube.

Sect. 3.

7

the axis towards the tube wall. The effect of turbulence in this pressure field is to create adiabatic mixing and tends to produce an adiabatic temperature distribution with the colder gas therefore being in the axial region of the tube A completely adiabatic distribution would however not be realized due to the effect of t~e thermal conductivity of the gas t ending to reduce the radial temperature grad1ent and due to the effect of non-uniform angular velocity. This latter effect has been described by VAN DEEMTER as follows: "When the angular velocity is not uniform a second mechanism is interfering, namely a flow of mechai1ical energy radially outward. By turbulent friction (eddy viscosity) the inner C'ompfY!Ssor layers of fluid try to force the outer layers to move with uniform angular velocity and therefore work is done by the fluid in the center on the fluid in the region of the wall. (~) Coolont out Due to this transfer of energy the outer region is heated at the cost of the inner in region. A non uniform angular velocity therefore involves a deviation from the adiabatic temperature distribution. f

v

p

N

Pz

I

lc ,,~...........

,,'

{J

\ '\.'' ......('-p)

Vortex Tube

(p,liz) a

'\Adiobotic tine

I

\ \

'

..,

.............

.......... '-' ......... 'Jfl) ....... ' .......... ' \ ' ------------\..-------p, d\ a c' f e '' ',

(p) e (p Te) 1

I

\

\

Colo' CMmber

'

\

v Fig. 6. Flow d iagram for possible open·cycle re· frigerating machine using Vortex Tube expansion.

Fig. 7. Indicator diagram of idealized operation of open-cycle refrigerating machine using Vortex Tube expansion.

3. The coefficient of performance of the vortex tube. In order to discuss the coefficient of performance of the vortex tube used as an open-cycle gas refrigerating machine, a possible mode of operation is given in Fig. 6, showing the circuit and in Fig. 7, giving the cycle on an indicator diagram. First the gas is adiabatically compressed from pressure p1 and t emperature Ta to pressure P2 and t emperature Tb, following the adiabatic path a-+b of Fig. 7. The work of compression Yt;, for one mole of a perfect gas is: y- 1

Yt;, = Cp Ta [(~~)_Y_-

1] = Cp(Tb-

Ta).

(3.1)

The gas then passes through the cooler, ideally at constant pressure P2 until it is cooled to the temperature Tc of the coolant. This gives the path b-H on Fig. 7. On leaving the cooler it enters the nozzle, N , of the vort ex tube. In the tube the gas divides, the cold fraction in amount 11 passing at a t emperature T, and pressure p1 int o the cold chamber. This is represented by t he path c - H of Fig. 7. Since the cooling process is not exactly adiabatic the point e on the indicator diagram is at a higher temperature than the point d, which is on the adiabatic

8

]. G.

DAUNT:

The Production of Low Temperatures.

Sect. 3.

through c at pressure p1 . The fraction (1 - ft) of heated gas leaves the valve, V, of the vortex tube at temperature T1 and pressure p1 and this is represented by the path C---+ f. Note that f is at larger volume than a, since T1> Tc '> Ta. This fraction (1 - ft), cooled to the temperature Tc by passing through the cooler, returns to the compressor input. This is represented by the path f---+c'. c' is not coincident with a unless Ta =Tc. This non-coincidence of c' and a results in a somewhat larger work of compression than that given in eq. (3 .1). The cold fraction ft of the gas entering the cold chamber absorbs heat, Q1 , isobarically at pressure p1 , and leaves at the temperature Ta of the material in the cold chamber. This is given by the path e---+a of Fig. 7. Also for one mole of total gas passing through the compressor, (3 .2) If one neglects the excess work of compression needed to change the fraction (1 - fl) of the gas from c' to a, then the coefficient of performance is:

~vortex = ·~ =

l!{a_-:J) •

(3.3)

The correct value of ~. allowing for the fact that c' and a are not coincident is smaller than the value of eq. (3. 3) by T-T 100 (1 - ft) -"-y ~percent (3.4) a

which for the application considered later is only a correction of less than 5 %. It is of interest to compare the value of (~vortex) for the vortex tube given by eq. (3.3) with the value for the coefficient of performance of an open-cycle gas refrigerating machine using an isentropic expansion engine. The latter is given in eq. (1.4). It is to be noticed that if in computing~ for the isentropic expander machine one neglected to use the work given out by the expander, then the value of~ would be (3.5) This is larger than ~vortex for the vortex tube cycle, (a) by a factor (1/ft), because the vortex tube only uses a fraction ft of the gas passing through the cold chamber and (b) by an extra amount due to the fact that Te> Td, because the vortex expansion is not isentropic. Table 2. Coefficient of performance etc., for actual vortex-tube refrigerating cycle using air. Take 7~ (Temperature of coolant) = (temperature of nozzle) = 20° C; p1 = 1 atm.; p2 = 11 atm. 1. Min . observed temperature T,, of cold gas from vortex tube No. 3)

(HILSCH's

Tube

2. !1. for min. T, 3. Temperature drop,

- 48° C 0.40

(~-

T,)obs. for vortex tube

4. Temperature drop, (Tc-Td), for ideal isentropic expansion

68° C 144° C

5. Coefficient of perf. ;vortex for vortex tube cycle. Assume Ta = 0° C. Eq. (3.3)

0.07

6. Coefficient of perf., ;', for gas refrigerating (adiabatic expansion) machine, not using work of expansion. Assume Ta = 0° C. Eq. (3.5)

0.45

7. Full. coefficient of perf. ;, for gas refrigerating (adiabatic expansion) machine. Assume Ta=0° C. Eq. (1.4)

0.96

8. Efficiency 1)rel of vortex tube cycle relative to gas refrigerating (adiabatic expansion) machine

7.3%

Sect. 4.

9

The closed-cycle gas refrigerating machines.

~vortex can be computed from HILSCH's experimental observations on his tube No.3. The results of this computation are given in Table 2. It is assumed that the vortex tube cycle is as in Fig. 6 with T, = 20° C, Ta = 0° C and P1 = 1 atm., p2 = 11 atm. Then the observed temperature, T,, of the cold gas emerging from the vortex tube would have a minimum value of -48° C for ,u = 0.40. For adiabatic compression, eq. (3-3) makes the coefficient of performance ~vortex =0.07. This is to be compared with the coefficients of performance ~~ and ~ for a gas refrigerating (isentropic expansion) machine operating between the same temperatures T, and Ta. ~~ is for the latter machine not using the work of expansion. One finds Compressor t = 0.45 and ~ = 0.97, showing that the vortex tube cycle has much poorer coefficient of pera formance than either of these. Its relative efficiency, 'Y}rel =~vortex/~, relative to the gas refrigCoo/qnf o//1 erating machine is then 7-3 o/o. Since the gas refrigerating machines described earlier have in poor coefficients of performance compared with, for example, compressed vapor refrigerating machines, it is unlikely that the vortex tube will have serious practical application in refrigeration, except where extreme simplicity is a vital requirement.

4. The closed-cycle gas refrigerating machines.

ti

AdiqbrJfio

The early development of closed-cycle machines p,Td Expqnsion using air as the working gas was due to GoRRIEl Engine 2 (1 845) and A. KIRK (1861) and later to ALLEN Su/Jsfrmoefube and to WINDHAUSEN 3 . The system, which is llef'rigerqfed essentially a STIRLING air engine reversed, is exactly analogous to that of an open-cycle lfeqf frqnsf'er J'urtu~ gas refrigerating machine, as described above. Fig. s. Flow diagram for closed-cycle gas The difference between the two types lies in the refrigerating machine. fact that in the closed-cycle system the same mass of gas is continually recycled, generally everywhere at pressures above atmospheric. One of the advantages of the closed-cycle is that dry air can be used, so avoiding the frosting difficulties encountered in the open-cycle machines. Also smaller compressors and expansion engines may be used, thus reducing frictional losses. A schematic diagram of the closed-cycle system is given in Fig. 8 which is identical with that of Fig. 1 for the open-cycle system except that the cold chamber is replaced by a heat transfer surface (generally a long pipe or pipes carrying the refrigerating gas) in contact with the substance to be refrigerated. In the system developed for example by ALLEN, known as the " ALLEN denseair system", air was the refrigerating gas and it worked between pressures P1 =4.5 atm. and P2 =16.5 atm. Although relatively low compression ratios could therefore be used in the closed-cycle system with at the same time reasonably small compressor and expansion engine, the coefficients of performance of such machines, prior to the 1 J. GoRRIE : US P a tent 8080. May 18 51. See also W. SIEMENS: Min. Proc. Instn. Civ. Engrs. 68, 1 79 {1882) . 2 A. KIRK: Min. Proc. Instn. Civ. E ngrs. 37, 244 {1873/74). 3 See J. A. EWING: Mechanical Production of Cold. Cambridge Univ. Press 1908 and R. PLANK: Handbuch der Kaltetechnik, vol. I. Berlin : Springer 1954.

10

J. G. DAUNT: Th e Production of Low Temperatures.

Sect.

s.

new Philips machines 1 , were very small. Their theoretical values are grven in eqs. (1.4), (1.6) and (1.9) being formally the same as for the open-cycle machines. The obsolescence of the traditional closed-cycle system therefore is due also to its poor performance. The Philips closed-cycle gas refrigerating machine described by KoHLER and joNKERS 1 has been able to approach more closely the ideal thermodynamical efficiency to be expected largely by carrying out the absorption of heat at the lower temperature approximately isothermally. This obviates the irreversibilities introduced by the isobaric expansion in the passage of the gas through the heat transfer surface. To enable such an isothermal expansion to be carried out the expansion engine must operate, both as to its input as well as its output, at the lower temperature. The possibility of low temperature operation of such engines was developed, initially by CLAUDE 2 , at a date much later than that of the traditional gas refrigerating machines described above. The early gas machines therefore owe their major inefficiency to the high temperature operation of the expansion engine. 5. The Philips gas refrigerating machine. A new development of the closedcycle has been described in detail by KoHLER and jONKERS 1 of the Philips Industries. Since this enables in a single stage expansion engine temperatures to be maintained which are low enough to liquefy air under atmospheric pressure with fair efficiency, a detailed description of it may be of value. In principle the significant difference between this and the earlier closedcycle refrigerating machines described above is due to the introduction of a heat exchanger between the compressor and the expansion engine. Fig. 9 is a basic diagram of this system and is to be compared with Fig. 8 of the earlier systems. By the use of the interchanger it is possible to maintain the expansion engine at the temperature T1 of the refrigeration and so enables almost all the cold produced by the expansion to be available essentially isothermally at T1 . It is possible to consider ideally the system illustrated in Fig. 9 as consisting of : a) Isothermal compression to pressure p2 of the gas at the temperature T2 , the isothermalism being effectively maintained by the cooler. (For this reason the compressor and cooler are grouped as one unit as indicated by the broken rectangular box marked T2 enclosing them.) b) Cooling of the high pressure gas from T2 to T1 by passage through the interchanger with consequent reduction in volume. c) Isothermal expansion from pressure p2 to p1 in the expansion engine with consequent extraction of heat from the substance to be refrigerated. (Here also the isothermalism of the combined processes of expansion and of extraction of heat from the substance to be refrigerated via the heat transfer surface is diagrammatically indicated by the enclosing of the expansion engine and of the heat transfer surface of Fig. 9 inside one broken rectangular box marked T1 .) d) Passage of the low pressure return gas back through the interchanger from temperature T1 to T2 • If one may assume the interchanger to be perfect such that no temperature differences exists between the ingoing and outgoing gas and such that no pressure differences are needed maintaining the gas flow through it, then this idealized cycle may be represented by the indicator diagram of Fig. 10. In this the path a --7- b represents the compression from p1 to p2 along the isotherm T 2 • Path 1 2

J. W. L. KoHLER and C. 0. JoNKERs: Philips Techn. Rev. 16, 69, 105 (1954). G. CLAUDE: Liquid Air, Oxygen and Nitrogen. Paris 1913. See also section G.

s.

Sect.

The Philips gas refrigerating machine.

11

b-+c represents the cooling form T2 to T1 in passage through the interchanger at constant pressure P2 , path c-+d the isothermal expansion in the expansion engine at temperature T1 , and path d-+ a the return path through the interchanger at constant pressure p1 . The heat extracted, Q1 , at T1 from the substance to be refrigerated rs then d given by: (5.1) Qt=fPdv r------ ----------------, I

I

I

I

I

>

I

:

I

Pz

:

Compr~ssor

I

'i

p

I

I

I

l

:

:

!

I

I

II

Coolon/ out

1

I I I

a................... ...__T.:.

lI

z

In

I

l _______t!_

---

. p,r,1'+---PI

v Fig.10. Indicator diagram for idealized operation of closedcycle refrigerating machine of Fig. 9. r ___

:

r,1

..f!J!i. fi

I

C

FtQ I 1

1

..----...,--- -.,---y---,

s

I

i

1 11

10

!!z.1 _____ .,

z

bptrnsion rc

•

~ngme

I

I

I I

: I

I Subsltrnce to be

l I

Jhfflpro~

I

!L- ----- - ---

I

o.oz

I

O.Of0~----~~0~---~~~~0~-~oK~JOO

I

I

--- - -- - -- ~

r,

lleut Tronsf'cr Surf'trcc

Fig. 9. Flow diagram for closed-cycle gas refrigerating machine using low temperature expansion engine and interchanger. Diagram represents basic features of the Philips gas re frigerating machine.

Tz

Fig. 1 t. Coefficient of performance of ideal (CARNOT) refrigerating machine, having upper temperature, T", of 300° I p1 ) and the light broken lines being typical constant temperature lines (T3 > T2 > Tr). The heavy broken curve is the boundary curve enclosing under it the twophase heterogeneous region. It is to be noticed that in the two phase region the isobars and the isotherms are linear and coincident. Moreover, since quite generally, (8Hj8 S)p = T, the slope of the isotherms (or isobars) in the twophase region gives the absolute temperature directly. Table 3 gives references for such diagrams (and for p-H diagrams) for ammonia and other working substances. ASRE Refrigerating Data Book, 7th Ed., Part II, p . 105 and ff. 1951. 2 Publications of Kinetic Chemicals Inc. Wilmington Delaware. (Colored Charts may be obtained from this company.) 3 R. and H. Chemicals Dept. E. I. du Pont de Nemours Co. Wilmington: Delaware. (Charts may be obtained from this company.) 4 ASRE Circular No. 12. Pub!. Amer. Soc. Refrig. Engng. 40 W . 40 St. New York, N.Y. 5 D. F. RYNNING a nd C. 0. HuRD : Trans. Amer. Inst. Chern. Engr. 41, 465 (1945). 6 Na t. Bur. Stand., Circular 1923, No. 142. 7 W. H. KEESOM and D. J. HouTHOFF: Leiden Comm. Suppl. 65 a, b (1928) . 8 DANA, JENKINS, BURDICK and TIMM: Refrig. Engng. 12, 403 (1926). 9 R. YoRK and E. F. WHITE: Trans. Amer. Inst. Chern. Engr. 40, 227 (1944). 10 R. W. WATERFILL: Industr. Engng. Chern. 24, 616 (1932). 11 H. J. MAciNTYRE and F. W. HuTCHINSON: Refrigerating Engineering. New York: J. Wiley & Sons 1950. 12 W. H. KEESOM, A. BIJL and L.A. J. MoNTE : Leiden Comm. Suppl. 108b (1954) and Appl. Sci. Res. 4, 25 (1954). 13 BARKELEW, VALENTINE a nd HURD: Trans. Amer. Inst. Chern. Engr. 43, 25 (1947). 14 C. S. MATTHEWS and C. 0. HURD: Trans. Amer. Inst. Ch ern. Engr. 42, 55 (1946). 15 Such diagrams were first introduced by R. MoLLIER: Z. VDI 48, 271 (1904). Diagrams of p - H, which are also of value in determining the characteristics of refrigerators, etc., and which were also introduced by MoLLIER, are referred to also as MoLLIER diagrams. 1

Sect. 8.

Use

of

the H- S

MoLLIER

21

Diagram.

The cycle of operations of both a dry-compression or wet-compression vapor refrigerator can be followed in detail by means of the H- S diagram of Fig. 21, on which only two isobars are represented, p1 corresponding to the pressure at which the working substance is evaporated in the evaporator (i.e. the compressor input pressure) and p2 corresponding to compressor output pressure. In a wetcompression cycle (corresponding to the cycle of Fig. 18) the state of the working substance at the compressor input is given by the point a on the isobar, p1 . Adiabatic compression takes the working substance to c, which is vertically above a on the p2 isobar, and which represents saturated vapor. Condensation in the condenser takes the substance to d, which being on the saturation curve represents saturated liquid. The isenthalpic expansion (constant H) through II

__ PJ ............. ----

/!--- ~t

II

p,

P;:

c/

-

-----Tz

.,,- - - .::i_'-~/

,I I

I

____ ,,

cfQ).........

.,_.~,

I

d' J.i -----'I

r, -----

s Fig. 20. Fig. 20.

'.!'

/sot. !iq.

I

s Fig. 21.

Schematic diagram of H- S MOLLIER diagram, showing typical isobars and isotherms. The heavy broken curve marks the boundary of the two~phase heterogeneous region.

Fig. 21. H- S MOLLIER diagram showing thermodynamic cycle of operations for idealized compressed vapor refrigerating machine. (The unprimed letters are for "wet" compression; the single primed letters are for "dry" compression and the double primed letters are for mixed compression.)

the throttling valve is represented by the horizontal path d -H, and the subsequent evaporation at constant temperature in the condenser is represented by the line e--+a. The coefficient of performance, ~. can be obtained, if it assumed that there are no losses, in the following manner. The heat, Q1 , absorbed in the evaporator at constant pressure and temperature must by definition be: (8.1) Ql=Ha-He and the heat, Q2 , given up to the condenser is:

(8.2) where the subscripts refer to the value of the enthalpy, H, at the appropriately lettered point. Remembering that Hd=H. and that the net work done on the machine, W, (neglecting dissipational losses), is given by W = Q2 - Q1 , we obtain for the coefficient of performance:

22

J. G. DAUNT: The Production of Low Temperatures.

Sect. 9.

These values of H can be immediately determined by consulting the MoLLIER diagram for the working substance of interest, once the working pressures are known. In the case of a dry-compression refrigerating machine, which is the one usually adopted in practice, the cycle as plotted on the MOLLIER diagram is essentially the same as that described above, except that now the starting point, a', must be on the saturated vapor line, and that the end of the compression must be represented by c' on the p2 isobar vertically above a'. The value of the coefficient of perP formance,~,isgiven byeq. (8.3) /s, with the appropriate primed / values substituted. Again note // that a knowledge of P1 and P2 is still sufficient to determine Po ~- If the machine is not an ideal wet-compression or dry compression type, but is a mixture of the two types, then the cycle of operations would be represented on Fig. 21 by a"-+ c"-+ d-+ e-+ a", where 11 the compression a" -+ c" is Fig. 22. Schematic diagram of p-H MOLLIER diagram, showing typagain assumed isentro1=ic. In ical isotherms and lines of constant entropy (S). Pc is the critical this case to calculate ~. either pressure of the fluid. the temperature corresponding p to the point c" must be known, {< or alternatively the "wetness" I of the vapor entering the comI I I pressor must be known in I I I order to fix the point a". In I I I practice , as is discussed in I I more detail in the below, the compression is never ideally isentropic and hence both the above facts must be known as well as P1 and P2 in order to II Fig. 23. p-H MoLLIER diagram showing thermodynamical cycle of operations for idealized compressed vapor refrigerating machine using "dry'' compression.

evaluate~-

9. Use of the p-H MoLDiagram. The p-H thermodynamic diagram, introduced also be MoLLIER 1 , is of great practical value in assessing the theoretical coefficients of performance of compressed vapor refrigerators. The diagrams, which plot the pressure p, against the enthalpy, H, include lines of constant temperature, constant entropy and constant volume. A simplified picture of such a diagram is given in Fig. 22. The full drawn dome-like line encloses the two phase (liquid and vapor) heterogeneous region, the maximum of which gives the critical pressure, Pc. The heavy broken lines are those for constant temperature; I;_ 5 3 . A dry-compression cycle of a compressed-vapor refrigerator can be followed on such a MoLLIER diagram as is shown in Fig. 23. Starting with saturated LIER

1 R. MoLLIER: Z. VDI 48,271 (1904). For references to such diagrams for common refrigerating substances see Table 3.

Sect. 10. Practical single stages systems, their working substances and efficiencies.

23

vapor at the evaporator temperature, T1 , and pressure, p1 , at the point a, adiabatic compression carries the substance along the isentrop from a to c, where c corresponds to the intersection with the isobar, p2 , which is the condenser pressure. The influx at constant pressure, p2 , to the condenser from the compressor is given by the horizontal path c-+c' and the subsequent condensation at constant pressure in the condenser at temperature, T2 , by the horizontal path c' -+d. The isenthalpic expansion through the valve is given by the vertical line d-+e and the subsequent evaporation at constant pressure p2 in the evaporator by the line e-+a. The fraction e liquefied in the evaporator is given by (9.1) To compute the theoretical coefficient of performance it is necessary to know only T1 and T 2 whence the values of Ha, He, and H. can be immediately read from the diagram. These values inserted into eq. (8.3) give ~theor· For further detail on the use of these p - H diagrams in practice, OPHULs' article in the ASRE Refrigerating Data Book 1 may be consulted. 10. Practical single stages !;ystems, their working substances and efficiencies.

In practice single stage compressed vapor refrigerators, as have been discussed above, may be put to a variety of uses for the extraction of heat at low temperature. For example the domestic refrigerator and air conditioning systems perform the extraction of heat at about 0° C; the so-called domestic "deep-freezer" at about -15° C. Extraction of heat at still lower temperatures, however, by such machines or in similar multistage and cascade types is of more interest here, as for example is performed in the liquefaction of other substances and in the precooling stages of air liquefiers, whether of the LINDE or Cascade types. Since in single stage compressed vapor refrigerators it is convenient to have the temperature, T 2 , at which heat is absorbed from the working substance by the condenser only a little higher than ambient temperature (20° C), the following criteria can be imposed in making a choice of working substance suitable for the machine: a) Ambient temperature (T2) must be less than the critical temperature, Tc, in order to liquefy the vapor by compression, and furthermore the pressure, P2 , required for this liquefaction must not be abnormally high. b) It is convenient to have the temperature of the evaporator, T1 , around the boiling point, Tn. of the working substance, so that the pressure, p1 , of evaporation may be approximately atmospheric. This is, however, not a strict requirement. The only essential requirement here is that T1 shall be larger than the temperature of the triple point, Tr,. in order to avoid solidification of the working substance. c) The substance must be chosen to give a small value for the compression ratio P2/P1 • This is necessary because the work of compression increases rapidly with the compression ratio. This question is intimately connected with the choice imposed on the value of the evaporator temperature T1 . d) Within the limitations set out in items a), b), and c) above, the choice of working substance is finally decided by a compromise between the thermodynamic properties (i.e. high coefficient of performance, etc.) and practical considerations governing the use of the machine (for example some substances with good thermodynamic properties are very corrosive or alternatively highly toxic when released from machine). 1 F. OPHuLs: ASRE Refrigerating Data Book, 7th Ed., Part. I, p. 11. 1951.

J.

24

G. DAUNT: The Production of Low Temperatures.

Sect. 10.

Table 4. Characteristics of refrigerants used in compressed vapor machines operating between the following temperatures: Evaporation -15° C; Condenser 30° C I. Carbon Dioxide

Ammonia

2. 34.

s. 6. 78. 9-

30.0° C (p 2 of Fig. 21) Evaporator pressure (atm.) at - 15° C (pi of Fig. 21) Compression ratio, r Enthalpy 2 (calfg.) of saturated vapor (leaving evaporator) at - 15° C (Ha of Fig. 21) Enthalpy 2 (calfg.) of liquid entering evaporator at - 15° C (He of Fig. 21) Net refrigerative effect, QI(caljg.) · [(Ha-He) of Fig. 21] Coefficient of performance ; (dry compression neglecting losses) 3 Refrigeration in cals per liter of pist on displacement P ractical hazard

Methyl Chloride CH,Cl

so,

co,

NH,

1. Condenser pressure (atm.) at

Sulphur Dioxide

Freon 12 CCl, F,

11.52

70.8

4.52

6.34

7-36

2.33

22.7

0.805

1.42

1.80

5.63 102.0

4.46 108.5

4.09 43.8

3.13 77.1

4.95 341.0 77-3

46.3

23.4

25.9

15.4

263.7

30.8

78.6

82.6

28.4

4.24

2.88

4.7 306

1880

I

toxic

high pressure

1 corrosive

517

4.82

Ijvery toxicji i

4.65

292

195

toxic , flamable

none

The above considerations led to the early adoption for machines operating with evaporator temperatures above about - 50° C of the following long used working substances: ammonia, carbon dioxide and sulphur dioxide 4 • A comparison between these substances can be made from Table 4 which sets out their various characteristics when used in compressed vapor refrigerators operating between 30° C and - 15 o C. From the table it will be seen that the coefficients of performance and the compression ratios for NH 3 and S02 , as well as for the more recent substances used, CH3Cl and CC12F 2 , are about the same, whereas for C02 their values are somewhat smaller. From the point of view of the smallness of the necessary piston displacement per unit of refrigeration, C0 2 is outstanding. C0 2 also has the ad vantage of being Ta ble 5. Theoretical coeffi cients of performance (i .e. losses neglected) in an ammonia vapor compression refrigerator (dry compression) with condenser temperature at 30° C for various values of the evaporator temperature, T 1 . Also given are the evaporator pressures, the compression ratio, and the efficiency, 11rel, relative to a CARNOT" cycle operating between the same temperatures. Temperatute of evaporator

Pressure in evaporator

T, oc

p, (a tm.)

0 -10 -20 -30 -40 -50

4.23 2.87 1.88 1.18 0.71 0-39

!

I

I

Compression ratio, r

Theoretical coefficient of performance

p,fp,

~

2.7 4.0 6.1 9-75 16.2 28.5

8.1 5.6 4.1 3-15 2.45 1.9

IT"-"'"'•W••~·, relative to CARNOT

cycle flrel

!

0.89 0.85 0.815 0.78 0-735 0. 685

-- - · · · - - -- · - -

Some of this data is t aken from L . S. MoRSE. 7th Int. Cong. R efrig. 3, 718 (1937). Enthalpy measured from - 40° C. 3 For ideal CARNOT cycle operating between these temperatures, ;max = 5. 7 5. A v alue of ; = 1 corresponds to an efficiency of 0.212 "tons" per h.p. 4 See E. GRIFFITHS and J. H. AsBERY: Proc. Brit. Assoc. Refrig. Mar. 1925 for references to early literature. 1 2

Sect. 10. Practical single stages systems, their working substances and efficiencies.

25

non-toxic and non-combustible. It is used therefore where these practical assets are of importance. However the high pressures needed in the C0 2 cycle and its poorer coefficient of performance compared with the other substances listed in Table 4 make it to be not of general application. For more detailed comparisons of various working substances and of their relative efficiencies see KEESOM 1 and PLANK 2 • The very high value of the calculated (neglecting losses) efficiency of the ammonia machine relative to the ideal CAR NOT efficiency ('i]rel = 0.82) is to be noted 3 . It is this factor that has made the compressed vapor machine preferable to old-types of gas (air) refrigerating rna8 1/ chines. 8 In reaching to temperatures lower than the example of -15° C given above with 7 single stage machines the coefficient of G performance, as noted above, diminishes and the compression ratio increases. Table 5 I and Fig. 24 show the way the theoretical ( (neglecting losses) value of the coefficient ~ 11 of performance, ~. of a compressed vapor ~ J refrigerator using ammonia in dry com~ ~ pression varies with the lowering of the z / !@ to C 0° from , T evaporator temperature, 1 ~ - 50° C. For these calculations, as for Table 4, the condenser temperature, T2 , o_so -'IO -Jo -zo -to o tO zo•cJo was arbitrarily fixed at 30° C , a figure cvuporufor Temp. commonly used in practice. It is clear Fig. 24. The theoretical (i.e. loss-free) value of the from the Table 5 that ~ goes to quite low coefficient of performance, ; , as a function of the temperature, T for a dry cycle ammovalues as T1 is reduced, and moreover the evaporator nia compressed vapor refrigerating machine. The theoretical efficiency, 'i]rel, relative to a condenser temperature, T2 , is taken to be 30° C. CARNOT cycle operating between the same temperatures also decreases as T1 decreases. For wet compression machines the values of ~ are slightly larger than those given here for the dry compression. It should be noted also from Table 5 that the compression ratio, r, for the lower temperatures of evaporator operation reaches values ( ~ 30 for T1 = - 50° C) which are much too high for single stage compressors. For operation therefore to - 50° C and below, multi-stage systems are necessary, as are described later. For single-stage systems the compression ratio must be less than about 8 or 9 for reasonable practical efficiency and consequently for evaporator operation below about - 20° C, substances more favorable than ammonia must be looked for. A further reason for choosing a more favorable substance lies in the choice of the evaporator pressure, p1 . If this is below atmospheric pressure, as it is for ammonia below - 33° C, there is the practical disadvantage of the possibility of air leaking into the apparatus and degrading its performance and of the necessity for large volume compressors, with consequent increased frictional losses. In order to overcome this second problem, at least partially, suitable substances must be sought which have lower boiling points · than ammonia. This is also

I

v

I

/

f

1,

W . H. KEESOM : Leiden Comm. Suppl. 76a (1933). R. PLANK : Z. ges. Kalteind. 47, 81 (1940). 3 In refrigerating engineering the common unit of refrigerating capacity is the "ton", derived from the average rate of heat absorption required to freeze 2000 lbs. (1 ton) of ice from water at the melting point every twenty four hours. The "ton" therefore is equivalent to 840 calsfsec. or 200 b.t.u.fmin. It is common also to express the coefficient of performance, $, in "tons" per horse-power. A value of ,; = 1 corresponds to 0.212 "tons"fh.p. 1

2

26

J.

G. DAUNT: The Production of Low Temperatures.

Sect. 10.

the criterion for choice of substances giving lower compression ratios, as has been pointed out by DAVIES\ as follows: The vapor pressure of the many substances of interest can be approximately expressed by L

lnp=A - -RT

(10.1)

where L is the latent heat of evaporation per mole. This equation is valid only over a relatively small range of temperature, over which L may be assumed constant. Near the critical temperature where L-+0, the equation cannot be satisfactorily applied. The compression ratio, r, therefore is given by:

( Tl T2

ln r = ln (b._)=-~ T1 - T2 ) . Pr

R

(10.2)

For a given T1 and T2 , therefore, r will be diminished by choice of substances with decreasing values of their latent heats of evaporation. Since, according to TROUTON's rule the ratio of L (calfmole) to the boiling point, T8 , in degrees K, is approximately constant and equal to the value 21, low L Table 6. Evaporator pressure, p1 ; compression ratios, r = P2.IP1 , for the lower boiling point substances used in values correspond to low boiling cou.tpressed vapor machines operating in dry comprespoints. Therefore, from eq.(10.2) , sion between 30° C and - 50° C. the compression ratios, r, should p, diminish with diminishing T8 . Substance (atm.) For the boiling points of various substances suitable for Methyl Chloride CH3CJ 0.27 23.7 compressed vapor refrigerators, Freon 12 CC12 F 2 0.375 19.5 Ammonia Table 3 should be consulted in NH3 0.39 29.5 Freon 22 CHCIF2 0.64 18.6 which the substances are arPropane 0.69 15.2 C3Hs ranged consecutively in order of Propylene 0.90 C3Hs 13.9 their boiling points, the highest Carbon Dioxide C02 6.73 10.5 Ethane boiling point being at the top of 8.4 5.45 C2Hs Nitrous Oxide N 20 9.8 6.45 the list. The six substances listed, all having boiling points above that of so2' are most suitable for relatively high temperature refrigeration, such as is encountered in air conditioning, refrigerated transportation, etc. For the others, Table 6 lists both the evaporator pressure, p1 , and the compression ratio, r, for operation in dry compression between 30° C and - 50° C. It is clear from this table that the lower boiling point substances have compression ratios which would allow single-stage machines to be succesfully operated. However, in practice, as is outlined below, two-stage machines are more economical for operation to - 50° C and below. It is noted that the last two substances listed in Table 3, namely ethylene and methane have critical points well below 30° C and hence cannot be included in the comparison of Table 6. These substances are of great value in cascade vapor refrigeration where they can be operated with condenser temperatures well below their critical points. Table 3 includes many working substances for relatively high temperature evaporation which have been introduced within the past two and one half decades. For example dichlorodifluoromethane (CC12F 2), known as Freon 12, 1 M. DAVIES: The physical principles of gas liquefaction and low temperature rectification. London : Longmans, Green & Co. 1949.

Sect. 10. Practical single stages systems, their working substances and efficiencies.

27

was introduced 1 in 1930; and many organic chlorofluoride refrigerants have been subsequently developed which have been given the generic name of Freon 2 • For evaporator temperatures down to about -15 o C. Methyl Chloride 3 and Freon 12 compare very favorably with ammonia, as is shown in Table 4. The virtue of Freon 12, besides its general lack of practical hazards, is that its thermodynamic properties are very similar to those of NH 3 and hence it can be used to replace NH 3 in existing machines with standard positive compressors without any radical changes being necessary. It is to be noted that Freon 22 (CHClF2) has a still lower boiling point than Freon 12 (see Table 3) and Freon 13 (CC1F3 ) (not listed in Table 3) has a boiling point of 192° K and both therefore are suitable in cascade vapor compression refrigeration for operation to about - 70° C. This discussion of single stage compressed vapor refrigerators, has been up to now a theoretical one in which losses have been neglected. Before closing, therefore, some comment should be made on the principal sources of loss, which reduce the coefficients of performance below the theoretical values, and which are as follows: a) Losses in the condenser. In order for the heat of condensation to flow to the condenser coolant a temperature difference must exist between the working substance of the refrigerator and the coolant. Hence the condenser temperature, T 2 , of the working substance must be higher than for an ideal reversible system. b) Losses in the evaporator. For similar reasons to those stated in a) above, the temperature, ~ of the working substance in the evaporator must be less than that of the material being cooled. This again means 'Y}rel is diminished. c) Loss in the piping etc. Frictional losses in the piping between the components of the machine and in the compressor passages must occur and these represent an extra load on the compressor. d) Unwanted heat transfer. In the piping, unwanted heat transfer may occur due to imperfect insulation which tends to reduce the refrigerative effort. e) Mechanical losses in compressor and motor. f) Non-ideality of compression. In compression heat transfer occurs between the gas and the compressor walls and this means that in the initial period of compression, when the gas is below ambient temperature, the gas receives unwanted· heat. This heating of the gas, in excess of that due to the compressional work done on it, in practice more than counterbalances the outflow of heat through transference to the compressor walls at the final period of compression. As a result the compression is not isentropic, but results in a net entropy increase of the gas. This is illustrated in Fig 19 in which the path a-+c represents the ideal adiabatic compression and in which the path a-+ci represents the irreversible actual non-isentropic compression. Since He >He, the coefficient of performance, as given by eq. (8.3) is reduced below the theoretical value. The losses due to non-ideality of compression can be reduced by multistage compression, which also allows a closer approach to thermodynamic perfection by the use of multiple expansion valves (see below). The other losses are inherent and in general reduce the actual coefficient of performance to between 0.6 and 0.8 times the theoretical value of ;. ~

1 T. MIDGLEY and A. L. HENNE: Industr. Engng. Chern. 22, 542 THOMPSON: Industr. Engng. Chern. 24, 620 (1932) for further early 2 See for example T. MIDGLEY: Industr. Engng. Chern., Feb.

J.

(1930). See also R. J. description. 1937, and publications

of Kinetic Chemicals (E. I. duPont de Nemours and Company). 3 See also publications of R. and H. Chemical Dept. E.I. duPont de Nemours and Company.

28

J.

G. DAUNT: The Production of Low Temperatures.

Sect. 11.

11. Multistage systems for operation to -50° C and below. One difficulty generally associated with single stage vapor compression refrigerators, certainly those using ammonia or similar working substances, was that for operation to - 50° C the compression ratio becomes uncomfortably high. It is convenient therefore to carry out the camp, r, pression in several stages, b Compressor. a Compressor b' by which one also gains the thermodynamical from c point of view 1. In a multistage compressor system, Coolon! in however, it is also possible out to gain greater thermodynamic efficiency by introducing also multiple expan,st [xponsion sion. It is at the isenthalpic v,. expansion where irreversibilities are inevitably introduced, and it is thermodynamically axiomatic that i! Tid [xponsion Vq/ve if these irreversibilities are Vz introduced by a sum of a succession of small temperature drops rather than by one large one, the total irreversibility is reduced. Fig. 25. Flow diagram for idealized two·stage compressed vapor refrigerat· In such a multi-stage coming machine using also two-stage expansion. pression, multiple expansion system, the vapor fraction appearing after each expansion is returned at its equilibrium pressure to a junction between stages of compression. p 1i 12 & The multi-stage com/ i ; 1 1 1 pression, multiple expan1 I I Sz 1 f / ,/ s, sian, vapor compression PJ{It,sAt) c/~--::7// machine is illustrated in I) b/ / principle by the two stage p3 (z,JJAt) ---------+system of Fig. 25. The whole p,(o,J9At) ---------~ n output of the second stage "' at pressure p3 at c is led into __J,--::'::--'-=--~;;---~ 11 the condenser at c' and liquefied at d at the condenser Fig. 26. P-H MoLLIER diagram showing thermodynamic cycle of operations for idealized two·stage compressed vapor refrigerating machine of Fig. 25. temperature, 'Fa. On passing the first expansion valve Ti, it is partly liquefied in the intermediate evaporator at p2 and T2 • The remaining vapor fraction passes via b' back to the input of the second stage of compression at pressure, p 2 • The liquid is further expanded through the second valve, 1!;, to P1 and I;. and the liquid fraction collected in the main evaporator where it can absorb heat from the material to be refrigerated. The evaporating liquid, leaving at a at pressure p1 is fed to the input side of the first stage of the compressor. The output from this first stage at pressure p2 is cooled to the saturation t emperature T 2 by passing it through the intermediate evaporator at b'.

----------f--i--#

1

I

. _ __

__.L_ _ _ _ _ _ _

1 See for example N.R. SPARKs: Theory of Mechanical Refrigeration. New York : McGraw Hill ( 1938) for a discussion of the relative efficiencies of multi-stage compression, single expansion, systems.

Sect. 11.

Multistage systems for operation to - 50° C and below.

29

The cycle of operations can be followed on the p - H MOLLIER diagram in Fig. 26 (here the lower case letters correspond to the equivalent points on Fig. 25). The cycle on this diagram is: Second stage adiabatic compression (all the gas) from b' -+C; condensation of all the gas in condenser from c-+c' -+d ; expansion through li;_ from d--+ e; return vapor at intermediate pressure, p2 , from e--+ b'; liquid in the intermediate evaporator corresponds to the point f. Next for the expansion at valve ~ the path is f--+ g; the return vapor is from g--+ a; the adiabatic compression of the fraction x of the gas in the first stage of the compressor is from a-+b and finally the gas comes back to b' by cooling in the intermediate evaporator. The numerical values of pressure and temperature noted on Fig. 26 are for a typical NH3 system. It should be noted that, whereas all the gas passes through the second stage of compression, only a fraction x passes through the first. Evaluation of x is made as follows: At the expansion at 1S_ from d-+e, a fraction, y, of liquid is formed given by: y = b' efb' f.

{11.1)

This however is not the fraction, x, of substance subsequently passing on through

Y; from f to g. This latter fraction, is given by the heat balance equation for the adiabatically isolated intermediate evaporator, which is obtained as follows: The heat developed in the intermediate evaporator, Q2 , by cooling of the fraction x from b-+b' is: (11.2)

and this must equal to the heat extracted by evaporation in the intermediate evaporator in amount: (11.3) Q2 = (y - x) (Hb'- H1). By combining eqs. (11.1), (11.2) and {11.3) we get Hb'- He

X= - ·-

---

Hb-Hf

Hb,- Hd -·· -·- . =-

{11.4)

Hb-Hf

In addition the fraction, z, of the substance passing through liquefied is

~

which is thereby {11.5)

so that the final fraction, e, of all the gas flow out of the compressor (second stage) which is liquefied in the main evaporator is: e

=

xz.

(11.6)

To compute the overall coefficient of performance, ~. of this two-stage machine, we note that the heat absorbed, Q1 , at the low temperature, ~, is: {11.7) and the heat, Q3 , finally given out in the condenser at T3 is : Q3= He- Hd .

(11.8)

The work done, W, therefore is: (11.9)

J.

30

G.

DAUNT:

The Production of Low Temperatures.

and the coefficient of performance, 1

,;=

W

Ql

~is

=

Sect. 12.

given by:

He- Hd 1 1 Ha-Hg ·--;-- .

(11.10)

For the conditions indicated in Fig. 26 for the two-stage multiple expansion NH 3 machine (i.e. T1 = - 50° C, T 2 = -15° C and T3 = 30° C) the following values are found for no loss conditions: X=0.735;

Z=0.89;

e=0.655;

~=2.26;

'l]re]=0.81.

(11.11)

It is of interest to compare this for a theoretical single-stage system, also using NH 3 , operating between the same end temperatures, namely -50° C and 30° C. Here it is assumed that the compression ratio of 29.5 can be maintained and that the system is also loss free. Such a system would be represented on Fig. 26 by the cycle a---+j---+d---+i---+a; and the fraction liquefied, e, at 1'r would be aifah. For this cycle: e=0.74;

~=1.9;

'l]re]=0.685 .

(11.12)

It is to be noted therefore that the fraction liquefied is greater here than in the two-stage system, indicating that per cal. of refrigeration at Tr more substance must be circulated per minute in the two-stage system than for the single-stage machine. However, the two-stage machine is about 20% more efficient theoretically and approaches closely to the ideal CARNOT cycle. Machines of this kind employing as working substance NH3 and C0 2 and Freon are used for refrigeration to - 50° C and below 1 , the first being due to LINDE in 1898. Three-stage compressor multiple expansion machines have been built for still lower temperature operation down to - 75° C.

12. Cascade compressed vapor systems and the liquefaction of air. In reaching for lower temperatures by vapor compression refrigeration historically the quest was to obtain final evaporator temperatures sufficiently low to liquefy air, nitrogen or oxygen by compression only. The critical temperatures of these socalled "permanent gases" are (cf. Table 8) respectively 132.5° K, 126° K and 154-3° K. Evaporator temperatures below -147° C were therefore desirable. As has been discussed above, to approach lower evaporator temperature working substances with lower boiling points than NH3 , S0 2 , etc. are required, such as are provided by ethylene and methane (cf. Table 3). Since these substances have themselves critical temperatures well below ambient, being 282.8° K for ethylene and 190.6° K for methane, it is necessary to provide condenser temperatures for their vapor compression cycles by the evaporators of other vapor-compression machines operating to higher temperatures. This is the so-called "cascade" system. To make the most general illustration of the cascade system of refrigeration and gas liquefaction, KEESOM's model 2 for N 2 liquefaction will be described. In this system four substances are operated in cascade, the fourth (substance D of Fig. 27) being N 2 . As shown in Fig. 27 the first cycle, using substance A, produces liquid A in evaporator A by the vapor-compression refrigeration process. The cold gas A leaving evaporator A exchanges its heat with the warm gas B leaving compressor B. Gas B is subsequently condensed by thermal contact with liquid A in evaporator A. Evaporator A therefore is also the condenser B 1 2

See for example H. E . REx: Refrig. Engng. 58, 566 (1950). W. H. KEESOM: Leiden Comm. Suppl. 76a (1933).

31

Cascade compressed vapor systems and the liquefaction of air.

Sect. 12.

for substance B. It is possible therefore to use as substance B a gas with a critical temperature only just greater than the temperature in evaporator A. After passing the throttling valve VB, substance B collects as liquid in evaporator B, which in turn serves as condenser C for substance C. Finally the temperature maintained in evaporator C is sufficiently low for substance D to be condensed therein by compression alone. After condensation in the condensing or precooling spiral Din evaporator C, liquid D collects at lower pressure in reservoir D, Svbslt7flce A

II

c

/}

Compressor Supp(y /J ifl

out

Fig. 27. Schematic flow diagram for idealized cascade compressed vapor refrigerating (and/or liquetying) machines using four different working substances in cascade.

wherefrom it may be withdrawn through the drain valve. It is to be noted that the efficiency will be enhanced greatly by use of the interchangers, B, C, and D whereby the incoming warm gases B, C, and D are cooled before reaching their respective condensers. Their introduction in the cascade process was due to KAMERLINGH-0NNES 1 .

It will be seen therefore that each cycle of the system is a vapor-compression refrigerator, the efficiencies of which, as we have seen above, approach closely those of ideal CARNOT machines. 1 H. KAMERLINGH-ONNEs : .Leiden Comm. 14 (1894); 87 (1903).- Leiden Comm. Suppl. 35 (1913). See also C. A. CROMMELIN: Leiden Comm. Suppl. 45 (1922).

32

]. G. DAUNT: The Production of Low Temperatures.

Sect. 12.

In the system investigated by KEESOM 1 , which was previously successfully operated in practice 2 , the substances A, B, C, and D were respectively ammonia (NH3), ethylene (C2H 4), methane (CH4) and nitrogen (N 2). One of the advantages of this choice was that no substance had to evaporate at less than atmospher~.; pressure, thus allowing standard type compressors to be employed. Details of the temperatures and pressures of condensation and evaporation of the various substances in this system are given in Table 7. Table 7 also shows the main features, giving working substances, temperatures und pressures (where known) of the historical landmarks in the development of the cascade process of air liquefaction 3 . PICTET's first attempt in 1877 4 is noted here. Using S0 2 and C0 2 in a double cascade system he claimed to have reached 13 3o K in the C0 2 evaporator. High pressure air at about 500 atm. passed in thermal contact with this evaporator and was expanded at a valve. No liquid was formed, only a jet of mist. It must be concluded, since the critical temperature and pressure for air are 132.5° K and 37.2° atm. (cf. Table 8), that the temperature of the C0 2 bath was much higher than 133° K. The Polish school under Table 7. Data on early cascade compressed vapor refrigerators for air, 0 2 or N 2 liquefaction. Designer

I

P ICTET 5

I

WROBLEWSKI

and

Substance A Condensed A at Temperature: °K Condensed A at Pressure: (atm.) Evaporated A at Temperature: °K Evaporated A at Pressure: (atm .)

so2 248 2.75 208

Substance B Condensed B at Temperature: °K Condensed I\ at Pressure: (atm.) Evaporated Bat Temperature: °K Evaporated B at Pressure: (atm.)

C02

~1

+

I

I

I

Condensed C at Temperature : °K Condensed C at Pressure: (atm.) Evaporated Cat Temperature: °K Evaporated Cat Pressure: (atm.)

i

+

195

~175 ~0.07

C2H4 I l-175 ~2.5 [ 1.4 145 ( ?) -123 ~1 -0.13

: ~208

Substance C

K - ONNES 8

OLSZEWSKI 8

C2H4 -195 ~3.7

-100

-so

I

0

so

100

Temperature

150

zoo

JOO

Z!iO

0

C

Fig. 30. The joU LE-THOMSON coefficient, a.0 for N 2 as a function of temperature for various pressures, as marked. For the lower curves the zero has been shifted, as shown, to clarify the reading. [After]. R. ROEBUCK and H. OsTERBERG: Phys. Rev. 48, 450 (1935).)

and MILLER: J. Arner. Chern. Soc. 64, 400 (1942) on C02 . - J OHNSTON, BEZMAN and H ooD : J. Arner. Chern. Soc. 68, 2367 (1946) on H 2 . - JoHNSTON, SWANSEN a n d WIRTH: J . Arner. Chern. Soc. 68, 2373 (1 946) on D 2 . - CHARNLEY, IsLES and TowNLEY: Proc. Roy. Soc. Land., Ser. A 218, 133 (1953) on N 2 , C2 H 4 , C02 , N 2 0 . - For a detailed review of the experimental work up to 1929 see H. LENZ, Handbuch der Experirnentalphys ik, vol. 9/1, p. 47, 1929 and A. EucKEN, Handbuch der Experirnentalphy sik, vol. 8/1, p. 511, 1929.

3*

36

J.

G. DAUNT: The Production of Low Temperatures.

Sect. 13.

on the absolute values of the input pressure and temperature. Curves showing the JOULE-THOMSON coefficient rxH, as a function of temperature for various pressures ' for air ' due to HAUSEN 1 and for N 2 due to ? IJ.,.ZB RoEBUCK and OsTERBERG 2 are shown in Figs. 29 and 30. ~0,28 will be seen that quite generally the cooling effect, It :il' ~ O,Z'I \\ i.e. the numerical value of rxH, decreases with increasing pressure. The behavior with variation of T js more comt!"" o,u plicated showing a maximurn at low pressures near the O,!JJ 1\ \ critical temperature Tc and indicating that the position ~ 0,18 of this maximum shifts to ~ \ ~\ l\ higher temperatures as the '\ l\ pressure is increased. It is be noticed (cf. Fig. 29) to I'\: i"- ~ for example in air at that ['-.,~ 1"'- ~ p = 200 atm., rxH is negat- -?Po"'-.... j ["---., ~ """" ["'-..," !"--, ~ tive (i.e. there is a heating ~ ...... ""-... """-- ~ C"- ~ ::::::-.. effect) if the initial tem........ .......... ............ perature is below about 1:::::::, '-... ::----.. ~ ::::::-.. 15 3o K. This behavior for -.......... t:=: ::-....- :::-.....::::--- ~ air is typical for all gases. o,oz r-- :::::::: t:::: t-- It is also observed that at 0 --r-::::: sufficiently high temper-o.ozo zo 110 &o ao 100 tzo 1110 tGO 1ao zoo zzo t'IO zGO zao atures, rxH is negative for oc all values of the pressures. Temper(J/ure Fig. 31. The jou LE-THOMSON coefficient, a.H' for air as a function of temper(This is illustrated by Fig. 31 ature for various pressures, as marked. This shows the high temperature gives the variation which behavior where rx9 tends to negative values. (After]. R. ROEBUCK: Proc. of rxH with temperature Amer. Acad. Arts Sci. 64, 287 (1930).] for air as measured by 3 p = 0, rxH changes from for where ROEBUCK .) The limiting temperature positive to negative values is called the "Inversion Temperature", 1'; . Above 1';, rxH is always negative fm; all values of the pressure p. The inversion temperatures, 1';, of many commonly used gases are given in Table 8 and from eq. (13.5)

\I\ 1\\

"'

""-('"'-

;.""

"' '""'

Substance

"

;;;

Boiling point

OK

r::::

I Critical tern-

perature

OK

::;:

P::

Table 8.

I Critical I pressure atm.

Inversion temperature

Triple point

RTc

r= Pet'c

OK

OK

I

Triple pressure Cm.Hg

!

He3 Helium He4 Helium Hydrogen H2 Deuterium Dz Tritium T2 Ne Neon Nitrogen N2 Air (21% 0 2 ) I Carbon Monoxide CO A Argon Oxygen 02 Kr Krypton Xe Xenon 1 2

3

*

3.2 4.2 20.4 23.7 25.0 27.2 77-3 80.1 81.1 8].4 90.1 121.3 164.0

3.34 5.19 33.2 38.8 43.7* 44.4 1·26.0 132.5 134.1 151.1 154.3 210.1 289.7

1.15 2.2G 12.98 17.4 20.8* 25.9 33.5 37.2 34.6 48.0 49.7 54.0 58.2

"'so. 5 ( ?) 204.6

3.27 3.13

621 603

3.25 3.42 3.42

723 893

Il I I

3.43 3.42 3.44 3.58

H. HAUSEN: Forsch. Ing.-Wes. 274, 1 (1926). J. R. RoEBUCK and H. OsTERBERG: Phys. Rev. 48, 450 (1935). J. R. RoEBUCK: Proc. Arner. Acad. Arts Sci. 64, 287 (1930). Calculated. See E. F. HAMMEL: J. Chern. Phys. 18, 228 (1950).

13.96 18.73 20.6 24.57 63.1 83 9 54.4 104 133

i

5.4 12.86 16.2 32.4 9.29 51.2 0.12

I

it will be seen that at

37

The inversion curve.

Sect. 14.

T;, (13 .7)

From Table 8, the values of T;, for the gases originally used by JouLE and THOMSON (air, 0 2 , N 2) except H 2 are seen to be well above room temperature. For H 2 and He for example, however, T;, = 204.6° K and T;, ~ 50,5° K 1 respectively, and hence heating effects would be observed in JouLE-THOMSON experiments at room temperature. 14. The inversion curve. A calculation of the inversion temperature and an analysis of the way in which rxy changes from positive to negative values for variations of the parameters p and T can be made from eq. (13.7), provided the equation of state for the gas under consideration is known. As a first approximation, it is instructive to make such calculations using the VAN DER WAALS equation of state as representative of the gas, namely:

(P + ; -) (v- b) = RT

(14.1)

where a and b are the terms due to the attractive and repulsive components respectively of the mutual interactions between the molecules. Now the condition that the JouLE-THOMSON effect should vanish, i.e. that rxu=O, is from eq. (13.7) given by

(14.2) where the value of (8Tf8v)p for the VANDER WAALS gas can be obtained directly by differentiation of eq. (14.1). This leads, after a little work, to the result:

p = ~RT- __1_Rrz 3b

3a

(14.3)

as a first approximation valid for T large. If "reduced" pressures, n, and temperatures, r, are used such that n=P/Pc; r = TfTc and r = RTc!Pcvc where p" vc and Tc are the critical pressure, volume and t emperature respectively, eq. (14-3) can be rewritten as :

(14.4) where again its validity is restricted to high values of r only. When r tends to the value 1, the relationship is considerably complicated. However, eq. (14.3) and (14.4) serve to show that the states of the gas, as represented by points on a p, T-diagram, for which the JouLE-THOMSON effect is zero, lie on an approximately parabolic curve. A graphical representation of this curve is given in Fig. 32 in which the broken curve is the locus of points for which rxy = 0 for a gas obeying a VAN DER WAALS equation. Every point beneath the curve represents a state of the gas for which a cooling takes place in the JouLE-THOMSON effect (rxy>O), whereas for every point above the curve a heating takes place. The intercept with the r axis at n = 0 at the high temperature side gives the inversion temperature, as previously defined. In reduced temperatures it will be seen that r; = 18/r for a VAN DER WAALS gas, as also is obtained from eq. (14.4). This serves to explain the curve of Fig. 31 for which, at t emperatures higher than the inversion 1

SOM:

Owing to the paucity of adequate data this value is not too accurate. See W. H. Helium. Amsterdam: Elsevier 1942.

KEE-

J. G. DAUNT: The Production of Low Temperatures.

Sect. 14.

temperature, rxH is negative for all values of p. Fig. 32 also shows that there is a lower inversion temperature given when r R:J 2.2/r for a VAN DER WAALS gas, a result which does not appear in 13 - - J 17cob eq. (14.4) due to its very approximate 10 ---- v!In der/¥(117/s nature at low r. For a VANDER 'vVAALS , ....----.............. 8 ',' Table 9. Values of the critical pressHre, Pc· ' fi / Region of ',,, critical temperature, T,, and r ( = RT,/Pc vc) //

/

/

used by JACOB [Phys. Z. 22, 65 (1926)] for his reduced inversion curve.

cooling on Jovle-Thomson ',, exp(Jnsion ',,

z ()

' ', Z

II

fi

8ttJtZt'lt6"

r-r--,

''

Substance

to

Fig. 32. The "reduced" inversion curve, giving the reduced pressure, n, as a function of the reduced temperature r. [After M. jACOB: Phys. Z. 22,65 (1921)]. The broken curve gives the reduced inversion curve for a VAN' DER WAALS Gas. (This fig. by courtesy of M. DAVIES: The Physical Principles of Gas. Liquefaction and Low Temperature Rectification. London: Longmans, Green & Co. 1949.)

C02 • CzH4 Oz air Nz H* 2 He**

Pc atm,

72.4 50.0 49.7 37.2 33.4 12.8 2.26

Tc oK

304 283 154 132.5 126 33.2 5-19

3.53 3.42 3.42 3.42 3.42 3.27 3.27

gas therefore held at any temperature between r = 2.2/r and r = 18/r there is a cooling in the ] OULE-THOMSON effect only for pressures less than some value determined by the "inver630 "K sion curve" of Fig. 32. zoo In practice,· ~real gases cannot be represented by a reduced VAN DER WAALS equation of state. The experimental values of the inversion of the ] OULETHOMSON effect for anumber of gases, namely C0 2 , C2 H 4 , 0 2 , air, N 2 , H 2 , have been collated by ]ACOB 1 and his results show thatall these gases quoted except H 2 are describable by one reduced inversion curve as shown by the full curve in Fig. 32. The actual values of p and T can be obtained from this curve by using m~---~r-----~----+-------+------~ the values of Pc, Tc and r 0 '--------,11:0':c---....,B p > p1 ), and the light broken curves are typical isotherms (T2 > T3 > T1 ). It is to be noticed that inside the heterogeneous region the isobars and isotherms are linear and coincident, having a slope equal to the absolute temperature. Now the cycle 1 If the compressed air is treated as a perfect gas, then for isothermal compression Jt;,ompr = R T2 In (pJpl)(e.

Handbuch der Physik, Bd. XIV.

4

so

J. G. DAUNT: The Production of Low Temperatures.

Sect. 18.

of the LINDE air liquefier is as follows: The point a represents the gas at T2 and p1 , i.e. at the input to the compressor. Isothermal compression takes the representative point to b at T2 and P2 • In practice T2 ~::::; 293 o K and P1 and P2 approximately 1 and 200 atm respectively. The path b to c represents the cooling of the input gas as it passes down the interchanger and at c the gas is isenthalpically expanded from p2 and T3 to p1 and T1 • This isenthalpic expansion at the valve is represented by the horizontal line (constant H), C--7-d. The point d therefore gives a measure of the amount of gas liquefied, the fraction c, liquefied at the expansion being (edjef). The liquefied fraction is represented by the point f, being saturated liquid at T1 and p1 . The fraction remaining gaseous is represented H by the point e, being saturated Pt p p1 vapour at T1 and p1 . This gaseous fraction returns via the interchanger to the compressor, lJ as represented in the diagram by the path e--7-a. Ji(393°K) If one supposes, as is implicitly assumed in the above graphical representation, that a) the thermal isolation of the interchanger and low temper!j (t70°K} ature parts of the machine is perfect, so that no heat is lost or gained therein and b) that the interchanger is perfect so that the gas emerging at low presFig. 4 s. H _ S diagram showing the thermodynamic cycle :f Sure regains the temperature]~, operations for the simple LINDE air liquefier of Fig. 43. then COnSidering the liquefication system, the enthalpy, H, is conserved. This is so since by hypothesis no heat is lost or gained and at the JOULE-THOMSON expansion H is invariant. Hence, the total enthalpy Hb of the gas entering the interchanger must be the same as the sum of the enthalpy Ha of the gas emerging from the interchanger and of the enthalpy H1 of the liquid produced. Since the liquid fraction is taken to be s, this enthalpy balance equation can be written: Hb = (1 - s) Ha + s H1 ) or (18.1) Ha-Hb

----1..-.//_.---------

f=~

·-~ .

Ha-Ht

This equation enables, s, the so-called "liquefaction coefficient" to be readily evaluated from a H- S diagram provided the working pressures and temperatures are known. In practice the temperature of compression T2 is fixed by considerations of convenience, being approximately a little above ambient temperature, and the input pressure p1 also fixed, being generally approximately 1 atm., a pressure at which it is convenient to supply make-up gas from a gasholder. In consequence Ha and H1 are predetermined. The liquefaction coefficient, s therefore, as is seen from equation (18.1), depends only on the value of Hb, becoming larger as Hb is made smaller. This result is of interest in that the liquefaction coefficient is independent of the conditions at the expansion and is determined instead by the conditions at the high temperature input end of the interchanger.

Sect. 18.

51

Use of the H- S diagram and conditions for maximum liquid yields.

The conditions at the input (high pressure) side of the interchanger (point b of Fig. 45) can be examined theoretically, since thermodynamically quite generally it is known that: (18.2) First this equation shows that for a perfect gas, for a mole of which pv = R T, (8Hj85)r = 0; i.e., isothermals on the H- 5 diagram are straight lines parallel to the 5 axis for perfect gases, a condition which is approximated at high values of 5 and T for real gases. For such a case therefore no liquefaction can be obtained since the enthalpies before and after isothermal compression would be the same [cf. eq. (18.1)]. Such a conclusion is supported by the previous considerations of the jOULE-THOMSON effect given in chapter C. Secondly this equation enables the condition for ~..> O,Zf ~--j-/1"::7"'--t-=!};~::t==::::::::J==---j maximum liquefaction coef- 1€ ·~ ficient, c., to be obtained. As ~ is indicated by eq. (18.1), c. ~ 0,15 r---t+T--7"'1-:: 7""''--t---+---+- ---l is a maximum when Hb is a .~ mm1mum. ~ 0,09 1----.'h'l-.7"'--- +---+---+---+-- --1 The isotherm at T2 , as ~ ·~ is shown in Fig. 45, has a ..., minimum at the point marked b1 , and at this point 170 atm zoq 1JG GB fOZ lnlfiq/ Pressure, p3 (8Hj85)r. =0. By eq. (18.2) therefore this minimum oc- Fig. 46. The computed liquefaction coefficient, •, for hydrogen liquefaction using a simple LINDE. circuit as a function of pressure, p The curs when: [Curves due to four curves are for four precooling temperatures, T 2•

2•

T 2 = v ( ~: )P.

(18.3)

JoHNSTON,

BEZMAN

and Hooo: ]. Amer. Chern. Soc. 68, 2367 (1946).]

If reference is now made to eq. (13 .4) it will be seen that the condition specified by eq. (18.3) immediately above is also the condition for zero JOULE-THOMSON coefficient. In other words for maximum c., T2 must be the inversion temperature corresponding to the pressure at the high pressure stage of the compressor. Another way of stating this conclusion, which was first arrived at by MEISSNER!, is that the maximum liquefaction coefficient is obtained when P2 and T2 correspond to a point actually on the inversion curve. For air this would mean that for T2 ~ 293 o K maximum liqu.efaction occurs for P2 = 440 atm. In practice values of p2 only about half of this value are employed. For hydrogen, using the data given by WooLLEY et al 2 and assuming T2 = 80° K one obtains p2 = 157 atm. For helium using the data of ZELMANOV 3 and assuming T2 = 15° K, one obtains p2 = 31 atm. In practice the choice of working pressure is not too critical since the maximum in the curve of c. versus p2 is a broad one. This is illustrated in Fig. 46, taken from the measurements of joHNSTON et al 4 on H 2 , where plots of the liquefaction coefficient, c., versus p2 are made for a number values of the temperature T2 • It will be seen from the curve for T2 = 80° K, that the maximum value of c.= 18.4% at p2 = 140 atm. On the other hand for P2 =100atm. c. is still as high as 16.6 %. W. MEISSNER: Z. Physik 18, 12 (1923). \VooLLEY, ScoTT and BRICKWEDDE: J. Res. Nat. Bur. Stand. 41, 379 (1948). a ZELMANov: J. Phys. USSR. 3, 43 (1940). JoHNSTON, BEZMAN and HooD: J. Amer. Chern. Soc. 68, 2367 (1946). 4*

1

2

J. G. DAUNT:

52

The Production of Low Temperatures.

Sect. 19.

19. Increasing the yield by precooling. Considerable increases in the liquefaction coefficient, s, and consequent decreases in the power requirement per liter liquefied, can be obtained by precooling of the high pressure gas before it enters the LINDE circuit. A diagram of such a system is given in Fig. 47. The high pressure gas leaves the compressor at P2 and T3 , enters the heat interchanger, Ev and cools in the refrigerator evaporator to a temperature T2 • This refrigerator evaporator in the case of air liquefaction might be the evaporator of an ammonia machine (cf. chapter B) or might be the lowest temperature evaporator of a cascade refrigerator. In the case of hydrogen liquefaction the refrigerator evaporator is a bath of liquid (N 2 + 0 2) mixture boiling under reduced pressure; whereas for helium liquefaction it would be a bath of liquid hydrogen also boiling under reduced pressure. The gas leaving the refrigerator evaporator at p2 and T2 then enters the LINDE circuit which consists of the interchanger E 2 and the JouLE THOMSON expansion valve, li;_. The remainder of the diagram is self-explanatory. O,f

Y, -Joule Thomson Vo!ve

O,f -0 .Fig. 47. Schematic flow diagram for simple LINDE liquefier with precooling.

1.--- ~ -!0

...---

-tfl -.Jfl

-w

...--- ...---fifl -6'fl

...---

-lfl

-Hfl

----{l(J

0

C-!flfl

Fig. 48. The theoretical liquefaction coefficient, e, for air liquefaction, using a simple LtNDE s ystem with precooling, as a function of the precooling temperature, T 2 , for constant input pressure, P2 , of 200 atm.

The states of the high and low pressure gas at the warm end of the interchanger £ 2 are connotat ed b and a respectively; then the liquefaction coefficient is given by eq. (18.1) where H1 is the fluid enthalpy at PI and TI . To obtain numerical values of e as a function of the temperature T2 of the gas emerging from the refrigerator evaporator, the enthalpy values, Ha and Hb must be taken from the thermodynamic charts (see Table 10 for references). We have computed e for a simple LINDE air liquefier with precooling for the following typical conditions, namely: PI= 1 atm., T1 = 80° K, p2 = 200 atm. The results are shown in Fig. 48 which plots s as a function of the precooling t emperature T2 . In practice the precooling is frequently performed in an air liquefier by an ammonia refrigerator so precooling the air to about - 45 o CI. With cascaded 1

R.

LINDE:

z. VDI 65, 1357 (1921).

53

Increasing the yield by precooling.

Sect. 19.

compressed vapor refrigerators, using for example Freon 22 and Freon 13, lower precooling temperatures can be reached. The high efficiency of these compressed vapor refrigerators allows a great gain to be made in the total power required for the air liquefaction. A table showing the computed (theor.) and some typical observed (pract.) efficiencies of precooled and non-precooled air liquefiers is given in Table 12. The difference between the theoretical and practical efficiencies Table 12. Air liquefaction. No. of kW-hrs per liter liquid 1 • Fixed data: p2 = 200 atm.; p1 = 1 atm; T1 = 80.1° K. Temperature

of precooling (T,) 'K

293 228 200

Simple

Ideal

reversible

process kW-hrfliter

Theor kW-hr/liter

0.167

1.05 0.61 0.50

I

Two-stage

LINDE

Pract kW-hr/liter

I I

2.04 1.09 0.83

I '

(percent)

Theor kW·hr/liter

9 20 24.5

0.71 0.40

•

I

LINDE

Pract kW·hr/liter

I

1.05 0.73

bein"g due to heat losses in imperfect insulation, imperfect heat interchangers and to non-isothermal compression. The latter source of loss can be minimized by multiple compression in many stages, but as mentioned previously it is difficult to get less than about 1.7 times the ideal isothermal values of work of compression. The losses due to imperfect insulation are strongly dependent on the physical size of the machine. For larger machines the surface to volume ratio diminishes and so larger machines suffer less loss from this source. As an example of this Table 13 (taken from "Industrial Gases" by H. C. GREENWOOD, Table 13. Outputs and efficiency of simple LINDE liquid air plants. (Taken from "Industrial Gases" H. C. GREENWOOD. Publ. Van Nostrand. 1919.) Liters of liquid air per hour without precooling Liters of liquid air per hour with precooling Power kW kW - hrfliter without precooling kW -hrfliter with precooling

0.75

5

2.6

14.2

3.45

2.85

12.5

35

70

20

so

100

39 3.12 1.95

78 2.22 1.56

142 2.04 1.43

pub. VAN NOSTRAND 1919) gives the outputs and efficiencies of various sizes of LINDE liquid air plants. It will be seen from the table that for the machines with precooling the 100 literfhr one is approximately twice as efficient as the 5 literfhr model. The question of interchanger efficiency is taken up in chapter I. In hydrogen liquefiers employing the simple LINDE circuit precooling is always necessary, in order, as explained earlier, to get the gas below its inversion temperature of 204.6° K. In practice, this is done by cooling the high pressure gas with a mixture of liquid 0 2 and N 2 boiling under reduced pressure. The minimum temperature possibly available ranges from 54.4° K for pure 0 2 to 6).1° K for pure N 2 , these being the triple points of 0 2 and N 2 respectively. H. LENZ: Handbuch der 1 Data taken from R. LINDE : Z. VDI 65, 1357 (1921). Experimentalyphisk, vol. 9(1, p. 127. 1929; see also M. DAVIES: Gas Liquefaction and Rectification. London: Longmans 1949. - M. RUHEMANN: Gas Separation. Oxford Press 1940.

54

J. G. DAUNT:

The Production of Low Temperatures.

Sect. 19.

Curves showing the liquefaction coefficient s for hydrogen, as computed from JouLE-THOMSON effect measurements of JoHNSTON et all, plotted as a function of the high pressure, p2 for various values of the precooling temperature, T 2 , have already been given in Fig 46. These curves, for T2 equal to 64° K, 69° K, 75 o K and 80° K show clearly the marked gain in the yield as the precooling temperature is reduced. Similar curves, shown in Fig. 49, for T2 between 70° K and 38° K, have been prepared by KEYES et al 2 from earlier thermodynamic data on H 2 by KEESOM and HouTHOFF 3 . These show more markedly the profound gain to be obtained 0,7,----,-----,-----,-----,- ---,

.......,

O,!i

.~ --- Uquio' Nz 8oth ... .....; !"-od4, namely prepurification of the H 2 - - -gas by means of a trap cooled with liqs uid air, boiling under reduced pressure, before the gas is passed into the low pressure gas holder. KAMERLINGHONNES estimated that by this means .lntercllonger Ez the impurity content is reduced to less than -.}0 %. 7 A quite different arreangement to deal with the purity problem was intro- Joule-Thomso~~ 1/u/ve v; duced by KAPITZA and COCKROFT in -L 5 1932 . Their hydrogen liquefier conCooling C'ondenser C sisted of two separate circuits, one a llzUq.uid 8oth B :'1 · ·-- -~closed circuit of very pure hydrogen f~~ - J'olio'if'ietf lmpurdies ~.91=; :;---:- Collect here which served as the refrigerant, and a - -1 .... ~ f=second of commercial technical hyC~ Pruin '8 drogen (purity 99.5%) which was con1/ulve liz densed in a low pressure circuit. A Fig· 59. Schematic flow diagram for the double circuit schematic diagram of the apparatus is hydrogen liquefier of KAPITZA and Co8KCROFT. [Nature, Lond. 129, 224 (1932).] given in Fig. 59. The closed high purity circuit, having a total content of about 0.7 m. 3 of H 2 at NTP, included the compressor which forced the gas at 160 to 170 atm. pressure at 1 into the first heat interchanger E 1 . This gas reached the temperature of liquid N 2 on leaving the liquid N 2 bath at 5. (For simplicity only one liquid N2 bath is shown in Fig. 59. In practice, two liquid N2 baths were used, the first one with the liquid N 2 boiling at atmospheric pressure at

I

1 2 3 4 5

W.·MEISSNER: Phys. Z. 29, 610 (1928). JONES, LARSEN and SIMON: Research 1, 420 ( 1948). C. B. HooD and E. R. GRILLY: Rev. Sci. Instrum. 23, 357 (1 9 52). H . KAMERLINGH-0NNES: Leiden Comm. 94 f. ( 1906). P. KAPITZA and J.D. CocKROFT : Nature, Lond. 129, 224 (1932).

Handbuch der Physik, Bd. XIV.

66

J. G.

DAUNT: The Production of Low Temperatures.

Sect. 25.

about 77° K and the second boiling under reduced pressure at about 65 o K. A heat interchanger was placed between the two baths. In this way the load on the pump reducing the pressure over the liquid N 2 was reduced well below that which would occur using only one liquid N 2 bath.) At 5 the pure H 2 entered the heat exchanger E 2 of the simple LINDE circuit and at 6 the isenthalpic expansion at the JOULE-THOMSON valve, J.i, took place. The liquid formed collected in the H 2 liquid bath, B, until it was half full; from here the remaining gas flowed through the spiral inside the cooling condenser C and back at 7 into the low pressure return circuit. Technical H 2 from cylinders with purity about 99.5% after passing through a reducing valve entered the second H 2 circuit at 3 at a pressure of from 3 to 4 atm. After being cooled by passing through interchanger E 1 and the liquid N2 baths, this H 2 passed into the cooling condenser, C, where it liquefied at 3 to 4 atm. This cooling condenser was filled with wire gauze to help liquefaction. The impurities solidified in C and fell into the bottom part of C which was of ample volume. To draw off the liquid H 2 the liquid flowed through the draw-off tube at 8, through a further cooling spiral 9 in the liquid H 2 bath B and out through the drain valve, Jt;. With this arrangement, by using relatively large diameter tubes in the condensing circuit, the impurities caused no blocking troubles and it was reported that this machine at Cambridge University produced about 4 liters liquid H 2 per hour. No detailed data on construction however were given. BLANCHARD and BITTNER 1 however have given a very full constructional description, including data on the heat interchangers, of a similar two-circuit H 2 liquefier which they constructed at Johns Hopkins University. This machine had a flow in the high purity H 2 refrigeration circuit of 34 m. 3 jhr at an input pressure of 150 atm., and yielded 8.4 liters liquid per hour when the precooling temperature was 63° K (cf. Table 14). The construction of a similar machine has been briefly reported by HUFFMAN 2 • 25. Liquid neon circuits. It was suggested by CLUSIUS 3 that many of the disadvantages of high pressure LINDE type liquefiers for liquid hydrogen could be avoided by using instead a neon liquefier and then liquefying the hydrogen in a separate low pressure condensing system. It is to be noted that the boiling point, under atmospheric pressure, of Ne is 27.2° K whereas the critical point of H 2 is 33.2° K (cf. Table 8). A somewhat similar system has been constructed and reported on by HooD and GRILLY 4 • It consisted of a high pressure LINDEtype neon liquefier, similar in essential details to the hydrogen liquefiers described above and a separate neon-hydrogen converter. The liquefier had an input flow rate of approximately 100 m. 3 N e per hour at from 140 to 170 atm. input pressure and produced about 17 liters liquid per hour with a precooling temperature of 71° K (t:=20%). This liquid Ne was then transferred into the "converter" which was a separate unit and at the same time H 2 gas at 6.6 atm. was condensed in the converter, thus producing 27.5 liters/hour of liquid hydrogen. In this way the plant produced about 20% more liquid H 2 per hour than it would do producing liquid hydrogen directly in the same LINDE-type liquefier. As the authors stated, it would have been preferable to make the two gas system all into one unit and this may account for their subsequent abandonment of the neon circuit. 1 2 3 4

E. R. BLANCHARD and H. W. BITTNER: Rev. Sci. lnstrum. 13, 394 (1942). H. M. HuFFMAN: Chern. Rev. 40, 1 (1947). K. CLusrus: Z. ges. Kalteind. 39, 94 (1932). C. B. Hooo and E. R. GRILLY: Rev. Sci. Instrum. 23, 357 (1952).

Sect. 26, 27.

67

Small hydrogen liquefiers.

26. Ortho-'>- para-Hz conversion. If normal-Hz, i.e. 75% ortho-Hz and 25% para-Hz, is liquefied and stored in storage Dewars, the ortho-'>-para conversion takes place relatively rapidly 1 • z and the heat of conversion would result in a rapid loss of the stored liquid hydrogen. Measurements of this effect have been made for example by LARSEN, SIMON and SwENSON 3 and GRILLY 4• The latter author for example found that a 25 liter capacity Dewar, which had a basic evaporation rate for fully converted liquid Hz of 22 cm. 3 liquid/hour, freshly filled with liquid Hz of 70% ortho-Hz content lost 10 liters liquid in the first 4 days; whereas the same vessel filled with liquid H 2 of 32% ortho-Hz content only lost 4 liters in the equivalent period. Very distinct gain in storage economy is to be made if ortho-para conversion is made to take place during or before the liquefaction process. In the Oxford University H 2 liquefier described by jONES, LARSEN and SIMON 5 provision for such conversion is made by having one of the precooling stages of the liquefier include a conversion catalyst. In this liquefier the high pressure gas passes over a charcoal catalyst contained in a vessel about 1.5 liters volume and maintained at 75° K (the charcoal was activated carbon grade C.S. supplied by the British Carbo-Norit Union Co.). In this way the exit hydrogen was converted to about SO% ortho-H2 content. In a system described by GRILLY 6 the catalytic system consisted of a circular bundle of seven brass cylindrical tubes, (each of 3,42 liter volume, 89 em. long, 7.00 em. o.d., 0.318 em. wall) connected in series and mounted in a Dewar at 78° K. The first two tubes contained charcoal and the others a chrome alumina catalyst (20% Cr20 3 on an Al20 3 carrier supplied by the Harshaw Chemical Co.). Approximately 18 m. 3jhr (NTP) normal H 2 at 4 atm. pressure could be passed through this and the effluent H 2 was of 51.8% ortho-H 2 content. In GRILLY's arrangement this effluent gas was fed into the low pressure feed line of the H 2 liquefier system. 27. Small hydrogen liquefiers. Many small hydrogen liquefiers have been made, used and reported on. They have proved of extreme value where liquid hydrogen is required only in small quantity, particularly since they can be operated from commercial high-pressure cylinders of H 2 without the need for expensive compressors. It would not be practical to list all those which have been described but the following gives a representative selection of them. First NERNST 7 used one producing approximately ~ liter liquid per hour. Subsequent descriptions, for example, have been given by LATIMER 8 ; LATIMER, BuFFINGTON and HoENSHEL 9 ; RuHEMANN 10 ; KEYES, GERRY and HICKS 11 ; AHLBERG, EsTERMANN and LuNDBERG 12 ; FAIRBANKs 13 ; DE SoRBO, MILTON and ANDREWS 14 . 1 E. CREMER and M. PoLANYI: Z. phys. Chern., Abt. B 21, 459 (1933). 2 SCOTT, BRICKWEDDE, UREY and WAHL : J . Che rn. Phys. 2, 454 (1934). 3 LARSEN, SIMON and SWENSON: Rev. Sci. lnstrum. 19, 266 (1948). 4 E. R. CRILLY: Rev. Sci. lustrum. 24, 1 (1953). 5 JONES, LARSEN and SIMON: Research 1, 420 (1948). 6 E. R. CRILLY: Rev. Sci. lnstrum. 24, 1 (1953). 7 W. NERNST: z. Elektrochem. 17, 735 (1911). See J. E. LILIENFELD: Z. kompr. fliiss. Gase 13, 165 (1911). 8 W. M. LATIMER: J. Amer. Chern. Soc. 44, 90 (1 922). 9 LATIMER, BUFFINGTON a nd HOENSHEL: J. Amer. Chern. Soc. 47, 1571 (1 925). 1o M. RuHEMANN: Z. Physik 65, 6 7 (1 930). 11 KEYES, GERRY and HICKS : J. Amer. Chern. Soc. 59, 1426 (1 937) . 12 AHLBERG, ESTERMANN and LUNDBERG: Rev. Sci. lnstrum. 8, 422 (1937). 13 H. A. FAIRBANKs: Rev. Sci. Jnstrum. 17, 473 (1946). 14 DE SORBO, MILTON and ANDREWS: Chern. Rev. 39, 403 (1946).

5*

J.

68

G.

DAUNT:

The Production of Low Temperatures.

Sect. 27.

It is considered sufficient to detail the description of only one of these machines, since they all are very similar. A diagram of the general arrangement of AHLBERG, EsTERMANN and LUNDBERG's H 2 liquefier is shown in Fig. 60. High pressure hydrogen from a number of cylinders initially at about 150 atm. pressure is first passed through a CaC1 2 drying tube and then into the purifying precooling unit A before passing into the liquefier B. The first heat interchanger I in A consists of 2 feet of f' o.d., 0.030" wall, copper tubing pushed inside approximately 2 feet of -f6 " i.d., -h" wall, lead tubing and wound into a helix of a few turns. The copper tube carries the high pressure gas; the annular region inside the lead tubing the low pressure. The four tubes containing the charcoal CaC~~

Manifold

tube

A

8

cxllousl lfydrogen cylinders /nlerchonger I

Cl>urco11/

- - lfi!Jit pressure hydrogen lines ·:·:.- tow pressure llydrq!Jcn lines Fig. 60. General arrangement of the small scale LINDE·type hydrogen liquefier of AHLBERG, EsTERMANN and LuNDBERG. [Rev. Sci. Instrum. 8, 422 (1937).]

are of !" o.d. heavy brass, 10" long, connected in series. The whole unit A fits in a Dewar 2~" i.d., 24" long containing liquid N 2 boiling under atmospheric pressure. The liquefier unit is shown in detail in Fig. 61. Interchanger II consists of 5 feet of double tubing exactly like interchanger I. The high pressure gas then passes through a pre-cooling coil, PC, made of 20 feet of ~" o.d., 0.030" wall, copper tubing. This coil is immersed in liquid N 2 contained in a closed space made with a German-silver tube 2t'' o.d., 3\z'' wall and 16" long; and the liquid N2 can be made to boil under reduced pressure by pumping through the exit tube, V P. After passing through the precooling spiral, PC, the H 2 gas enters the final LINDE circuit at the interchanger, TT. This is a "Twisted tube" interchanger following the design of G. F. NELSON, as described by BICHOWSKY 1 , and was made as follows: a 20" length off' o.d., 0.015" wall, annealed German-silver tube was rolled flat (with about 1~ mm. 1

F. R.

BICHOWSKY:

J. Ind. Chern. Soc. 14,

62 (1922).

Sect. 28, 29.

69

Comparison of isentropic with isenthalpic expansions.

inside clearance) and twisted around its axis to form a spiral of about 1 em. pitch. This was then inserted in a thin lead tube, l 6 " i.d., 0.015" wall, and this sheath formed the low pressure return. After passing the TT interchanger the gas expands at a needle valve and returns via the low pressure circuit to the atmosphere. The liquefier unit is contained in a Dewar approximately 2f' i.d. which serves to collect the liquid H 2 formed on expansion at li the valve and a vacuum jacketed syphon, 5, is s t = provided for withdrawal of the liquid H 2 • IV = In operation, if the liquefier Dewar was first b precooled with liquid N 2 , the machine began liquefying about 5 min after N 2 temperatures r-had everywhere been reached. For liquefaction 3 the flow was maintained at about 5 m. (NTP) I ~ -lnl!fr'cMn~er II per hour and yielded about 1 liter per hour when I T'). Then heat will flow from fluid F to fluid F', at a rate say q per cm.2 of wall surface. There will be temperature drops, L1 T, and L1 T', at the wallfluid boundaries associated with this flow of heat. Also there will be a temperature drop, LITw across the wall given by:

(42.1) where Aw and t are respectively the heat conductivity and thickness of the wall. The transfer of heat from fluid to wall, or vice versa, is described in terms of a heat transfer coefficient, oc, such that for small temperature differences:

q=

oc L1 T = rx' L1 T'

(42.2)

where oc and oc' are the heat transfer coefficients which give the rate of heat transference per unit temperature difference per cm. 2 of transfer surface 3 . The term rx, the evaluation of which is discussed in Sect. 46, is a function of a number of parameters of the fluid, such as the thermal conductivity, kinematic viscosity, density, specific heat at constant pressure. It is also a function of the shape and R. SPOENDLIN: J. Res. CNRS. 15, 1 (1951). See, for example, the design of W. F. GIAUQUE used by J. G. DAUNT and H. L. JOHNSTON. [Rev. Sci. Instrum. 20, 122 (1949).] 3 In this section and in Sect. 46 some basic information on heat transfer between fluids and solids is given. For more detail the reader is referred to the many texts on this subject, including for example W. H. McADAMs : Heat Tra nsfer. New York: John Wiley & Sons. 1942. - R. C. L. BoswoRTH : Heat transfer Phenomena. New York: John Wiley & Sons 1952. - M. JACOB and G. A. HAWKINES : Elements of Heat Transfer and Insulation. New York: John Wiley & Sons 1950. - Also reference is recommended to the very complete monograph by H. HAUSEN: Warmeiibertragung in Gegenstrom, Gleichstrom und Kreuzstrom. Berlin: Springer 19 50. 1

2

93

The temperature distribution in simple interchangers.

Sect. 43.

size of the boundary wall and of the fluid velocity past the wall. In the latter connection it should be emphasized that for all conditions of heat transfer treated here it will be assumed that the fluid flows are turbulent. The fluid mixing accompanying turbulent flow assures that all parts of the fluid contact the wall, so improving the thermal transfer. In general, differences of the material (if all metal) and of surface state of the wall have a negligible influence on oc.. However, if the wall should get "plated" with impurities condensed out of the gas flow, IE(inpuf} IE'(oufpuf) such a oil vapor, C0 2 , etc., the thermal trans(pT,) f fp'lf') fer is reduced. This is a serious problem in the practical operation of heat interchangers.

l

tTemp.

,-------r:~Z"'

l

t

_____l

til

-T--•

_1______

LIT'

T'

~--------j_________F':\~"'~"'"'"~

T'

I I I

-Cold End

(l-o}

I I

' "----------------~-«-' T2 ) at the entrance and exit of the input stream and the temperatures T; and T; (T; > T;) at the exit and entrance respectively of the return stream are constant. For simplicity suppose: a) The outer wall of tube a is perfectly lagged so that no heat may flow through it. b) That the conduction of heat down the material of the walls of tubes a and b is negligible. c) That the overall transfer coefficient, x, across tube b between the two gas flows is the same along the whole length of tube b and that the tube b is of uniform perimeter along its length. After the preliminary calculations have been carried out, consideration will be given to the effect of relaxing conditions a), b) and c) above. At a distance l from the colder end of tube a, let the bulk gas temperatures of the input, and output flows in the steady state be T and T' respectively. Let m and m' be the mass flow rates of the input and output gases and CP and CP' , the specific heats at the constant pressures p and p' respectively. Then define quantities ~ and ~' such that: ~

=m cp;

~~

=m' cp'·

(43.1)

The heat flow, dq, between the two gas flows over the length l to l + dl is: dq

= x (T - T') P dl

(43.2)

where x, as before, is the overall heat transfer coefficient [eq. (42.6)] and where P is the effective perimeter 1 for heat transfer of tube b. Also considering the gain or loss of heat in the gas along the length dl, we have: dq =

~ ~~ . dl = ~ d T,

dq =

~~

}

dd:'- dl = ~~ dT'.

(43.3)

Integrating eq. (43.2) between l=O and l=l, we have:

f T ~q l

xp l=

T' .

(43.4)

0

Also under the restrictions a) and b) above concerning external heat fluxes, one gets from the heat balance between the two gas streams between points l = 0 and l=l: (43.5) and from the heat balance over the whole length, L, of the interchanger (43.6) Eqs. (43.5) and (43-6) implicitly assume~ and~', i.e. Cp and CP'' to be independent of temperature. Such an assumption is in general only a crude first approximation. It is made here for simplicity, the more detailed treatment being left to the reader. 1

For further discussion of the term "effective" see Sect. 45.

Sect. 43.

The temperature distribution in simple interchangers.

95

One may re-express eq. (43. 5) in the following way, which will prove convenient (T- T')

=

(T2 - T;) -

(!'i-~-) (T2 -

T)

(43.7)

which by differentiation yields :

~) dT. -~,d(T- T) = (~'-

(43.8)

I

Combining eq. (43.8) with eq. (43-3), one gets: dq =

~~~~~

(43-9)

d (T- T') li!mp.

li!mp.

r,

r,

T,' T.' f

Ti t

Tz' length of Intercllllnger

t

Fig. 85. Temperature along length of interchanger of Fig. 84 as measured from cold end. Case of (!: > (!:'

tengtll of' lntercllonger Fig. 86. Temperature along length of inter· changer of Fig. 84 as measured from cold end. Case of(!:= (:£', is given in Fig. 85. As shown by the figure, (T - T') diminishes exponentially as L increases. To reduce the difference in temperatures, (T1 -T{), at the warmer to zero, i.e. in order that the emergent gas stream should take exactly the input gas input temperature, clearly an infinitely long exchanger would be necessary. If the total heat capacities per sec. of gas flow, (:£ and (:£' are equal, then the exponent of e in eqs. (43.10) and (43.11) is zero, whence

(43.12) a situation illustrated in Fig. 86, in which the temperature difference (T- T') between the two streams remains constant.

J.

96

G.

DAUNT:

The Production of Low Temperatures.

Sect. 44.

44. Some practical considerations of gas flows in interchangers used in low temperature producers. In general, the output flows which we have to consider in low temperature technology are smaller than the input flows, measured in gjsec. In liquefiers for example this is certainly true, for some fraction of the input flow is liquefied and this fraction does not therefore return back through the interchangers. There may be situations, however, where the two mass flows, mand m', are equal, such as may occur in a refrigerator. To determine therefore, in the extreme case of whether (;£: is greater or less than (;£:' [c.f. the definitory eq. (43.1)], a study must be made of the variation of Cp with pressure.

m=m''

t

Cp 0,11

·o,zc.~~--~--~~~~--,.~w~~M~~---~~~~-z~~~~--~~~~~-z~w~~z~~~~~z~w~-J.~~o~~~~

r-

Fig. 87. Specific heat, Cp, at constant pressure for air as function of absolute temperature for various isobars. [Curves due to H. HAUSEN: Warmeiibertragung in Gegenstrom, Gleichstrom and Kreuzstrom. Berlin: Springer 1950.)

For a perfect gas Cp is independent of p. For a real gas, well above the critical temperature where its equation of state can be adequately described in terms of the first two virial coefficients only, it can be shown that for small pressures:

(.§._) op

T

= _

r(o2B). oT 2

(44.1)

This term is generally positive, i.e., Cp increasing with increasing p. This can be seen in the graphs of Cp versus T for air at various pressures given in Fig. 87. From this figure it will be seen also that the effect of pressure on Cp becomes much more complex near the critical temperature (Tc for air = 132.5° K). Since interchangers are generally used at temperatures well above ~. one may assume that in general Cp> Cp'• where Cp applies to the input (high pressure) gas and Cp' to the output (low pressure) gas. At temperatures above about 160° C, helium is an exception to this general rule 1 : for example, at 275 ° C (C 2ooAtfC1 At) = 0.99. Below 150° C, (8Cpj8p)r for helium is positive but small. For example, for - 50° C < T < 150° (C4oAt/C1 At) < 1.001. 1

W. H. KEESoM: Helium, p. 86. Amsterdam: Elsevier 1942.

97

Interchanger efficiency.

Sect. 45.

Assuming Cp>Cp' however, for most practical applications, one may further assume therefore that for T ~ I;, (44.2) since m:: : : m'. The condition (44.2) means that the exponent of e in eqs. (43.10) and (43.11) is negative (or in the extreme case zero). This in turn means that, as is illustrated in Fig. 85, interchangers are convergent, i.e., are such that (T- T') diminishes as one passes from the cooler to the warmer end. Eq. (43 .11) shows that to maintain a given value of the temperature difference, (T1 -T;), at the warm end of the interchanger for given T2 and at the cold end, the quantity

r;

(~,- ~) PL

(44.3)

must be held constant. As [' approaches [, therefore, the total length of the interchanger must be increased. This fact illustrates the difficulty of constructing highly efficient interchangers when the total heat capacities per second of gas flow, [ and [', are nearly equal. This fact, pointed out by jACOBS and CoLLINS 1 , explains why the interchanger requirements on a LINDE type gas liquefier, which have relatively large liquefaction coefficients, are less critical than those, for example, on a CoLLINs-type helium liquefier which has a much smaller value of coefficient of liquefaction. It also illustrates the fact that the constructional requirements on interchangers for low pressure air liquefiers are more stringent than for high pressure systems, since the ratio Cp(Cp' for the low pressure system is smaller. 45. lnterchanger efficiency. The efficiency of an interchanger can be defined as the ratio of the t ot al heat transfered per second from one stream to the other to the maximum possible heat transferable per second in an infinitely long interchanger. Confining ourselves to the practical case where

this means that the efficiency,

Q

.- = 'YJ = Qmax

Eq. (45.1) means that 'YJ

=

1-

'YJ,

is given by :

(§;'(T{-T;) (§;' (1:1 - 7:'2 ) = 1 -

(T1 -T{) (1:1- 7:2' ) .

(45 .1)

is given by:

'YJ

t emp. diff. between flows at warm end max. t emp. diff. bet ween ends of interch an ger

(45.2)

By use of eqs. (43.5). (43.7) and (43.11), one may obtain from eq. (45 .1) an alternative implicit expression for 'YJ in terms of parameters which are more readily pre-calculable, as follows: (§;

- 'T]Cf']

ln [ (1 -'T])Cf. or, putting [' = x [,

1

ln (

=

((§;-(§;')

(45.3)

~ xPL

1-_ 'T]'TJX) = (1 1

x)

l-----, Top cop

Ti"onsfer tube vtJ!ve Liqllld hydrtJgen - - - t - r t

Coil (a) - - - -t--n

--®Vacuum Jocket- --H-11 Ltould Helium - --++fH,_, Transfer Tube 1\t::_i:::=:=:::!..J) Exterior Case (fiTted w1¥!1 Lvcife windows)

Fig.14. Sectional view of Leiden liquefier. 1

J. G. DAUNT and H. L.

Fig. 15. Sectional view of helium liquefier at Ohio State University.

JoHNSTON:

Rev. Sci. lustrum. 20, 122 (1949).

129

Steady flow with hydrogen cooling.

Sect. 13.

IIi!

10im.

He 38otm. Totm

!SOatm.

~

P.friple

Fig. 16.

A s HMEAD

liquefier at Camhridge .

llydroge.?

Silica Cd Cliorcool of 90°/Ltm is apparent value of e for Tfe ""1. data, we ought presumably to restrict ourselves, in the first place at least, to the simple monovalent metals (Li, Na, K, Rb, Cs; Cu, Ag, Au) and to temperatures such that T/8 ::s 1. If the GRUNEISEN-BLOCH formula were strictly valid the parameter 8 would of course be found to be strictly constant, just as in the corresponding specific heat problem if the DEBYE model were exact. Otherwise we may regard as a temperature-dependent parameter, and KELLY and MAcDONALD 6 have considered in some detail the analysis of resistive and calorimetric data in this way. Using experimental results on the alkali metals from the experiments of MAcDONALD and MENDELSSOHN 7 together with further data obtained since in our laboratory, Fig. 25 has been drawn showing on normalized scales the behaviour of 8R as determined from electrical resistance measurements, by comparison with the BLOCH-GRUNEISEN formula. It is immediately evident that sodium appears to behave rather "ideally" since eR is found to be essentially constant from Tf8R ~ 0.8 to below Tf8R ~ 0.1'

N

E!JT

m 2 e2

1

x 5 dx

{(k!Ji) 2 + (x kT) 2 }2(e'"-1)(1 - e X)

(19.1)

0

where d is the density, c the velocity of sound in the metal, and if> an effective "screening temperature" which we define by: if>= Abo 8, where A 0 is the lattice n 2 parameter, and b appears in the screened CouLOMB potential - !___ e-r/b. This may be written alternatively

( e )4 Q oc ¢) .

1

V" T5 JV[ . (96

r

f

Ei/T

{1

xs dx

+ x2 (Tj$)2}2 (e'"- 1) (1- e

'") .

(19.2)

0 1 Cf. N. F. Morr: Proc. Cambridge Phil. Soc. 32, 281 (1936) or N. F. MorT and H. JoNES ([2], p. 86). 2 W. V. HOUSTON: Phys. Rev. 34, 279 (1929). 3 W. V. HousToN : Phys. Rev. 88, 1321 (1952). 4 L. NoRDHEIM: Ann. Phys., Lpz. 9, 607, 641 (1931). 5 F. SAUTER: Ann. Phys., Lpz. 42, 110 (1942). 6 C.-A. BussE and F. SAUTER: Z. Physik 139, 440 (1954). 7 J. M. ZIMAN: Proc. Roy. Soc. Lond., Ser. A 226, 436 (1954). 8 R. PEIERLs: Ann. Phys., Lpz. 4, 121 (1930). 9 E. H. SoNDHEIMER: Proc. Roy. Soc. Lond., Ser. A 203, 75 (1950).

'177

Later developments and the influence of electron screening.

Sect. 19.

For very good screening we have @jcfJ:7.5 kgauss (rT""'QT/9273oK). (b) Variation of magnetoresistance in tin crystal.

0·2 r---

r=.95xto-~ T~ZO·¥-'K./1=7-skG

_.,

€~ 0·10

.90

J '

~ 270 160

'

36'0°

(Angle of rotation)

b

J

a

71 I

0·6

I

f /

0·6'

4

/l

I IJV

Jr

V. ' ' I I! '' ' .

J 1/ ' '

..1 / If "' _,ti?"

..........

/

--

5 '

I

/

''

100

'

-

''

I

3,;,'

0·6'

''

20Q

/v I

'

2

!..---' p.--

3 ..... f.-"

kb

JrlQ

0

I

'

I

v

I

I

L

,:j

...-" ~

h'eltl-

'

1--

I

I

'

v I

I

1:--!-"'

~

0

1

I

.0·6

]I

;/

I

b

.1

v ~

I

I

1--

,{I

,.......... ,_.-

1

'/

,-- f-""'"

2~

200

100

fi'eltl-

v

_.. f.kb

J(JfJ

Fig. 32 a and b. Magneto-resistance of cadmium (a) and gallium (b). a: Curve 1, H perpendicular to I, Cd1 , temperature of liquid nitrogen;curve 2, H perpendicular to I, Cd 1 , temperature of solid C0 2 and ether; curve 3, H perpendicular to I, Cd11 room temperature; curve 4, H perpendicular to I , Cdn, temperature of liquid nitrogen; curve 5, H parallel to I, Cd1 , temperature of liquid nitrogen. b: Curve 1, H perpendicular to I, temperature of liquid air; curve 2, H perpendicubr to 1, temperature of solid C0 2 and ether; curve 3, H parallel to I, temperature of liquid nitrogen (after KAPITZA).

182

D. K. C. MAcDoNALD: Electrical Conductivity of Metals and Alloys.

Sect. 20.

period by charging up a bank of condensers 1 , or secondary storage cells as KAPITZA did earlier, and then discharge them rapidly through a coil. By placing the coil itself in the refrigerant bath the resistance of the coil may be kept very low, so increasing the current on discharge. These techniques have also been used by SHOENBERG 2 in measurements of the DE HAAS-VAN ALPHEN effect (the periodic variation with field strength of magnetic susceptibility observed in certain metals). OLSEN used a bank of electrolytic condensers of 3000 [LF capacity charged to 360 volts discharged through a coil 1. 5 em. in length with a mean diameter of 0.7 em., having 750 turns of 0.2 mm . copper wire. No external strengthening 7.6 was found necessary for the coil for the fields generated of about 150 kilogauss. About 50 cm. 3 of liquid helium 7.5 was evaporated by each discharge through the coil.

7.2

7.7

''

'

I ,I 5

I 10

I TS

I

I

20

II (Kilogovss-Approx)

25

30

Fig. 33. Transverse magneto-resistance of sodium. T=4.2° K; '; :.:::::::2 X 10-".

(J

25

50

700

fi(KG}-

150

Fig. 34. Magneto-resistance of copper at low temperatures (after OLSEN).

Many questions remain as yet unanswered in magneto-resistance of metals 3 . In particular, the linear variation of !JeHle with H as first found by KAPITZA (see Fig. 32; compare also Fig. 33 for sodium by MAcDoNALD, and OLSEN's results, Fig. 34, on the longitudinal effect in copper) remains unexplained. Theory .de

AH2

generally predicts -eH ~ i3+cH2. (e.g. SONDHEIMER and WILSON 4 ; (HAMBERS 5 ) and a quasi-linear section will exist for C H 2 ,......, B /3, but this region appears inadequate to account for the observations. It is perhaps worth remarking that in sodium, which we have concluded approximates rather closely to the ideal free-electron model, linearity of !JeHle with H (transverse field) does not make its appearance until lfR;:; 10 (cf. Fig.28}, while in cadmium, (hex. c.p. structure), on the other hand, lfR ~ 0.1 seems to be adequate. We notice also in OLSEN's work on copper that with a longitudinal field, linear behaviour starts (apparently rather abruptly) at about lfR ~ 2, while there appears no sign of this in the transverse magnetoresistance up to lf R,......, 7. It would evidently be very desirable to extend the work on transverse magneto-resistance in copper to higher values of lfR (which might Cf. W. J. DE HAAS and J. WESTERDIJK: Nature, Lond. 158, 271 (1946). D. SHOENBERG: Nature, Lond. 170, 569 (1952); see also Nature, Lond. 171, 458 (1953): Physica, Haag 19, 791 (1953). 3 See e.g. D. K. C. MAcDONALD and K. SARGINSON : Rep. Progr. Phys. 15, 249 (1952). (A Review with Bibliography). 4 E. H. SONDHEIMER and A. H. WILSON: Proc. Roy. Soc. Lond., Ser. A 190, 435 (1947). 5 R. G. CHAMBERS: Proc. Phys. Soc. Lond. A 65, 903 (1952). 1

2

Sect. 21.

The size-effect in metals.

183

be done with existing magnetic fields by using purer metal of greater mean free path) and to make longitudinal measurements on sodium over the available range of lfR. Although when lfR> 1 it may be necessary to consider specifically the influence of quantisation of electron-orbits in the magnetic field the example of cadmium just quoted in comparison with sodium suggests that some more "elementary" principle may perhaps be involved which we have not yet appreciated. 21. The size-effect in metals. A further consequence of the almost "ideal" behaviour of sodium is found in the influence of size on the magneto-resistance effect in that metal (MACDONALD 1 , MACDONALD and SARGINSON 2 , SONDHEIMER 3 , CHAMBERs 4); we shall first discuss the size-effect itself, and the information we can gain from experiments in this field. If a physical dimension, a, of a metallic specimen is made comparable with the electron mean free path, l, then the observed properties will differ from those of a conventional "bulk" sample. The scattering of the electrons at the surface, which we normally neglect in comparison with that in the volume of the metal, can no longer be ignored. At ambient temperatures, with z,.._,w-6 em., we require extremely thin specimens in order to observe such effects and indeed it may be questioned whether we are entitled to consider the structure then as identical with "bulk" metal. At low temperatures in pure metals, however, l is sufficiently great that no such doubts can arise when we are dealing with samples of perhaps 1'r; mm. in thickness. SoNDHEIMER 5 in a valuable review of size-effects in metallic conduction has pointed out that "since the calculation of the mean free path (for electron-scattering) from fundamental principles is highly complicated and involves many drastic approximations, it is desirable to have methods by which l may be estimated directly from observational data". It appears indeed desirable in general that in the theory of metals as many parameters as possible should be determined at least semi-" operationally", in BRIDGMAN's sense, and that every opportunity should be taken to reduce the number of "paper-and-pencil concepts" which occur basically in the theory. SoNDHEIMER observes that there are essentially three broad classes of size-effect conduction phenomena. First, we have the simple case where we observe the increase in resistivity over the value in "bulk" metal of a thin wire or plate ("film") arising from the limitation of normal mean free path, l, by the boundaries of the specimen; such measurements can then be used to deduce the ratio of l to the physical size, say a, of the specimen. Secondly we have the new effects referred to above which arise when a magnetic field is applied to such a specimen. In this case the radius, R, of the electron-orbit in the magnetic field is now involved and we are concerned essentially with the geometrical (and hence classical) relationship of R, l, and a. Not only information about l should now be available but also about R and hence the free-electron momentum. If we are not to be forced to extremely thin specimens of metal and very high magnetic fields, then low temperatures must be used to provide as large a mean free path as possible. Finally there is the so-called "anomalous skin-effect" 6 in metals, which might more appropriately be called the high frequency size-effect. In this case a D. K. C. MAcDONALD: Nature, Lond . 163, 637 (1949) . D. K. C. MAcDoNALD and K. SARGINSON: Nature, Lond . 164, 920 (1949). - Proc. Roy. Soc. Lond., Ser. A 203, 223 (1950). 3 E. H . SoNDHEIMER: Nature, Lond. 164, 920 (1949) . Phys. Rev. 80, 401 (1950}. 4 R. G. CHAMBERS: Proc. Roy. Soc. Lond ., Ser. A 202, 378 (1950}. 5 E. H. SoNDHEIMER: Adv. Physics 1, 1 (1952). 6 Dr. R. G. CHAMBERS has kindly advised me that the word anomalous should here be taken to mean simply "a new kind of". 1

2

184

D. K. C. MAcDoNALD: Electrical Conductivity of Metals and Alloys.

Sect. 21.

dimension, o, is introduced which corresponds to the depth to which a high-frequency electromagnetic field can penetrate into the metal. So long as ljo~ 1, classical theory applies and the high-frequency resistance due to the "skin-effect" is calculated in the normal way. When, however, l jo ,...._,1 the mean free path intrudes directly so that a situation broadly similar to the "normal" size-effect develops, and again low temperatures are essential to a study of this phenomenon. The theoretical study of these problems is now always based on the BOLTZMANN equation for the staa: tistical distribution reached by the electrons under the o"'-------x('tx) influence of the applied fields and the collisions 1 ; Fig. 35 . Thin plate analysis. the size-limitation enters directly through the appropriate boundary conditions imposed on the solution 2 • If we consider the case of a thin metal plate, thickness a in the x, y plane (see Fig. 35), the BoLTZMANN equation reads:

r

(21.1)

where f(vx, vl" vz; x, y, z) is the electron distribution function. Terms such as _d;}[ do not occur since we are only considering an electric field in the x-direction; similarly only offoz enters since in the other directions, where the film is supposed "infinite", the distribution function must be uniform. If now we assume we may set (ddf) where the " relaxation-time"

(:!-;) ·

T=ljv, and write

t

coil.

= -(!-I•), T

/ = /0 +/1 (v, z) then:

.

(21 .2)

which has the general solution: ExT · a:;;ofo { 1 + 'P (v ) e -Z'

R::;

3

eo

1

+ _}__!_ .

(21.4)

8 a

and if lfa> 1. (very thin films):

eo-

4

(!

l

1

(21 .5)

-~ log(i;--;;} .

The possibility of some degree of "elastic" scattering may also be included by assuming that a fraction, p, of the electrons undergoes specular reflexion at the surface, the remaining fraction, (1 - p), being diffusely reflected. This is of course only an interpolation device and cannot be considered as necessarily corresponding to any physical reality. Equations (21.4) and (21.5) would now read :

-~ = 1 + ~ _!_ (1 - p) 8 a

llo

and

1 e_411-P log (!f a) . a .1

e-:: - -j '

-tp'

In Fig. 36, e!eo is shown plotted against lfa for p = 0 and t· In thin rods 1 of diameter a ,the expression corresponding to (21.3) is now:

·e;; = (!

(

12

1 - ---;-

f1

(1- t2 )fi dt

I

oo

e

atx 1 {

- -

1

-:;s -

1 ~ xs }- dx

)-1

(21.6)

1 R. B. DINGLE Proc. Roy. Soc. Land, Ser., A 201 . 545 (1950) . - The theory for a wire of rectangular cross-section has also been given by MAcDoNALD and SARGINSON: Proc. Roy. Soc. Land. , Ser. A 203, 223 (1

if

w 1), then l' = l~(ak)- 2 (8.2) for low frequency waves (ak ~ 1), whf're a is of the order of the average distance between the molecular groups forming the glass (e.g. Si-0 tetrahedra in the case of quartz glass). This explains why l' oc T - 2 at low temperature, but if the observed value of l' is compared with the mean free path at high temperatures, which is identified with l~, it is found that the observed conductivity at low temperatures is too large by a factor of the order of 50 to 100, depending upon the choice of a. 1 F. SIMON: Ann. Phys., Lpz. 63, 278 (1922). 227 (1926).

F. SIMON and F. LANGE: Z. Physik 38,

Sect. 8.

217

Thermal conductivity of amorphous solids.

This discrepancy can be resolved by assuming that l~ is considerably greater than l~ 1 , though both are given by an expression of the form (8.2); that is l{=Aa(ak)- 2 if

ak( -

-

·

3

·-····.

10:n:a

K3Ma gs -- · T , 1i3y2

---·-

(9.7)

but did not derive an absolute value for r,. at Low t emperatures. The scattering by st atic imperfections has been treated in Sect. 6. It is seen from (9.1) that if roc w - n, xoc p -n. For point imperfections roc w- 4 ; for cylindrical imperfections without a long-range strain field roc w- 3 , but for dislocations rex: w - I because of their strain field; for grain boundaries r is independent of

220

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 9.

frequency. It is thus possible, in principle, to distinguish between different types of imperfections by the temperature dependence of the thermal conductivity. The external boundary of a crystal also causes scattering, because the real boundary is different in nature from the idealised boundary assumed in deriving the normal modes of the crystal. In general the mean free path due to boundary scattering is of the order of the smallest linear dimensions of the crystal. Since boundary scattering is the only process for which the absolute value of the mean free path can be estimated with reasonable certainty, some attention has been paid to the calculation of the effective mean free path. CASIMIR [11] has calculated the conductivity of an infinitely long cylinder on the assumptions that there are no interaction processes in the interior of the crystal, and thermal equilibrium is attained only at the boundaries, where phonons are absorbed, and re-emitted isotropically. The number of phonons at an interior point and of a given direction is governed by the temperature of their point of emission. This distribution, integrated over all directions, gives the local heat current, and integrating over the cross-section the total heat current is obtained. Thus

x=tSvL

(9.8)

where for an infinitely long cylinder of radius R L

=

2R L (vi)-2/(2.: (vi)-3)~. 1

1

(9.9)

If the phonons are in part reflected specularly at the surface, the effective value of L is increased; it is independent of frequency as long as the coefficient of specular reflection is frequency independent. BERMAN, SIMON and ZIMAN [46]

have made this generalisation, and have also calculated the reduction in the effective value of L when the rod is of finite length only. It is doubtful, however, whether the quantitative results of any of these calculations should be applied literally except possibly at very low temperatures, because these calculations assume the absence of three-phonon interactions conserving k, while over an appreciable range of the region where the imperfection mean free path l' is larger than L, l1 and l 2 are still smaller than L, so that the calculations are not necessarily valid. There is as yet no treatment of the effect of the ordinary three-phonon interactions on the thermal conductivity in the size-dependent region. If there are more than one of the above interaction processes, their combined relaxation time is given by 1

-, (w) =

.z: IX

1

T o: (w)

(9.10)

where r"'(w) is the relaxation time due to interaction processes (1X). Since this additivity applies only to each frequency separately, the overall thermal resistance W is not the sum of the thermal resistances due to each process, but in general, if llfi;. = 1/x"' is the thermal resistance for process IX acting alone, then

W>l.:llfi;.

(9.11)

(IX)

and the deviation from the additive resistance law is the stronger, the greater the difference between the frequency dependences of the r"''s, and is greatest when the two resistances are of comparable magnitude. The additive resistance rule is thus a useful qualitative rule, but when the two resistances are comparable, the exact integrals must be evaluated, or a cut-off approximation [20] used.

Sect. 10.

Thermal conductivity of crystals (Observations) .

221

The thermal conductivities at very low temperatures (T ~ 8) in the presence of the following scattering mechanisms, each acting alone, are as follows: a) External boundary or grain boundaries:

f (;x_ 00

x=

1

3 SvL

4nK4 T 3

= -- ~ L

x 4 ex

1) 2

dx

(9.12}

0

where L is of the order of the shortest linear dimensions of the specimen, or L ,.._,zoc 2 , l being the average distance between grain boundaries and oc the average angle of tilt. b) Umklapp-processes:

(9.13} as follows from (9.6), and xis given by (9.7) if T > 8 . The absolute value of x at low temperatures has not yet been estimated. c) Point imperfections:

(9.14} where a3 is the volume of a unit cell, G-1 is the concentration of imperfections per unit cell, and 5 is the scattering parameter defined in (6.4}; 5 2 has been estimated for a number of model imperfections [21]. d) Dislocations: Substituting (6.8) into (9.1) (9.15)

where b is the magnitude of the BuRGERS vector and A -1 is the number of dislocations per unit area. Eq. (9.12} to (9.15} apply only to isotropic materials, and vis the average value over all polarizations. 10. Thermal conductivity of crystals (Observations) 1 • EucKEN [25] measured the thermal conductivity of a number of dielectric solids down to oxygen temperatures and in a few cases down to hydrogen temperatures; he found generally for crystals that x varied roughly as T - 1 , in agreement with (9.7), and that the conductivity was larger for crystals with large 8-values. When measurements were extended to temperatures well below 8 the following classes of behaviour were observed: (a) x increases faster than T-1 with decreasing T, until a maximum is reached; at lower temperatures x is roughly proportional to the specific heat; this is interpreted in terms of Umklapp-resistance and, at lowest temperatures, boundary-resistance. (b) x varies as T -1 or more slowly, with decreasing T a maximum is reached, at lower temperatures boundary resistance predominates; the resistance above the temperature of the maximum is thought to be due to imperfections. (c) In polycrystalline solids the boundary resistance is enhanced, and the maximum is shifted to higher temperatures. I

See also

BERMAN's

review [5].

222

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 10.

Intrinsic or Umklapp-resistance. A variation of x faster than T-1 , indicating Umklapp-resistance, was observed by BERMAN in quartz 1 and sapphire [39], in the case of a very pure alkali halide [51] and in rutile (private communication), by WILKS, WEBB and WILKINSON in solid helium [42] to [45], and by WHITE and WooDS [121] in bismuth-see Sect. 23. In the case of diamond [43], [46] and germanium [50], [121] there is only slight evidence. Solid helium is particularly interesting, because by varying the density it is possible to vary f), SO that the dependence Of X on f) can be compared with the theoretically predicted variation (9.13)- Such a comparison can only be very rough, because the variation due to the factor (8jT) 2 is swamped by the factor / 9 /aT, and the theory in its present form makes no certain predictions about the value of 1)(. For the various helium specimens, x can be expressed as a unique function of 8fT, but the conductivities of other materials do not fit the same curve [44], [24]. But since these differences are not very large, it appears that the present theory is qualitatively correct. In the vicinity of the maximum it has been found in all these cases that the conductivity is very much less than would have been expected from a combination of boundary and Umklappresistances alone - for example see Fig. 6. This has been interpreted as additional resistance due to static imperfections. At first sight it seems a suspicious coincidence that this should be so JtF''-2.!..,_-~5l--..-----'ro=-~.a-:':v,------.f.:'::0"'°KT:"":'ra? in all the cases of class (a) so far observed. It TBmperolureshould be remembered, however, that crystals Fig. 6. The thermal conductivity of sapphire '11 f · ·h · single crystals of different diameters accordWI orm a continuous range Wit vanous amounts ing to [39], [5 ]. Full curve: measOf imperfectiOnS; if the imperfection resistanCe iS ured values; dashed curve: resistances combined according to the cut-off approximation high, they belong to class (b); if the resistance [ 2 0]; dotted c~rr:;~;::~iJ:~~~~s combined by is lower, they belong to class (a) with observable resistance at the maximum-say class (a'), and only for very low imperfection resistances would the belong to class (a) proper. But since Umklapp-resistance decreases very sharply with decreasing temperature, so that the maximum in the curve of x against T would be very sharp in the case of class (a) proper, class (a') corresponds to a very wide range of imperfection concentrations, so that it is understandable that with present techniques of crystal growths no case of class (a) proper has yet been observed. Boundary resistance. Boundary resistance in the case of large crystals was first observed in the liquid helium region by DE HAAS and BIERMASZ [30] in the case of quartz, diamond and KCl, and has since been observed for all dielectric solids measured at these temperatures. If this is the only resistive process acting, the following requirements should be satisfied: (I)() the conductivity should be proportional to the lattice specific heat, which means in most cases that is should vary as P, ({J) the mean free path deduced from the conductivity by substituting into (9.8) should be of the magnitude of the shortest linear dimension of the crystal, and (y) the conductivity BERMAN

1 DE HAAS

paid to it.

and

BIERMASZ

[29] also observed this variation, though no attention was

Sect. 10.

Thermal conductivity of crystals (Observations).

223

should vary linearly with the size of the specimen. In the presence of additional resistive processes some or all of these requirements are not fulfilled. In the case of additional scattering by internal or grain boundaries (ex) is satisfied, but the observed mean free path is less than expected from the external dimensions and of course less sensitive to changes in them. Improvement of the agreement with (/3) and (y) is not achieved by lowering the temperature, but by decreasing the external dimensions. In the case of additional scattering processes whose mean free path increases with decreasing frequency, neither (ex), (/3) or (y) is fulfilled, but agreement with (9.8) is improved both by lowering the themperature as well as by decreasing the size of the specimen. The measurements of DE HAAS and BIERMASZ [30] indicate an additional scattering mechanism with a frequency dependent mean free path. Even at their lowest temperatures (,....._,2o K) u varies more slowly than T 3 , and the discrepancy is greater the larger the crystal, also u varies more slowly than linearly with the diameter of the specimen. In the case of K Cl the deviations from (9.8) have been shown [20] to be consistent with the scattering by point imperfections which must be assumed (see below) to account for the thermal resistance at liquid hydrogen temperatures; since the frequency dependence of boundary and pointimperfection scattering is so different, the effect of the latter processes is appreciable at temperatures surprisingly far below the temperature of the maximum. The deviations from (ex), (/3) and (y) in the case of quartz [30], [20], synthetic sapphire [39] and solid helium [44] are probably caused by the same mechanism which prevents the attainment of the theoretical value of the maximum conductivity, discussed above. BIJL [34] and GARRETT [37] have measured potassium chrome alum from 1.4 to 3.9° K and from 0.16 to 0.29° K respectively. In the former region u varies as T 2 • 3 , in the latter as T 3 . However, the phonon mean free path calculated from GARRETT's values was only about 0.05 em, while the diameter of his crystal was 1.5 em, suggesting that in this case scattering by internal boundaries took place. There has been no experiment on size dependence in this case; presumably u should be independent of the external dimensions. Similar results were obtained previously by KuRTI, RoLLIN and SIMON [31] for potassium chrome alum and iron ammonium alum. Exhaustive tests of the CASIMIR theory were recently made by BERMAN, SIMON and ZIMAN [46] on diamond. Because of its high DE BYE temperature and the high temperature of its conductivity maximum, it was possible to eliminate other scattering processes in the liquid helium temperature range. On the other hand, the specimens were not long, and the finite length had to be corrected for. It was found that u oc T2·8 below 6° K; yet the observed mean free path exceeded the value calculated from the size of the specimens and the conductivity varied with specimen size. This was taken to indicate that the scattering at the external boundaries was not completely diffuse, but partly specular. The deviation from the T 3 dependence could thus be explained in terms of an increase with decreasing frequency of the coefficient of specular reflection, which seems reasonable. BERMAN, FosTER and ZIMAN [52] have also investigated the size-dependent conductivity of synthetic sapphire crystals-see also [46]. They found a mean free path, slightly smaller than expected from the theory, for rough crystals, but the conductivity was proportional to the diameter, and varied as T 3 . For crystals of smooth surface, however, they found a longer effective mean free path and a slower temperature variation, interpreted as in the above case of diamond.

224

P. G. KLEMENS: Thermal Conductivity of Solids at Low Temperatures.

Sect. 10.

Imperfection resistance. DE HAAS and BIERMASZ [29] measured the thermal conductivity of KCl and KBr crystals. They found at liquid hydrogen temperatures W ocT, instead of the exponential variation expected for intrinsic resistance. KLEMENS [20] suggested that the resistance arises mainly from scattering by point imperfections, in accordance with (9.14). At higher temperatures it was found that W increases more slowly with T; this is consistent with the expected departure from RAYLEIGH scattering (l oc w- 4) in the case of higher frequencies. At higher temperatures an appreciable fraction of the total resistance is probably intrinsic. The KCl specimen used by DE HAAS and BIERMASZ had impurities of Na + and Mg++ in concentration of somewhat less than 10- 4 per atom; also there should be a vacancy in the crystal for every divalent impurity to ensure electric neutrality. Substituting the observed resistance into (9.14), it is found that I: (5 2/G), summed over all types of imperfections, is about 1.2 x 10- 4 , in rough agreement with what is known about the impurity content. The interpretation of the thermal resistance in terms of point imperfection is supported by the fact that BERMAN [51] has measured the thermal conductivity of LiF crystal of such purity that its resistance was intrinsic, as indicated by an exponential temperature dependence; other alkali halide crystals showed a W ocT behaviour. There has been evidence of thermal resistance by static imperfections in the case of a number of crystals of class (a); however, the temperature dependence of the additional resistance is usually not determined sufficiently well to draw conclusions about these imperfections. However, in the case of diamond [43], [ 46] the extra resistance seems to be temperature independent; hence KLEMENS 1 has tentatively suggested that this resistance is due to disordered regions of diameter of the order of 50 A. EucKEN and KuHN [26] measured the thermal resistance of mixed crystals of KCl and KBr at comparatively high (oxygen) temperatures. The additional resistance due to admixture was approximately independent of temperature. This was also observed in the same temperature region by DEVYATKOVA and STILBANS [47] for KCl crystals with known F-center concentration; the latter experiment would have yielded extremely interesting results if the measurements had been made at liquid hydrogen temperatures. BERMAN, SIMON, KLEMENS and FRY [40], [39], [20] have studied the thermal conductivity of a quartz crystal after successive dosages of neutron irradiation, as well as the effect of annealing. Neutron irradiation produced an additional resistance which seems composed of two parts: one part increases with temperature, and this has been ascribed to isolated displacement defects; the other part varies as T-n, when n lies between 1 and 3; this has been ascribed to large disordered regions which arise when an energetic displaced atom produces an avalanche of displacements, in accordance with the theory of radiation damage 2 . Since quartz is a glass-forming substance, the material in the region of such an avalanche has probably become vitreous 3 . Polycrystalline Solids. The thermal conductivity of a solid consisting of many closely packed small crystals is, apart from a temperature independent factor which depends upon the density of packing, the same as the conductivity P. G. KLEMENs : Phys. Rev. 86, 1055 (1952). F. SEITZ: Disc. Faraday Soc. No.5, p. 271 (1949). For a review of radiation damage in solids see, for example, G. H. KINCHIN and R. S. PEASE: Rep. Progr. Phys. 18, 1 (1955). 3 Thermal conductivity measurements of neutron irradiated diamond and synthetic sapphire have now been reported by R. BERMAN, E. L. FosTER and H. M. RosENBERG: Report of Conference on Defects in Crystalline Solids, p. 321. London: Physical Society 1955. In the case of sapphire the interpretation is similar to the case of quartz. 1 2

Sect. 10.

Thermal conductivity of crystals (observations) .

225

of each small crystal. Since the dimensions of each crystal are small, boundary resistance is correspondingly enhanced, the conductivity is proportional to the specific heat, and the temperature of the maximum is shifted to high temperature s. PoMERANCHUK [14] had suggested that the scattering from a small-angle grain boundary varied as w 2 ; this introduced the difficulty that it was not certain where the difference lay in principle between such a grain boundary and the boundaries of a polycrystalline solid, which scatter independent ly of frequencysee for example [5]. It is now known [21] that a low frequencies the disordered region in the immediate vicinity of the grain boundary, which contributes to scattering as discussed by PoMERANCHUK, 2,'0 deg ~ gives rise only to a small part of the total W/cm50 h scattering probability; the major part comes ., 5 from the strain field at large distances, so that 'L 70 of ly a grain boundary scatters independent ,/ ~ ) 57 frequency. There is thus no difference, in prinI ,'f ciple, between a small angle grain or mosaic _\\ r- J .? r---/If ~-" \ boundary in a coherent crystal and a bound7 .? former the that /j except \ ary of a crystallite, I .; has a smaller scattering probability. This exll .? 5 plains the results of GARRETT [37] and of ld I to referred [31] KuRTI, RoLLIN and SIMON .? above. 5 In the case of compressed powders the I 5 .? size of the crystallites can be estimated and, -2 1/ :J assuming unit scattering probability at each II 5 J(J -J boundary, the boundary resistance can then be estimated. 5 .? I" Ill -.1 KURTI, RoLLIN and SIMON [31] and VAN 2 1 .? .!" 7tl i'tl ikl 7a7 .W°K DIJK and KEESOM [32) found that the conTemperalvr'l' ductivity of a compressed powder of iron am- Fig. 7. Thermal conductivity of aluminium oxide monium alum was about 1/10 of the conduc- according to BERMAN [ 41 ]. Dashed curve: 3 mm. diameter single crystal synthetic sapphire. Open tivity of the large crystal [31] which itself circles: same crystal ground to 1.5 mm. diameter. Full circles: sintered alumina. apparently had a phonon mean free path of only .....,0.05 em.; the grain size was not given. HunsoN [35] also obtained values for compressed powder of the same salt with crystallites estimated between 10-3 and 10- 2 em. in size. As pointed out by BERMAN [5) his average phonon mean free path was about 10- 3 em. and agreed with the size of the crystallites. Extensive measuremen ts on polycrystalli ne solids were made by BERMAN [41]. In Fig. 7 his results for sintered alumina are compared with the results for synthetic sapphire. The size of the crystallites was measured and found to range from 5 to 30 X ro- 4 em. At high temperature s the conductivity is. proportional to, and about half of, the conductivity of the large crystals; this factor ! is thought to arise from the packing geometry. At lowest temperature s the conductivity is about 1o-2 of the conductivity of the 1. 5 mm. specimen, and the phonon mean free path is thus of the order of 20 to 30 X 10-4 em., in rough agreement with the size of the crystallites; the conductivity varies as T 2·7, so that the phonon mean free path increases slowly with decreasing frequency, exceeding the geometrical mean free path. Similar results were obtained for sintered beryllia. BERMAN [41] also measured the thermal conductivity of various graphite samples. The interpretatio n is not so straight-forw ard as in the other cases, as

IV

I

I

I

I

I

v

I

Handbuch der Physik, Bd. XIV.

15

226

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 11.

the lattice is highly anisotropic. A remarkable feature is that in some cases the conductivity increases faster with temperature than the specific heat. This can be interpreted in terms of a scattering probability which increases with decreasing frequency. This variation is in the opposite sense of what is found usually, and is also theoretically unacceptable. KLEMENS 1 has discussed the specific heat and thermal conductivity of graphite on the basis of a DEBYE theory, modified for high anisotropy. He concludes that a variation of the conductivity faster than of the specific heat is possible in the case of flake-shaped crystallites. SMITH [48] measured the thermal conductivity of small flakes of natural graphite. He found the conductivity roughly proportional to the specific heat. Since each flake was crossed by slip-lines, so that each perfect region was of the same dimension in each direction, this result is consistent with the above interpretation; however, further work must be done before one can decide which is the correct explanation. 11. Other factors influencing heat transfer. It has been shown that the thermal conductivity can be written in the form x=

i 2: S; v;l;

(11.1)

i

where the summation is over all normal lattice modes, S; being the contribution of each mode to the specific heat, V; the velocity with which energy is transferred, and l; a mean free path. Eq. (11.1) should apply not only to conduction by lattice waves, but also to all other unlocalised excitations in the solid. If in general an atomic system is capable of excitation, and if this atomic system is regularly repeated through the volume of the crystal, any property of the atomic system (now a function of position) can be described by plane waves of wavevector k as in Sect. 3-see for example [10]. To every discrete energy level of the isolated system now corresponds a band of energy values E (k), the broadening depending upon the degree of interaction between neighbouring systems. The velocity of energy transport is the group velocity (n) - 1 8Efok. The system of waves will not be completely stationary, but there will be interactions among the waves and between the waves and excitations of a different character. Thus a relaxation time and a mean free path are defined, though sometimes this concept must be used with caution, as in Sect. 1. Particular cases of such excitations are the lattice waves, discussed previously, and the outer electrons of the atoms in a metal, to be treated in III. and IV. In addition the following excitations can contribute significantly to the specific heat, and may thus contribute appreciably to thermal conduction: spin, magnetic moment, rotation and orientation of molecules and other order-disorder effects, and movement of atoms from site to site. In all these cases the effect on the thermal conductivity can be twofold: there may be an additional component of thermal conductivity, but in so far as the additional excitations interact with other excitations (e.g. lattice waves) they will act on these excitations as an additional mechanism of scattering. This is exemplified in the case of conduction electrons in a lattice. In III. the additional conductivity due to the conduction electrons will be discussed; in IV. it will be shown that the conductivity by the lattice waves is reduced due to interactions with the conduction electrons. There have been some theoretical discussions on the role of spin interactions and interaction between magnetic moments in the case of paramagnetic solids. FROHLICH and BEITLER [16] have resolved the spins into progressive waves as above and have calculated the resulting conductivity, considering the mutual 1

P. G.

KLEMENS:

Austral. ]. Phys. 6, 405 (1953).

Sect. 12.

Thermal conductivity of liquids.

227

interactions of the spin waves as the only resistive processes. The spin waves are assumed to obey BOLTZMANN statistics. They found that X oc rn, where - 2 < n < -1.5, if T "-'0.01 o K, and at 0.06° K for potassium chrome alum x ~ 3 X 10- 7 watt. em. - 1 • deg-1 • This is considerably less than the observed conductivity at that temperature. The same problem has been treated in greater detail by PoMERANCHUK [17], while PoMERANCHUK and AKHIESER [18] treated conduction by the interaction of magnetic dipoles. The formal theory is the same in both cases; the excitations are quantized waves, and for reasons which are not clear to the present writer, they are assumed to be fermions. The formal theory of thermal conductions is thus analogous to conduction by electrons (see Sect. 13 and 14), and their effect on the lattice conduction is treated in a manner analogous to Sect. 19. They find xm, the conductivity of the magnetic excitations, to vary as T at lowest temperatures (scattering by imperfection), pass through a maximum and vary as T-1 at higher temperatures (scattering by phonons). The mean free path of phonons due to interactions with the magnetic excitations varies as T-1 , so that at lowest temperatures the lattice conductivity is unaffected and varies as P. They estimate roughly 1 the intrinsic value of xm and find that xm should exceed the lattice conductivity below about 0.02° K, provided, of course, that xm is not seriously reduced by static imperfections. Neither the experiments of KURT!, RoLLIN and SIMON [31] nor those of GARRETT [37] show any signs of a component of conductivity other than the lattice conduction, but this is not at variance with these theories. On the other hand REZANOV and CHEREPANOV [73] calculated the thermal conductivity of ferromagnetic metals by treating the spin waves as bosons. The role of the spin waves is mainly to scatter electrons, reducing the electronic thermal conductivity. Their theory is formally similar to the theory of Sect. 14. While the existing theory of conduction by magnetic interactions lacks experimental material to which it can be compared, there exists experimental material of thermal conduction in cases of anomalies in the specific heat, which has not yet been treated theoretically. Thus EucKEN and ScHRODER [27], GERRITSEN and VAN DER STAR [33] and VON SIMSON [53] have measured the thermal conductivity of hydrogen bromide, methane and ammonium chloride respectively. These solids have anomalies in their specific heat due to rotational or orientational states, and corresponding anomalies have been found in their thermal conductivities. These can be understood in terms of (11.1); thus in the case of methane reasonable agreement is found if vis taken to be Edf1i"-'Efkn1i, and l ""'d, where d is the intermolecular distance and E is the energy of the excitations, which is determined by the t emperature of the peak in the specific heat. While pure germanium, as measured by RosENBERG [50] and by WHITE and WooDs [121], shows the same behaviour as dielectric solids, a strongly impure specimen, measured by EsTERMANN and ZIMMERMANN [49], showed additional resistance which may be due to the scattering of lattice waves by electrons in an impurity band. 12. Thermal conductivity of liquids. The transport properties of liquids can be treated either on the basis of the kinetic theory of gases, extrapolated to the case of high densities and short molecular mean free paths 2 or on the basis of The crudeness of their numerical estimate was commented upon by BERMAN [5]. See, for example, J. 0. HIRSCHFELDER, C. F. CURTIS and B. R. BIRD: Molecular Theory of Gases and Liquids. New York : J. Wiley and Sons 1954. 15* 1

2

228

P. G. KLEMENS: Thermal Conductivity of Solids at Low Temperatures.

Sect. 13.

the theory of conduction in solids, extrapolated to the case of high disorder, with a possible additional contribution due to the ability of the molecules to migrate. The second approach would, of course, be similar to the treatment of amorphous solids in Sect. 8. The thermal conductivities of the following liquefied gases have been measured at low temperatures: liquid argon and nitrogen by UHLIR [54], liquid oxygen in a narrow temperature range by PRaSAD [55] and liquid helium I by GRENIER [56] and by BowERS [57]. The thermal conductivity of liquid helium II between about 0.6° K and the A-point is determined by two-fluid circulation and presents special problems 1 . Between the A-point and the boiling point the thermal conductivity of liquid helium was found to be proportional to temperature and to be roughly in agreement with the gas kinetic theory equation (12.1)

being the viscosity and Cv the specific heat per unit volume. FAIRBANK and WILKS [58] have measured the thermal conductivity of liquid helium II below 1o K. It is known that at very low temperatures the thermal energy of liquid helium resides almost entirely in the phonon gas; below 0.6° K the thermal conductivity could be expressed as 'YJ

~=tSvL

(12.2)

where L is fairly constant (increasing slightly with decreasing temperature) and of the order of the diameter of the tube which contained the liquid (0.03 em.). This suggests thermal conduction by the phonons, their mean free path being limited by the external dimensions of the specimen. This interpretation is also supported by ZIMAN's discussion 2 of the propagation of heat pulses below 0. 5o K. It is, however, remarkable that the mean free path does not appear to be limited by scattering arising from the disordered structure of the liquid, analogous to the scattering of phonons in amorphous solids; the phonon mean free path is given (8.3) by l =A a(ak}- 2 • (12-3) In quartz glass A "-'200 for longitudinal waves. With the same value of A, l should range from 10- 2 to 10-3 em. for the important frequencies in the temperature range covered. If the value of A in liquid helium is larger than in quartz glass, this would indicate a higher degree of local ordering in the former. It is not clear why this should be so.

III. Thermal conductivity of metals and alloys: Electronic component. 13. The free electron theory 3 • The present theory of electronic conduction in solids is based on the treatment of BLOCH [59], who in the first instance regards each electron as moving in a periodic potential produced by the metal ions and the other electrons, and then considers the deviation from periodicity due to the vibrations of the lattice as a perturbation. See the article on liquid helium by K. A. G. ME"DELSSOHN in vol. XV. J. M. ZIMAN: Phil. Mag. 45, 100 (1954). 3 For a detailed treatment see SoMMERFELD and BETHE [1), WILSON [4), MoTT and JoNES [3], and also the companion article by H. JoNES, vol. XIX of this Encyclopedia. 1 2

229

The free electron theory.

Sect. 13.

The wave-function of an electron in a periodic potential is of the form

1p=Uk(x)eik · :r, where uk(x) has the same periodicity as the potential, that is,

the periodicity of the crystal lattice. To each k-value correspond two possible electron states (of different spin) extending through the crystal; their energy E (k) is an eigenvalue of the ScHRODINGER equation

l7 2

u+ 2i(k·V)u + [~:- (E- V)- k ]u = 0. 2

(13.1)

The possible values of the wave-vector k are the same as for the lattice waves-see Sect. 3-and depend· only on the periodicity and size of the crystal. Additive changes of k by multiples of the inverse lattice vectors b, defined by (3.2), leave 1p invariant, and the k-space is separated into BRILLOUIN zones by planes which satisfy the condition for BRAGG reflection (13.2)

(k+nb)·b=O.

The energy E (k) depends upon the form of the potential. It is a continuous function of k within each zone, but is discontinuous across a zone boundary (13.2). Also, since E(k)=E(-k), the normal derivative of E with respect to kat a zone boundary vanishes~ If the potential is constant, E = 1i2 k 2f2m, and this is continuous across the zone boundaries, which in this case have no physical meaning. If the periodic potential departs only slightly from constancy, E (k) will depart only slightly from the free electron value except at the zone boundaries, where the normal derivative vanishes, and a discontinuity is formed. When allocating a k-value to a particular electron state of energy E, there is an arbitrariness, but one can choose a particular zone so that the free electron value 1i2 k 2f2m matches E as closely as possible. It may be necessary to then adopt an effective electron mass value which differs from the usual mass, and may even be negative. The values of E (k) in a zone trace out a "band" of energy values. At any point at the zone boundary there is discontinuity in energy. Thus two bands may be separated by a "forbidden" region of energy values, but on the other hand the existence of an energy gap at each point of the zone boundary does not exclude the possibility that the bands may overlap. According to the PAULI principle, the maximum occupation of each state is one. The number of k-values in each zone is G, the number of unit cells in the crystal, so that there are 2 G states. If the number of electrons per unit cell is odd, then at absolute zero at least one zone is only partly filled, and the substance is a metal. If the number of electrons per unit cell is even, it may be an insulator or a metal, depending on whether there is a gap between the highest full band and the next band, or whether there is overlap between these bands. The equilibrium occupation probability of a state of energy E is jO =

[ e(E- C)/KT

+ 1] -1 =

(e'

+ 1)-1

(13-3)

where C is a parameter, the FERMI energy, given by the condition

J/0 (E, C) n(E)dE =

N

(13.3a)

where n(E)dE is the number of states in the energy intervalE, dE, and N the number of electrons. This makes C t emperature dependent; in particular, if K T ¢;._ Cand N and n are kept constant (13 .4)

230

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 13.

The density of states per unit volume can be expressed as

n

(E)

2

=

(2n) 3

f

dS

(13-5)

!gradk El

where the integration is over the contour E (k) = E in k-space. If E (k) is a function of I k I only, the contour is spherical and _

8n

2

dk _

dk

n(E)- (2n)a k dE - n(k) dff•

The modulated plane waves are eigenstates only if the potential is purely periodic; in real crystals there are transitions of particles between the eigenstates due to deviations from the perfectly periodic potential. These processes maintain equilibrium. On the other hand, an electric field F or a temperature gradient 17T tend to disturb equilibrium. The BoLTZMANN equation, which is the condition that the actual occupation probability f of states is stationary, becomes _!!___

1i

F · _!!_ ok

+ _!__1i ~ ·17 T _!}__ = ok dr

})_] at

(13 .7)

where the right-hand side denotes the rate of change due to interaction processes. In the usual approximation f is taken to be / 0 on the left-hand side, and the rate of change due to interactions is described by a relaxation time of(k)]- / 0 - /

--at -

g(k)

_

(1).8)

T(k)--- T(k)

where it is assumed that T is independent of the deviations g of the state k and of all the other states. For an isotropic material and spherical energy contours the BoLTZMANN eq. (13.7) has the simple solution g(k) = - fk T

:~ ~ ~~

{eF -17T ( ~

+ :~)}

(13.9)

where fk is the direction cosine of k relative to F, and 17 T is positive if in the same sense as F. The electric current density is

. f e-y:; 7ik

J=

1 8E

2

(13.10)

g(k) (2 n)a dk

and the heat current is

(13.11) The electrical conductivity is defined as a= jJF if 17T = 0, and is obtained by substituting (13.9) into (13.10). The thermal conductivity is x = - Q/17T if j=O. Putting j=O into (13.9) and (13.10), a relation is derived between eF and 17T (EfT +dCJd T). Substituting this into (3.11) and (3.9), an expression for the thermal conductivity is obtained. It is noteworthy that (1).10) and (13.11) involve integrals of the following type, which can be expanded if KTfJ; similarly 12; (T) oc P if T fJ; thus one can intercompare the four magnitudes W; (Ti)fT12 and (!; (T1 )/Tl, where T1 8. It will be found that there are discrepancies in the relationships between these quantities. There are also descrepancies in respect of the temperature dependence of W; and (!; at the intermediate temperatures, but these are less important effects; the reasons for the discrepancies between the conductivities at the extreme temperatures will presumably also provide an explanation at intermediate temperatures. It has been shown that the WIEDEMANN-FRANZ law (13.15) should hold if a relaxation time can be defined independent of the form of g(k), and in Sect. 14 it was shown that this is the case at high temperatures; hence at high temperatures irrespective of the band structure

T2 W; (T2) =

_e;f_ l 2

,

(T2 > fJ).

(15.1)

This is indeed fulfilled for all metals except for those which have an appreciable lattice component of thermal conductivity, and can be regarded as a verification of the general principles of the free electron theory. Thus for purposes of testing the band model only three quantities are independent out of f2 (T1 ), e(T2), W (T2) and W (T1 ), and there are only two independent relations between them. Any two of the following three are independent T)

(!; (

1

=

497.

6

e(T2) rls -t(94 ' 2

W; (TI) = 64.0 Nc} W; (T2) ( ~~ . ~ (T1J_ _ _?4.o T12 497.6

m

_e2

a L

r,

Gt

_(Jj T15

•

(15.2) (15.3) (15 .4)

Eq. (15.2) is obtained from (14-30), and (15.3) from (14.28) and (14.29c). Eq. (15.4) is best derived by eliminating (!; (T2) and W; (T2 ) from (15.1), (15 .2), (15-3) ; it could, however, have been derived directly from the theory of Sect. 14 without reference to the conductivities at high temperatures. The thermal conductivities of a number of monovalent metals have now been measured on specimes of sufficiently low residual resistivity Wo and to sufficiently low temperatures that the ideal thermal resistivity W; can be deduced with confidence for the limit of low temperatures. Thus BERMAN and MAcDONALD [83], [84] measured sodium and copper; MENDELSSOHN and RosENBERG [85], [87] measured copper, silver and gold, in addition to various other metals discussed in Sect. 16; WHITE [88], [89], [90] carried out very extensive measurements on a number of specimes each of gold, silver-see Fig. 10-and copper, and MAcDoNALD, WHITE and WooDs [91] measured potassium, rubidium and caesium. In all these cases it was observed that, if N,. is taken to be of order unity and f) of the order of the DEBYE temperature, then the observed values of W; (T1 ) are much lower, relative to W;(T2 ), than predicted by (15.3). A similar effect

240

P.

G.

Thermal Conductivity of Solids at Low Temperatures.

KLEMENS:

Sect. 15.

has been noted by HULM [92), [93] and by ANDREWS, WEBBER and SPOHR [95] for non-monovalent metals, where, however, the quasi-free electron theory does not apply. However, to test the temperature dependences of (!;and W; directly by means of (15.2) and (15.3) introduces the difficulty, pointed out by KLEMENS [70] and by ZrMAN [71], that the magnitude of (!; (T2 ) - hence of W; (T2 ) -may not be given correctly by the BLOCH theory of Sect. 14, because that theory disregards Umklapp-processes (14.2b), dispersion of the phonon velocity at high frequencies, and a possible variation of the electron-phonon interaction parameter C (q) with phonon wave-number q. Eq. (15.4), however, is independent of these effects, deg ,. I I

\

~

!.fO

t

I

.

I

-x- = AgJ expcmded .swle

-o- =Agt

I \

"

"

1 II I

I

~ at~-x

I I I

O·Ag1

I~

If, n! I

I

I I

I

I

I

'

X

~

''

X

''

' "'

'

)C.--x-

~-

25

•·Ag5 -

l 1 in all cases, which implies that for some reason horizontal diffusion is relatively more effective than vertical diffusion in producing resistance. This can only be explained in terms of non-spherical FERMI surfaces [70]; once this is accepted, it will also explain the interaction between electrons and low-frequency transverse waves. The effect of deviations of the FERMI surface from spherical shape on the conduction properties will be two-fold: the integrals (13 .13) and (13 .14) will be affected even if -r is kept fixed, because v and gradk E depend on the function E (k), and secondly the relaxation times themselves will be affected. We are not concerned with the former effect, since this is common to all conductivities and disappears in (15.2), (15-3), (15.4), but with the effect on -r. If determined by vertical movement, as in the case of W; at low temperatures, -r depends only on the local properties of the FERMI surface, and will be relatively unaffected by changes in the shape of the FERMI surface. If, however, -r is determined by horizontal many-step diffusion, as in the case of l!i at low temperatures, it will depend markedly on the shape of the FERMI surface. The deformations of that surface will be such that the volume enclosed is unchanged, they will have the symmetry of the BRILLOUIN zone and they will be outward along those directions where the zone boundary is closest to the 1 M. BLACKM AN: Proc. Phys. Soc. Lond. A 64, 681 (1951). Cf. also panion article in vol. VII/1.

Handbuch der Physik, Bd. XIV.

BLACKMAN

16

s com-

242

P.

G.

KLEMENS: Thermal Conductivity of Solids at Low Temperatures.

Sect. 15.

centre of the zone. It was suggested [70] that such deformations reduce the average path length of an electron diffusing to an opposite point on the surface, hence they enhance horizontal diffusion relative to vertical movement and cause a deviation D 4 > 1. This effect should be particularly marked if the outward deviations are so large that the FERMI surface touches the zone boundary. This is more likely to happen for face-centred than for body-centred cubic metals, since the zoneboundary approaches the centre of the zone more closely. In face-centred structures there would be eight points of contact, of octahedral symmetry. An electron can reach a "neutral" region (i.e. one for which g = 0) not only by diffusing to a direction perpendicular to the direction of the field, but also by diffusing to a point of contact. In the latter case the average distance of travel is halved, so that the resistance due to such paths is four times the ordinary resistance, and since the ordinary paths also contribute to the resistance, D 4 " ' 5. While for body-centred structures it requires a stronger deformation for the FERMI surface to touch the zone boundary, the change in resistance will be larger if this does occur, for then there are 12 points of contact, the average path length to a point of contact is only one third of the average path lengths to the "neutral" region of a spherical surface, so that D 4 "'1 0. Table 1. Conduction properties of monovalent metals. Element I

Li [80]. 5 Na [83] 3.8 K [91]. 1.2 Rb [91] 9 .2 Cs [91] 2.2 Cu [84], 185] , [87]. [90 J 2.55 Ag [85], [87]. [89] . . . 1 6.4 Au [85]. [8 7], [88] . . . . 1.3

1

I

I

I

-

D,

I

X 10-4 X 10-4 X 10- 3 3 2

x wx w-

0.7 0.73 0.7 1.7 1.7

I 3.55.37 4 .5 6.5

x w-n

I

150 100 60 25(?i

x w-5

0.26

2.64

x w-16

!

315

330

sos I

6.2

5.4

x w-

5

0.24

1.11

X 10-15

215

220

340

5.2

4.2

x w-s

0.64

3.9 x 10-15

170

170

270

5.8

4.5

I

-

~350

X 10-15 X 10-13 X 10-12

~260

200 70 ~so ~25

~0.7

260 170 100

-

5.3 3.7 3.3

1 ~8

-

1.8 16 9 ~10

Remarks : L = 2.45 X w - 8 wa tt-ohm-deg 2, t hermal resistivities are given in watt-un its, electrical resistivities in ohm-units, T 1 ~ Bv is 0-value derived from low-temperature specific heat, €JL is the DEBYE temperature of the longitudinal polarization branch, (9R is the 0-value required to fit (15.2), D 3 and D 4 are defined in (15.5) and (15.6).

e.

It appears from inspection of the values D 4 in Table 1 that the FERMI surface touches the zone boundary in the case of the noble (face-centred) metals, and in the case of the two heavy alkali (body-centred) metals rubidium and caesium, while it does not touch in the case of sodium. This agrees with the above conclusion that the FERMI surface is least likely to touch the zone boundary in the case of the light alkali metals. The case of potas:.ium is anomalous, as it shows the highest value of D 4 . A close inspection of the electrical resistance curve 1 reveals that (!; rx T 3 above 6° K, and falls off sharply below that temperature. It is possible that the FERMI surface comes close to, but does not touch, the zone boundary. The low DEBYE t emperature and the close approach would result in Umklapp-processes being frozen out only at very low temperatures, apparently below 6o K. What has been taken to be a P variation of (!; below 6° K may really be the exponential 1

D. K. C. MAcDoNALD and K. MENDELSSOHN: Proc. Roy. £oc. Lond., Ser. A 202, 103 ( 1950).

Sect. 15.

243

Comparison with experiments: Monovalent metals.

variation due to the freezing out of Umklapp-processes, and the variation (!; oc T 5 may only be attained at lower temperatures and with a value of eJT5 much lower than the value assumed in the table. The residual resistance, of course, prevents observations of the low values of(!;· The thermal conductivity of lithium has been measured by BIDWELL [80] down to liquid hydrogen temperatures; W; varies as T2 at those temperatures, and it is found that D 3 ,_,0.7. The electrical resistance at very low temperatures varies as T4.5, instead of P as expected theoretically, so that a comparison between (!; and W; at low temperatures cannot be made by (15.4). Nevertheless it appears that e;/W; is larger than expected from theory. The anomalous behaviour of lithium is quite different from the more regular behaviour of sodium, and the reason for this difference is at present obscure. There is little independent evidence in support of the hypothesis that the FERMI surface touches the zone boundary, because it seems that electrical resistivity at low temperatures is more sensitive to this eventuality than other observed properties. There is qualitative evidence from the thermoelectric properties of monovalent metals 1 that the band structure deviates violently from the simple model in the case of the noble metals, and in decreasing degree, in the cases of caesium, rubidium and potassium. The change of the electrical re3istance in a magnetic field is also sensitive to the geometry of the FERMI surface. According to KoHLER 2 the change of electrical resistivity of a monovalent cubic metal in a strong transverse magnetic field should be isotropic (constant as the transverse field is rotated about the direction of the electric current), provided the FERMI surface does not touch the zone boundary. But the magneto-resistance of gold in a strong field is markedly anisotropic 3 so that in this case, and probably also in the case of other monovalent metals, it appears that the FERMI surface does touch the zone boundary. The discrepancies D 3 can be understood in terms of Umklapp-processes, which increase the high-temperature resistance relative to W; (T1 ) by a factor of up to 2, and in terms of the dispersion of the lattice waves. The latter effect, decreasing the frequency of the high-energy phonons by a factor of up to 1. 5, increases the high-temperature resistance by a factor ,_;(1.5) 2 because of the factor in (14.19). Thus D 3 can range up to about 4 to 5, apart from any effects due to a variation of the interaction parameter C (q). The absence of a maximum in the observed W; can be understood in t erms of the relative increase in (!;: to a rough approximation W; = Wv + WH, where Wv and WH are the resistances arising from vertical and horizontal movement respectively. At low temperatures Wv p WH, at high temperatures WH p Wv; also WH T = (!;/L. At intermediate temperatures Wv and WH are comparable, and in that temperature range Wv passes through a maximum, while WH increases monotonically with t emperature. With the ratio of WHfWv as determined by the theory of Sect. 14 it so happens that Wv + WH has a maximum value: it is easily seen that if WH is increased by a factor of order 4 to 5, this maximum will be greatly reduced and possibly eliminated, so that the absence of the resistance maximum and the high value of D 3 and D 4 are related. WHITE has observed for silver [89] and copper [90] that W; oc Tn, where n WH oc P, and is slightly larger than 2. This is explained as follows: for T ~

ve

e

1 See the companion article by D. K. C. MACDONALD; also D. K . C. MAcDoNALD and S. K. Rov : Phil. Mag. 44, 1364 (1 953). - P. G. KLEMENs: Phil. Mag. 45, 881 (1954) D. K. C. MAcDoNALD and W. B. PEARSON: Proc. Roy. Soc. Lond., Ser. A 221, 534 (1 954) , 2 M. KoHLER: Ann. Phys., Lpz. 5, 99 (1949). 8 E . }usn and H. ScHEFFERs: Phys. Z. 37, 383, 475 (1936). 16*

244

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 16.

+

Wv ex: P, so that W: ex: T2 (1 ,8 P). Now WH = e;/T L. If (!;, relative to W:, is given by (15.4), the term in ,BP is negligible except at temperatures where Wv already varies more slowly than P, but if(!; is increased by a factor 5, then the term in ,8 is enhanced. Thus one finds for T = 0/20 that dW:fd T = 2.4 W:/T, as against dW;fdT=2.1 W:/T if f3 is given by the BLOCH theory. The former value is in approximate agreement with WHITE's observations. WHITE [88], [89], [90] has also carefully measured deviations from the additivity of Wo and W: (13.18 b), expressing this as an apparent variation of T¥; with Wo. He found that at low temperatures W: apparently increased with increasing Wo. This deviation from (13.18 b) is in the same sense as predicted by SoNDHEIMER [64] from his computations, and they can be qualitatively understood as follows: if Wo />W;, WILSON's solution (14.29a) is appropriate, but if W;~J.Vo, the solution is (14.29c), that is W; is reduced by over 30% . Since in practice W; is unobservable if W; ~ Wo, the observed range of values of W; will be less, and WHITE observed a reduction of only about 20% for a reduction in J.Vo. Furtherfore the index n in W; ex: Tn decreases with increasing J.Vo; this can be understood in terms of the above theory, since Wv is increased, but WH is hardly altered. 16. Comparison with experiments: Other metals. There is at present no detailed theory of the conduction properties of non-monovalent metals which takes account of their band structure, in particular the temperature dependence of (!; at low temperatures is not understood. Therefore it is not possible to quantitatively test the theory for these metals, as was done in Sect. 15. However, some conclusions of the BLOCH theory are independent of the detailed band structure, and can be applied to all metals; these are: W; oc T2 if T ~f), eo should be independent of temperature and J.Vo T = eo/L; also if T > e, (!;ex: T and W; T = e;/L. The residual thermal resistivity will be considered in Sect. 17. MENDELSSOHN and RosENBERG [85], [86], [87] have made extensive measurements of elemental metals, and in addition to Cu, Ag and Au they measured the conductivity of the following metals below 90° K. Group II: Be, Mg, Zn, Cd. Group III: Al, Ga, In, Tl. Rare Earth: La, Ce. Group IV : Ti, Zr, Sn, Pb. Group V: V, Cb, Sb, Ta. Group VI: Mo, W, U. Group VII: Mn. Group VIII: Fe, Co, Ni, Rh, Pd, Ir, Pt.

HULM [92] measured Hg, In and Ta; Bi was measured by a number of authors and will be discussed in Sect. 23. In addition to the survey measurements of MENDELSSOHN and RosENBERG, the following metals were studied in some detail: Al by DE NOBEL [94] and by ANDREWS, WEBBER and SPOHR [95], Mg and Pd by KEMP, SREEDHAR and WHITE [96], [100]. The measurements mentioned here are only the most recent ones extending to liquid helium temperatures; a comprehensive list of all thermal conduction measurements below room temperatures has been compiled by PowELL and BLANPIED [7], covering work until early 1954. The most notable of these earlier measurements are those of GR0NEISEN and collaborators (for example [81], [82], [103]), which extended only to liquid hydrogen temperatures, but on specimens which in many cases had a lower residual resistivity than the specimens of MENDELSSOHN and ROSENBERG.

245

Comparison with experiments: Other metals.

Sect. 16.

In the case of Be, Ti, Sb, La, Ce, U and Mn the residual resistivity andjor

the lattice component of thermal conductivity were so large that W; could not be reliably determined at low temperatures. In those cases for which W; could be determined at sufficiently low temperatures, it was verified that W; varied roughly as P, as required theoretically. It is possible to express W; in the form U';(T)

(16.1)

(T ~ e)

T2

and values of W;/P and A for various metals are given in Table 2. It is remarkable that for the various elements from different groups the range of variation of A should be so small. Table 2 . Conduction properties of polyvalent metals. Element

Mg. Zn Cd Hg. AI Sn Pb

w

Fe Pd Pt

W1 (I;)

Woo

e

T'1 8.7 3 4 2 2 .7 3-9 2 .5 9 1 3.5 4 .3

0.62 0.92 1.08 3-5 0.40 1.7 2.8 0.6 1.2 1.3 1.4

x 10- 5 X 10- 4 X 10- 4 X 10- 2 X 10-5 X 10- 4 X 10- 3 X 10- 5 X 10-4 X 10-4 X 10- 4

330 250 160 60 ( ?) 400 200 100 300 420 275 230

A

16 20 10 20 11 9 9 13 14 20 16

Remarks: Woo and W; in watt·-1 · cm.deg., C}, and a' isgreaterthan, or of the order of, rp,: the transverse waves do not interact as strongly (or not at all) with the conduction electrons and are only loosely coupled to the longitudinal waves. It should be noted that cases (a) and (b) lead to the same value of the thermal conductivity, if the latter is expressed in terms of Cl. In the case (c) there arises an additional component of conductivity, analogous to the component ~~ in the case of non-metals, because of the coupling relaxation time a'. In this case, however, there are no divergence difficulties at low frequencies. The additional component is thus easily derived, though we shall not do so here, because all the metals and alloys studied so far appear to belong to case (a). If it is assumed that scattering by electrons is the only important resistive process, then it is easily seen, by substituting r of (19.5) and (19.3) into (4.8) and neglecting a' and a, that the lattice component of the thermal conductivity is r~e.

{20.1)

Note that this expression is independent of 8. Since the electron-phonon interaction constant is not known, MAKINSON [61] eliminated it by expressing WE in terms of ~"", the electronic ideal thermal conductivity at high temperatures: (20.2) where Na is the number of electrons per atom. It has been shown in Sect. 15 that the BLOCH theory does not reproduce correctly the temperature dependence of ~i, the ideal electronic thermal conductivity, and that this discrepancy arises mainly because it fails to take account of Umklapp-processes and the dispersion of the lattice waves, which are important in determining ~oo, while the expression for ~i at low temperatures is not influenced by these effects It thus seems better to compare WE with the low-temperature limit of ~;. as was done by KLEMENS [72]. In this way one

254

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 20.

compares two magnitudes which are determined by the same processes, and one also eliminates the effect of any slow variation of C with q. From (15.2) and (20.2) one obtains in the case of a spherical FERMI surface (20.))

In the derivation of (20.2) and (20.3) it was assumed that coupling is of case (a), the conduction electrons interacting equally with waves of all polarizations. This has two consequences. Firstly, since all modes interact with the electrons, thee-value in the expression for"; is an average over all polarizations, and is thus approximately equal to eD, the DEBYE temperature deduced from the low-temperature specific heat. Now WE of (20.1) is independent of e, so that the appropriate 8-value in (20.)) is t9v. Secondly, Cl=C}=C2/), where C2 is the value of the coupling constant appearing in the expressions for X; in Sect. 14. This has been taken account of in the derivation of (20.2) and (20.3). Consider now coupling according to case (b). Now C}=O and Cl=C2 , where C2 is the value of the coupling constant in Sect. 14. This introduces a further factor 3 into (20.2) and (20.)), the lattice conductivity being reduced relative to the electronic conductivity. Furthermore the appropriate 8-value in (20.)) is now eL> the DEBYE temperature of the longitudinal branch. Thus for case (b) (20.4)

Since the value of WE relative to X; differs by a factor of about 15 to 20, depending on whether (20.)) or (20.4) is the appropriate comparison formula, it is possible to discriminate experimentally between cases (a) and (b) by measuring "g relative to "; . It now remains to discuss what are the appropriate values of N,.. Since x8 of (20.1) varies as kf 2 (dEfdk)~, while xCX)ock~(dEfdk)~ and if T W;; secondly, if W0 is increased by alloying, the electronic band structure and possibly the lattice properties are changed, so that all electronic conduction properties are altered, including TV; of course. This difficulty is discussed elsewhere [119].

Sect. 22.

Separation of electronic and lattice components.

261

It is possible to deduce x 0 = 1/Wo without electrical resistance measurements if it is assumed that x"ex T2, as it would be if WE were the principal resistance. This will often be the case at liquid helium temperatures for reasonable grain size, so that WB ~ WE; since WE ex T- 2 other resistive processes will become important only at higher temperatures. It is then possible to express x in the form aT+ b T2, where the first term describes the electronic and the second term the lattice component. This method was used by KEMP, KLEMENS, SREEDHAR and WHITE [118] and also by SLADEK [145]. It does, however, require accurate measurements of x and T, and for specimens of low overall thermal conductivities discrepancies were found between x 0 thus obtained and eo measured electrically; these discrepancies may be the result of small heat leaks which become important for specimens of low overall conductivity [119]. There are other methods of deducing x. which depend upon the fact that xe is reduced in a magnetic field, but that is unaltered 1 . This method is only useful when xe can be appreciably reduced by moderate magnetic fields, that is mainly in the case of anisotropic metals. Even then there is the difficulty that the variation of w;, with H is not known. The following methods have been proposed and applied in some cases: (a) Since W. increases with H, so that xJ should become very small for large H, the curve of W against H shows saturation, and the saturation value of W would be ~. However, since for some models w;, (H) shows saturation (18.10 b) one cannot be sure that the observed saturation value of W(H) should be identified with ~· According to KoHLER [76] xe ex H- 2 for divalent metals in strong fields, while other metals would show saturation. Thus for divalent metals a plot of x (H) against 1/H2 should be a straight line, and its extrapolation to H- 2 = 0 should give xg . One can check this extrapolation for anisotropic single crystals by varying the orientation of the magnetic field, keeping the direction of h eat flow fixed. Using the results for Be of GRUNEISEN and ADENSTEDT [103] and of GRuNEISEN and ERFLING [104] he found discrepancies in the extrapolated values of xg which could be explained in terms of inaccuracies in the extrapolation procedure. (b) If the electrical and thermal conductivities are measured simultaneously in magnetic fields, the deduction of xg is more certain. If L e= xefa T were independent of H, the separation of x, would then be easy. This method was proposed by GRuNEISEN and ADENSTEDT, but it was soon seen that it was not reliable, since Leis not independent of H, as was later deduced theoretically [74]. Only in the limiting case J¥o > W;, where Le = L, would Le (H) be independent of H. This method could be useful in cases where Wo > W; but W, > Wo. In divalent metals, however, where xe and a both vary as H - 2 , Le should tend to a constant value for strong fields, and a plot of x against a for various values of H should, when linearly extrapolated to a = O, give xg. KoHLER [76], using measurements on Be, obtained approximately the same value of xg by this method as by plotting xe against 1/H2 • (c) In anisotropic metals one can plot x against a for various values of H, and by rotating H about the direction of flow, the value of xg thus derived can be checked for consistency. This method was used by GRuNEISEN, RAUSCH and WEISS [106] for Bi at liquid oxygen temperatures. It is seen that there is no unique solution to the problem of evaluating X g . Clearly the more ancillary measurements are made, the more reliable is the

x:

1 The improbability of and SoNDHEIMER [74].

Xg

being changed by a magnetic field has been discussed by WILSON

262

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 23.

deduced value of Xg; but for different cases there are different degrees of difficulty: while in some cases it is sufficient to measure x as a function of T, there are other cases where measurements of thermal and electrical conductivities in various magnetic fields for various orientations could still leave appreciable uncertainties in x 5 . 23. Comparison with experiments: Pure metals. Knowledge of the lattice component of pure metals is obtained in three ways: (a) in some cases "• is clearly negligible and the lattice component is the measured thermal conductivity, (b) the lattice component is derived '~ i I ~I I I I I I po:o, by extrapolation from the lattice deg o ~ Ge 1 ~· ~ component of dilute alloys, and (c) o • Bi 'folymslollin{') + 'S/17§ cnsi'es to behave similarly. It now remains to be shown that (25.5), with the value Cn of (25.4) and appropriate values of ls, does indeed describe the electronic thermal conductivity. Let us consider the derivation of (25.5) for normal metals. The thermal conductivity is defined as - Q/V T, Q being the heat current, under the supplementary condition that the electric current j = 0. In superconducting metals j = jn js, where jn and js are the contributions to j from the n-regions and s-regions respectively of the FERMI surface. If Qn is the heat current due to the normal electrons under the condition jn = 0, there now arises an additional heat flow Qs due to the fact that j = 0, but jn =f= 0. The existence of this heat flow due to circulation was suggested by GINSBURG 1 and the idea was later revived by MENDELSSOHX and OLSEN [132]. The additional heat flow Qs has been estimated by KLEMENS [124] as follows: Qs consists of two parts, the change in Qn because jn =f= 0, and the heat transported by virtue of the fact that energy is required to raise an electron from an s-region to ann-region and is given up in the reverse process. It is the latter effect which has been disregarded in (25.6) by using en in (25.5), and the following justifies this procedure. Since j = 0, Ijn I = Ijs I , and since in a superconductor in the steady state F = 0 or is at least small 2 , Ijn I is of order L11 ( K Tg ) K J7 T, where L11 is the transport coefficient (14.11) extended to include scattering by static imperfections, but modified to consider only contributions to the current from the n-region. The

+

J.

1 N. L. GINSBURG: Phys. USSR. 8, 148 2 GINSBURG assumedF-17~=0 ; this would

(1944). change Qs, but not its order of magnitude.

Sect. 26.

269

Thermal conductivity in the superconducting state.

heat transported by in is of order L11 K 2 T (K TfC)2 VT, which is smaller than Q, by a factor of order (KTfC) 2 • The second contribution to Q5 is of order fsKTc, since the latent heat per electron for the transition from normal to superconducting is of order K Tc- The second contribution to Qs is thus of order L11 (K T(C) K 2 Tc VT, and is greater than the first . Thus Qs is smaller than Q" by a factor of order KTcfC, the circulation mechanism does not contribute appreci3.bly to the total electronic therm3.l conductivity 1 , and in (25. 5) only the component C, of the specific heat contributes to the heat transport. Since the electrons in the s-region of the FERMI surface cannot be thermally excited into a continuum of states, it follows that lattice waves can be scattered only by the electrons of the n-region, and not by the s-regions of the FERJ\II W/cmdeg 9 .;ol-+- - I surface. Thus

_i_

-;;:: = x =

(-~-r

(25.10)

t

1

2 and if ~xg,, so that, depending on the circumstances, x 5 may be smaller or larger than x, . dJ 10 Templ!rolure 26. Thermal conductivity in the su14. The thermal conductivity of a lead single crystal, perconducting state. The thermal con- Fig. according to RosENBERG [87]. The inset shows the curve for the superconducting state on a larger scale. ductivity of superconductors has been measured both in the superconducting and in the normal states. The latter measurements are made in a magnetic field above the threshhold value and, if necessary, are reduced to zero field strength by extrapolation (e.g. HULM [92]) . Phenomena associated with the transition from the normal to the superconducting states will be discussed in Sect. 27. Observations on the thermal conductivity in the superconducting state can be classified into (a) cases where Xg < xe and W; > W0 , (b) cases where xg < xe and Wo > W;, (c) cases where x g is negligible in the normal state but appreciable in the superconducting state, and (d) cases where Xg is appreciable both in the normal and in the superconducting states. There are, of course, cases intermediate between any of the above classes, and their interpretation is correspondingly uncertain. rx) The ideal resistance in the superconducting state. In order that W; > W0 below Tc, the specimen must be pure and the transition temperature must be reasonably high. This condition has so far been fulfilled only in the case of lead and mercury. Lead has been measured by BREMMER, DE HAAS and RADEMAKERS [126], [129] , [130], by MENDELSSOHN and PONTIUS [131] and by MENDELSSOHN, OLSEN and RosENBERG [132], [133], [86], [87]. Mercury has been measured by DE HAAS and BREMMER [127] and by HULM [92]. Fig. 14 shows the thermal conductivity of lead according to RosENBERG [87], and is similar to the curve of DE HAAS and RADEMAKERS [129], except that 1

However, this argument has been criticized by

MENDELSSOHN

[136].

270

P. G.

KLEMENS:

Thermal Conductivity of Solids at Low Temperatures.

Sect. 26.

the maximum of '"'s at about 3o K is somewhat higher in the latter case. Fig. 15 shows a plot of xsfxn against T fTc for a number of mercury specimens of HULM [92]; his curve of xs against T for pure mercury is similar in form to the curve of Fig. 14 for lead. According to the considerations leading to (25.9), '"'s should be independent ofT just below Tc, until scattering by imperfections becomes important, and in general '"'s should be of the form 1 (26.1) --- = W. = lfis w;,s Us

+

where lfis = w;n (Tc) is independent of temperature and Wos is related to w;,n by (15 .8), while w;,n oc T-1 . For the present purposes it is important to note thas w;,s should increase monotonically with decreasing temperature, and so should vv;. The observed behaviour of xs does not conform to these predictions. Immediately below Tc '"'s oc P, so that g = (TjTc) 5 , in contrast to (25.9). At lower temperatures '"'s does not decrease steadily with decreasing temperature, but increases again and then decreases at a temperature such that w;,n is com· parable to w;nIn interpreting the observed behaviour of xs, various points of view are Hg2 possible. The one favoured by the pres00':---""_"---f!.::::~---:::o/':-/ --:::45~--a-=-w=----="rc ent writer is the following: w;s is apT/ 0 proximately described by (25 .9), but Fig. 15 . Ratio xlJfxn plotted against reduced temperaturr. for reasons which are not known, and TfTc of a number of mercury specimens, numbered in presumably outside the scope of the order of increasing residual resistivity, according to HuLM [92]. as well as the lead specimen (dott-e:._~~=,~~ possible to explain this by assuming ~ some imperfections to be present in the more dilute alloys and not in the more

Sect. 16.

Group I a.

303

I. Lattice and electronic atomic heats of the elements 1 • 16. Group Ia. The only results in the liquid helium temperature region which have been reported for elements in this Group are measurements on sodium. Sodium. PICKARD and SIMON measured the atomic heat between 2 and 25° K and found a peak around 7° K. Fig. 4 gives a plot of the DEBYE f) versus the absolute temperature T; the minimum in f) at 7° K corresponds to the peak in the atomic heat. Unpublished results of HILL and SMITH and of PARKINSON (see reference 16.2), however, do not confirm this peak. RAYNE measured the atomic heat below 1o K and found a peak between 0.8 and 0.9° K. He suggested that this might be connected with a transformation from the body-centered cubic lattice, stable above this transition temperature, to the face-centered cubic latroo tice. This, or another common OK mechanism, might conceivably account as well for the peak Q) observed around 7° K. That t 120 the two peaks occur at differNa 11JQ ent temperatures in different samples might be a consequence of different external 1S 20 10 s 0 'K parameters such as impurities, -r strains or cooling. Fig. 4. GJ versus T for sodium. 0 experimental points (reference 16.3). The theoretical curve is due to BHATIA (reference 16.1). The result of a calculation by BHATIA of the lattice atomic heat alone is also plotted in Fig. 4. He concludes that there is excellent agreement except for the peak. The difference between the two curves below 4° K is very likely due to the electronic atomic heat contribution. Because of its relatively simple electronic configuration, the electronic band structure of sodium has been the subject of extensive theoretical investigation. BARDEEN found that the free-electron treatment with effective mass ratio of 0.95 was a reasonable approximation to his calculated band structure. PINES calculated the effect of taking the electron exchange energy into account by means of the "collective electron" approach and found the corresponding CE to be 0.82 of the free-electron value. BucKINGHAM and ScHAFROTH estimated the value of the interaction parameter occurring in their theory of lattice-electron interaction from the unpublished results of PARKINSON referred to above, which they quote as indicating CE to be about twice the free-electron value. Unambiguous data in the liquid helium temperature range and below would thus be very useful in assessing the applicability of these various theoretical treatments. 16.1 A. B. BHATIA, Phys. Rev. 97, 363 (1955) : Na; vibration spectrum. 16.2 M. J. BucKINGHAM and M. R. ScHAFROTH, Proc. Phys. Soc. Lond. A 67, 828 (1954): electron-lattice interaction. 16.3 G. L. PICKARD and F. SIMON, Proc. Phys. Soc. Lond. 61, 1 (1948): Na, Hg; 3 to 90° K, Pd; 2-22° K. 1 References to data listed in the Tables and to data which are commented on in the text are collected at the ends of the respective sections except for theoretical calculations of y, references to which will b e found in [11], [12], [13]. For convenience in cross-reference, each item is identified by a number giving the section in which it appears as well as the order of listing in that section, e.g. 16.3. The lists are not exhaustive, especially for work above 10° K. Data in this temperature range are referred to, in general, only when they can be compared with data below 10° K. Extensive lists of references covering all temperatures can be found in the compilations of SHIFFMAN [141 and of SHULL and SINKE [15].

304

KEESOM and PEARLMAN: Low Temperature Heat Capacity of Solids.

Sect. 17.

16.4 D. PINES, Phys. Rev. 92,626 (1953): "Collective electron" treatment, including application to electronic atomic heat. 16.5 J. RAYNE, Phys. Rev. 95, 1428 (1954): Na, Cu. Ag, Mo, W, Pd, Pt; below 1° K. 16.6 F. SIMON and W. ZEIDLER, Z. phys. Chern. 123, 383 (1926): Na, K; 14-280° K , Mo, Pt; 16-300° K.

17. Group lb. (J.) Copper. Agreement among the different investigators appears to be satisfactory. The results of EsTERMANN et al. are higher than the average, but their sample was not very pure and their results have a rather large scatter. The results of KEESOM and KoK are about 5% higher than those of CoRAK et al., but the scatter in the former is also of about this order. Ex"K perimental values of @ ( T) JSDf are given in Fig. 5 together ~ with the curve calculated by Cu ~ ~ LEIGHTON from his theoretiJ20 cally determined vibration ~ 230 ~ spectrum and the agreement ~ is excellent. 2()()'---------------_!E iiii._="-"---The elastic constants of ":si~'flo~ copper have recently been measured by OvERTON and OQ:Joo--~ Au GAFFNEY from the normal o'-----s':-'----~:':-~----~:':-&---.:"K:----:'zo boiling point of helium to room -r temperature. The values of the Fig. s. e versus T for Cu, Ag and Au. The theoretical curves are due to elastic constants extrapolated LEIGHTON (reference 17.13). O: CLUsms and HARTECK (reference 17.2); to 0° K together with DE O: GJAUQUE and M EADS (reference 17.7). LAUNAY's theoretical calculations (mentioned above in Sect. 9) give @0 equal to 340° K which is in excellent agreement with the calorimetric values. The value of y for free electrons in a univalent metal with the atomic volume of copper as calculated from (8.12) is only 0. 502 millijoulesfmole degree2 , so that the eE is 1.4. . The calculated energy band for copper valence electrons has approximately the "normal" form given by (8.7) at the FERMI level, so that this number may also be taken as roughly the effective mass. The shape of the energy band has been calculated approximately by JONES, from ·the form of the BRILLOUIN zone. A more detailed calculation has been carried out by KRUTTER and others on the basis of the 3d and 3 s wave functions. The former gives ga (C)= 0.308 and the latter gives ga (C)= 0.316; by (9.3) these correspond to y = 0.73 and 0.75 millijoulesfmole degree 2 respectively, so that there is excellent agreement with the calorimetric values of KEESOM and KoK and of RAYXE. Those of ESTERMANN et al. and of CoRAK et al. are higher and somewhat lower, respectively. fJ) Silver. The electronic contribution to the atomic heat was first discovered by KEESOM and KoK in their measurements on silver. They found that (T) computed from C, the total measured atomic heat, decreased sharply in the helium temperature range, but that C could be represented well by the sum of terms proportional to T 3 and to T. Their measurements extended into the intermediate temperature region between the helium and hydrogen ranges as well, but they reported that difficulties with thermometer calibration in this temperature region made their results there largely qualitative. Nevertheless, attempts have been made by others to find a theoretical explanation for an apparent peak in @(T) from their data at about 5° K. KATZ discussed the effect on @(T) of superimposing EINSTEIN peaks on a DEBYE vibration spectrum,

l

f.

e

305

Group lb.

Sect. 17.

while BHATIA and HORTON recently investigated the possibility of getting such a peak in (T) by suitable choice of elastic constants. Neither these latter authors nor LEIGHTON found, however, that such a peak occurred in their calculated (T) curves for the actual elastic constants of silver. Later measurements by KEESOM and PEARLMAN and by CoRAK et al. make it appear probable that such a peak does not exist. Fig. 5 also gives fJ(T) for silver from 1 to 20° K. The experimental curve and that computed by LEIGHTON from his theoretical vibration spectrum are in excellent agreement. Another apparent anomaly, connected with the electronic heat capacity, was reported by KEESOM and PEARLMAN in 195 2. In a more recent publication (1955) this was traced to deviations of the accepted 1948 temperature scale from the absolute scale. A careful discussion of the effects of such deviations is given by CoRAK et al. Their value of y is somewhat lower than the free electron value, 0.644 millijoulesjmole degree 2 , while the other y values reported are somewhat higher. The ratio (!E is much closer to unity for silver than for copper.

e

e

y) Gold. Results are now available from 1o K upwards. The ratio 1.16, which is between those for silver and copper.

(!E

is

Table 2. Low temperature atomic heat of the Elements-Group I b. Element

e,('K)

Cu

334 352 344

range ('K)

1-20

i i

1~;

------ ___ I_ Ag

230 225 225 229

Tn(' K)

7

1-10 1-4 1 >10 > 12

>16 1-5

reference

0-73 0.92 0.72 0.69

17.12 17.4 16.5 17-3 17.7

0.67 0.66 0.68 0.61 0.65

17.10 1 7.11 16.5 17-3 17.16 17.14,

l_ _ !

- - - -- - - - I·-·--··--····

Au

. I

1- 5

> 1o_

. .. Y degree') I (mtlhjoules/mole

I 20

I

i

l--------i~=~7---~ I 17.2 0.74

I

17-6 17.3

17.1 A. B. BHATIA and G. K HoRTON, Phys. Rev. 98, 1715 ( 1955): theoretical vibration spectrum for face-centered cubic lattice, with application to silver. 17.2 K CLusrus and P. HARTECK, Z. phys. Chern. 134, 243 (1928): Ag, Zn; 12-200° K, Au, Ga; 15-200° K. 17.3 v.·. S. CORAK, M.P. GARFUNKEL, C. B. SATTERTHWAITE and A. WEXLER, Phys. Rev. 98,1699 {1955): Cu, Ag, Au; 1-5° K. 17.4 l. ESTERMANN, S. A. FRIEDBERG and j. E. GOLDMAN, Phys. Rev. 87, 582 ( 1952): Cu, Mg, Ti, Zr, Cr; 2-4° K 17-5 A. EucKEN, K. CLusrus and H. WoiTINCK, Z. anorg. allg. Chern. 203, 39 (1931): Ag; 12 - 200° K 17.6 T. H. GEBALLE and W. F. GIAUQUE, J. Amer. Chern. Soc. 74, 2368 (1952): Au; 16 -300° K 17.7 W. F. GIAUQUE and P . F. MEADS, J. Amer. Chern. Soc. 63, 1897 (1941): Cu, AI; 15 - 300° K. 17.8 H . JoNES, Proc. Phys. Soc. Lond. 49, 250 (1937): Electronic energy levels for face-centered and body-centered cubic lattices; application to Cu. 17.9 E. KATZ, J. Chern. Phys. 19, 488 (1951): superposition of EINSTEIN peaks on DEBYE vibration spectrum, derived from empirical 0 ( T) curves; application to Ag. Handbuch der Physik, Bd. XIV.

20

306

KEESOM and PEARLMAN: Low Temperature Heat Capacity of Solids.

Sect. 18.

17.10 W. H. KEESOM and J. A. KoK, Proc. Kon. Ned. Akad. Wetensch. 35, 301 (1932). also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 219d: Ag; 1-20° K. Physica, Haag 1, 770 (1933), also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 232d: Ag, Zn; 1-5° K. 17.11 P. H. KEESOll! and N. PEARLMAN, Phys. Rev. 88, 140 (1952); 98, 548 (1955): Ag; 1-4° K. 17.12 J. A. KoK and W. H. KEESOM, Physica, Haag 3, 1035 (1936), also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 245a: Cu, Pt; 1 - 20° K. 17.13 R. B. LEIGHTON, Revs. Mod. Phys. 20, 165 (1948): Theory of lattice atomic heat for face-centered cubic lattice; application to Ag. 17.14 P. F. MEADS, W. R. FoRSYTHE and W. F. GIAUQUE, J. Amer. Chern. Soc. 63, 1902 (1941): Ag, Pb; 15-300° K. 17.15 W.C.OvERTON Jr. and ].GAFFNEY, Phys. Rev. 98,969 (1955): Measurement of elastic constants of Cu between 4 and 300° K; calculation of @0 . 17.16 B. YATES and F . E. HOARE, private communication: Ag, Pd; above 1° K.

18. Group IIa. ct.) Beryllium. The older work of CRISTESCU and SIMON indicated a peak in the atomic heat around 11 o K, but the newer results of HILL and SMITH do not confirm this. Since the 2s energy band can accomodate just two electrons, the fact that divalent beryllium is a metal rather than an insulator implies that the 2s and 2 p bands must overlap. HERRING and HILL have calculated the electronic band structure, which is shown in Fig. 6. The minimum following the peak is an indication of the overlap and since Co falls near the minimum it would be expected that eE is less than unity. This expectation is borne out; the observed value is 0.47 while that calculated from the curve is 0-39. The two 8e parabolas in Fig. 6 are respectively the extrapolation of the normal portion of the band near the lower edge (upper parabola, corresponding to f.l = 1.6) and the free elec-t tron level density (lower parabola). Since Fig. 6. Electronic band structure of Be, after HERRING and HILL, Phys. Rev. 58, 132 (1940). The two parabthe band is far from normal at the FERMI olas correspond to normal bands with I'= 1.62 for the upper, I' = 1.00 for the lower. level, it would be invalid in this case to interpret eE as an estimate of f.l · f3) Magnesium. The spread in the data in the whole temperature range is large. It would appear to be desirable to have more experimental data, especially measurements on one sample covering the whole temperature range. y) Calcium. The values of 8 andy fit the data between 10 and 20° K. Without results at lower temperatures, however, it is impossible to judge if the true T3 region has been reached. Table 3. Low temperature atomic heat of the Elements-Group I I a. Element

range ('K)

e,('K)

!

~ :.~-~-0_

.. ___B _ e___ __ Mg -----

Ca

(330 - 390)

, .. Y (m•lhjoulesfmole degree')

Tn('K)

reference

4_;:_~_g_o_ _:_l----~-226____ -I !~:~

:__ 'I

2-4 > 10

I

>10

i

(1.35)

---- ~ ---- - - ; - - - - - - - -

(219)

I

I

0.38

' 17.4 '18.1, 18.2 ----

18 1

Group lib.

Sect. 19.

307

18.1 K. CLUSIUS and J. V. VAUGHEN, J ..\mer. Chem. Soc. 52, 4686 (1930): Ca; 102000 K, Mg; 10-300° K, Tl; 10-250° K. 18.2 R. S. CRAIG, C. A. KRIER, L. W. CoFFER, E. A. BATES and W. E. WALLACE, J. Arner. Chern. Soc. 76,238 (1954): Mg, Cd; 12-320° K. 18.3 S. CRISTEScu and F. SIMON, Z. phys. Chern. Abt. B 25, 273 (1934): Be; 10-300° K, Ge ; 10-200° K, Hf; 13-210° K. 18.4 R. W. HILL and P. SMITH, Phil. Mag. 44, 636 (1953): Be; 4-300° K.

19. Group lib. a) Zinc. Around 4° K the results of SILVIDI and DAUNT are slightly higher than those of KEESOM and VAN DEN ENDE. The corresponding JOO OK ~2GO

LM '~o~------~5--------~~~------~~---.~--~zo

-r

Fig. 7. EJ versus T for Zn.

eo values are 296° K and 321

o K respectively. There is good agreement at higher temperatures among the results of different investigations. It is striking that is nearly constant between 12 and 20° K (see Fig. 7), but with a very different

e

O,Y

0,2

0

-rz

0,6

Fig. 8. C/T versus T' for Cd below I' K, after S.utotLDV (reference 19.5).

value from that in the helium region. This is thus an excellent example of a "pseudo- T 3 region" in constrast to the true P region below 5o K. !00

'K ~

t

80 80 -

X

0

fOO Fig. 9 .

e versus T

5

-

70

r

!5

20

for Hg. Smoothed cu:rve and Q : PICKA RD and SIMO N (reference 16.3) ; x : HOLST (reference 19.2); O : BusEY and GIAUQUE (reference 19.1).

KAMERLINGH ONN E S

and

(3) Cadmium. The only measurements below hydrogen temperatures appear to be those of SAMOILOV below 1 o K. His results are given in Fig. 8 as a plot of CJT versus P. The values in the hydrogen region are less than half the value of determined by SAMOILOV. y) Mercury. PICKARD and SIMON's smoothed results are given in Fig. 9. We also include two points measured by KAMERLINGH ONNES and HoLsT as these are the first calorimetric data in the liquid helium range. In order to separate the electronic and lattice terms data at lower temperatures are necessary.

eo

e

20*

308

KEESOM and PEARLMAN: Low Temperature Heat Capacity of Solids.

Sect. 20.

Table 4. Low temperature atomic heat of the Elements-Group II b. y

Element

e,('Kl

range (' K)

Tn('Il and KoK (reference 21.6); Q: CLUS!US and VAuGHEN (reference 18.1). made a careful analysis of their results using for the lattice contribution the sum of a term proportional to T 3 and a term proportional to T 5 . They found the coefficient of the P term to be small, so that @ does not vary appreciably in the helium range. In Fig. 11 the solid line gives their values of@ while the results of CLusrus and ScHACHINGER are plotted separately as points. There appears to be some disagreement, except at 20° K. CLEMENT and QuiNNELL found (L1C)r, to be 9.75 millijoulesjmole degree. By (20.1) this corresponds to y = 1.44 millijoulesjmole degree 2 which again is lower than the calorimetric and magnetic value. For indium T8 > T0 but the magnetic threshold curve is not parabolic in the neighborhood of T0 • lJ) Thallium. The variation of @ with T is given in Fig. 12. KEESOM and KoK found the value 6.19 millijoulesjmole degree for (L1C)r,, the jump in the atomic heat at the normal transition point, T0 = 2.36° K. By (20.1) this corresponds to y = 1. 3 millijoulesjmole degree 2 which agrees excellently with the value obtained from magnetic threshold field measurements. The calorimetric value which we obtained from a least squares fit to the points of KEESOM

en

311

Group IVa.

Sect. 22.

and KoK above T0 is much larger, however (see Table 6). There appears to be no obvious reason for this discrepancy since T8 > T0 and the magnetic threshold curve is approximately parabolic so that both of the assumptions on which (20.1) is based are essentially valid. Table 6. Low temperature atomic heat of the Elements-Group III b. y

Element

e,(' KJ

range (' K)

(millijoulesfmole degree')

Tn( ' K)

calorimetric

B AI

41 8

-

> 15 I

Ga -

109

- - - - - --- -- -

Tl

I

> 15

- - ·---

In

--

- --

- -- - - ------

89

1-20 > 12

- - - --

4

> 10

- ·--·-1 1.81 1.44 1

- 4

I

-

-

-

-

3.1 1. 31

1

1.2 0.3 8

[10]

17.2. 21.1 [10 ]

21.2 21.2 1.8

I

[10]

21.3 21. 6 21. 6 18.1, 21.4

I I !

21.7 21.7 17. 7 ·----···

I

i

1 - 20

21.5

i

1.46 0.85 1 iI

20

1-20

-

-

-

reference

magnetic

I

> 13

--------- - - - -

-

j

1.3

[10]

21 .1 G. B. ADAMS Jr., H. L. JOHNSTON and E. C. KERR, J. Amer. Chern. Soc.' 74, 4784 (1952} : Ga ; 15-320° K. 21.2 J. R. CLEMENT and E . H . QuiNNELL, Phys. Rev. 92, 258 (1953) : In; 1- 20° K. 21.3 K. CLUSIUS and L. SCHACHINGER, Z. angew. Phys. 4, 442 (1952) : In; 12-273° K. 21.4 J. F . G. HicKs Jr., ]. G. HooLEY and C. C. STEPHENSON, J. Amer. Chern. Soc. 66, 1064 (1944): Tl; 14 - 300° K. 21.5 H. L. JOHNSTON, H . N. HERSH and E . C. KERR, J. Amer. Chern. Soc. 73, 1112 (1951}: B; 13-305° K. 21.6 W . H. KEESOM and J . A. KoK, Physica, Haag 1, 175. 503, 595 (1934) , also in Commun . Kamerlingh Onnes Lab. Univ. Leiden, nos. 230c, 230e, 232a: Tl; 1-4° K. 21.7 J. A. KoK and W. H. KEESOM, Physica , Haag 4, 835 (1937). also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 248e: AI ; 1-20° K . 21.8 W. DE SoRBO, ]. Chern. Phys. 21, 168 (1953): Ga; discussion of T2 dependence. 22. Group IVa. This Group has been the subject of extensive study for a variety of reasons. Two of the elements each occur in two different crystal lattices at low temperatures. Two become superconducting and especially for tin, the thermodynamic consequences of the transition between the superconducting and the normal states have been studied in detail. Finally the lattice atomic heat of the diamond lattice has been investigated theoretically, and also that of the graphite modification of carbon. ex) Carbon-Graphite. Graphite crystallizes in a typical layer lattice, in which the atoms in certain planes have smaller interatomic distance, and are more strongly bound by interatomic forces , than is the case for atoms in adjoining planes. Much theoretical work has been done towards elucidating the effect of this anisotropy on the low temperature heat capacity. In addition, the electronic band structure has been investigated. 1

Calculated from (20.1), using observed v alues of (Ll C)r,.

312

KEESOM

and

PEARLMAN:

Low Temperature Heat Capacity of Solids.

Sect. 22.

The assumption of isotropic distribution of lattice vibration momentum vectors used in the discussion of Sect. 5 is no longer tenable except at very low frequencies. The anisotropic distribution appropriate to the graphite lattice seems to give rise to the expectation of a quadratic dependence of the lattice atomic heat on temperature, over a certain temperature range. Various authors, however, are in disagreement as to the form of the resulting vibration spectrum and thus of the temperature range covered by the T2 dependence. On the other hand, there seems to be agreement that at sufficiently low temperatures a transition to cubic dependence should be observed, but again, the location of this transition seems to depend on the manner in which the calculation is performed. There is the further difficulty 3 O,J that quantitative prediction from theoretical results demands knowledge of elastic properties (in order to evaluate the interatomic forces) which have not been measured for graphite. Data obtained by DE SoRBO and TYLER between 13 and 300° K when plotted as Log C versus Log T show a region of T 2 dependence between 13 and 54° K. In a similar plot, the data of Fig. 13. CfT, C and CL versus T' for graphite. After KEESOM and PE.13

- - - -- - - - -

C (diamond)

(2000)

Si

658

1-100

Ge

366

1- 4 2-4 4-170 > 20

Sn (gray)

2

> 70 > 20 drop method

- --- --

I

-1 -

--- - ~----- ---- ··- -- ·- ------------- - - - - --1

Pb

94.5

1-4 3.5-3-9 1-20 1-4 > 9 1-70 1 - 20 6.7-].7 > 15

---

3

4

- --

22.19 22.22 22.2 22.18

22.13 22.4 22.6 22.5, 18.3

4

(200)

189

- - --

- -- --l-- - -1

7-110 >9

Sn (white)

22.14 22.1 22.23

0.031

1.82 1.4 1

'

- --

-- -

(see text) 22.6 22.17 22.11 22.10 19.3 22.20 22.17

1.86 [10] - - -1--- - - --

3.0

22.7 22.9

3.64 1

22.3

17.14 [10]

contribution is directly proportional to the effective mass ratio and proportional to the cube root of impurity concentration [see (9.7)] so that it will be about the same in both substances. The '·-...... •.._ IJJilmonrlflheorJ lattice component, however, ·- ~ -----------·is almost six times larger in germanium than in silicon Q:, let? so that measurements would qB have to be made at temperatures below 1 o K for the electronic contribution to be detected. ~o~---~0.~"----~-----~---~a s) Gray tin. Tin exists o, " o,ro o,ts ,zo T in several allotropic forms . -a; The oc-modification, gray tin, Fig. 14. fi!j€! 0 versus Tf€! 0 for elements having the diamond lattice. The has the diamond crystal theoretical curve for diamond is due to SMITH (reference 22.21) and structure and is stable belo"'· those for Ge and Si are due to HsiEH (reference 22.8). 18° C. Since the density of gray tin is less than 80% of that of metallic tin ({1-tin, white tin) which is stable between 18 and 170° C, massive crystals of metallic tin decompose into

!

1

Calculated from (20.1), using observed values of (LI C) T,.

Sect. 22.

315

Group IVa.

a fine powder upon transforming into gray tin. The usual problems connected with calorimetric measurements on powder in a calorimeter vessel containing exchange gas prevented HILL and PARKINSON from obtaining accurate heat capacity data below 7° K. C) The diamond lattice. Except for diamond itself, the substances which have the diamond crystal structure show very similar temperature variations in their low temperature molar heats. The dependence of 8(8 0 on T(8 0 is plotted in Fig. 14 for germanium, silicon and gray tin. The molar heat of indium antimonide has been measured 200 between 1 and 20° K (unpub'Kt-----..... 180 lished results of KEESOM and SnfwhiteJ PEARLMAN), and 8 0 found to Q:) be 200° K. This substance also 116'0 ~ has the diamond crystal struc0 ture and its lattice constant is almost identical with that ~~~------~5~----2-~m~----~--~;~-.K~--~M of gray tin (6.45 A compared -r to 6.46 A). The atomic masses Fig. 15. E) versus T for white Sn, after K E ESOM and VAN DE:S ENDE (reference 19~3). of In and Sb are close to that of Sn. If 8 0 is assumed to be 200° K for gray tin as well, then 8j8 0 versus T/8 0 for InSb and gray tin are identical up to T(8 0 equal to 0.10. The curves for silicon and germanium are also similar up to the common minimum at about T(8 0 = 0.06. Corresponding curves have been calculated on the basis of the BoRN-VON KARMAN theory, by SMITH for diamond and by HsiEH for germanium and silicon, and are shown in Fig. 14 for comparison (dot1(}() 'K dash lines) . The experimental points of PITZER for the atomic 95 heat of diamond lie fairly close to the theoretical curce. 'YJ) Metallic tin. Since the normal superconductive tran85 sition point of metallic tin is conveniently located in the ~0~-------s~------~m~------~~~--~c---~M helium temperature region (T0 = 3.73 ° K) its thermo-r dynamic consequences have Fig. 16. e versus T for Pb after HoROWITZ et al. (reference 23.7) been studied extensively at Leiden. The value of y which follows from (L1C)r, as observed by KEESOM and KoK (10 millijoules ( mole degree) and (20.1 ), is 1.4 millijoules(mole degree 2 . This is somewhat lower than the calorimetric value, which is to be expected in this case since T8 < T0 so that C,. ( T0 ) is larger than is assumed in the derivation of (20.1). Hence (L1C)r, will be too small and the same should be true of y calculated from (20.1). For further details concerning calorimetric investigations of the superconductive transition of tin we refer to Sect. 31 to 33 below and to the article on superconductivity in the next volume of this Encyclopedia. The relation between 8 and Tin the normal state up to 20° K is shown in Fig. 15 . {}) Lead. Following HoROWITZ et al. we give in Fig.16 the variation of 8 with T, together with a theoretical curve deduced from LEIGHTON's work (see reference 17.13). The results of the different investigators have a considerable spread but fall around the curve. 0

Nt

0

00 ~ ~~

316

KEESOM and PEARLMAN: Low Temperature Heat Capacity of Solids.

Sect. 23.

It is clear from Fig. 16 that T8 is less than T0 for lead (T0 = 7.23° K). The argument given above for tin concerning the relation between y determined from (LIC)r, and (20.1) and that directly observed from c.. should therefore apply in this case as well. The measurements of CLEMENT and QUINNELL in the neighborhood of T0 give the value 52.7millijoulesjmole degree for (LIC)r, which corresponds to y = 3.64 millijoulesjmole degree 2 • However, this is higher than the value calculated from the Cn data of HOROWITZ et al. This discrepancy may be attributable to the departure from parabolic form of the threshold magnetic field curve near T0 • 22.1 U. BERGENLID, R. W. HILL, F. J. WEBB and J. WILKS, Phil. Mag. 45, 851 (1954): graphite; 1. 5-100° K. 22.2 R. BERMAN and J. POULTER, J. Chern. Phys. 21, 1906 (1953): diamond (drop method); difference in heat content between 90 and 290° K and between 4 and 90° K. 22.3 J. R. CLEMENT and E. H. QuiNNELL, Phys. Rev. 85, 502 (1952): Pb; 6.7-7f K. 22.4 I. EsTERMANN and S. A. FRIEDBERG, Phys. Rev. 85, 715 (1952): Ge; 2-4° K. 22.5 l. ESTERMANN and J. R. WEERTMAN, J. Chern. Phys. 20, 972 (1952): Ge; 202000 K. 22.6 R. W. HILL and D. H. PARKINSON, Phil. Mag. 43, 309 (1952): Ge; 4-170° K, gray tin; 7-110° K. 22.7 M. HoROWITZ, A. A. SILVIDI, S. F. MALAKKER and J . G. DAUNT, Phys. Rev. 88, 1182 (1952): Pb; 1-17° K. 22.8 Y. C. HsiEH, J. Chern. Phys. 22, 306 (1954): Ge and Si; theoretical vibration spectra. 22.9 W. H. KEESOM and J. N. VAN DEN ENDE, Proc. Kon. Ned. Akad. Wetensch . 33, 243 (1930) and 34, 210 (1931), also in Commun. Kamerlingh Onnes Lab. Univ. Leiden nos. 203d and 213c: Pb, Bi; 1-20° K. 22.10 W. H. KEESOM and J. A. KoK, Proc. Kon. Ned. Akad. Wetensch. 35, 743 (1932), also in Commun. Kamerlingh Onnes Lab. Univ. Leiden, no. 221e: Metallic tin; 3-5-3 -9° K. 22.11 W. H. KEESOM and P. H. VAN LAER, Physica, Haag 5, 193 (1938), also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 252b: metallic tin; 1-4° K. 22.12 P. H. KEESOM, K. LARK-HOROVITZ and N. PEARLMAN, Science, Lancaster, Fa. 116, 630 (1952): neutron irradiated and annealed Si; 1-4° K. 22.13 P.H.KEESOM and N.PEARLMAN, Phys. Rev . 91,1347 (1953): Ge; 1-4°K. 22.14 P. H. KEESOM and N. PEARLMAN, Phys. Rev. 99, 1119 (1955) : graphite; 1-20° K. 22.15 K. KoMATSU and T. NAGAMIYA, J. Phys. Soc. Japan 6, 438 (1951): graphite; theoretical calculation of lattice and electronic atomic heat. 22.16 K. KoMATSU, J. Phys. Soc. Japan 10, 346 (1955): graphite; lattice atomic heat theory, including references to earlier work. 22.17 F . LANGE, Z. phys. Chern. 110, 343 (1924): gray and metallic tin; 9-280° K, W; 20-91° K. 22.18 N. PEARLMAN and P. H . KEESOM, Phys. Rev. 88, 398 (1952): Si; 1-100° K. 22.19 K. S. PITzER, J. Chern. Phys. 6, 68 (1938): diamond; 70-300° K. 22.20 K. G. RAMANATHAN and T. M. SRINIVASAN, Phil. Mag. 46, 338 (1955): metallic tin; 1-4° K. 22.21 H. M. J. SMITH, Phil. Trans. Roy. Soc. Lond., Ser. A 241, 105 (1948): diamond; theoretical vibration spectrum. 22.22 W. DE SoRBO, J. Chern. Phys. 21 , 876 (1953): diamond; 20-300° K. 22 .23 W. DE SoRBO and W. W. TYLER, J. Chern. Phys. 21, 1660 (1953) : graphite; 133000 K. 23. Group IVb. rx) Titanium. Measurements have been reported between 2 and 4° K and above 14° K, but the two sets of data do not appear consistent. Table 8 includes only the helium range data. It would be desirable to have data on one sample covering the entire temperature range.

(3) Zirconium. Data are also available between 2 and 4° K and above 14° K, but in this case they are in reasonable agreement. The DEBYE decreases from 270° K in the helium region to 254° K in the hydrogen region.

e

Sect. 24.

317

Group Va. Table 8. Low temperature atomic of the elements-Group IVb. 0 0 (°K)

Element

range (°K)

y (millijoulesfmole degree')

TB(°K)

Ti

2-4 >15

278

4

3-34

l I

--·-·

reference

magnetic

calorimetric \

17.4 23.1

0.46

[10]

i

Zr

2-4 >14

270

4

2.95

I I

17.4 23.2 1.64

[10]

23.1 C. W. KoTHEN and H. L. JoHNSTON, J. Arner. Chern. Soc. 75, 3101 (1953): Ti; 15-305° K. 23.2 G. B. SKINNER and H. L. joHNSTON, J. Arner. Chern. Soc. 73, 4549 (1951): Zr; 14-300° K.

24. Group V a. Of the elements in this Group only bismuth has been measured in the helium temperature region; antimony has been measured down to 13° K. 120

~

I s

.900 Fig. 17.

e

versus T for Bi after (reference

KEE SOM 24.1);

10

TS

-T and

Q:

(reference 24.2). - 0 - : ARM STRONG and and VAN DEN ENDE (reference 22.9).

PEARLMAN KEESOM

GRAYSON-SMIT H

I

rx) Bismuth. The variation of 8 with T, which is shown in Fig. 17, appears to be normal. The results of KEESOM and VAN DEN ENDE are about 10% higher than those of ARMSTRONG and GRAYSON-SMITH in the hydrogen region. Table 9. L ow temperature atomic heat of the E lements-Group V a. Element

0 0 (°K)

Bi

117

I

I I I

I

range (°K)

1-4 1-20 > 14

I TB(°K) 2.3

I (millijoulesfmole degree') y

1

0.078

reference

II

24.2 22.9 24.1

24.1 L. D. ARMSTRONG a nd H. GRAYSON-SMITH, Canad. J . R es. A 27, 9 ( 1949): B i : 14-22° K. 24.2 P. H. KEESOM and N . PEARLMAN, Phys. Rev. 96, 897 (1954): Bi; 1-4° K. 24.3 W. DE SORBO, Acta rnetallurgica 1, 503 (1954): Sb; 13-70° K.

)18

KEESOM

and

PEARLMAN:

Low Temperature Heat Capacity of Solids.

Sect. 25.

25. Group Vb. These elements are of interest for investigation of the thermodynamics of the phase transition from the normal to the superconducting state. They have the advantage that their values are high, as well as their normal transition temperatures. The former correspond to low values of CL, so that the electronic atomic heat, which is of major interest in the transition, is a large fraction of the total. High transition temperatures permit measurements down to low values of the reduced temperature T JT0 , where T 0 is the normal transition temperature. A slight disadvantage, however, is that T0 for vanadium and niobium falls in the "intermediate" temperature region between 4 and 10° K where no cooling bath is available. One difficulty that is apparent with measurements on metals in this Group is that different samples give different results without apparent reason, even if measured by the same group of investigators. rx) Tantalum. The data of WoRLEY et al. on tantalum in the normal state agree with those of KEESOM and DESIRANT above T2 = 6, but the former find higher values of CJT below this (see Fig. 26). In a discussion of these results at the Low Temperature Conference held in Houston in December 1953, it was suggested that the magnetic field used by WoRLEY et al. might not have been high enough, so that the sample perhaps was partially in the intermediate state at the lowest temperatures. The results of KEESOM and DESIRANT above the boiling point of helium appear to be too high, probably due to difficulties in extrapolating their resistance thermometer calibration. These measurements and those of DESIRANT and MENDELSSOHN (see references 25 .3 and 25.5 for the latter) give values of (L1C)r, of 34 and 40 millijoulesfmole degree at T0 equal to 4.0 and 4.4 o K, respectively. By (20.1) these correspond to y values of 4.2 and 4 millijoulesfmole degree 2 , respectively, which are lower than both the calorimetric and magnetic values. f3) Vanadium . WoRLEY et al. observed positive deviations in Cf T at their lowest temperatures similar to those mentioned for tantalum. The spread in their value of refers to measurements on two different samples, the lower value corresponding to the purer sample. SATTERTHWAITE et al. do not report deviations in CfT versus T 2 . They do not give an analysis of the impurities in their sample, which had a T0 value about 0.2° K higher than that of the purer sample of WoRLEY et al. An interesting indirect method was used by CoRAK et al. to obtain their earlier estimate of y, since they had not measured Cn and therefore could not evaluate (L1C)r, directly. They calculated Sn (T0 ), the entropy of the normal state at T0 on the assumption that C, had the normal form below T 0 • By ()2.2) this can be set equal to s. (T0 ), the entropy of the superconducting state at T0 , since He, the threshold magnetic field vanishes at T0 • In the resulting equation which also assumes 5,(0° K) = S.(0°K) (=0 by NERNST's Theorem),

eo

eo

(25.1)

eo

they estimated rx from (15.2) and the average of the values reported by WORLEY et al. Since they were able to calculate s. (T0 ) from their measurements of c., the atomic heat of the superconducting state below T0 , they were able to solve fory. as

y) Niobium. Above 12° K the results deviate from the usual behavior of CL,

e seems to increase above eo without any intervening decrease 1 ZEMANSKY

1.

indicated in private discussion that these data may need revision.

319

Group VI b.

Sect. 26, 27.

Table 10. Low Temperature Atomic Heat of the Elements-Group Vb. y

0,( 0 1{)

Element

range (°K)

(millijoules jmole degree')

TB(°K)

calorimetric

v - - --

{273-321) 338

1-5 1-5

Nb - -- - Ta

252

2-20

246 {213-225)

1-20 2-5 3-5-5-5

(8.6-8.9) 9.26 8.8 1

5

- -

-

10

--5

I

6.28

-

reference

magnetic

25.8 25.6 25.2 [10]

25.1

8. 5

--~ 1 (5.0-5.4) . I 42 I 4.2 2

I

25.4 25.7 25.3. 25.5 25.4

8.0

[10]

25.1 A. BROWN, M. W. ZEMANSKY and H. A. BooRSE, Phys. Rev. 86, 134 (1952): Nb; 2-20° K. 25 .2 W. S. CoRAK, B. B. GooDMAN, C. B. SATTERTHWAITE and A. WEXLER, Phys. Rev. 96, 1442 (1954): V (superconducting); 1-5° K. 25.3 M. DESIRANT, Report of an International Conference on Fundamental Particles and Low Temperatures (The Physical Society, London, 1947), Vol. II, p. 124: Ta; 3-5 to 5-5° K. 25.4 W. H. KEESOM and M. DESIRANT, Physica, Haag 8, 273 (1941), also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 257b: Ta; 1- 5° K. 25.5 K.MENDELSSOHN, Nature, Lond. 148,316 (1941): Ta; 3-5-5-5°K. 25.6 C. B. SATTERTHWAITE, W. S. CORAK, B. B. GooDMAN and A. WEXLER, Bull. Amer. Phys. Soc. 30, No.4, 34 (1955): V (normal) ; 1-5° K. 25.7 R. D. WoRLEY, M. W. ZEMANSKY and H. A. BooRsE, Phys. Rev. 91, 1567 (1953): Ta; 2-5° K. 25.8 R. D . WoRLEY, M. W. ZEMANSKY and H. A. BoORSE, Abstracts of the Third International Conference on Low Temperature Physics and Chemistry (The Rice Institute, Houston, Texas, 1953, unpublished), p. 57: Ta and V; 2-5° K. Further details concerning these measurements are given in J. EISENSTEIN, Rev . Mod. Phys. 26, 277 (1954).

26. Group VIa. No measurements in the helium temperature range are available for elements in this Group. 26.1 C. M. SLANSKY and L. V. CouLTER, J. Amer. Chern. Soc. 61, 564 (1939): Te; 143000 K. 26.2 W.DESORBO,

J. Chern. Phys. 21, 1144 (1953): Se; 15 -300°K.

27. Group Vlb. a) Chromium. If the method of least squares is applied to the data of EsTERMANN et al. in the liquid helium range a slightly lower value of 0 0 is obtained than that given by these authors. Neither value is very accurate, however, as the electronic contribution to the atomic heat is large and there is considerable spread in the experimental points. The hydrogen range data are reported only in an abstract from which quantitative conclusions cannot be drawn. versus T (3) Molybdenum. HoROWITZ and DAUNT obtained a curve of (see Fig. 18, curve I) by using a plot of Cf T versus T2 to determine CE, finding from each value CL = C- CE for each measured point and then calculating of CL. Above 18° K they used the data of SIMON and ZEIDLER. The one point shown in Fig. 18 at about 1.5° K is an average of all their values below 4° K, which show a considerable spread. If, however, the slope of the least squares line

e

e

e

1 2

Indirect method-see text. Calculated from (20.1), using measured values of (.d C)r0

320

KEESOM

and

PEARLMAN;

Low Temperature Heat Capacity of Solids.

Sect. 27.

fitted to their values of CjT versus Pis used to find 8 0 , we get the value given in Table 11, which corresponds to curve II in Fig. 18. This curve resembles the behavior of 8 versus T of most other elements more closely than does curve I. y) Tungsten (Wolfram). The data of SrLvrm and DAUNT (see reference 19.6) have been superseded by the newer results of HoROWITZ and DAUNT. A variety of alternatives is available in interpreting these data, which are plotted in Fig. 19 up to 20° K, with the liquid helium temperature range points plotted separately on a larger scale on the inset. The least squares fit for all the points is shown as a solid line on both the main graph and Mo the inset. This corresponds to the second entry under Win Table 11 (80 = 379° K, y = 1. 2 millij oules j mole degree 2). The dashed line in the inset has the same slope as the solid line with an inter7{/ cept corresponding to RAYNE's value -r Fig. 18. e versus T for Mo. o: HoRowirz and DAuNT of y (1.48 millijoulesjmole degree2). N ei(reference 27.2); 0: SIMON and ZEIDLER (reference h 1. f. h . . h . 16.6). For solid lines see text. t er me 1ts t e pomts m t e mset very well, but the least squares fit to these points alone, shown as the dash-dot line in the inset, corresponds to much lower 8 0 and y values (80 = 279° K, y = 0.8 millijoulesjmole degree2 ) which are given as the third entry under W in Table 11. HoROWITZ and DAUNT estimated 8 0 as 250° K by the method described above in the paragraph on molybdenum; their estimate of y is 0.75 millijoulesjmole degree2 (first entry in Table 11). o • 20 These low values of 8 0 would ............ • _...,< , result in a B(T) curve which 2 ~ o,_-----rises immediately after the true T 3 region rather than falling as is the case with 10 other elements. As mentioned below the calculated vibration spectrum for tungsten also corresponds to a much L----~-----::''::-----::!:;:--;;;:;,-----::!.o higher value of 8 0 . These _ - r2 observations, and the scatter IJ • ~ of the points below 4° K lead Fig. 19. CJT versus T' for W. Q: HoROWITZ and DAUNT (reference 27.2); for lines see text. us to the conclusion that the best available estimates of 8 0 and y are }79° K, corresponding to the slope of the solid line in Fig. 19, and RAYNE's value, 1.48 millijoulesjmole degree 2, respectively. FINE has calculated the vibration spectrum for the body-centered lattice and applied the results to tungsten. The parabolic portion of the spectrum at the low frequency end corresponds to a 8 0 of 367° K, which is in reasonable agreement with the value given in Table 1 (p. 302). FINE was able to compare his calculation only with experimental data above 26° K, from which it appeared that the calculated heat capacity was too low. The experimental data above 26° K are higher than would follow from a simple DEBYE spectrum with equal to 367° K, so that 8 apparently decreases below 8 0 at temperatures above 20° K. MANNING and CHODOROW calculated the electron distribution and found ga (C) equal to 0.858 levelsjeV atom. The value calculated with RAYNE's value of y in (9.3) is 0.627. By using the same distribution with one electron less, they

e

321

Group VII and VIII.

Sect. 28, 29.

estimated ga (') for tantalum to be 1.09, whereas the value calculated from the tabulated y (see Table 10 in Sect. 25) is 2.3. Table 11. Low temperature atomic heat of the Elements-Group VI b.

-

Element

@0 (°K)

Cr

402

~~-- -

-- ··--

Mo - ---- ----- - -

w

425

range (°K)

I

1-4 >10

4

1-10 16

10 - ···

----~----

250 379 279

I reference y . .. fmole I To(°K) I (mllllJOules degree')

20

--··--

?

2-20

I

I

-

1.54

17.4 27.4

2.09 2.10

27 .2 16.5 16.6

- - --

0.75 1.2 0.8 1.48

27.2 (see text) (see text) 16.5 22.17

27.1 P. C. FINE, Phys. Rev. 56, 355 (1939): W; calculation of vibration spectrum. 27.2 M. HoROWITZ and J. G. DAUNT, Phys. Rev. 91, 1099 (1953): Mo; 1-10° K , W; . 2-20° K. 27.3 M. F . MANNING and M. I. CHODOROW, Phys. Rev. 56, 787 (1939): Wand Ta; calculation of distribution of electrons. 27.4 J. WEERTMAN, D . BuRK and J. E. GOLDMAN, Phys. Rev . 86, 628 (1952): Cr; 103000 K.

28. Group VII a and b. Manganese . Of the elements in these Groups only manganese has been measured in the liquid helium range. GuTHRIE et al. have recently reported the value 13.8 millijoules(mole degree 2 for y, with a probable error of about 5%, but t.!ley gave no 8 0 value. The earlier data above 10° K indicate a 8 value of 417° K, but it is doubtful if this can be extrapolated to lower temperatures.

reference

Element

Mn

2-4 >14

13.8

I 28 .1,28.328.2

28.1 L. D. ARMSTRONG and H . GRAYSON-SMITH, Canad. J. Res. A 27, 9 (1949): Mn; 14-22° K. 28.2 R. G . ELSON, H . GRAYSON-SMITH and J. 0. WILHELM, Canad. J. Res. A 18, 83 (1940): Mn; 16-40° K. 28.3 G . GuTHRIE, S. A. FRIEDBERG and J. E . GoLDMAN, Phys. Rev. 98, 1181 (1955): Mn (electronic contribution only); 2-4° K.

29. Group VIII. These elements occur in sets of three at the ends of the three transition series, which correspond to the filling of the 3d, 4d and 5d electron shells, respectively. Many of the properties of the transition elements, such as the ferromagnetism of iron, cobalt and nickel, have been related to the structure of the d-levels. The information which can be gained about this structure form the y values of all the transition elements will be discussed in the next section. r:t..} Iron. The results of DUYCKAERTS and of KEESOM and KURRELMEYER are in excellent agreement, but those of EucKEN and WERTH appear to be about 10% too low. KEESOM and KuRRELMEYER noticed a small irregularity around Handbuch der Physik, Bd. XIV.

21

322

KEESOM

and

PEARLMAN:

Low Temperature Heat Capacity of Solids.

Sect. 29.

13.5° K (just below T2=200 in Fig. 20) and attributed this to a minute quantity of hydrogen in the cavities of their sample. MANNING found from his calculation of the band structure the value 1.9 millijoulesjmole degree 2 for y, compared with 5.0 from the atomic heat. {J) Cobalt. In the overlapping temperature region the results of DuvcKAER TS and of CLusrus and ScHACHINGER agree excellently. The 6J(T) curve is normal; after a constant @0 value in the true P region it decreases. In their analysis of the higher temperature data, CLusrus and ScHACHINGER conclude that the electronic contribution is noticeable up to room temperature. y) Nickel. Agreement between the different sets of data is reasonable; the results of CLusrus and GoLDMAN in the hydrogen range are about 2% higher than those of KEESOM and CLARK. The true P region seems to extend up to 20° K and above this temperature begins to de,--------, crease. :0 ~I6 -c FLETCHER' S value of I 1 0'bo-~l{ 7.1 millijoules /mole degree 2 ~~..JS I a I L __ L_j for y from his band structure 0 10220 calculation agrees very well r with the value 7.4 obtained calorimetrically. o) Palladium. There is ( ) 100 '100 reasonable agreement among JOO the different sets of data in Fig. 20. C/ T versus Ti for F e : 0 : KEESOM and KuRRELM EYER thehydrogen region, with the (reference 29.9) ; 0 : DUYCKAERTS (reference 29.5); 1::,: EucKEN and WERTH (reference 29.7). true T 3 region extending to about 17° K. There is some spread in the reported y values, with that of PICKARD and SIMON being rather higher than those of RAYNE and of YATES and HoARE. s) Platinum. All the data for the whole temperature range are in excellent agreement. The y values of KEESOM and KoK above 1 o K and of RAYNE below 1 o K agree within the experimental error.

e

0

_l _ _ _

Table 13. Low temperature atomic heat of the Elements-Group VI I I.

e

Element

I

Fe -

---- - -

0

I

(°K)

1-20 1-20 >15

469 466

---- ~

Co Ni ·-

-------

I

I -

20 20

5.0 5.1

reference

I

--

2-18 > 15

20

459 456

10-26 1-19 >10

19 19

275 273

2-22 >1 14

17

229

1-20 17

20

II

-

29.5 29.9 29.7

---

5.0

29.6 29.4

7.6 7-3

29.2 29.8 29.1, 29.7

13 9-30 10.7

16.3, 29.10 17.16 16.5 29.3

-- -~

Pd

Pt

I TB('K) I (millijoulesfmole y degree')

445

- - - - --

-

range (°K)

I I

- --

I

I

I

I

6.6 6.9

I

17.12 16.5 16.6

Sect. 30.

Electronic heat capacity in the transition metals and their alloys.

323

29.1 R. H. BusEY and W. F. GIAUQUE, J. Amer. Chern. Soc. 74, 315 7 ( 19 52): Ni; 13300° K. 29.2 K. CLUsws and J. GoLDMAN , Z. phys. Chern. Abt. B 31, 256 (1936): Ni; 10-26° K. 29.3 K. CLusws and L. ScHACHINGER, Z. Naturforsch. 2a, 90 (1947): Pd; 14-270° K. 29.4 K. CLUSIUS and L. SCHACHINGER, Z. Naturforsch. 7a, 185 (1952): Co; 15-270° K. 29.5 G. DuYCKAERTS, Physica, Haag 6, 401 ( 1939), also in C. R. Acad. Sci. Paris 208, 979 (1939): Fe; 1-20° K. 29.6 G. DuYCKAERTS, Physica, Haag 6, 817 (1939): Co; 2-18° K. 29.7 A. EucKEN and H. WERTH, Z. anorg. u. angew. Chern. 188, 152 ( 1930): Fe, Ni; 15 - 205° K. 29.8 W . H. KEESOM and C. W. CLARK, Physica, Haag 2, 513 (1935). also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 235e: Ni; 1-19° K. 29.9 W. H. KEESOM and B. KuRRELMEYER, Physica, Haag 6, 633 (1939). also in Commun. Kamerlingh Onnes Lab. Univ. Leiden no. 257a: Fe; 1-20° K. 29.10 G. L. PICKARD, Nature, Land. 138, 123 (1936): Pd; 2-22° K. 30. Electronic heat capacity in the transition metals and their alloys. A striking disparity is evident in Table 1 between the y values of the transition metals and those of the nontransition metals. The average of 15 entries in the former group is 5. 8 milli joules j mole degree 2 while that of 14 in the latter group is only 1.2. If, as is reas- ~!oJ 1Q onable in terms of the interpre- t""'"' I tation to be discussed later, the I I I I I I values of the three ferromagnetic I I I I I I I I elements Fe, Co and Ni are I I I I I I I I I I I II I I 1 doubled, the disparity becomes I I I I I greater; the average for the tran- - .B -.J -.o -.1 -.J -.¥ -.2 -.5 0 sition metals so weighted is 7.2. -t:eV The relationship between the Fig. 21. g versus • for the 3d and 4s bands, after SLATER, Phys. Rev. 49, 537 (1936). high electronic atomic heat of the transition metals and the dbands was first pointed out by MoTT 1 . The value of ga (C) and therefore of the electronic atomic heat would be expected to be high in d-bands for two reasons. In the first place, the spatial extension of the d-electron wave functions is not as large as that of the valence s-electron wave functions so that there is less overlap between the d-electron wave functions on adjacent atoms. Hence the d-bands should be narrower than the valence s-bands. Furthermore, the former must accomodate ten electrons while the latter accomodate only two . Thus if it is assumed that in the transition metals the d-band and the valence s-band overlap and that the FERMI level falls within the d-band these two factors could account qualitatively for the observed difference in y values between the transition and non-transition metals. For a quantitative discussion information is required on the actual structure of the d-band. This structure has been calculated for the 3d-band by KRUTTER 2 and later by SLATER 3 and the latter's results are given in Fig. 21. The vertical dotted lines indicate the points at which the 3d-band accomodates the indicated number of electrons. The overlapping 4s-band is also shown, and it is clear that this band makes only a small contribution to ga (C) when the FERMI level falls in the 3d-band. This curve has been the basis of an extensive recent discussion

r-r

:

J h ti'

-

4 (•)

1 N. F. MaTT: Proc. Phys. Soc. Land. 47, 571 (1935). 2 H. M. KRUTTER: Phys. Rev. 48, 664 (1935). 3 J. C. SLATER : Phys. Rev. 49, 537 (1936). 21*

324

KEESOM

and

PEARLMAN:

Low Temperature Heat Capacity of Solids.

Sect. 30.

by DAUNT [12] who plots it with nv, the number of valence electrons per atom, rather than energy as abcissa (see Fig. 22). The ordinate is y which by (9.3) is proportional to ga ('). This equation must be modified for application to the ferromagnetic metals Fe, Co and Ni. The d-band is presumed to split into two sub-bands, one for each direction of spin. In the ferromagnetic metals one of these sub-bands is completely filled with five electrons and shifted to lower energy than the other. The valence electrons in excess of five are distributed between the higher of the d-sub-bands and the overlapping s-band. Since the filled sub-band makes no contribution to the heat capacity we have instead of (9.3)

ga('h·3d+ga(')4s=0.424y. (30.1) As was mentioned above ga (C) 4 s is small compared to ga (C) 3 d; and smce (30.2) we have approximately for the ferromagnetic metals

ga (')3d = 2

X

0.424 y.

(JO. 3)

Hence the values plotted in Fig. 22 for these metals are twice the calli" orimetric y values. ~Zr llo The comparison in Fig. 22 be Htbx· presented by x(T + 5) =C. OO

0.280 0 +0.0132 1.090

278 + 0.0116 1.098

0 .248 643 + 0.0038 1.13 7

0.174 1448 -0.0452 1.306

0.144 1811 -0.091 1.430

- 0.0218

0.342 0 -0.0204 0.849

0.324 334 -0.0223 0.862

0.280 737 -0.0312 0.918

0.175 1611 -0.087 1.138

0.144 1972 -0. 136 1.259

-- 0.0509

0.418 0 -0.0422 0.700

0.383 407 -0.0447 0.718

0.304 885 -0.055 0.769

0.175 1759 -0.118 1.016

0.144 2119 -0.172 1.140

- 0.0781

0.554 0 -0.059 0.564

0.449 512 - 0.063 0-595

0.305 1030 -0.076 0. 664

0.170 1881 -0.144 0.897

0.140 2232 -0.201 1.038

- 0. 113

0.930 0 -0.075 0.286

0.529 754 -0.080 0.366

0.280 1273 -0.098 0.560

0.140 2012 -0.174 0-777

0.124 2326 -0.235 0.902

- 0.128

1.332 0 -0.079 0.222

0.540 870 -0.085 0.292

0.265 1384 -0. 106 0.497

0.140 2112 -0.186 0.737

0.124 2425 -0.249 0.852

The procedure for the calculation of the absolute temperatures was the following. The differences in internal energy U (see Sect. 21) of the several isentropics at H = 0 were derived from the caloric measurements described in Sect. 43. The variation of U along each isentropic could be calculated from the relation [cf. Eq. (8.1)]: H

LlU=-(MdH.

(52.1)

0

In this way the U values of Table 24 were calculated with an arbitrary zero point. 1

T. H .

GEBALLE

and W. F.

GrAUQUE:

J.

Amer. Chern. Soc. 74, 3513 (1952).

125

Other salts.

Sect. 52.

Subsequently, the entropies of the first column were calculated; not from the magnetization curve at the initial temperature (the behaviour of the salt does not obey a BRILLOUIN function), but from the relation: LlS =

1- dU, f -T

(52.2)

which is valid at constant magnetic field [cf. Eq. (8.1)]. It was applied to the U values at the field of 8500 oersteds, where the absolute temperatures could still be measured with the help of a carbon thermometer (see Sect. 43). Finally, all the temperatures of Table 24 could be calculated applying:

T=(~~)H·

Table 25. Cobalt sulphate (FRITZ and GrAUQUE). Adiabatic magnetization curves for a powdered ellipsoid. (oMfoH)s is given in e.m.u. per mole. H; and 7i are the initial field and temperature of the demagnetization, H is the applied field. -

Tt

0 800 1600 2400 6000

(52-3)

Tt

- - =3200 T,

- =1990

-- -·---- -·

I (-~~)8

H

HI

H1

_HI_ =1280 H

2.92 0.81 0.21 0.18 0.13

0 800 1600 2400 6000

I

oM ) s bli-

H

I

6.21 1.40 0.365 0.152 0.036

0 776 1185 2373 5390

I (~Z-Js 11.47 2.17 1.15 0.231 0.028

Measurements on cobaltous sulphate were made by FRITZ and GIAUQUE 1 . A powdered sample was used (see Sect. 45) and the applied field was parallel to the measuring field. Values of (8Mj8H) 5 on three isentropics are given in Table 25. The zero field susceptibilities can be compared with those of Table 14. Table 26. Gadolinium phospho-molybdate (GrAUQUE and MAcDouGALL, and GIAUQUE, STouT EGAN and CLARK). Adiabatic magnetization curves. The first number in each block gives the quantity (oMfoH)s in e.m.u. per mole; the second number is the magnetic moment in e.m.u. per mole, the third number is the temperature difference from the zero-field temperature. (oMfoH)s, M and .dT (mill:dei(Tees) TH=O

H=O

0.798 H;=1633 1i = 1.429

9.81 0 0

I

·-

H=SO

9-78 490 0 .94

I H=I OO I H=250 I I

8.84 9-72 2375 978 19 3.8

H=SOO

7-03 4380 80

I H=IOOO I H=2000 I H=4000 I H=8000

I

I

1.42 0.38 0.48 3-95 12460 10980 9400 7020 289

0.664 H; = 2040 11 .79 :Ii= 1.419 0 0

11 .76 10.76 11.65 2880 1174 589 20 3.8 0.94

0.42 1. 52 4.44 0.31 8.09 5240 13 550 10860 112410 8240 82 738 265

0.487 H;=2760 16.06 = 1.430 0 0

13.83 15.45 15-95 801 3800 1 587 21 0.95 3-9

10.10 5.25 6820 10370 234 78

r.

22.20 18.67 22.95 0.339 H;= 3750 23.09 2282 1152 5 312 :Ii = 1.423 0 21 0 .96 0 3-9

0.19 0.34 1. 73 15120 13460 15860 620 -

0.08 0.34 6.07 12.60 1.99 18 860 17010 19390 9222 13460 504 205 72

0.02 0.46 2.32 6.52 0.202 H;= 5660 38.30 15.00 25.84 35.10 37-90 22080 18160 24890 24340 13280 8320 3 744 1909 7i=1.451 0 21 4.1 169 1.04 0 393 57 0.02 2.38 15.42 0.53 28.25 40.00 6.55 44.00 0.172 H;=6420 45.60 26690 26100 23640 19680 14720 9480 1i = 1.444 0 2247 4350 144 1.14 343 5.0 0 23 57 0.1 0 2.11 6.52 0.49 15.72 31.20 51.60 0 .126 H; = 8000 62.20 60.00 29 250 28330 26 250 22500 17470 11 940 5881 7i = 1.433 0 . 3081 132 I 299 24 5.6 1.42 1 0 I 58 1

J. ]. FRITZ and W.

F. GrAUQUE :

J.

Amer. Chern. Soc. 71, 2168 (1949).

126

D. DE KLERK: Adiabatic Demagnetization.

Sect. 53.

We shall not give the absolute temperature values as calculated by FRITZ and GIAUQUE, since they were based on wrong assumptions concerning the entropy of the salt, see Sect. 45. Gadolinium phospho-molybdate was investigated by GIAUQUE and MAcDouGALL!, and later by GIAUQUE, STOUT, EGAN and CLARK 2 • The results are given in Table 26. TheM values were calculated from the measured (8Mj8H)s data. The zero field temperatures were derived from caloric experiments (see Sect. 47), the temperatures in fields from the magnetic data.

D. Magnetic investigations at the lowest temperatures. I. Cooperative effects. 53. Introduction. As was stated in Sect. 28 most paramagnetic salts used for the demagnetization process show a maximum in the susceptibility; below this maximum the magnetic properties undergo radical alterations. The present chapter deals with the phenomena occurring in this region. In general the situation is much more complicated and less surveyable than at the higher temperatures. A theoretical interpretation of the phenomena is far from complete. Moreover the quantitative results obtained in different experiments may be different by an order of magnitude and it even happens that the data found with the same sample in the same apparatus are noticeably different on subsequent helium days. The general course of the results may be described in the following way. Near and below the temperature of the maximum, the susceptibility as measured with a ballistic inductance bridge (denoted by x. whereas x' and x" are the real and imaginary components of the dynamic susceptibility, see Sect. 23) depends on the value of the measuring field. Measurements with an a.c. bridge show that x" becomes important in this region. It has already a noticeable value somewhat above the temperature of the maximum of x. but a steep rise takes place near this maximum. Here x' is markedly lower than the ballistic susceptibility. The value of x' decreases with increasing frequency, whereas x" increases with increasing frequency. This behaviour suggests the occurrence of a relaxation effect. Usually, however, it is impossible to describe the curves quantitatively with one relaxation time and the experimental values are not in agreement with the extrapolation of the spin-spin or spin-lattice relaxations found in the liquid helium region. The a.c. experiments suggest relaxation times of the order of 10- 3 sec., but beside these there are also much longer times involved, which influence the ballistic measurements, giving rise to double deflections as described in Sect. 24. For several salts x" increases with decreasing temperature, but b a few cases x" shows also a maximum at a temperature somewhat below that of the maximum of x'· Both x' and x" depend on the value of the measuring field. In most cases x" is very small as compared with x', only a few percents at the lowest temperatures. For manganese ammonium sulphate, however, x"/x' reaches an appreciable value, viz. 0.4. If the susceptibility is defined as MfH•._1 (see Sect. 7) it is obvious that it depends on the shape of the sample. In some cases, however, the results obtained with ellipsoids of different excentricities do not give the same values for M/Hint. W. F. GIAUQUE and D.P. MAcDouGALL: J. Amer. Chern. Soc. 60, 376 (1938). W . F. GIAUQUE, J. W. STOUT, C. J. EGAN and C. W. CLARK: J. Amer. Chern. Soc. 63, 405 (1941). 1

2

Sect. 53.

Cooperative, effects, introduction.

127

The susceptibilities are strongly influenced by external magnetic fields, but the curves for different salts are widely different. In the case of chromium potassium alum, for instance, at the lowest temperatures, the susceptibility, as measured for a sphere, decreases to about half its value in a field smaller than 50 oersteds. In the case of chromium methylamine alum, however, after a slight decrease, a steep increase is found, followed by a pronounced maximum. If lines of constant magnetic field are drawn in a susceptibility versus entropy diagram the low field curves show a maximum in X· The maximum moves to lower entropies with increasing field strength, and for quite moderate fields it vanishes below the region accessible to the experiments. (In the case of the chromium alums it was found that a second maximum occurs at higher fields, but the origin of this maximum is completely obscure and in the present section we leave it out of the discussion.) Below the locus of the maxima the quantity (oMjoS)H is positive so that, according to Eq. (9.7), the temperature on an isentropic magnetization curve decreases with increasing field. On the locus of the maxima, T passes through a minimum. If the field is increased to the initial value of the demagnetization, the temperature must go up to the initial temperature. It can easily be shown that, if T on an isentropic shows a minimum, S on an isothermal exhibits a maximum. In a T versus H plot, lines of constant entropy and magnetic moment both show minima. In general it is found that the temperature drop on an isentropic in low fields is small, so that effectively there is not much difference between the isothermal and the adiabatic magnetization curves in this region. Hysteresis effects have been found at the lower temperatures in several salts. The shape of a hysteresis loop can be measured by switching a field on and off in a number of steps in both directions observing the ballistic galvanometer deflections, see Sect. 23. If we are only interested in the remanent magnetic moment (e.g. as a thermometric parameter, see Sect. 11) a very simple loop of only four deflections is sufficient. During such experiments it was found that the value of the remanent moment may depend somewhat on the number of steps in which the loop is passed through. The general tendency is that the remanent moment increases with increasing maximum field of the loop and with decreasing temperature. There are, however, exceptions from this rule. In the case of chromium potassium alum the remanent moment passes through a maximum when the field is increased; in the case of chromium methylamine alum the remanent moment shows a maximum with falling temperature. If the hysteresis loops are plotted as a function of Hext they are very narrow and the remanent moments are exceedingly small. If, however, the loops are plotted against Hint (see Sect. 7) they show a more familiar shape. The coercitive fields are very small under all circumstances. The hysteresis effects start at a well defined temperature Tc, which is slightly higher than the temperature of the susceptibility maximum. If a magnetic field is applied parallel or perpendicular to the small measuring field, the hysteresis phenomena decrease rapi:lly and they vanish already in fields of some tens of oersteds. A transition curve where the hysteresis effects disappear can be constructed in a T versus H diagram. This curve does not coincide with the locus of the maxima of the susceptibility mentioned above, the region of the hysteresis effects being somewhat wider. The two curves, however, are rather close together and the difference is neglected in most theoretical work.

128

D.

DE KLERK:

Adiabatic Demagnetization.

Sect. 54.

The occurrence of a x" in the hysteresis region must be partly due to the hysteresis effects thEm:>elves. A proof that also relaxation plays still an important role follows from the fact that x" is not frequency independent, as it should be in the case of pure hysteresis losses. But also the relaxation effects decrease rapidly outside the hysteresis region. This follows both from the smallness of x" and from the absence of double deflections in ballistic measurements somewhat outside the region (see Sect. 24) . The specific heat of all paramagnetic salts shows a steep increase in the neighbourhood of the susceptibility maximum. In the following sections we give first a survey of theoretical work dealing with the phenomena described here; after this we discuss in some detail the experimental results obtained with various salts. We will present the data in the form of graphs and not in tables (contrary to what we did in Chap. C), since we believe that at the present time most of the data are of qualitative interest only. 54. The theoretical problem. The phenomena mentioned in the foregoing section must be due to interactions between the magnetic ions giving rise to cooperative effects. The only way to come to a satisfactory theoretical description is to give a very rigorous discussion of both the magnetic dipole and exchange interactions. In general, the dipole interaction presents more difficulties than the exchange since the forces are of long range. No complete theoretical picture is available at the present time, but there are two methods to come to approximate solutions. One is to derive formulae valid at high temperatures and to extrapolate them to the lower region. This is the method of Sect. 32. It is hardly probable, however, that the formulae obtained in this way will keep their validity below Tc (see Sect. 53). The other method starts from the other end. It consists of finding the configuration of the magnetic ions with the lowest free energy at absolute zero (as a function of an applied magnetic field) and then introduce the influence of the temperature as a perturbation. It is possible, however, that formulae obtained in this way are only valid at temperatures low as compared with Tc, hence for each salt at temperatures essentially below the region that can be obtained with it by the magnetic cooling method (see Sect. 1). Let us first investigate whether the formulae discussed in Sect. 32 lead to the occurrence of a transition temperature. The LORENTZ formula [Eq. (7.15) J predicts a CuRIE point : (54.1) so that the value depends on the axial ratio of the sample, and no cooperative effects can occur in the case of a sphere. This is in distinct disagreement with experimental evidence. ONSAGER's formula [Eq. (7.20)] does not predict a CURIE point at all. VAN VLECK's relation (32.5), if the term with 1J and all higher terms are neglected, gives cooperative effects for a sphere in the case of non-vanishing exchange interaction only. Since, however, it must be expected that Tc is of the same order of magnitude as -r, so that the denominator of Eq. (32.5) converges only very slowly or not at all, it is not justified to derive conclusions from the first terms of VAN VLECK's formula alone. It follows that the theories of Sect. 32 do not give satisfactory results for the region near and below Tc. Before going into the details of other interaction theories we give a discussion on the basis of the WErss molecular field.

Sect. 55.

The molecular field theory of antiferromagnetism.

129

55. The molecular field theory of antiferromagnetism. WEISS 1• 2 gave a phenomenological theory of the interactions in a magnetic substance, as long ago as 1907. It was based on the assumption that the interactions can be accounted for with the help of a virtual magnetic field at the positions of the magnetic ions, proportional to the magnetization of the substance per unit volume (see Sect. 7). Though, at first sight, this assumption is rather crude it has yielded some remarkably successful results. If only the magnetic dipole interaction is taken into account the theory leads in first approximation to the LoRENTZ formula of Sect. 7. HEISENBERG 3 showed that, if the WEISS field is a consequence of the exchange interaction between neighbouring ions, it leads to an explanation of ferromagnetism as it occurs at normal temperatures. NEEL 4 - 6 , and later BITTER 7 and VAN VLECK 8 investigated the consequence of accepting a negative ratio between the WEISS field and the magnetization. The system of magnetic ions was split into two sublattices, A and B, in such a way that each ion of A is surrounded by ions of B only, and vice versa. It was assumed that the ions of each sublattice experience a WEISS field proportional to the magnetization of the other sublattice, but in the opposite sense. Under these conditions it is found that a transition temperature Tc occurs, above which the salt is essentially paramagnetic, following a CURIE-WEISS law:

c

x = r-=7J·

(55.1)

but below this temperature the sublattices have spontaneous magnetizations in opposite directions. This configuration of the magnetic ions is denoted as "antiferromagnetic". The direction of spontaneous magnetization of the two sublattices is determined by the crystalline anisotropy. The course of susceptibility with temperature depends on the orientation of the measuring field. If the field is parallel to the direction of the spontaneous magnetization, the susceptibility below the transition point decreases to zero with falling temperature, so that X shows a maximum at T.. If the field is perpendicular to the spontaneous magnetization, the susceptibility below Tc is independent of temperature. In the case of a powdered sample the susceptibility at absolute zero is two thirds of that at ~. The influence of large magnetic fields was studied by GARRETT 9 , NAGAMIYA 10 , YosmA 11 , GORTER and HAANTJES 12 , and Mrs. VAN PESKI and GoRTER 13 . For a field perpendicular to the direction of spontaneous magnetization the susceptibility is independent of the field strength. If it is applied parallel to the spontaneous magnetization, the susceptibility increases with increasing field strength, P. WEiss: J. de Phys. 4, 661 (1907). P . WEiss: Ann . Phys., Paris 17, 97 (1932). 3 W . HEISENBERG: Phys. Z. 49, 619 (1928) . 4 L. NEEL: Ann. Phys., Paris 18, 5 (1932). 5 L. NEEL : Ann . Phys., Paris 5, 232 (1936). 6 L. NEEL: Ann. Phys. , Paris 3, 137 (1948). 7 F. BITTER: Phys. Rev. 54, 79 (1938) . s J. H . VAN VLEcK: J. Chern . Phys. 9, 85 (1941). 9 C. G. B . GARRETT : J. Chern . Phys. 19, 1154 (1951) . 1 0 T. NAGAMIYA: Progr. Theor. Phys. 6, 342 (1951) . 11 K. YosmA: Progr. Theor. Phys. 6, 691 (1951). 12 C. ] . GoRTER and J. HAANTJES: Leiden Commun. Suppl. No. 104b; Physica, 's-Grav. 18, 285 (1952). Leiden Commun. Suppl. No. 109a; 13 T . VAN PESKI-TINBERGEN and C.]. GoRTER: Physica, 's-Grav. 20, 592 (1954). Handbuch der Physik, Bd. XV. 9 1

2

D.

130

DE KLERK:

Adiabatic Demagnetization.

Sect. 55.

but an anomaly is found at a field He, where the magnetizations of the two sublattices orient themselves perpendicular to the field. If the field is increased further, another anomaly is encountered at a field Hg, where a transition takes place from the antiparallel configuration perpendicular to the field to a parallel orientation in the direction of the field. The relation between Hg and T plotted in the H, T-plane is called the" critical field curve". The shape is somewhat similar to the transition curve of a superconductor, see Fig. 40. GARRETT derived for the relation between Tc (the intersection with the T-axis) and Hg0 (the intersection with the H-axis):

kTc=f.tHg0 ,

(55.2)

1,0r-~-=====~---v~--~,-----;-~r~----~~~/~

/

~

'

~

O,!i ------

/

/

/

---

0

7,f

11/lf!J Fig. 40. Temperature versus magnetic field diagram for an antiferromagneticcrystal {according to GARRETT).-.-.- LinC's of constant magnetic moment. ---Lines of constant entropy.

where It is the magnetic moment of the dipoles. The more rigorous, strictly threedimensional, theory of GORTER and HAANTJES leads to:

(55-3) This was derived for a lattice of rhombic symmetry. The case of a crystal with cubic symmetry was discussed by Mrs. VAN PESKI and GoRTER. Only the behaviour at absolute zero was discussed, and the influence of a field parallel to a cubic axis. It was assumed that the preferred orientation for the spontaneous magnetization of the two sublattices is parallel to a cubic ax1s. If a field is applied parallel to one of the cubic axes, the sublattices orient themselves immediately parallel to one of the other cubic axes, so that the first transition field, ~, does not occur, only Hg being found. Under these conditions the choice of the cubic axis for the zero field orientation depends on the direction in which a field has been applied previously, and in practical cases this may lead to hysteresis effects in small fields. Let us suppose that the field is applied in the x-direction and that the zero field orientation of the sublattices is parallel to the y-axis. Then the susceptibility as measured in the x-direction is the quantity x11 introduced in Sect. 50.

Sect. 55.

131

The molecular field theory of antiferromagnetism.

For X.t there are two possibilities; it can be measured parallel to the zero field orientation of the sublattices or perpendicular to it. These two possibilities may be indicated by X.t (y) and X.t (z). The difference vanishes for fields larger than Hg where both sublattices are parallel to the x-axis. Three expressions could be introduced into the HAMILTONian for the cubic symmetry. They lead to three solutions for the X versus H diagram at absolute

.')(.((

Hg

Hg

If-

If-

Fig. 41. Susceptibilities x 11 and XJ.. for an antiferromagnetic crystal with cubic symmetry at absolute zero (according to Mrs. VAN P.EsKt and GoRTER). The field is applied parallel to a cubic axis. For further description see text.

Fig. 42. Susceptibilities x11 and XJ.. for an anti ferromagnetic crystal with cubic symmetry at absolute zero (according to Mrs. VAN PESKI and GoRTER). The field is applied parallel to a cubic axis. For further description see text.

zero, represented in Figs. 41, 42 and 43. The X.t curve for fields below Hg is the average of X.t (y) and X.t (z). This is what is probably measured in an actual experiment. The case of Fig. 41 represents a preference of the two spin systems for the cubic axes independently of one another. Fig. 42 is concerned with an anisotropic interaction between the two sublattices. The curve for x11 is the same as in Fig. 41, but X.t is essentially different. The solution of Fig. 43 is f somewhat of a mixture of the two ~'..: foregoing cases. Hg proves to be larger, x11 is constant below Hg and 'XI/ zero above it. The curves ·for X.t are somewhat similar to those of Hg IfFig. 42. Fig. 43. Susceptibilities x11 and Xl. for an anti ferromagnetic ANDERSON 1, 2 discussed the possi- crystal with cubic symmetry at absolute zero (according to Mrs. VAN PESKI and GoRTER). The field is applied parallel to bility that an antiferromagnetic a cubic axis. For further description see text. crystal should consist of several pairs of antiparallel sublattices of different orientations, t aking into account nextto-nearest-neighbour interaction. In this case it is no more true that the susceptibility of a powder at absolute zero is two thirds of that at the transition temperature. . . . . If only two antiparallel sublattices occur wtth mterachons between wns of different sublattices alone the derived from measurements in the paramagnetic region [Eq. (55.1)] is related to Tc by : (5 5.4) 8 = - Tc.

e

1 2

P. W. P. W.

ANDERSON: ANDERSON:

Phys. Rev. 79,350 (1950). Phys. Rev. 79, 705 (1950).

132

D. DE KLERK: Adiabatic Demagnetization.

Sect. 56.

If also interactions between the ions of one sublattice are introduced this is no more true and 8 may even become positive 1 . The above considerations indicate that almost any susceptibility curve can be explained if the proper theoretical assumptions are made. Though this is a somewhat unsatisfactory state of affairs, it may be in agreement with the fact that fairly large differences are found in the results obtained with different salts. N:EEL 2 introduced the hypothesis that antiferromagnetics may be split into domains, like the WEISS domains of a ferromagnetic. Each domain consists of two antiparallel sublattices, the orientations of neighbouring domains being different. Since the antiferromagnetic domains have a very small net magnetic moment, their shapes are much more irregular than in the case of ferromagnetic domains and there is very little direct correlation between the orientations in neighbouring domains. If a field is applied, the domains tend to orient themselves perpendicular to the field and, apart from this, the walls may undergo a reversible or irreversible shift. A direct experimental proof of the occurrence of antiferromagnetic domains is difficult. The complication is encountered, in the theoretical interpretation of the measurements, that one has to distinguish between the magnetic properties of the sample as a whole and those of each domain, separately. It is very difficult to give a satisfactory explanation of the hysteresis effects in an antiferromagnetic. It is well possible, since the remanent moments are very small, that they are only secondary effects caused, for instance, by impurities in the crystal. If a domain structure occurs, however, hysteresis may also be due to irreversible phenomena in the walls 3 . 56. Interaction theories. The first effort to find the state of lowest energy of a magnetic crystal at abolute zero was made by SAUER 4 . He considered a cube of 125 ions and calculated the field at the central lattice point, for various possible arrangements of the ions. LUTTINGER and TISZA 5 made calculations for cubic lattices with magnetic dipole interaction, also at absolute zero. The free energy of the crystals could be computed for various configurations of the dipoles. The danger of the method is apparently that the configuration with the really lowest energy may be overlooked in the calculations. For a face-centred cubic lattice (the case of the alums) the energetically most favourable configuration proved to be one with dipoles aligned parallel in chains, neighbouring chains being antiparallel. For a spheroid of excentricity larger than 6: 1, however, the free energy is lower for the para:Ilel orientation. This is due to the contribution of the demagnetizing energy of the spheroid. SAUER and TEMPERLEY 6 considered the influence of non-zero temperature with the help of the BRAGG-WILLIAMS approximation, hence assuming long-range order. As in the case of the molecular field theories (see Sect. 55) the lattice was split into two sublattices of antiparallel orientations. Parameters r1 and r2 were introduced giving the fractions of dipoles with wrong orientation in each sublattice. For any temperature equilibrium, values of r1 and r 2 can be calculated as a function of the applied magnetic field by minimizing the free energy of the crystal. 1 J. S. SMART: Phys. Rev. 86, 968 (1952). L. NE:EL: Proc. Conf. Theor. Phys. 1953 Tokyo, p. 703. J. A. BEUN, M. J. STEENLAND, D. DE KLERK a nd C. J. GoRTER: Leiden Commun. No. 301 a; Physica, 's-Grav. 21, 767 (1955). 4 J. SAUER: Phys. Rev. 57, 142 (1940). 5 J. M. LuTTINGER and L. TrszA: Phys. Rev. 70, 954 (1946); 72, 257 (1947). 6 J. SAUER and H. N. V. TEMPERLEY: Proc. Roy. Soc. Lond., Ser. A 176, 203 (1940). 2

3

Sect. 57.

Chromium methylamine alum.

133

It is found that a critical field curve occurs, like in molecular field theories, inside which the antiparallel configuration is stable. For the case of a spherical (56.1) k Tc = 2 fl Hgo, sample it follows;

as was also derived by GoRTER and HAANTJES, see Eq. (55.3). The transition on the critical field curve for temperatures below 2 Tc/3 is first order with a latent heat and a discontinuity in M. This has not been confirmed by the experiments. Variation of the shape of the sample leaves Tc unaltered, but the transition curve moves toward the T-axis with increasing excentricity. For an axial ratio of about 6:1 the region of antiferromagnetic order vanishes and the behaviour of the salt becomes ferromagnetic. ZrMAN 1 applied the BETHE method (hence assuming short range order) to the case of magnetic dipole interaction. His results were rather similar to those given above. The ratio k ~fpHg0 is of the order of unity and depends on the number of nearest neighbouring ions. KITTEL 2 demonstrated the possibility that the magnetic dipole interaction may lead to a domain structure. This, however, is not the antiferromagnetic domain structure as proposed by N:EEL (see Sect. 55), but a configuration of antiparallel domains of the order of 10-4 cm., the orientation being parallel inside each domain. The occurrence of such domains gives rise to appreciable complications. The transition from the antiferromagnetic to the ferromagnetic order in a spheroid of axial ratio approximately 6: 1 (see above) is ruled out; and in the interpretation of experimental results we have to make a sharp difference between the "technical" magnetization curve of the sample as a whole and the behaviour of the individual domains. Conclusions about the question whether this domain structure occurs or not might be derived from measurements of the BARKHAUSEN effect. At the end of this section, attention should be drawn to the possibility that the spin wave method might provide a suitable way of approach to the solution of the problem. The method has been worked out successfully for exchange ferromagnetism3-5 and antiferromagnetism 6 -11, but, as far as we know, it has not been applied to the problem of magnetic dipole interaction. Since, however, the results obtained from it are most reliable if the number of excitation waves is small, the possibility exists that the formulae derived for a given salt are only valid at temperatures below the region that can be reached by demagnetization of the salt itself, see Sect. 54.

II. Results obtained with individual salts. 57. Chromium methylamine alum. The relation between entropy and susceptibility as determined by BEUN, STEENLAND, DE KLERK and GORTER 12 is shown in Fig. 44. The measurements were performed with single crystals of spherical

J. M. ZIMAN: Proc. Phys. Soc. Lond. A 64, 1108 (1 9 51) . C. KITTEL: Phys. Rev. 82, 965 (1951). 3 F. BLocH: Z. Physik 61, 206 (1930). 4 F. BLocH : Z. Physik 74, 295 (1931). 5 T. HoLSTEIN and H. PRIMAKOFF: Phys. Rev. 58, 1098 (1940). 6 L. HULTHEN: Proc. Kon. Acad. Wetensch. Arnst. 39, 190 (1936). 7 G. HELLER and H. A. KRAMERS: Proc. Kon . Acad. W et ensch. Arnst. 37, 3 78 (1 934). 8 J. M. ZIMAN: Proc. P hys. Soc. Lond. A 65, 540 (1952). 9 J. M. ZIMAN : P roc. P h ys. Soc. Lond. A 65, 548 (1 9 52). 10 J. M. ZIMAN: P roc. P hys. Soc. Lond. A 66, 89 (1 9 53). 11 R. Kuso: Phys. R ev. 87, 568 (1952). 12 J. A. BEUN, M. J. STEENLAND, D. DE KLERK and C. J. GoRTER: Leiden Commun. No. 301a; Physica, 's-Grav. 21, 76 7 (1955) . 1

2

134

D.

DE KLERK:

Adiabatic Demagnetizati on.

Sect. 57

shape, mounted in such a way that one cubic axis was parallel to the small measuring field. Two samples were investigated , and systematical differences up to several percents were found in the susceptibilities. The results at the highest entropies are those discussed in Sect. 34- A sudden rise in susceptibility occurs at 5 = R ln 2, followed by a maximum near 5=0.5R. Below this maximum the a.c. susceptibilit y x' is noticeably smaller than the ballistic 0 zo e.m.u. 6'0-708 susceptibilit y X· It was found that x' depends only very little Fig. 44. Entropy versus susceptibility diagram for two spherical on the frequency and the amplisingle crystals of chromium methylamine alum (according to BEuN, tude of the measuring field. STEENLAND, DE KLERK and GoRTER). D. x' for first sphere, • = 225 c/sec. The susceptibilit y in the vi0 x for first sphere, measuring field 1.10 oersteds, cinity of the maximum as measfree period of ballistic galvanometer 1.3 sec. 'V x' for second sphere, v=225 cfsec:. ured by HuDSON and McLANE 1 x for second sphere, measuring field 1.10 oersteds, free period of ballistic galvanometer 1. 3 sec. is shown in Fig. 45. These authors also investigated two spherical single crystals and found differences of several percents in the susceptibilities. (Only the data of one sample are represented in Fig. 45.) Taking this into account, the agreement between the Leiden and Washington results is not bad. HUDSON and McLANE found that the susceptibilit y values near the maximum are largely influenced by the rate of precooling the crystal from room temperature to the liquid nitrogen region. This must be connected with the transition in the crystalline structure mentioned in Sect. 34- By cooling very slowly (several hours) O,J 11...L0-------~~.L,----e-.m-.-u-._ ___js-0 .70-tJ it was possible to obtain reproX/!1 _ ducible results. There is an indication that x' Fig. 4 5- Entropy versus susceptibility diagram for a spherical single crystal of chromium methylamine alum in the region of the sushas a double maximum, one peak ceptibility maximum (according to HuosoN and McLANE). occurring at 5 = 0.541 R, the + x', measuring field 0.46 oersteds, v = 210 cfsec. X x', measuring field 0.46 oersteds, v = 150 cjsec. other at 0.562R. Also the balJi. x, measuring field 1. 72 oersteds, free period of baUistic galvanometer 5.6 sec. listic susceptibilit y shows a double e x, measuring field 3.43 oersteds, free period 5.6 sec. maximum, but only for weak • x, measuring field 6.86 oersteds, period 5.6 sec. + x, measuring field 10.92 oersteds, free free period 5.6 sec. measuring fields. The lower maxiL __ _ _

1 R. P . HuosoN and C. K. McLANE: Phys. Rev. 95, 932 (1954).

Chromium methylamine alum.

Sect. 57.

135

mum decreases with increasing field and vanishes already at about ten oersteds, see Fig. 45. It is also the Leiden experience that only near the maximum X depends on the measuring field. The ballistic measurements performed at Oxford with an ellipsoidal sample 4:1 of compressed powder lead to a different result. GARDNER and KuRTI 1 found that x is constant from 5 = 0.50 R down to 5 = 0.36 R, the lowest entropy reached in the experiments. The imaginary part of the a.c. susceptibility, x", as measured by HUDSON and McLANE is shown in Fig. 46. The x" values are quite small over the whole region with the exception of a sharp peak near the maximum of X and x'. Similar results were obtained in Leiden. It is not absolutely sure that the field de- 0,6 pendence as shown in Fig. 46 is real. HuDSON and McLANE even suggested the possibility that x" might show a double maximum like x'· A striking difference between the t Leiden and Washington experiments"' Mf------:?1=------t-------1 is found in the frequency dependence ~ of x". In Leiden it was observed that x" increases with increasing frequency. ( From this it was concluded that therelaxation times occurring in the salt are very short. The Washington people 0,~10l:------::'-:c---c-::'::---=-=:-::----;;' e.m.u. 0,6'-1(}-a 0/l o,z found that x" is nearly independent of X.'i'Rfrequency and this leads to long relaxa- Fig. 46. Entropy versus imaginary part of a.c. susceptition times below the maximum. In bility diagram for a spherical single crystal of chromium alum (according to HuosoN and McLANE). accordance with these conclusions hard- methylamine \J Measuring field 0.30 oersteds. 0 Measuring field 0.4 5 oersteds. ly any double deflections (see Sect. 24) were found in the Leiden ballistic measurements (only occasionally close to the maximum) but very pronounced time effects were found in the Washington experiments. The supposition seems to be justified that the relaxation times of the Leiden and Washington samples were really strongly divergent. This is the more remarkable, since one of the Leiden samples was of the same origin as one of the Washington samples. Maybe, again, this is connected to the rate of cooling to liquid air temperature. Due to the very steep course of x" with entropy near the maximum the variation with time during a heating period is very fast. Consequently, it is difficult to extrapolate x" to the time of the demagnetization so that, for absolute temperature determinations, x" is impracticable as a thermometric parameter (see Sect. 11). The remanent magnetic moment as a function of the measuring field and the entropy, according to BEUN, STEENLAND, DE KLERK and GORTER, is shown in Fig. 47. The results are in reasonable agreement with those of HUDSON and McLANE. In Leiden the starting point of the hysteresis was found to be 5 = 0.54R. HuDSON and McLANE gave 5=0.53 R, GARDNER and KuRn 5=0.50R. The remanent moments are considerably smaller than those found for chromium potassium alum (see Sect. 58) . 1

W. E.

GARDNER

and N.

KuRT!:

Proc. Roy. Soc. Lond., Ser. A 223, 542 (1954).

D.

DE KLERK:

Adiabatic Demagnetization.

O.Gr---------------~--------------,

0

0

Z/l

1/l

.z:-

Fig. 47. Entropy versus remanent magnetic moment diagram for a spherical single crystal of chromium methylamin• alum (accord-

ing to BEuN, STEENLAND, DE KLERK and GoRTER). Free period

of ballistic galvanometer 1. 3 sec.

6 Measuring field 1.10 oersteds.

0 Measuring field 2.20 oersteds.

0 Measuring field 4.39 oersteds. \} Measuring field 8.78 oersteds. 0 Measuring field 21.95 oersteds.

0,11

0,08

/

0.011

0 -0,01

/ .......

......_

I

i

Sect. 57.

It appears that the remanent moment, for a given field Hext• shows a maximum as a function of entropy. Until now, chromium methylamine alum is the only salt showing this behaviour. A consequence is that the remanent moment is unfeasible as a thermometric parameter. The magnetic method of absolute temperature determination is difficult for this salt_ z, x', x" and .I: all are unsatisfactory as thermometric parameters below the susceptibility maximum (see above). x" is rather small (even at its maximum it is much smaller than for chromium potassium alum) so that it is difficult to distinguish between the heat absorption from the alternating field and the a.c. losses in the bridge (see Sect. 12). Moreover, the rapid variation of x" with time during a heating period is

a

I

I

V'

./

1O,G

1,/l

0

J'/1?-

(/,1

O,Z

(/,J

T-

"K 0,11

Fig. 48. Fig. 49. Fig. 48. Heat content versus entropy diagram for chromium methylamine alum (according to GARDNER and KuRTI). The zero point of Q/R is arbitrary, see text. Different symbols refer to different helium runs. The triangles represent points for which QJR was determined by heating into the region above the maximum. Fig. 49. Entropy versus absolute temperature diagram for chromium methylamine alum (according to GARDNER and 0 T• thermometer. X The Ct•RtE point.

+I

KURT!),

thermometer. - - Experimental curve.

6 Hysteresis heating. ---Theoretical curve for

o= 0 .27° K. another source of inaccuracy. After many tedious experiments HUDSON and McLANE came to the conclusion that the absolute temperature of the CuRIE point (defined as the starting point of the remanent moment) lies most probably in the region between 0.015 and 0.020°K.

Sect. 58.

Chromium potassium alum.

137

The best absolute temperature determinations are those by GARDNER and They used gamma radiation for heat supply. Above the maximum X (or T*) was the parameter. The quantity c* = d QfdT* was calculated and from this the total heat content Q = Jc* dT* could be computed as a function of entropy (with an arbitrary zero). The results are given in Fig. 48. Below the susceptibility maximum the method was modified as described in Sect. 48. After the demagnetization the total energy was determined, necessary to heat the sample to a temperature well above the susceptibility maximum, where X could be used as a parameter. The data obtained with this method are also shown in Fig. 48 (the triangles). The absolute temperature for each value of the entropy is equal to the slope of the curve of Fig. 48. The results are plotted in Fig. 49, together with some additional measurements made with the remanence thermometer and by applying hysteresis heat. The temperature of the CuRIE point was found to be 0.020° K. Recent Leiden measurements with a.c. heating gave T = 0.020° K for the CuRIE point and T = 0.002° K for the lowest temperature (S = 0.26 R). Experiments with gamma ray heating, however, gave de- tM '""""----,------~----,--------, finitely higher values for the lowest temperature. KuRTI.

58. Chromium potassium alum. Measurements below the susceptibility maxi- ""tnzt-----~--~,---+-----+--------j mum of this salt were per- -.::,. formed in Leiden, Oxford and Washington. The results are usually somewhat different for different same.m.u. 6YJ 110 0 zo ples. Fig. 50 gives susceptiX/R-bility values obtained in 50. Entropy versus susceptibility diagram for four spherical single crysLeiden 1 with four sam- Fig. tals of chromium potassium alum (according to BEUN, STEENLAND, DE KLERK GORTER). S > 0.40 R: a.c. measurements. S < 0.40 R: ballistic measples. They all were spheri- and urements with galvanometers of various free periods. Curve a : samples 1 cal single crystals mounted and 2, measuring field 1.08 oersteds, free period 7 sec. Curve b: sample 3, field 1.08 oersteds, free period 0.2 sec. Curve c: sample 4, July 19 51, with one cubic axis paral- measuring measuring field 0.33 oersteds, free period 1.3 sec. Curved : sample 4, ] anuary 1953, measuring field 1.08 oersteds, free period 1.3 sec. lel to the field of the magnet and one parallel to the measuring field. At the higher temperatures, where x and x' are equal, the x' values have been plotted since they can be measured with a higher precision. At the lower temperatures we have plotted X· The differences between the four curves are partly due to the different measuring techniques (see the subscript of the figure). It is sure, however, that also part of the differences is caused by the properties of the samples themselves; for instance, the differences at the higher temperatures give rise to the different values of the splitting parameter quoted in Sect. 35. The results obtained with one of the samples 2 • 3 are shown in more detail in Fig. 51. Both X and x' show a sudden rise just before the maximum. The 1 J. A. BEUN, M. J. STEENLAND, D . DE KLERK and C. J. GORTER : Leiden Commun . No. 300a; Physica, 's-Grav. 21, 651 (1955) 2 D. DE KLERK, M. J. STEENLAND and C. J. GoRTER: Leiden Commun. No. 278c; Physica, 's-Grav. 15, 649 (1949). 3 M. J. STEENLAND : Thesis, Leiden 1952.

138

D . DE KLERK: Adiabatic Demagnetization.

Sect. 58 .

rise is steeper than in the case of chromium methylamine alum (compare Figs. 44 and 51), and it takes place at a smaller entropy value, well below R ln 2. The susceptibility maximum is noticeably higher than in the case of the methylamine alum; to such an extent that, in the maximum, the correction for the demagnetizing field [see Eq. (7.5)] is very large, Xint being at least a factor twenty higher than Xext. This is distinctly not true in the case of the methylamine alum. For the latter salt Xint at the maximum is still of the same order of magnitude as C / Tmax; for the potassium alum Xint at the maximum is much larger than C/Tmax. Below the maximum, X is smaller than x', as in the case of the methylamine alum, but the difference is more pronounced. The value of x' is very little influenced by the frequency of the measuring field, but x depends on the free period of the ballistic galvanometer. This was demonstrated by making measurements with different galvanometers alternately in the same heating period. Double deflections were found (see Sect. 24) in the neighbourhood of the maximum, especially if a galvanometer with a short free period was X/I? X/I? used (e.g. 0.2sec.). These phenomena in0 zo 1/0 80' 10-9 dicate the occurrence of GO e.m.u. X/1?relaxation effects and Fig. 51. Entropy versus susceptibility diagram for a spherical single crystal of this is confirmed by the chromium potassium alum (according to data of DE KLERK, STEENLAND and GoRTER). x" JR is plotted on a tenfold magnified scale. x/R: measuring field behaviour of x". 1.08 oersteds, free periodofballisticgalvanomete r7 sec. x'fRandx"fR: amplitude of measuring field 0.183 oersteds, v=225 c /sec. Some values of x" are shown in Fig. 51 1 . It is already perceptible above the maximum of x', where no hysteresis effects were found. x" depends on the frequency of the measuring field. Experiments with several frequencies during the same heating period gave a proportionality to

yl.7±0.15.

The x" values of Fig. 51 show a maximum at an entropy somewhat below the maximum of X and x'. The values are noticeably larger than those obtained for the methylamine alum, and the maximum is less pronounced (see Sect. 57). Still x" is much smaller than x', the ratio x"/x' never exceeding 0.03 (it should be noticed that x" in Fig. 51 is plotted on a tenfold magnified scale). Though the behaviour of both X and x" indicates the occurrence of relaxation effects, it proves to be impossible to describe all the phenomena with one relaxation time. The time derived from the ballistic experiments is of the order of 10- 2 sec., whereas the a.c. measurements suggest times shorter than 10- 3 sec. Measurements by AMBLER and HUDSON 2 with a spherical single crystal gave slightly higher values for X and x' near the maximum than the Leiden experi1 D. DE KLERK, M. J. STEENLAND and C. J. GoRTER: Leiden Commun. No. 278c; Physica, 's-Grav. 15, 649 (1949). 2 E. AMBLER and R. P. HuDSON: Phys. Rev. 95, 1143 (1954).

Sect. 58.

Chromium potassium alum.

139

ments. DANIELS and KuRTI 1 , who used an ellipsoid of compressed powder with axial ratio 6: 1 found somewhat lower values for X; they made only ballistic measurements. The main qualitative difference between the Leiden results and those of AMBLER and HuDSON is that the maximum of X found by the latter authors is less sharp; the maximum of x' has about the same shape in both experiments. AMBLER and HuDSON found no double deflections with a ballistic galvanometer of 5.6 sec. Moreover x" did not show a maximum; it approached a constant value at about the entropy of the Leiden maximum, but at lower entropies it increased again. Some data on the remanent magnetic moment, J:, obtained by STEENLAND, DE KLERK and GoRTER 2• 3 are given in Fig. 52. The CuRIE point, defined as the point where a remanent moment appears first, was found at S = 0.40R, slightly higher than the entropy of the susceptibility maximum. This is in good agreement with t the value given by AMBLER and ~ o,J f-------+-----~------1 H UDSON (5 = 0.42 R), and with old ~ measurements of KuRTI, LAINE and SIMON 4 ( S = 0.44 R). The values of J: found for this salt are appreciably larger than o,ao~-------~-----------8~g-au-ss--c-m~m-o7l~J those for the methylamine alum E(see Sect. 57), but still they are Fig. 52. Entropy v ersus remanent magnetic moment diagram for single crystal of chromium potassium alum (according only a few percents of the moment atospherical STEENLAND). Measuring field 1.08 oersteds, free period of balwhich, oersted 1.08 of field the in listic galvanometer 7 sec. Different symbols refer to different helium runs. itself, is about one percent of the saturation moment. The field dependence of the remanent moment was also investigated by STEENLAND, DE KLERK and GoRTER. For a given entropy J: increases first with the measuring field, then it decreases. The maximum moves to higher field values with decreasing entropy. The values of the remanent moment are not too well reproducible. For one sample it was found that J: increased by about 20% in the course of a year; the data obtained with different samples may be different by even a factor of three. This, together with the smallness of J:, may indicate that the remanent moment is entirely due to spurious effects, such as impurities in the crystal or lattice defects. Complete hysteresis loops were measured with maximum fields of 4.32 and 12.95 oersteds. Each loop was described in 24 steps. Some data on the loops of 4-32 oersteds are collected in Fig. 53. If the loops are plotted against Hext they become very long and narrow (see the lower righthand block of Fig. 53). In ferromagnetism, however, it is standard practice to "shear" the loops, i.e. plot them as a function of Hint• see Eq. (7.5). The sheared loops have a more familiar shape than the unsheared ones, as can be seen from Fig. 53- The difficulty in these experiments is, however, that the term eMjV in Eq. (7.5) is of the J.M.DANIELS and M.KURTI : Proc. Roy. Soc. Lond., Ser.A 221,243 (1954). M. J. STEENLAND, D. DE KLERK and C. J. GoRTER: Leiden Commun. No. 278d; Physica, 's-Grav. 15, 711 (1949). a M. J. STEENLAND: Thesis, Leiden 1952. 4 N. KURT!, P . LAINE and F. SIMON: C. R. Acad. Sci ., Paris 204, 675 (1937). 1

2

140

E

DE KLERK:

Adiabatic Demagnetization.

Sect. 58.

same order of magnitude as Hext• so that small inaccuracies in the density of the sample, or small deviations from the spherical shape, may introduce appreciable errors into the shapes of the sheared loops. It was found that the remanent moment calculated from the loops of 12.95 oersteds was always somewhat larger than the directly measured remanent moment. This, again, is probably due to relaxation effects. Such differences were not found in the loops of 4.32 oersteds. JOO

;:;-.. !l zoo .;,

."

D.

J'/I?= O,Z7Z

S/H-O.JJG

0.79R a.c. measurements. 5 < 0.79R ballistic measurements. o o oersteds. o 130 oersteds. e1 34o oersteds. A survey of the results b. 20 oersteds. cp 170 oersteds. 18! 430 oersteds. · · · th (5 ) \1 80 oersteds. B 260 oersteds. x 540 oersteds. IS g1ven Ill e ' Xdiagram of Fig. 70. Some measurements performed with a. c. showed that the course of x' with field is very similar to that of X· The x" showed a pronounced maximum at the same field value as x'; there it was about a factor two larger l.f.!o-e than in zero field. The remanent moment decreases very steeply with the applied field. At S = 0.36R, in a field of ten oersteds, it is already smaller than one tenth of its initial value. The occurrence of strong anisotropies was demonstrated by rotatFig. 71. Polar diagram of susceptibility X.L versus orientation of the transverse ing the applied field field for a spherical single crystal of chromium methylamine alum (according to around the axis of the BEuN, STEENLAND, DE KLERK and GoRTER) . The straight lines are the directions of the cubic axes. 0 Experimental points, 5 ~ 0.20R, H J. ~ 170 oersteds. The measuring field. One of experimental curve may be split into a diagram with quaternary symmetry and _ _ _ __ _ ..:.a..:.n excess curve with twofold symmetry. the polar diagrams ob1 J. A. BEuN, M. J. STEENLAND, D. DE KLERK and C. J. GoRTER: Leiden Commun. mQ~~----~---------,-----------,

! mz~----~~~~~~---t------__,

No. 301 a; Physica, 's-Grav. 21, 767 {1955).

Chromium methylamine alum.

Sect. 64.

tained in this way is shown in Fig. 71. The most surprising result is that the symmetry is not quaternary, as might be expected around a cubic axis, but binary. It is possible, however, to split the curve formally into two parts, one with the expected fourfold symmetry (with its sides parallel to the cubic axes), and an excess curve with binary symmetry. The orientation of the excess curve proved to be different during subsequent helium runs (compare Figs. 71 and 72). Maybe this is due to processes during the precooling to liquid air temperature, see Sect. 57. For a given field the anisotropy is the more pronounced the lower the entropy is, see Fig.72. The anisotropy is small for relatively small fields (seethecurvefor 42.Soersteds in Fig. 73), but at higher fields it becomes more pronounced and increases relatively with increasing field. One polar diagram was measured above the CURIE point (at S = 0.70R and in a field of 13 0 oersteds) ; no anisotropy was observed there. Also HunsoN and McLANE 1 made investigations in perpendicular fields, with the applied field parallel to a cubic axis. The experiments were carried out with 1

R. P.

HUDSON

and C. K.

153

Fig. 72. Polar diagram of susceptibility XJ.. versus orientation of the transverse field for a spherical single crystal of chromium methylamine alum (according to BEuN, STEENLAND, DE KLERK and GoRTER), The straight lines are the directions of the cubic axes. Curves for H J.. = 170 oersteds at various entropy values. 0 S=0.44R. V 5=0.33R, b. S=0.26R, 0 S=0.20R,

Fig. 73. Polar diagram of susceptibility Xl. versus orientation of the transverse field for a spherical single crystal of chromium methylamine alum (according to BEUN, STEENLAND, DE KLERK and GoRTER). The straight lines are the directions of the cubic axes. Curves at S = 0.20R for various values of the field. b. H1. = 425 oersteds. 0 H1. = 170 oersteds, 0 H1. =42.5 oersteds, McLANE :

Phys. Rev. 95, 932 (1954).

154

D.

DE KLERK:

Adiabatic Demagnetization.

Mr------------, ------------,---- --------.

~r---------~~---------+----------~

oL-----------~2~oo~----------~~o-----O~e----~ooo /{-

Fig. 74. Susceptibility x 11 as a function of a longitudinal field for a spherical single crystal of chromium methylamine alum (according to BEUN, STEE NLAND, DE KLERK and GoRTER). The fields were applied parallel to a cubic axis. 0 S=0.318R, 1::, 5 = 0.253R, 0 5=0.200R.

Sect. 64.

a.c. at 210 cfsec. The susceptibility curves were similar to those ofFig.69, but the height of the maximum increased with falling entropy, to such an extent that, in the x versus S diagram, the curve for 180 oersteds exhibited a higher maximum than the zero field curve, contrary to the Leiden results of Fig. 70. A possible explanation is that in the sample of HUDSON and McLANE the excess curve with twofold symmetry happened to be parallel to the cubic axis. Further it was found in HuDSON and McLANE's experiments that the small decrease of the susceptibility in fields below 20 oersteds did not occur at the lowest entropies, below S = 0.48R.

Susceptibility curves obtained in Leiden from experiments in longitudinal fields are shown in Fig. 74. The applied field was parallel to a cubic axis. The measurements were performed ballistically; the small measuring field was 1.08 oersted, and the free period of the galvanometer was 1. 3 sec. Qualitatively the reoL--------!2o=--- -----q-::L0,..----e-.m-.u-.----:!60 . 10 -8 sults are similar to those ~/Rin transverse fields (compare Fig. 69), but the Fig. 75. Entropy versus susceptibility diagram in longitudinal fields (x 11 ) for a spherical single crystal of chromium methylamine alum (according to BEuN, maxima are much higher STEENLAND, DE KLERK and GORTER). Lines of COnstant field strength, and the rise and fall of Ballistic measurements. 0 0 oersteds. the curves are almost B 71 oersteds. + 230 oersteds. 1::, 9 oersteds. cp 106 oersteds. x 318 oersteds. vertical. They take place 'V 27 oersteds. IS! 141 oersteds. ).. 424 oersteds. at 60 and 210 oersteds. 0 53 oersteds. Ell 177 oersteds. "( 531 oersteds.

Sect. 64.

Chromium methylamine alum.

155

An indication is found for a double maximum, but it disappears again at the lowest temperatures. AMBLER and HuDsON 1 made measurements in longitudinal fields, also by the ballistic method, the measuring field being 1.72 oersted, the free period of the galvanometer 5.6 sec. The susceptibility versus field curves were very similar to the Leiden results, though the agreement was not quantitative. Indications for a double maximum were found below 5 = 0.45 R. The Leiden entropy versus susceptibility diagram is shown in Fig. 75. Here, contrary to Fig. 70, the susceptibilities in moderate fields reach values noticeably higher than the maximum of the zero field curve. Since the susceptibility in a longitudinal field is equal to (8Mj8H) 5 (see Sect. 50) the magnetization curves can be obtained by integration. Some Leiden data are given in Fig. 76 (the TOOOO'.------r------.------,-=------, ., figure contains also a few § curves above the CuRIE point). Due to the increase t ~ of the susceptibility in small I fields the curves well below /'1 5000 1-----+---:;;b~-t---~--1"'::::..._---i the CuRIE point start with a concave part. The sharp fall in the susceptibility at 210 oersteds results in a Oe .JOO 200 sharp bend of the magneIItization curve (very pro- Fig. 76. Magnetization curves for a spherical single crystal of chromium methylamine alum (according to BEuN, STEENLAND, DE KLERK and Goa· nounced in curve E). Above TER). Fields applied parallel to a cubic axis. The value of the saturation magnetization is 16 700 gauss cm.' jmol. this bend the course of M Curve A: 5 = 1.111R, Curve B: 5 = 0.738R, Curve C: 5 = 0.529R, with H is convex. At the CurveD: 5 = 0.372R, Curve £: 5 = 0.200R. bend, the magnetic moment is about one half of the saturation moment and the internal field in the sample was calculated to be 98 oersteds. Also AMBLER and HUDSON calculated the magnetization curves from their measurements in longitudinal fields. They compared them with the curves computed from the experiments of HuDSON and McLANE in transverse fields on the assumption that anisotropy is absent [hence applying Eq. (50.1)]. Large deviations were found in strong fields, but reasonable agreement was obtained below about 100 oersteds. From this the conclusion was derived that the anisotropy in low fields is small and this is in agreement with the shape of the curve for 42.5 oersteds in Fig. 73. The magnetic moment data may also be plotted in an M versus S diagram with lines of constant magnetic field. The values of AMBLER and HUDSON are shown in Fig. 77. Up to 120 oersteds the curves show a maximum and the locus of the maxima is indicated by the dotted line. Inside the region bounded by the dotted line (oMjoS)H is positive so that, according to Eq. (9.9) application of a magnetic field gives a decrease in temperature. Outside this region the magnetocaloric effect has the normal sign. Some of the Leiden data on the variation of temperature with the applied field are collected in Fig. 78. An explanation of the susceptibility curves of Fig. 74 may be given on the basis of the occurrence of antiferromagnetic domains as introduced by NEEL 2 , see Sect. 55 . It seems 3 that each curve may be derived into four intervals:

r

t

1 E . AMBLER and R. P . HuDSON: Phys. R ev. 96, 907 (1954). 2 L. NEEL: Proc. Conf. Theor. Phys., Tokyo 1953, p. 703. 3 J. A. BEUN, M. J. STEENLAND, D. DE KLERK and C. J. GoRTER: Leiden Commun. No. 301 a; Physica, 's-Grav. 21, 767 (1955).

156

D.

DE KLERK:

Adiabatic Demagnetization.

Sect. 64

(a) The region up to about 10 oersteds. Here the susceptibility shows a modest but steep fall with increasing field while the weak hysteresis phenomena are rapidly reduced. (b) The region between about 10 and 60 oersteds. Here the susceptibility is approximately constant though, as a matter of fact, there is a flat minimum between the fall below 10 oersteds and the spectacular nse at about 60 oersteds. (c) The interval between 60 and 210 oersteds. At about 60 oersteds the susceptibility rises to a level well above the zero field value and a decrease occurs near 210 oersteds. In between, it is more or less constant. In this region a marked crystalline anisotropy shows up in x1 05 with a binary superposed on a 5/Rquaternary symmetry. Fig. 77. Magnetic moment versus entropy diagram for a spherical single crystal of chromium methylamine alum (according to (d) The region above 210 oerAMBLER and HuosoN). The magnetic fields were applied parallel to a cubic axis. The dotted line is the locus of the maxima. steds. Here XII has a low value ¢ 20 oersteds. 1> I 00 oersteds. D 200 oersteds. while x1 decreases smoothly. The /:; 40 oersteds. S~0.043R. in the case of the chromium methylamine alum (cf. Fig. 74). We do not know whether a small decrease of x11 occurs in weak fields, since no experiments were made in fields of the order of 10 oersteds. 1

1

C.

G.

B.

GARRETT:

Proc. Roy. Soc. Lond., ,Ser. A 206, 242 (1951).

The achievement of thermal contact.

Sect. 68.

165

No maxima were found in the K 2 and K 3 directions, nor were there in investigations with a powdered sample. The values of the magnetic moment could be integrated from the data of Fig. 88; from the results the variation of temperature with field on the isentropics was evaluated, see for instance Sect. 64. Since the zero field temperatures were known from caloric experiments with a.c. heating (see Sect. 63) a T versus H diagram could be composed with lines of constant 5 and M. It is given in Fig. 89 ; the largest values of Mat the O.IJ .---.--,-.--........,--r---rr~---.r-r---r----.. right hand side being about two thirds of the saturation °K moment. The T versus H diagram, as pointed out in Sect. 64, is 0.70 equivalent to the M versus 5 diagrams given in Figs. 77, 83 and 86 for other salts. The dotted line of Fig. 89 is the locus of the minima of the two sets of curves [they must T coincide for both sets according to Eq. (9.7)]. The locus proves to be not very much different from that of the maxima of the susceptibility curves of Fig. 88. GARRETT supposed 1 that the dotted line of Fig. 89 is the critical field curve of Sect. 55, hence the borderline between the paramagnetic and antiferromagnetic regions. It 1100 Oe soo 300 200 TOO 0 IIwas pointed out in Sect. 64 Fig. 89. Temperature versus magnetic field strength diagram with lines that this is not necessarily cor- of constant entropy and magnetic moment for a spherical single ccys~ rect. In the case of chromium tal of cobalt ammonium sulphate (according to GARRETT). The fields applied parallel to the K axis. The dotted line is the locus of the methylamine alum, this border- were minima. The values of 5 and M are given with the curves, the latter in gauss cm.'fmol. line was identified by BEUN, STEENLAND, DE KLERK and GoRTER with the locus of the steep fall in the x11 versus H curves of Fig. 74. The absence of a steep fall in the curves of Fig. 88 is probably due to the fact that the crystalline symmetry of the cobalt ammonium sulphate is completely different from that of the alums. The Tutton salts have two ions in the unit cell, each with a tetragonal symmetry axis, see Sect. 40. No theoretical picture is available for the interactions between the magnetic ions in such a crystal. 1

E. Other investigations below 1o K. I. Heat transfer and thermal equilibrium. 68. The achievement of thermal contact. The aim of demagnetization work is not only to investigate the magnetic, caloric and thermometric properties of paramagnetic salts, but also to cool down other materials with a salt in order 1

C.

G.

B.

GARRETT:

Magnetic Cooling, p. 75 Cambridge (Mass.) 1954.

166

D. DE KLERK: Adiabatic Demagnetization.

Sect. 68.

to make investigations on them. In the latter kind of experiments the salt is the thermostat, often also the thermometer, and special techniques had to be developed for achieving good thermal contact between the salt and the substance under investigation. Since heat transfer takes place through the thermal vibrations of the lattice it must be expected that the problem becomes the more serious the lower the temperature is. Suppose the substance under investigation (for instance a metal wire, the resistance of which is to be measured) is connected to the paramagnetic salt by means of a "transfer medium" (e.g. liquid helium or some kind of glue) then the thermal equilibrium is accomplished in the following steps: 1. the heat transfer from the spin system to the lattice of the salt. 2. the establishment of thermal equilibrium in the salt itself. 3. the heat transfer from the lattice of the salt to the transfer medium. 4. the heat conduction of the transfer medium. 5. the heat transfer from this medium to the substance under investigation. 6. the establishment of equilibrium in the substance under investigation. Experimental values for the times involved in each of these processes are hard to obtain; usually the experiment yields only the sum of several of them (for instance of 3, 4 and 5). Little is also known about each step from theory. The equilibrium time between the spin system and the lattice of a paramagnetic salt should be closely related to the spin-lattice relaxation time as determined from paramagnetic relaxation experiments. This relaxation time has been determined in two different ways; (1) with the help of the paramagnetic saturation method, and (2) by placing the salt in a field H = H,, + heiwt, where w is an audiofrequency. In the first set of experiments, performed with diluted paramagnetic alums, EscHENFELDER and WEIDNER 1 found relaxation times of the order of 10- 3 sec. in the liquid helium region; they were proportional to T-1 • The second method 2• 3 gives relaxation times of the order of 1o- 2 sec. and the temperature dependence varies between T- 2 and T- 5 . An explanation of this discrepancy was suggested by GoRTER, VAN DER MAREL and B6LGER 4 • The long relaxation times of the second method are due to the heat transfer of a small band of the system of lattice oscillations, excited by energy transition in the spin system, to the helium bath. Consequently, the times found in the first method are the ones of interest in the present considerations. It should be noticed that in the relaxation experiments quoted here the salt was in direct thermal contact with the helium of the cryostat, making the lattice specific heat effectively infinite. CASIMIR 5 pointed out that, if the salt is thermally insulated, the relaxation time is smaller by a factor cLf(cL cH), where cL is the specific heat of the lattice. Since this factor is approximately proportional to T 5 it follows that down to very low temperatures the lattice must follow the temperature of the spin system in a negligible time. The thermal conductivity of chromium potassium alum, as measured by KURT!, ROLLIN and SIMON, and by GARRETT, was already discussed in Sect. 19. It decreases steeply with falling temperature; below 0.14 o K the equilibrium time becomes too long to be measured. Investigations with iron ammonium alum, also by KuRT!, RoLLIN and SIMON, gave very similar results.

+

A. H. EscHENFELDER and R. T. WEIDNER: Phys. Rev. 92, 869 (1953). H . C. KRAMERS, D. BIJL and C. J. GoRTER: Leiden Commun . No. 280a; Physica, 's-Grav. 16, 65 (1950) . 3 D. BIJL: Leiden Commun. No. 280b; Physica, 's-Grav. 16, 269 (1950). 4 C. J. GoRTER, L. C. VANDER MAREL and B. BoLGER: Leiden Commun. Suppl. No. 109c; Physica, 's-Grav. 21, 103 (1955). 5 H. B. G. CASIMIR: Leiden Commun . Suppl. No. SSe ; Physica, 's-Grav. 6, 156 (1939). 1

2

Sect. 68.

The achievement of thermal contact.

167

Experiments on the thermal conductivities of other materials will be discussed in Sect. 78. In the case of a non-superconducting metal the heat conductivity is reasonably good down to very low temperatures. Since it is mainly due to the free electrons 1 it is proportional to T. For metals in the superconducting state, however, the heat conductivity is much smaller. Liquid helium has a very good thermal conductivity at 1o K. It follows from the experiments, however, (see Sect. 70) that it decreases strongly with falling temperature. At about 0.1 o K or 0.2° K it is of the same order of magnitude as that of He I. Here, under normal experimental conditions, thermal equilibrium can still be reached in a short time through a thin layer of liquid, not through a long narrow capillary. Very little is known about the heat conductivities of adhesives and glues. Probably they are not too good, but if a really thin layer is applied the equilibrium time may be reasonably short. Residual helium gas in a sample may act as a transfer medium as long as the pressure is well above 1o-6 mm. Extrapolation of the vapour pressure curve suggests that this may be true for temperatures above 0.4° K. Data on heat transfer from one medium to another are hard to obtain. It is difficult to separate them from the heat conductivities of the media themselves. MENDOZA 2 could explain his heat conduction and superconductivity results by introducing an empirical coefficient of heat transfer between a salt and a metal (68.1) d Q = (3 A p d T proportional to T2, hence where Q is the heat flow per second and A the area of contact. (3 was of the order of 300 ergs sec.- 1 cm.- 2 degree-a. GooDMAN a, in later experiments, found (3 = 4 x 104 ergs sec. -1 em.- 2 degree- a. It is not surprising that this coefficient is widely different for different experimental conditions, depending, for instance, on the tightness of the contact between salt and metal. At 0.2° K GooDMAN's value leads to a surface layer conductivity of 1.6 X 10a ergs sec.-1 cm.- 2 degree-1, whereas the thermal conductivity of chromium potassium alum at this temperature is 4 X 10a ergs sec.-1 cm.- 1 degree- 1 , that of copper is 4 X 106 ergs sec.-1 em. - 1 degree-1 and that of liquid helium 105 ergs sec.-1 cm.-1 degree-1. It follows from the above data that the best transfer media are non-superconducting metals and liquid helium; but it also follows that the main sources of troubles at the lower temperatures are the large resistance in the contact layer between two media and the small heat conductivities of the salts themselves. The heat transfer between two media may be improved by achieving an intimate contact over a large area. The consequence of the bad heat conductivity of the salts is that , even if a piece of salt is in good thermal contact with a transfer medium, only the outer layer is active as a coolant. In some cases this is not too serious. If the heat capacity of the substance under investigation is much smaller than that of the salt a reasonably low temperature is still reached. But if the specific heat of the substance is large, or if appreciable amounts of heat are developed in it (e.g. in the case of experiments on electric or thermal conductivity) a noticeable difference from the temperature of the bulk salt may occur. In this case it is impossible to derive the temperature of the substance from a thermometric parameter of the salt. An improvement is obtained by powdering the salt and embedding it in a transfer medium with a good thermal conductivity. This can be easily done 1 2 3

C. J. GORTER et al.: Progress in Low Temperature Physics, p. 187. Amsterdam 1951. E. MENDOZA: Ceremonies LANGEVIN-PERRIN, p. 61. Paris 1948. B. GooDMAN : Proc. Phys. Soc. Lond. A 66, 217 (1953).

168

D. DE KLERK: Adiabatic Demagnetization.

Sect. 69.

with the help of liquid helium. In the case of a metal the best solution is to have thin sheets or wires not too far apart in the salt powder. Good heat transfer is achieved by compressing the sample hydraulically, 0 usually after addition of a binding agent, see Sect. 70. 69. Liquid helium as a transfer medium. Liquid helium, as was pointed out in Sect. 68, is a feasible transfer medium. The main problem of an apparatus containing liquid helium is the film creeping out of the sample tube, which may cause a heat leak to the bath. The first solution was given by KuRTI, RoLLIN and SIMON 1 . A thick walled metal capsule is partly filled with powdered salt. Helium gas of about 120 atmos-

c

A

Fig. 90. Liquid helium capsule (accord-

ing to HULL, WILKINSON and WILKS).

Fig. 91. Liquid helium valve (according to DEKLERK).

Fig. 92. The capillary technique (according to HuosoN, HuNT and KURT!).

pheres is admitted and the capsule is sealed off. At low temperature the helium is condensed and covers the salt completely. The substance under investigation may be soldered to the outer wall of the capsule. A capsule is shown in Fig. 90. It consists of an alloy with low electric conductivity, e.g. cupro nickel or phosphor bronze, in order to keep heating by eddy currents as small as possible. After introduction of the salt the screwed plug is soldered in while the other end of the capsule is kept cool, so that the salt is not deteriorated by the heat. Helium is admitted through the capillary A. In the bore of this capillary is a wire of soft solder B. The gas is sealed in the 1

N. KuRTI, B. V. RoLLIN and F. SIMON: Physica, 's-Grav. 3, 266 (1936).

Sect. 69.

Liquid helium as a transfer medium.

169

capsule by hammering the capillary flat and then applying heat so that the solder runs 1 • Finally the capillary is cut above the solder seal. Good results have been obtained with these capsules, but sometimes they leak and under certain circumstances it may be undesirable to have large amounts of metal in the sample. It is practically impossible, for instance, to make sus,.____.. ceptibility measurements with a.c. A solution in which the high pressure filling of helium at room temperature was avoided was given by DE KLERK 2 • A valve was constructed as shown in Fig. 91. The seat consisted of chrome-iron, both ends being sealed to glass tubes. The plug was made of steel. After filling the cryostat the appropriate amount of helium H gas was condensed into the sample and then the valve was closed by means of a long metal rod whkh could be lifted afterwards. The measuring coils for the mutual inductance bridge were wound in such a way that the field was zero at the position of the valve. 6 H---+-F The difficulty with these valves is that the use of grease is impossible. The conical end of the plug must be so well centered in the seat that the helium film of about 3.5 X 10- 6 em. thickness does not creep through. This is a very high demand and one is never sure whether a valve that has worked satisfactorily during one run will be good during the next one. Under the best circumstances the heating up time from about 0.05 to 1 o K was roughly two hours. If a container with liquid helium II is sus8 pended from a tube the film flow through the A tube is roughly proportional to its circumis leak heat of ference and the main source Fig. 93. Apparatus for the measurement of recondenzation into the container a. According sound velocity in liquid helium (according to CHASE and HERLIN). to these considerations HUDSON, HUNT and KuRTI 4 constructed an apparatus in which the valve was replaced by a long narrow capillary as shown in Fig. 92. The capillary was made of german silver, it was 7 em. long and had an internal diameter of 0.2 mm. The upper end was connected to a diffusion pump in order to prevent the film evaporating above the capillary from recondenzation into the sample. With this simple arrangement the heat leak proved to be of the same order of magnitude as in the case of a valve, and for this reason the valve technique has been abandoned in recent years. Glass capillaries of about the same dimensions were used in the Leiden experiments on the specific heat of liquid helium and the propagation of second sound below 1o K, see Sect. 70. 1 2 3

4

R. A. HULL, K. R. WILKINSON and J. WILKs: Proc. Phys. Soc. Lond. A 64, 379 (1951). D. DE KLERK : Leiden Commun. No. 270c; Physica, 's-Grav. 12, 513 (1946). B . V. ROLLIN and F. SIMON: Physica, 's-Grav. 6, 219 (1939). R. P. HUDSON, B. HUNT and N . KURT! : Proc. Phys. Soc. Lond. A 62, 392 (1949).

170

D. DE KLERK: Adiabatic Demagnetization.

Sect. 70.

An interesting apparatus which may be considered as a combination of the valve and the capillary techniques was recently described by CHASE and HERLIN 1 ; it was originally designed by AsHMEAD. It is represented in Fig. 93- A is the sample tube and B the salt container. The space in between is evacuated through the pumping line H. E and F are thin walled stainless steel cones, machined and lapped to fit together as closely as possible. If the inner cone is lifted helium flows from the bath into B and the liquid between the cones provides good thermal contact. If F is seated the heat flow is reduced to a very low value, and the thermal insulation is sufficient to keep the experimental chamber cold for more than an hour. No exchange gas is needed in this apparatus, so that pumping the vacuum space when the field is on is eliminated. This reduces the magnetization time appreciably. The apparatus was used for measurements of the velocity of sound in liquid helium (see Sect. 71). The necessary equipment was mounted in the experimental chamber D. The supply wires were brought through the vacuum space by means of the stainless steel tubes G which were filled with vaseline. The vaseline freezes at low temperatures and prevents the helium from flowing into the sample. 70. Heat transfer between solids. Thermal equilibrium between a salt and a metal at the lower temperatures is inadequate if the metal is glued to the surface of the salt sample. The occurrence of large inhomogeneities in the salt's temperature is clearly K demonstrated in Fig. 94. This represents a heating curve at 0.}5° K obtained by VAN DIJK 2 during his specific heat measurements on gadolinium sulphate F (see Sect. 46). A phosphorbronze thermometer and a heating wire were wound on the sample together, alter-

C'

II

r*

_T ________ _ dT"'=0.0136°

s

_1_ -----

0

time-

8

min

12

Fig. 94. Local overheating during specific beat measurements (according to VAN

DIJK).

Fig. 95. Apparatus with vane tecbni· que (after GooDMAN and MENDOZA).

nat ely a turn of each . The course of the t emperature as derived from the phosphor bronze thermometer shows a local overheating which is only slowly equalized. 1 2

C. E. CHASE and M.A. HERLIN : Phys. Rev. 97, 1447 (1955). H. VAN DIJK: Leiden Commun. No. 270a; Physica, 's-Grav. 12, 371 (1946).

Sect. 70.

Heat transfer between solids.

171

A good solution for a metal-to-salt contact, as was indicated in Sect. 68, was first given by MENDOZA 1 , 2, SC'e Fig. 95. Thin copper sheets or vanes F were soldered to a copper rod C. The spaces between the sheets were well filled with powdered salt mixed with a binding agent and then the sample was compressed hydraulically under a pressure of 2000 atmospheres. The total contact area of the vanes in MENDOZA's apparatus was 30 cm.2. The binding agent was a solution of a plastic cement in acetone. The latter proves to evaporate fastly and completely from the compressed sample. In Fig. 95, 5 is the substance under investigation, a superconducting ellipsoid gripped in a cup at the lower end of the copper rod. H is a cylindrical shield of copper foil in good thermal contact with the salt K, protecting 5 from stray heat. Vertical slots were cut in H and in the cup holding 5 in order to reduce eddy currents. It was found that thermal equilibrium between the salt and 5 is reached five minutes after demagnetizing the salt to 0.1 o K. The basic idea of MENDOZA was developed and modified by several investigators. The general trend in recent years has been to enlarge the area of contact between the salt and the metal. DARBY, HATTON, RoLLIN, SEYMOUR and SILSBEE 3 in their two-stage demagnetization experiments (see Sect. 80), replaced the vanes by six copper wires of 0.2 mm. diameter. In later experiments this number was appreciably increased. In a recent Leiden experiment initiated by WHEATLY (unpublished) 500 wires were soldered to copper frames. They were embedded in a mixture of equal quantities of chromium potassium alum and silver chloride and the whole sample was compressed to 2000 atmospheres. The total contact area was 100 cm. 2. Since silver chloride is very plastic it provides an intimate contact between the wires and the salt. Thermal equilibrium between two pills of this type, connected by a copper rod, was reached in about three quarters of an hour at 0.06° K. In recent experiments in Oxford 4 the sample was not compressed at all. Vanes with a large total area (larger than in the above Leiden experiment), were used and powdered salt was inserted between them by shaking the apparatus vigorously. Glycerin was used as a binding agent. Thermal equilibrium between such a sample of chromium potassium alum and a similar one of cerium magnesium nitrate was obtained at 0.025 ° K within an hour. DABBS, RoBERTS and BERNSTEIN 5 made an experiment on the polarization of indium nuclei. Twenty sheets of the metal were soldered to silver wires of 12 em. length and iron ammonium alum was crystallized around the other ends of the wires. The whole unit was mounted on rigid insulators in a silver cage which was cooled by another sample of iron alum. The salts were cooled magnetically to 0.03 5° K and the temperature reached with the indium as derived from the fractional change in the transmitted neutron intensity was 0.043° K. HEER, BARNES and DAUNT 6 , in their magnetic refrigerator (see Sect. 81), used a finned copper shaft as shown in Fig. 96 surrounded by a brass cylinder. Iron ammonium alum, mixed with small pieces of copper wire and with silicone vacuum stopcock grease, was compressed inside this unit under 200 atmospheres. Satisfactory contact was reached down to at least 0.1 o K. 1 E. MENDOZA: Ceremonies LANGEVIN-PERRIN, p. 53. Paris 1948. 2 B. B. GooDMAN and E . MENDOzA: Phil. Mag. 42, 594 (1951). 3 J. DARBY, J. HATTON, B. V . RoLLIN, E. F . W. SEYMOUR and H. B . SILSBEE: Proc. Phys. Soc. Lond. A 64, 861 (1951). 4 F. N .H. RoBINSON: Thesis, Oxford 1954. 5 J. W. T. DABBS, L. D. ROBERTS and S. BERNSTEIN: Phys. Rev. 98, 1522 (1955). 6 C. V. HEER, C. B. BARNES and J. G. DAUNT : Rev. Sci. Instrum. 25, 1088 (1954).

172

D. DE KLERK: Adiabatic Demagnetization.

Sect. 70.

STEELE and HEIN 1• 2 , in their experiments on carbon thermometers and on the superconductivity of titanium, pressed chromium potassium alum without a binding agent around a single copper fin B mounted on a brass base C as shown in Fig. 97. The copper specimen holder E was screwed tightly to C. The thermal contact was satisfactory the first time the apparatus was brought to liquid helium temperature. It was found, however, that it deteriorated to some extent once the apparatus was allowed to warm up to room temperature. This was ascribed to the salt's breaking away from the copper fin, possibly due to the difference in thermal expansion. If a new salt pill was used for each helium run the results were quite reproducible. ----l-1f-+-.4

Fig. 96. Fig. 96. Construction of the working unit of the magnetic refrigerator (according to

Fig. 97.

and DAUNT). Fig. 97. Apparatus for the investigation of carbon composition thermometers and superconductors (according to CLEMENT QuiNNELL, STEELE, HEIN and DoLECEK). HEER, BARNES

A strongly simplified vane technique has been in use for many years in investigations on the occurrence of superconductivity. It was introduced by KuRTI and SIMON 3 in 193 5. Small grains of the metal are mixed with powdered salt and compressed to a solid pill. After demagnetization the susceptibility of the pill is followed during the warming up. The disappearance of superconductivity is evidenced by a fairly sudden discontinuity in the susceptibility curve, since a superconductor behaves like a completely diamagnetic substance with volume susceptibility -1/4n, whereas the susceptibility of a normal metal can be neglected. The transition curve may be derived by observing heating curves in magnetic fields. The method has some obvious disadvantages: 1. In pressing the pill the metal is subjected to considerable stresses. 2. If particles of large size are used the thermal contact may become bad owing to the difference in thermal expansion. This is particularly dangerous in investigations on the threshold curve because of the caloric effects occurring during the transition (see Sect. 6). 1 M. C. STEELE and R. A. HEIN: Phys. Rev. 92, 243 (1953) . z J. R. CLEMENT, E. H. QuiNNELL, M. C. STEELE, R. A. HEIN and R. L. DoLECEK: Rev. Sci. Instrum. 24, 545 (1953). 3 N. KURTI and F. SIMoN: Proc. Roy. Soc. Lond., Ser. A 151, 610 (1935).

Sect. 71.

Investigations on liquid He•.

173

3· Application of a magnetic field influences the temperature of the salt somewhat . 4. The magnetic field inside the pill is different from that outside, the difference is unknown if the superconducting particles are of irregular shape and distributed at random. These disadvantages are avoided by using an apparatus like Fig. 95. On the other hand the method described here is very simple. It is very useful if one is mainly interested in the question whether a substance becomes superconductive or not. Finally we want to mention an interesting apparatus developed by CooKE 1 , in which the susceptibilities of two salts can be compared. A single crystal sphere of cerium magnesium nitrate is covered by a spherical shell of a different salt, the latter being powdered and mixed with some grease. Cerium magnesium nitrate, as was stated in Sect. 48, is highly anisotropic, to such an extent that parallel to the symmetry axis the susceptibility is almost zero. Moreover it has a very small specific heat and CURIE's law is obeyed down to a few thousandths of a degree. If, after demagnetization, thermal equilibrium is reached, a susceptibility measurement in the direction of easy magnetization of the cerium magnesium nitrate gives the sum of the two susceptibilities, whereas a measurement perpendicular to it gives only the susceptibility of the shell. In the following sections, we give a survey of non-magnetic investigations that have been performed in the demagnetization region. We restrict ourselves to the experimental details as far as they are of interest for work below 1o K, and to a short description of the results. For the theoretical discussions of the results we refer to other chapters of the Encyclopedia. II. Experimental results. 71. Investigations on liquid He4 • Preliminary experiments on the heat conductivity of liquid helium were performed by KuRTI and SIMON 2 and by DE KLERK 3 . KuRTI and SIMON used a twin capsule connected by a capillary of 18 mm. length and 0.5 mm. diameter. DE KLERK applied two glass spheres filled with powdered salt connected by a glass tube 10.5 em. long and 3.2 mm. internal diameter; the apparatus was closed by a helium valve, see Fig. 91. In both experiments a temperature difference was set up between the two salt pills, and the heat conductivity of the liquid was derived from the course of the temperatures of the two samples with time. It was found that the heat conductivity decreases rapidly with falling temperature. At about 0.2° K it was of the same order of magnitude as that of He I. More recent experiments were made by FAIRBANK and WILKS 4 , also with a capsule technique. Complete experimental details have not yet been published, but the measurements were made with a german silver tube of 0.29 mm. internal diameter. Heat was supplied at one end and the heat conductivity was derived from the course of temperature of two thermometers soldered to the tube. Since the heating of the whole sample to 1o K took several hours, it was possible to obtain good equilibrium conditions for each measurement. The results are shown in Fig. 98. A pronounced break occurs in the curve between 0.6 and 0.7° K. Below, the heat flow is normal, i.e. proportional to the A. H . CooKE: Housto n Low T e mpera ture Conference programme, 1953. p . 26. N . KuRT! and F. SIMON: Nature, Lond. 142, 207 (1938). a D. DE KLERK: Leiden Commun. No. 270c; Physica, 's-Grav. 12, 513 (1946). 4 H. A. FAIRBANK and J. WILKS: Phys. Rev. 95, 277 (1954).

1

2

174

D. DE KLERK: Adiabatic Demagnetization.

Sect. 71.

temperature gradient. Above, this is probably no more true. Here the curve is very steep; it can be extrapolated to the values obtained above 1o K. The first investigations on the specific heat of liquid helium below 1 o K were performed by PICKARD and SIMON 1 and by KEESOM and WESTMYZE 2 . The latter authors found a proportionality to TG down to 0.6° K, whereas PICKARD and SIMON obtained a T 3-law below 0.8° K. More recent investigations were made by HuLL, WILKINSON and WrLKS 3 with the help of a capsule filled with iron ammonium alum. A manganin heater was connected to the outside of the capsule and the temperature values were derived from the susceptibility of 3 k the salt. Between 1.4 and 0.6° ~~ K a proportionality to T 6 ·2 was ~ found, below 0.6° K the specific 7 r-heat of the liquid helium proved •• to be so small as compared with that of iron ammonium alum that it could not be measured 0.3 r-with a reasonable precision . • This difficulty was avoided in experiments of KRAMERS, WAS••• SCHER and GORTER 4 by replacing 0.! r-• the iron ammonium alum by .,.·,.. copper potassium sulphate. The latter salt has a specific heat which is roughly one twentieth 0.03 r-of that of iron ammonium alum and the measurements could be '-------7-::-----::!";;-1----;~1--;!-;:_~,_-;:..l.-;;;;v;: extended with a reasonable preci0.3 O.!f. 0.5 0.0 0.7 0.8oKo.9 sion to 0.25° K. A consequence rof the smaller specific heat is a Fig. 98. Thermal conductivity of liquid helium as a function of temperature (according to FAIRBANK and WILKS). smaller cooling capacity, but it proved to be possible, by choosing the correct helium to salt ratio, to obtain a sufficiently low temperature with a reasonable field value. The authors used 14 grams of copper potassium sulphate and 1.8 grams of helium and a temperature of 0.1 o K was reached with a field of 12 500 oersteds. A difficulty encountered with the copper potassium sulphate is that the experimental values for the CURIE-WEISS and the coefficient of the 1/T2-term in the specific heat were noticeably different in different investigations, see Sect. 42. This introduces some uncertainty into the interpretation of the measurements at the lower temperatures. The salt used by KRAMERS, WASSCHER and GoRTER was of the same origin as that investigated by DE KLERK, and the calculations performed with his constants gave the most satisfactory results. A glass apparatus with a capillary was used, see Sect. 69. Heat was supplied with a carbon resistor and the temperature was obtained from the susceptibility of the salt.

l

t

..

....

e

1 G. PICKARD a nd F. SIMON: Abstracts of papers communicated to the Royal Society of London 1939. p. 521. 2 W. H. KEESOM and W. K. WESTMYZE: Physica, s-Grav. 8, 1044 (1941). 3 R. A. HULL, K. R. WILKINSON and J. WILKS: Proc. Phys. Soc. Lond. A 64, 379 (1951). 4 H. C. KRAMERS, ]. D. WASSCHER and C.]. GoRTER: Leiden Commun. No. 288c; Physica, 's-Grav. 18, 329 (1952).

175

Investigations on liquid He'.

Sect. 71.

The results are shown in Fig. 99. A rather sharp bend occurs between 0.6 and 0.7° K, like in the heat conductivity curve of Fig. 98. Below it is the region where only the phonons make a contribution to the specific heat. Here the slope of Fig. 99 gives a proportionality to P. If we substract this contribution from the values above 0.7° K no unique power of Tis found for the excess curve. LANDAU derived for the roton contribution to the specific heat:

c, = B f(T)

(71.1)

e-.1 /kT,

where f (T) is a slowly varying function of temperature and L1 is the energy gap between the lowest energy levels of rotons and phonons. Since both B and L1 are unknown it is difficult to estimate whether the results above 0.7° K are in agreement with this formula or not. 10 7

/

erg g.•K

/ /

TO 6

c

TO

./o

10 ~ 0

TO 3

/

/ a2

~

v

v

~

0

aJ

a11

o.o

r-

aa

lO

l2 °K 1.s

Fig. 99. Specific heat of liquid helium (according to KRAMERS, WASSCHER and GoRTER). The points and the full line were obtained with DE KLERK's data on copper potassium sulphate, the broken curve with those of GARRETT.

Experiments on the propagation of sound in liquid helium were made by CHASE and HERLIN 1 using the apparatus of Fig. 93. The sound velocity as measured in cavities of 3.94 and 1.96 em. length depended only little on temperature. At 0.1 o K it was (240 ± 5) mjsec., whereas the value extrapolated from measurements above 1o K was (239 ± 2) mjsec. The attenuation is represented in Fig. 100. A pronounced double maximum occurs near 0.9° K. This is in agreement with KHALATNIKov's prediction that the attenuation is determined by two relaxation times, one due to phononphonon interaction, the other to phonon-raton interaction. The attenuation falls smoothly to zero at absolute zero. Above 0.3° Kit is proportional to T2·8 , below 0. 3o K it is somewhat steeper. The first experiments on the propagation of heat waves in liquid helium ("second sound") below 1o K were performed by PELLAM and ScoTT 2 and by ATKINS and OsBORNE 3 . Though in both experiments the thermal insulation was very poor, so that no good equilibrium was reached between the helium C. E. CHASE and M.A. HERLIN: Phys. Rev. 97, 1447 (1955). J. R. PELLAM and R. B. ScoTT: Phys. Rev. 76, 869 (1949). a K. R. ATKINS and D. V. OsBORNE: Phil. Mag. 41, 1078 (1950).

1

2

176

D. DE KLERK: Adiabatic Demagnetization.

Sect. 71.

and the salt, it was demonstrated that the velocity increases rapidly below 1o K and that the pulses are noticeably broadened at the lower temperatures. More recent experiments by DE KLERK, HuDSON and PELLAM 1 and by KRAMERS, Mrs. VAN PESKI, WIEBES, VAN DEN BURG and GORTER 2 showed that the theoretical speed limit at absolute zero as predicted by LANDAU: (71.2) (where v2 is the velocity of second sound and v1 that of ordinary sound) is exceeded in such a way that v2 seems to become equal to v1 . A glass apparatus

1-:.

~:·t

l·· ! ~·

2

:lo i'

I

. .... ...

:- I

·'·'~ ••., ••

... ..

':

-;

o:

...

'(

g:> ...

8

a

0.

o o/ oo, o•

oo:

.

...

~

o''

oo••

..

oo,o Oo (9• • •

0 0

oo 00 . .

oo'

oo.~

;~

Oo 01

A the magnetic field penetration is equivalent in effect to removing a sheath of thickness A from the surface of the specimen. As a result, 2!. p. . = 1 - -----·P.oo

and

p.(T)- p.(T0 )

- -- -- -----

P.oo

a '

2 [J.(T)- J.(T0 )] a

(16.2)

We see from (16.2) that measurements f.llf.loo as a function of T determine [A ( T) - A(T0 }] in this experiment. Combining these measurements with the colloid measurements, which determine A(T) /A(T0 ), it is possible to deduce the absolute value of A(T0 ). For mercury, this method yielded Ao =A (0° K) = 7.6 x 10-6 em. This value is now regarded as being nearly twice the true one, probably because the wires in the composite specimen had a ao10 slight spread of transition temperatures. r-b) Thin films. LocK 1 measured the mag~ aoos netization in a longitudinal field of thin films of tin, indium and lead. The films were de- Jl.ifl~ \ posited on thin sheets of mica by evaporation \ 5 in vacuum. Their thickness was about 10- em., 3 2 T(°K)and composite specimens consisting of a few 50 hundred mica sheets were used. The magneI tic moment per unit area of film is [BARDEEN, 40 / eq. (10.7)] 30

!

f.l

2aH [1- (A.fa) Tan ajA] ' = ·-- ---~ 4n

11/lfc 20 -

where 2 a is the film thickness, so that

_I!__= 1- (A./a) Tan a/A. Taking,

P.oo

A= A0 (1 -

t4)-1,

-

1"'-

-

~

2

(16.3) (16.4)

3

T(°K)Fig. 23. p.fp.00 and hfH, as functions of temperature for a mercury colloid (after SHOEN BERG).

LocK found that the data for a number of film thicknesses taken at several temperatures could be fitted very well to the expression (16.3) by appropriately choosing A0 • The values of Ao obtained in this way are given in Sect. 18. The range of film sizes was too small, however, uniquely to establish (16.3) as the correct law of penetration; other penetration laws can be found which fit the data equally well.

s) Large cylinders. 1. Low frequenc y methods. Due to the change in penetration depth with temperature, the inductance of a coil with a superconductive core should change very slightly with temperature. CASIMIR 2 first tried to observe this change, but because of a technical flaw in experimental design, his measurements gave a negative result. The experiment was repeated recently by LAURMANN and SHOENBERG 3 at a frequency of 70 cfsec. with cylindrical specimens of tin and mercury of about 1 em. diameter. They managed to overcome the many technical difficulties inherent in this experiment and obtained 1 2

3

J. M. LocK: Proc. Roy. Soc. Lond., Ser. A 208, 391 (1951). H. G. B. CASIMIR: Physica, Haag 7, 887 (1940).- Leiden Comm. 261c. E. LAURMANN and D. SHOENBERG: Proc. Roy. Soc. Lond., Ser. A 198, 560 (1949). 16*

244

B. SERIN: Superconductivity. Experimental Part.

Sect. 17, 18.

results for the temperature variation of [A (T) -A (T0 ) J and for Ao in good agreement with those obtained by the other methods. SHALNIKOV and SHARVIN 1 performed an interesting variant of this experiment, in which the temperature of a tin specimen in a constant magnetic field was varied at 4 cjsec. Due to the change in field penetration with temperature, an alternating emf was induced in a coil surrounding the specimen. The values of A near Tc deduced from this investigation are several times greater than those found by LAURMANN and SHOENBERG. The excessively large values are probably attributable to the poor surface condition of the specimen. 2. High frequency reactance. As part of this extensive series of measurements of the surface impedance of metals at microwave frequencies, PIPPARD 2 determined the change of A with temperature. The simplest observations to interpret are those dealing with the change in resonant frequency of a system on passing from the superconductive to the normal phase. At sufficiently low temperatures, the frequency change is proportional to

b- A,

o

where b is the electromagnetic skin depth of the normal phase. In tin, is independent of temperature, so that once the constant of proportionality is determined, the observations may be used to evaluate [A(T) - A(T0 )]. The results obtained in this way agree well with those given by the other methods. 17. Non-penetration of the static electric field. The theory, as originally formulated, left to experiment the task of answering the question of whether the electric field penetrates to a depth A into a superconductor or terminates on surface charges. The answer was obtained by H. LoNDON 3 , who looked for a small change in the capacity of a condenser when its plates became superconducting. The plates were made of mercury separated by a thin mica sheet. Although there were several technical difficulties, the expected effect if penetration occurred was four times the error of measurement. No change in capacity was detected, so that the present form of the theory was adopted in which no static electric fields can exist inside a superconductor. 18. Temperature dependence of A.. In Sect. 160 we remarked that the data of LocK can be fitted very well to theory, if it is assumed that the temperature variation of the penetration depth is Table 2 . Values of the penetration given by A=-___&____ depth at 0° K. (18.1) 4 .1., (em)

I

(1 - t )~

•

Reference

The results of all the other investigations discussed in Sect. 16 are also in accord with (18.1), within the limits of experimental error, so that In ~:6 X 10-6 I it is now generally accepted that to a first Sn approximation, A varies with temperature in Hg 3.8(11); 4.5(j_) Pb 3-9 this way. The values of Ao for several metals are given in Table 2. Comparing (18.1) with (14.6) which gives the temperature dependence of the order parameter, w, in GoRTER's two-fluid model, we see that Metal

(Footnote of p. 243)

(1-Y =w . 1 2 3

(18.2)

A. I. SHALNIKOV and Yu. V. SHARVIN: J. Theor. Exp. Phys. USSR. 18, 102 (1948) . A. B. PIPPARD: Proc. Roy. Soc. Lond., Ser. A 191, 385 (1947). H. LoNDON: Proc. Roy. Soc. Lond., Ser. A 155, 102 (1936).

Magnetic field dependence of A.

Sect. 19.

Furthermore,

LONDON's

theory gives [cf.

BARDEEN,

245

eqs. (7.13) and (10.3)] (18.3)

where m is the mass, and n5 is the density of superconducting electrons. Adopting the convention of the two-fluid model in which all the electrons are "superconducting" at 0° K, we see that (18.2) and (18.3) are in accord with the view that w is the fraction of "superconducting" electrons at any temperature, t. Conversely, (1-w) is the fraction of "normal" electrons at any t. The quantity, m, in (18.3) must be taken to be the "effective" mass of the electrons. From the measurements of A0 , it is thus possible to use (18.3) to evaluate mfnsO> where nso = ns (0° K). The ratio, r~;!!!~] (where m. is the normal electron nso ne mass and n., the electron density determined from the valence) is about 0.3 for indium, tin and mercury and 0.7 for lead. From the foregoing very naive standpoint, these results may be taken to mean that only a small fraction of the electrons became "superconducting" or that the effective mass is large. In the previous chapter we noted that the data for the temperature variation of both the threshold field and the specific heat of superconductors are not in complete agreement with the GORTER model. On the other hand, the determinations of the temperature variation of the penetration depth are in apparent agreement with the model. This discrepancy can probably be explained away on the basis that penetration depth measurements are extremely difficult and the precision of measurement required to reveal small deviations from t 4 -behavior was not attained. 19. Magnetic field dependence of l. PIPPARD 1 measured the variation of the penetration depth in tin with magnetic field. The experimental results are shown ooJ.---,----,.-----.--,---.--. in Fig. 24; A(H,) is the value of A at the I threshold field and A(0) is the value in I no field . The remarkable fact to emerge from this investigation is the very small J: ao2 change in A (at most a few percent) 1---+----+-~--+-·---Jf- 1 produced by fields near the threshold $ 001 : - ... value. I PIPPARD points out that if the ther1 modynamic effect of the field were con07.5 3. 5 Tc z.o r fc,~ 1 _ 30 fined to the penetration depth at the Magneticfielddependenceofthcpenetrationdepth 24. Fig. density entropy in SUrface, the change (after PIPPARo '). produced by the field in this small layer is enormous. (For example, near the critical temperature, the change in entropy density in the layer at the threshold field would be about one quarter of the difference between the entropy densities of the normal and superconductive phases.) Such a situation seems quite unrealistic. PIPPARD therefore concludes from the almost negligible change in A that the entropy change must be distributed in a layer of thickness a which is considerably larger than the penetration depth, thereby resulting in a much smaller change in entropy density. According to the two-fluid model, the increased entropy density is associat ed with an increase in the fraction of normal electrons in this layer. PrPPARD therefore considers a simple model in which the order parameter in the surface layer, a,

+!

'-cJ

1

A. B. PIPPARD: Proc. Roy. Soc. Lond., Ser. A 203, 210 {1950).

Handbuch de£. Physik, Bd. XV.

16a

246

B. SERIN: Superconductivity. Experimental Part.

Sect. 20.

is constant, but depends on the field at the surface as well as the temperature. As a result, the value of w in the layer differs from its value in the remainder of the specimen. It is assumed that in the bulk of the specimen the order parameter has its usual equilibrium value. Minimization of the free energy calculated from the foregoing model determines w in the layer, and this result combined with (18.2) serves to determine the dependence of A on field. It is found that a must be about 20A0 (or about 10- 4 em.) to obtain qualitative agreement between the predictions of the model and the measurements. BARDEEN (Sect. 30) gives a detailed discussion of this experiment. Implicit in the foregoing approach is the assumption that there is a long range order in the superconductor which prohibits the parameter, w, from changing rapidly in distances small compared with 10-4 em. Speaking qualitatively, this means that the density of superconducting electrons (and therefore the associated wave-functions of the superconductive state) also must be slowly varying within the same distance (cf. BARDEEN, Sect. 6). It should be remarked that the foregoing range of order must be a maximum occurring in bulk materials rather than a minimum necessary for establishing the ordered superconductive phase. This observation is the result of the realization that thin films and colloids having dimensions at least two orders of magnitude smaller than 10- 4 em. do exhibit superconducting properties and have the same transition temperatures as large specimens. 20. Dependence of .1.. on purity. The variation of penetration depth in tin with the mean free path for an electron in the normal phase as determined by PIPPARD 1 , is shown in Fig. 25. The free path was decreased by alloying indium with the tin. A small amount of impurity rapidly decreased the normal electrical conductivity of tin and the maximum indium concentration used in these measurements was 3%. As can be seen from Fig. 25, this small amount of indium increases the penetration depth in tin by more than a factor two. The thermodynamic parameters of the impure specimens, however, differ only slightly from those of pure tin. For example, the transition temperature of the 3% indium specimen was 3.63° K, whereas pure specimens have Tc= 3.73° K. There are corresponding small changes in the threshold field values. PIPPARD concludes that these observations of large changes in the penetration depth unaccompanied by correspondingly large changes in the thermodynamic parameters constitute a contradiction to LoNDoN's theory. According to the theory [as can be seen from (18.3)] it depends only on mjn5 , i.e. the effective mass and density of the superconducting electrons. Since the latter are very little affected by impurity, as is evidenced by the small change in the thermodynamic parameters, the observed large changes in A with impurity are unexplained. PIPPARD therefore developed his non-local description of the relationship between the magnetic field and current in superconductors which is discussed in great detail by BARDEEN (Sects. 16 to 26). According to this theory, the relationship between current density and field is not a point relation, but an integral relation involving the field in a region of linear dimensions about equal to the range of order surrounding the point. The effect of impurity scattering is to decrease the range of order. The data are in good agreement with the theory. The foregoing experiments also served to clear up somewhat the mystery of the observed anisotropy of it in tin 2 . The dependence of the penetration 1 2

A. B. PIPPARD: Proc. Roy. Soc. I.ond., Ser. A 216, 547 (1953). A. B. PIPPARD: Proc. Roy. Soc. I.ond., Ser. A 203, 98 (1950).

Sect. 21.

247

High frequency resistance of superconductors.

depth on the angle between the current and the tetragonal axis is shown in curve (b) of Fig.26. SHOENBERG 1 has shown that if LoNDON's equations are generalized to describe anisotropic bodies, the predicted anisotropy in A is contradicted by the observed anisotropy in tin. However, on the basis of the nonlocal theory, the anisotropy in A in pure specimens is a reflection of the anisotropy in the FERMI surface for the electrons of the metal 2 , rather than a direct reflection of crystalline anisotropy. By contrast, as can be seen from curve (a) in Fig. 26, in impure specimens the observed anisotropy is very small. This result is also a qualitative consequence of the de(a) I 10 f------;;---->----'-~ 0 -----'f crease in the range of order by imi purity. This set of experiments seems to 9 1 - -- -+--- - - +i- - -- -1 indicate clearly that LONDON's equations I (a) s~ +J%In must be modified. However, they will (b_J_JJ+-~r_e_sn_ _ ]: B l---undoubtedly continue to serve as a useful qualitative guide to penetration I I effects. .:t 7 - ·- - ---+- - ----+-- - - --i 0

0

li

0

-+1------1

~

I

v

61 - - -! (bJ/

~

'-

. ~\.~

5~----~r----r---~

i

i

4 0L---~3~ 0 ---~30 ° 0 ---6~0" 0~

Fig. 25. Variation of A0 with mean free path, l (after PIPPARD) .

o-

Fig. 26. Variation of A0 with the angle, IJ, between the current and the tetragonal axis of pure tin, and tin + 3% indium {after PIPPARD 2 ).

21. High frequency resistance of superconductors. According to LONDON's equations, a changing electric field can exist in a superconductor, and the metal then exhibits resistance. It can readily be shown 3 that, in the case of alternating fields, the normal current density, in, is related to the superconducting current density, is, by 1\ :ln (21.1) . -: ~T = -!5

I· ' ( ')2

Here,

1 / The discrepancy

V2.

1 2

3

E . R. ANDREW: Proc. Roy. Soc. Lond., Ser. A 194, 80 (1948). E. R. ANDREw: Proc. Roy. Soc. Lond., Ser. A 194, 98 (1948). 0. S. LuTES and E. MAXWELL: Phys. Rev. 97, 1718 (1955).

Sect. 24.

253

Intermediate state.

between the observed value of 0.67 and this larger value is, at present, not understood. Because of the delay in appearance of the intermediate state, the magnetization curves of small specimens deviate from the ideal macroscopic curves illustrated in Fig. 11. Fig. 29a shows a typical magnetization curve calculated for a small cylinder in a transverse field on the basis of a detailed model of the intermediate state 1 . D:EsiRANT and SHOENBERG 2 made a careful study of the magnetization curves of transverse cylinders of various radii, and qualitatively verified the existence of the unusual features shown in Fig. 29a. A typical meas/.0 - - - - - ured curve is shown in Fig. 29 b. The measurements reveal that in increasing fields, the magnetization drops abruptly with the appearance -'fJr/1as of the intermediate state at fields ffc · - - small cylinder for which e is appreciably greater ---- Iorge cylinder than one half. Moreover, the magnetization vanishes at fields noticeably 0.5 smaller than the threshold value, as fla/llc determined from measurements in a longitudinal field . The difference between He and the field at which the magnetization disappears increases with decreasing specimen diameter. -M/1 When the field is reduced from above the threshold value, the curve is retraced for the most part, except for a slight hysteresis which has two 0.5 0 features. We note first that the Hallie"horn" of the magnetization curve Fig. 29a and b. a) Magnetization curve of a small transv€'rse (a- b) is not retraced. Secondly, the cylinder as calculated by ANDREW 1, contrasted with the beof a large cylinder. b) Magnetization measurements of return transition to the intermediate havior DtsJRANT and SHOENBERG 2 on a 1.3 X 10- cm . diameter tin state (c-d) occurs discontinuously cylinder at 3.0° K. at a field considerably less than the value at which the magnetization vanishes. The latter feature is a consequence of the supercooling of the specimen; we discuss supercooling in the next section . The foregoing resistance and magnetization data were analyzed in considerable detail in an attempt to obtain values of Ll .. At best, only qualitative agreement was obtained between the data and the theoretical calculations which were based: for the most part, on the branching model. Since this model seems no·w to be out of date, and recent work (see Sect. 26) has provided more reliable information about Ll, we do not discuss these calculations here 3 . y) SILSBEE effect and paramagnetic effect. A current, 1

(24.2) flowing in a long cylindrical wire of radius a, produces a magnetic field equal to the threshold value on the surface of the wire. Soon after the discovery of 1 2

3

See footnote 2, p . 252. M. DE:siRANT and D. SHOENBERG: Proc. Roy. Soc. Lond., Ser. A 194, 63 (1948) . See ANDREW [Proc. Roy. Soc. Lond., Ser. A 194, 98 (1948) ] and D . SHOENBERG [1] ,

pp. 11 7- 120.

254

B. SERIN: Superconductivity. Experimental Part.

Sect. 24.

the threshold field, SILSBEE suggested that the magnetic field of the current itself is responsible for restoring the resistance of superconductive wires carrying currents in excess of i, (cf. Sect. 3). This case differs from the corresponding phenomenon which occurs in superconductive rings. In a ring, when the field at the surface tends to exceed the threshold value, the persistent current adjusts itself so that the field just equals H" and the specimen remains superconductive (cf. Sect. 5). However, when the current in a wire is maintained constant by an external source, resistance is restored for all currents greater than i,. For currents exceeding this threshold value, a) the wire is in an intermediate state. LoNDON 1 derived a model for the intermediate state in the presence of a \ current which is illustrated in Fig. 30. According to the model, the resistance f, 05 is zero for i < i, and jumps discontinuously to Rn/2 when i = i,, where Rn is 0 o ....

lO

R Rn

b)

1.0

4011Ul

-ooos Fig. 30.

20mo. !Omo.

(T-fc)°K

0

Fig. 31.

Fig. 30. Intermediate state of a wire carrying a current (after

LONDON).

Fig. 31 a and b. a) Resistance of a tin wire as a function of currPnt. The experiment by ScHUBNIKOW and ALEKSEYEVSKY 1 was on a O.Ot em. diameter wire at 1.95° K; He= 218 Oe. b) Theore tical resistive transitions of a 0.01 em. tin wire carrying various currents.

the resistance in the normal phase. As i is increased further, the resistance approaches Rn asymptotically. The data for a thin tin wire 2 , as illustrated in Fig. 31a, are only in qualitative agreement with these predictions. The resistance at ic jumps to about 0.8Rn rather than to Rn/2, and the approach to Rn is more rapid than predicted by the model-all resistance being restored when i,.......,2ic. ScoTT 3 found that the fraction of resistance discontinuously restored at i, varied with wire diameter in indium wires. The smaller the wire, the larger was the initial jump. However, for increasing currents SILSBEE's condition (24.2) still holds, independently of wire size 4 • KuPER 5 attributes the larger size of the jump at i, to the scattering of the conduction electrons at the normal-superconducting interfaces in the intermediate state. The scattering contributes an additional resistance. The theory is in fair agreement with the data. F. LoNDON [4], pp. 120-124; also BARDEEN, Sect. 33. L. W. ScHUBNIKOW a nd N. E. ALEKSEYEVSKY: Nature, Lond. 138, .804 (1936). 3 R. B. ScoTT: J. R es. Nat. Bur. Stand. 41, 581 (1 948). 4 See A. B. PIPPARD [Phil. Mag. 41, 243 (1950)] for a theoretical justification of this finding. 5 C. G. KuPER: Phil. Mag. 43, 1264 (1952). 1

2

Sect. 25.

255

Supercooling and superheating.

LONDON's model also may be used to determine how the resistance of a wire carrying a fixed current approaches zero as the temperature is reduced below the transition point. Theoretical curves are shown in Fig. 31 b; ic increases as the temperature is lowered, so that the resistance decreases. When ic exceeds the current in the wire, the resistance drops discontinuously to zero. The theory agrees only qualitatively with experiment (cf. Fig. 1), but it is clear that, because of the magnetic field of a measuring current, the resistance of a superconductor can vanish abruptly at Tc only in the limit of zero current. STEINER and SCHOENECK 1 first observed that a superconductive rod in a weak longitudinal magnetic field can exhibit unusual magnetic properties when carrying large currents. When the current exceeds a certain minimum value, the longitudinal magnetic flux in the rod I 35a 11=8..30e instead of being smaller than the flux in \ d=-H mrrL the normal phase, actually is greater. This phenomenon is termed the "paramagnetic ~ \ 25a effect", because the rod apparently behaves like a paramagnetic substance. The effect 1---t-+---t-'1\'il\ --\-------t "n from 0.5 to 2° K, with a maximum in "s occuring at about 1o K. The increase in can be very well explained as due to the relatively large contribution of "gs in this temperature region 4 • . The conductivity of pure mercury, though still electronic, is mainly determined by the scattering of electrons by lattice waves. The ratio, xsf"n• for the purer specimens shown in Fig. 39, decreases roughly as t5 , as the temperature becomes less than Tc. The effect of adding impurity to mercury is to make the ratio curve tend to approach the curve for tin. In the very impure specimens,

"s "n.

"s

W. HEISENBERG: Two Lectures, Cambridge University Press (1949). J. K. HULM: Proc. Roy. Soc. Lond., Ser. A 204, 98 (1950). 3 S. J. LAREDO: Proc. Roy. Soc. Lond., Ser. A 229, 473 (1955) . 4 K. MENDELSSOHN [see e.g., K. MENDELSSOHN and J. L. OLSEN: Proc. Phys. Soc. Lond. A 63, 2 (1950)] suggested an ingenious circulation mechanism, analogous to the superfluid flow which occurs in liquid He II, to account for the large values of x 5 fxn which occur in some alloys. Although the matter is not completely clear, recent evidence does not seem to favor this mechanism. 1

2

Sect. 28.

Thermal conductivity.

26S

however, the electronic conductivity is so much reduced, that the lattice conductivity becomes important and causes the ratio to increase at low temperatures. No explanation has been given for the t 5-variation of x,fxn in superconductors (such as mercury) in which lattice scattering predominates. Experiments by DAUNT and HEER 1 and GooDMAN 2 gave the first qualitative indication that x,fxn becomes extremely small in pure metals below 1o K. For example, in pure tin, x,/xn,..._,10- 3 at 0.2° K. OLSEN and RENTON 3 overcame several of the technical difficulties inherent in the foregoing work, and clearly established that x, in pure lead varies as T 3 between 0.4 and 0.9° K. LARED0 4 also observed the cubic dependence in tin, and, moreover, he found that specimens of widely differing impurity content had about the same thermal conductivity below 0. So K. The same T 3-law has been observed at sufficiently low temperatures in specimens of lead, thallium, tin, indium, niobium and tantalum by MENDELSSOHN and RENTON 5 • The cubic temperature dependence is a consequence of the fact that all the electrons are superconductive at these very low temperatures. As a result, xe,-o, and the lattice conduction is impeded only by boundary scattering. This scattering mechanism is unaffected by either the superconductive transition or impurities, with the result that

if T is small enough. The values found for the mean free path, C, for boundary scattering are about equal to the specimen diameters of tin rods. LAREDO 4 finds that in the temperature range t = 0.1S to 1.0, the electronic contribution to the thermal conductivity, xes, is in only fair agreement with the predictions of the HEISENBERG-KOPPE two-fluid model. He also finds that xes is anisotropic in tin, the conductivity along the tetragonal axis being greater than perpendicular to it. In concluding this section, we mention that lead wires have been used with success as thermal switches below 1o K. Owing to the small value of x, , the switch is "open" when the wire is superconductive; it is "closed" when made normal by a magnetic field exceeding He. y) Intermediate state. DE HAAS and RADEMAKERS 6 showed that when the superconductivity of a lead rod at So K was destroyed by a transverse magnetic field, the thermal conductivity increased linearly from its superconducting to its normal value as the field was increased from k He to He. This linear change is in accord with a model of the intermediate state in which the layers are perpendicular to the cylinder axis, with the n-layers having conductivity, xn, and the s-layers, x,. HuLM 7 showed that the magnetic transitions of tin rods in a longitudinal field are sharp ,and occur at He. An entirely new and unexpected transverse transition was first observed by MENDELSSOHN and 0LSEN 8 in a slightly impure lead specimen at about 3° K. Instead of increasing monotonically from the superconductive to the normal value with increasing field, the thermal conductivity first decreased sharply at 1 H" and then, after passing through a minimum, increased to the normal 1 2

3 4

5

6 7 8

J. G. DAUNT and C. V . HEER : Phys. Rev. 76, 854 (1949) . B . B . GoODMAN: Proc. Phys. Soc. Lond., Ser. A 66,217 (1953) . J. L. OLSEN and C. A. RENTON : Phil. Mag. 43, 946 (1952) . S. J. LAREDo : Proc. Roy. Soc. Lond., Ser. A 229, 473 (1955) . K. MENDELSSOHN and C. A. RENTON: Proc. Roy . Soc. Lond., Ser. A 230, 157 (1955). W . J. DE HAAS and A. RADEMAKERH: Physica, Haag 7, 922 ( 1940). - Leiden Comm. 261 e. J. K. HULM: Proc. Roy. Soc. Lond., Ser. A 204, 98 (1950). K. MENDELSSOHN and J. L. OLSEN: Phys. Rev . 80, 859 (1950).

266

B. SERIN: Superconductivity. Experimental Part.

Sect. 28.

value at H,. At 5° K, the specimen exhibited the usual linear behavior. Minima have since been observed in pure lead 1 and in tin and indium 2 at about 2° K. The depth of the minimum seems to increase with decreasing temperature and a very large effect has been found recently in tin at 0.5° K by LAREDO and PIPPARDs. The same deepening of the minimum is observed in lead down to 1 a K, but OLSEN and RENTON 4 found that the relative depth began to decrease below this temperature. In general the transition is irreversible, the minimum being less deep when the field is reduced from above H, than in the initial transition from zero field, and in many investigations no minimum at all is found in the transition from high fields. The data of MENDELSSOHN and OLSEN are shown in Fig. 40. SLADEK 5 observed minima in the longitudinal field transition of rods of alloys of indium containing more than 15% thallium. He showed that this T.O effect was associated with the persistence in the specimens of thin superconductive filaments in fields exceeding the threshold value. Recently, it has become clear that the '.)0 minimum in the transverse transition can be explained without invoking new mechanisms of thermal conduction. CoRNISH and 0LSEN6 0 considered a crude model in which it was 500 H(Oe) assumed that the electrons and lattice waves Fig. 40. Thermal conductivity of a slightly impure had virtually independent temperature dislead specimen in a transverse magnetic field (after MENDELssoHN and OLsEN). tributions, and on this basis, obtained fair agreement with experiment. The effect, at least at low temperatures, has since been treated with considerable rigor by LAREDO and PIPPARD 3 • They use a laminar model of the intermediate state of the rod with the layers transverse to the axis. As in any such model, the major impedance to the flow of heat results from the poor conductivity of the superconductive layers. Heat is conducted in them, for the most part, by the lattice. By an ingenious analysis, LAREDO and PIPPARD demonstrate that the mean-free path, C, for lattice scattering is much smaller in a layer than in the bulk superconductor. In the latter, C about equals the specimen diameter, whereas in the former, it equals the lamina thickness. Thus, the effective lattice conductivity of the superconductive layers is reduced by a factor of about five below the value in a bulk specimen. This increased impedance of the s-layers results in the rapid decrease of thermal conductivity when a specimen enters the intermediate state. As the field is further increased, the s-layers become progressively thinner, so that their impedance is gradually reduced and the thermal conductivity of the specimen slowly increases. The lattice conductivity in the layers is so much reduced that the electronic component of conductivity cannot be neglected. Taking this into account, the final calculated result is in very good agreement with the data obtained in increasing magnetic field. The reduction in depth of the minimum in decreasing field is not explained, although it is suggested that the difference arises because the structure of the intermediate state depends on whether the magnetic field is increasing or decreasing.

/

1 2

3 4

5 6

R. T. WEBBER and D. A. SPOHR: Phys. Rev. 84, 384 (1951) . D.P. DETWILER and H . A. FAIRBANK: Phys. Rev. 88, 1049 (1 952). S. J. LAREDO and A. B. PIPPARD: Proc. Cambridge Phil. Soc. 51, 368 (1955). J. L. OLSEN and C. A. RENTON: Phil. Mag. 43, 946 (1952). R. ]. SLADEK: Phys. Rev. 97, 902 (1955). F. H. J. CoRNISH and J. L. OLSEN: Helv. phys. Acta 26, 369 (1953).

Sect. 29.

Thermoelectric effects.

267

29. Thermoelectric effects. Many experiments have shown that no thermoelectric emf, E, is developed in a circuit containing two superconducting metals with junctions at different temperatures (see e.g., STEINER and GRASSMANN 1}. This means that the absolute thermoelectric power, e = dEjdT, of all superconductors is zero. As a result, the absolute thermoelectric power of a normal metal can be determined by measuring the emf developed by thermocouples formed between the metal and a superconductor. In their measurements on the couple, indium-lead (the lead being superconducting), KEESOM and MATTHIJS 2 found evidence to suggest that e did not fall abruptly to zero at the transition temperature of indium, but decreased gradually over a temperature interval of about 1o K, reaching zero at Tc. CASIMIR and RADEMAKERS 3 repeated the experiment more carefully with a tin-lead couple, and reported that the thermoelectric power of tin showed an unusual decrease at about 0.15 o K above the transition 1.0 temperature, which, theysuggested, "foreshadowed" the onset of superconductivity. o.a This matter has been settled recently ~ by PULLAN 4, who unambiguously demon- ~ao .. Sn. strated that the thermoelectric power of ~ M tin drops abruptly to zero at the transition ~ temperature in contradiction to the earlier 0.2 observations. A tin-lead couple was used, 0 .3.7 3.8 3.9 ?.3 but the temperature difference between the junctions in these experiments was o K Th h h Fig. 41. Thermoelectric power of tin as a function of On1Y ab OUt 0.01 · US, t e t erm0temperature (after PULLAN'). electric power could be found by dividing the measured emf by this small temperature difference, rather than by the usual less desirable procedure of differentiating the experimental curve of E as a function of T. The extremely small emf's developed were measured with a superconducting galvanometer. PuLLAN's results are shown in Fig. 41; it is clear that the thermoelectric power falls to zero in a temperature interval of about 0.01 o K, which is just the spread in the mean temperature of the specimen. PULLAN also found that the thermoelectric power of tin is unaffected by small magnetic fields, in agreement with the more extensive measurements of STEELE 5 • The latter experiments have been analyzed in considerable detail by SHOENBERG 6 • The difference between the THOMSON heats of two metals is given by

where e12 is the thermoelectric power between them. Since de12 jd T = 0 between superconductors, this relation shows that the THOMSON coefficients of all superconductive metals are equal. Furthermore, NERNST's theorem requires that a be zero at 0° K, so that it is reasonable to expect the THOMSON heat to vanish over the whole superconducting temperature range. This expectation was 1 2

250d. 3

270d. 4

5 6

K. STEINER and P . GRASSMANN: Phys. Z. 36, 527 (1935). W. H. KEESOM and C. J. MATTHIJS: Physica, Haag 5, 1 (1938). -

Leiden Comm.

H . B. G. CASIMIR and A. RADEMAKERS: Physica, Haag 13, 33 (1 94 7). -

Leiden Comm.

G. T. PULLAN: Proc. Roy. Soc. Land., Ser. A 217, 280 (1953). M. C. STEELE: Phys. Rev. 81, 262 (1951). D. SHOENBERG [J], pp. 86-94.

268

B. SERIN: Superconductivity. Experimental Part.

Sect. 30.

confirmed experimentally by DAUNT and MENDELSSOHN 1 , who showed that the induction of a large persistent current produced no detectable change in the temperature distribution in an unequally heated lead ring. They found that a for lead was less than 4 X 10-9 VtK.

VII. Superconductive alloys and compounds. 30. Many alloys and compounds become superconductive at low temperatures. Alloys of the superconducting elements, either with each other or with nonsuperconducting metals, are known to be superconducting in a large range of concentrations. SHOENBERG 2 lists about 40 alloy systems which fall into this category. Before World War II, many chemical compounds were also found to be superconductive 3 . Since the war, the properties of compounds have been intensively investigated and numerous new superconductors have been discovered. In Table 3 we present some examples of compounds as they have been classified and summarized in recent papers. No attempt has been made to achieve completeness or a logical scheme of classification; the main purpose is to convey the large number and diverse types of compounds which become superconductive above 1° K. Table 3. Superconductive compounds above J0 K . Class

Example

Bi-compounds . . . . . . . . (Metals) + (non-metallic elements) Ni As-structure . Mo and W alloys . . . . . . .

LiBi NbN PdSb Mo 30s

Approximate No. in class

I References

10

30 4

10

Recently, MATTHIAS 7 found the empirical correlation shown in Fig. 42 between the transition temperature of a superconductor and its number of valence electrons per atom. This suggests that optimum conditions for the occurrence of superconductivity seem to exist for 5 and 7 valence electrons per atom. A good many compounds exhibit reasonably sharp magnetic transitions in a longitudinal field. Alloys and the "hard" superconductors, on the other hand, tend to exhibit diffuse transitions (cf. Sect. 7{J). Small magnetic fields begin to penetrate gradually into the specimen and some superconductive threads persist in very large fields. This persistence of threads makes resistance measurements unreliable, because the specimen exhibits zero resistance in very high fields when the bulk of the specimen is in the normal phase. Since the threads are thin, the resistive transition is very sensitive to the strength of the measuring current. Furthermore, when the field is lowered from a high value, the specimen is left with a large frozen-in magnetic moment. The foregoing paragraph accurately describes the properties of the majority of cases. However, recent work has revealed important exceptions in particular 1 2 3

4

5 6 7

J. G. DAUNT and K. MENDELSSOHN: Proc. Roy . Soc. Lond., Ser. A 185, 225 (1946) . D. SHOENBERG [1], pp. 230-231. D. SHOENBERG [1], pp. 228-229. B. T. MATTHIAS and J. K. HULM: Phys. Rev. 87, 799 (1952). G. F. HARDY and J. K. HuLM: Phys. Rev. 93, 1004 (1954). B. T. MATTHIAS: Phys. Rev. 92, 874 (1953). B. T. MATTHIAS: Phys. Rev. 97, 74 (1955).

Sect. 30.

269

Superconductive alloys and compounds.

systems and in dilute alloys, and has also provided clues as to the cause of the non-ideal behavior. We briefly review this work below. WEXLER and CoRAK 1 determined the B-H curves of several specimens of vanadium which differed from each other in mechanical hardness. In the softest specimen at any given temperature, the initial penetration of the magnetic field occurred at a sharply defined value, indicating that the superconductivity of a substantial fraction of the material was destroyed at a well defined field. As the field was further increased, the induction gradually rose to B = H, showing that superconductive regions tended to persist in the specimen. WEXLER and CoRAK suggest that the sharp penetration fields correspond closely to the equilibrium fields which would be manifested by pure, unstrained specimens. With increasing specimen hardness, the point of initial penetration became less and less well defined and the transition temperature was lowered. Increased specimen hardness could be correlated with an increased content of absorbed nitrogen and oxygen. As a result, it is proposed that the magnetic properties exhibited by the hard superconductors are due to mechanical strain arising from either mechanical work or interstitially located Tc I impurities such as carbon, nitrogen and I I I I I I oxygen. SHOENBERG 2 had earlier obI I served ideal magnetic behavior in a very " 70 5 pure thorium wire, even though thorium 0 / ofom elecfrons volence #o. is in the "hard" group. It is suggested Empirical correlation between the transition that the structure of thorium and its Fig.42. temperatures of compounds and their number of valence electrons per atom (after MATTHIAs). large atomic volume reduce the strains impurities. interstitial by introduced The superconducting properties of indium-thallium solid solutions in the composition range from pure indium to SO% thallium were studied by STOUT and GuTTMAN 3 . The magnetic induction and electrical resistance of long cylindrical specimens in longitudinal fields were measured at various temperatures. Up to 10% thallium, the magnetic induction in the specimens at a given temperature jumped from zero to H at a sharply defined field, and resistance was restored at substantially the same field. Thus, these specimens exhibited ideal superconductive properties. As the concentration of thallium was increased above 10%, the flux penetration occurred over a wider range of field , and resistance appeared only after practically all the flux had penetrated, indicating that alloy effects became evident only at fairly high concentrations. The transition temperatures of the alloys were all smaller than T, for pure indium. Along these same lines, LoHMAN and SERIN 4 investigated the transition temperatures of dilute solid solutions of antimony, bismuth, cadmium, indium, lead, mercury and zinc in tin. All specimens exhibited sharp transitions, and in all cases the effect of the impurity was initially to lower the transition temperature. Recently, LYNTON, SERIN and ZucKER 5 have extended the measurements on the tin solutions to low temperatures. They find in addition to the initial decrease of transition temperature, that the electronic specific heat constant, y,

If\

I I

1 2 3

4 5

I I

A. WEXLER and W. S. CORAK: Phys. Rev. 85, 85 (1952). D . SHOENBERG: Proc. Cambridge Phil. Soc. 36, 84 (1940) . J. W . STOUT and L. GuTTMAN: Phys. Rev . 88, 703 (1952). See B. SERIN [3], Chapter VII. E . A. LYNTON and B. SERIN: Int. Conf. on Low Temp. Phys., Paris 1955.

270

B. SERIN: Superconductivity. Experimental Part.

Sect. 31.

of the normal phase shows a smalt gradual increase for all impurities, of valence both higher and lower than tin. In the discussion of Sect. 26 of the transition from the normal to the superconductive phase of tin, we stated that a superconductive sheath initially formed on the surface of a rod . The sheath only occupies about 1/5 of the total volume of a specimen, and we indicated that the flux trapped inside the sheath slowly escapes through small gaps in it. PIPPARD 1 has suggested that the superconductive metal on either side of a gap is prevented from coalescing by the necessarily large value of the range of coherence, ~ ~.il, in pure tin. In impure tin, when ~ "'.il, the gaps are assumed to be able to close, thereby permanently trapping flux (.il is the penetration depth). To check the effect of impurity on flux trapping, PIPPARD performed an extensive series of experiments on the amounts trapped in rods of pure tin and of tin alloyed with indium up to 3%. The 100 procedure was to place a rod in a transverse magnetic field which was increased above the threshold value and then reduced to zero. The specimen was then Sn+2.5%ln ) rapidly rotated through 180°, and the emf induced in a search coil was observed in Sn+l.B%/n order to obtain a measure of the amount """'\ of flux trapped. Significant results were obtained only af0.1 ter the specimens had been annealed for at least about 20 days, apparently to homoge0.6 t-T/Tc fl.B T.O nize the alloys. The proportion of trapped Fig. 4J. Trapped flux as a function of the temperaflUX in pure tin WaS only about 0.1 %, and it ture of tin-indium alloys (after PIPPARo '). increased steadily as the indium concentration was increased. In addition to this effect, it seemed that a change of behavior occurred between 2.3 and 2.5% of indium, which is illustrated in Fig. 43. For indium concentrations less than 2.3%, the proportion of trapped flux tends to zero as the temperature approaches Tc. For greater indium concentrations, the trapping rises to a value of about SO% at the transition temperature. PIPPARD suggests, therefore, that the apparent change of behavior at 2. 3 % indium marks the beginning of spontaneous coalescence. This interpretation of the experiments has recently been questioned by BuDNICK, LYNTON and SERIN 2• They find that as a result of continued annealing (up to about 100 days) the rise in trapped flux in impure tin specimens near Tc becomes progressively less pronounced. It appears that even the most impure specimens would show no rise after sufficient annealing. They therefore conclude that there is no direct evidence for the existence of spontaneous coalescence, but that in well annealed specimens, any flux trapped can, at all temperatures, for the most part escape when the external field is reduced to zero.

VIII. Diverse properties unchanged in the superconductive transition. 31. X-ray and neutron diffraction patterns. KEESOM and ONNES 3 observed the X-ray diffraction patterns exhibited by lead above and below the transition temperature and found no change in the patterns. Thus any changes in the 1

2 3

A. B. PIPPARD: Phil. Trans. Roy. Soc. Lond. A 248,97 (1955). J. I. BuDNICK, E. A. LYNTON and B. SERIN: Phys. Rev. 103, 286 (1956) . W. H. KEESOM and H. KAMERLINGH 0NNES: Leiden Comm. 1924, 174 b .

Sect. 32, 33.

Interaction with radiation, and field emission.

271

lattice spacing upon passing between the normal and superconductive phases are extremely small. Recently, WILKINSON, SHULL, ROBERTS and BERNSTEIN 1 measured the coherent and incoherent scattering of neutrons by the electrons in vanadium, lead and niobium above and below their transition points, and found that in no case was there a change of the coherent scattering or the diffuse background. This result clearly indicates that there is no detectable change in the electronic distribution with advent of superconductivity. Examination of the nuclear scattering in lead and niobium showed that there were no pronounced changes in the atomic lattice vibrations at the transition temperature 2 • These same authors report that the total thermal neutron cross section for tin in the normal and superconducting states is the same within one percent. 32. Interaction with electrons. McLENNAN, McLEOD and WILHELM 3 measured the absorption by thin lead films of electrons having energies of a few Mev. The advent of superconductivity produces no change in the absorption coefficient. MEISSNER and STEINER 4 determined that the transmission coefficient of tin foils for electrons of about 10 ev energy is the same in the superconductive and normal phases. Recently, STUMP and TALLEY 5 measured the lifetime of positrons in superconductive lead and tin. The lifetime in lead appreciably increased when the lead passed from the normal to the superconductive state, but no change in lifetime was found in tin. At the present writing, the interpretation of these meassurements is obscure. 33. Interaction with radiation, and field emission. The photoelectric current coming from lead illuminated with ultraviolet light was measured by McLENNAN, HuNTER and McLEon 6 , and it was found that the superconductive transition caused no detectable change in current. Within the error of measurement of 0.2%, GoMER and HULM 7 observed no difference in the field emission current from tantalum at temperatures above and below Tc. DAUNT, KEELEY and MENDELSSOHN 8 measured the reflectivity of lead and tin for infra-red radiation of about 10(L wavelength. There is no measurable difference in the reflectivities of the superconductive and normal phases. This result has been verified for tin by RAMANTHAN 9 • WEXLER 10 found that the amount of visible light transmitted by thin lead films is unaffected by the advent of superconductivity. I should like to thank warmly my colleagues, Drs. E. A. LYNTON and P. LJNDENFELD and my wife, BERNICE SERIN for their many h elpful comments on this article. 1 M. K. WILKINSON, C. G. SHULL, L. D. ROBERTS and S. BERNSTEIN: Phys. Rev. 97, 889 (1955). 2 This is equivalent to finding no change in the DEBYE temperature, 13. 3 J. C. McLENNAN, J. H. McLEOD and J. 0. WILHELM: Trans. Roy. Soc. Canada (3) 23 (III), 264 (1929). 4 W. MEISSNER and K. STEINER: Z. Phys. 76, 201 (1932). 6 R. STUMP and H. E. TALLEY: Phys. Rev. 96, 904 (1954). 6 ]. C. McLENNAN, R. G. HUNTER and J. H. McLEOD: Trans. Roy. Soc. Canada (3) 24 (III), 3 (1 930). 7 R. GoMER and J. K. H uLM : J. Chern . P hys. 20, 1500 (1952). 8 J. G. DAUNT, T. C. K EELEY and K. MENDELSSOHN: Phil. Mag. 23, 264 (1937). 9 K. G. RAMANTHAN: Proc. Phys. Soc. Land. A 65, 532 (1952). 1o A. WEXLER: Phys. Rev. 70, 219 (1946).

272

B. SERIN: Superconductivity. Experimental Part.

Bibliography. [1] SHoENBERG, D . : Superconductivity, 2nd ed. Cambridge: University Press 1952. [2] LAUE, M. voN: Theory of Superconductivity. New York : Academic Press 1952. [3] GoRTER, C. J. : Progress in Low Temperature Physics. Amsterdam: North-Holland Publishing Company 1955. [4] LoNDON, F.: Superfluids, vol. 1. New York: John Wiley and Sons 1950.

References Appended in Proof, March 1956. Chap. II. Sect. 24y. (1) BEDARD, F., and H. MEISSNER: Measurements of contact resistance between normal and superconducting metals. Phys. Rev. 101, 31 (1956).

Chap. III. Sect. 14y. (2) WoRLEY, R. D., M. W. ZEMANSKY and H. A. BooRsE : Heat capacities of V and Ta in the normal and superconducting phases. Phys. Rev. 99, 447 (1955) . Sect. 15. (3) GRENIER, C.: The anisotropy of the effect of the elastic deformation on the superconductivity of tin. C. R. Acad. Sci., Paris 238, 2300 (1954); 240, 2302 (1955). (4) GARBER, M., and D. E. MAPOTHER: Effect of hydrostatic pressure on the superconducting transition of tin. Phys. Rev. 94, 1065 (1954). (5) MuENCH, N. H.: Effects of stress on superconducting Sn, In, Tl and AI. Phys. Rev. 99, 1814 (1955). (6) HATTON, J.: Effect of pressure on the superconducting transition of thallium. Phys. Rev. 100, 1784 (1955) . (7) MAcKINNON, L.: Relative absorption of 10 Mcjsec. longitudinal sound waves in a superconducting polycrystalline tin rod. Phys. Rev. 100, 655 (1955). (8) B6MMEL, H. E.: Ultrasonic attenuation in superconducting and normal-conducting tin. Phys. Rev. 100, 758 (1955). (9) PIPPARD, A. B. : Ultrasonic attenuation in metals, Phil. Mag. 46, 1104 (1955). (10) PIPPARD, A. B. : Thermodynamics of a sheared superconductor. Phil. Mag. 46, 1115 (1955).

Chap. IV.

Sect. 16e (2). (11) FABER, T. E., and A. B. PIPPARD: The penetration depth and conducting AI. Proc. Roy. Soc. Lond., Ser. A 231, 336 (1955).

hf resistance of super-

Sect. 21. (12) GREBENKAMPER, C. J., and J.P. HAGEN: High frequency resistance of metals in the normal and superconducting state. Phys. Rev. 86, 673 (1952). ( 13) GREBENKAMPER, C. J.: Superconductivity of V at 24,000 Mcjsec. Phys. Rev. 96, 316 (1954). ( 14) GREBENKAMPER, C. J.: H-f resistance of Sn and In in the normal and superconducting state. Phys. Rev. 96, 1197 (1954). ( 15) FAWCETT, E. : The surface resistance of normal and superconducting tin at 36 kMcjsec. Proc. Roy. Soc. Lond., Ser. A. 232, 519 (1955) . (16) BLEVINS, G. S., W. GORDY and W. H. FAIRBANK: Superconductivity at millimeter wave frequencies. Phys. Rev. 100, 1215 ( 1955). (1 7) BIONDI, M. A., M. P. GARFUNKEL and A. 0. McCouBREY: Millimeter wave absorption in superconducting aluminum. Phys. Rev. 101, 1427 (1956). N. B. The foregoing two references [(16), (17)] report a new observation . For frequencies, such that hv 5 k T, (i.e. frequencies > 77 kMcjsec. in the case of Sn, and > 22 kMcjsec. in the case of AI) the metal seems to have the same residual resistance as in the normal state, at temperatures at which there is already complete de superconductivity. For a given frequency, the temperature has to be reduced below the usual transition point before the surface resistance begins to decrease. The higher the frequency, the lower the temperature to start the decrease in surface resistance.

273

Bibliography.

Sect. 22. ( 18) LEWIS, H . W. : Search for a HALL effect in a superconductor II. Phys. Rev. 100, 641 (1955) .

Chap.V. Sect. 24rx.. (19) SCHAWLow, A. L.: Structure of the intermediate state of superconductors. Phys .Rev. 101, 573 (1956). Sect. 24y. (20) GRASSMANN, P., and L. RINDERER: Critical values of the current in superconducting Pb-Bi alloy in external magnetic field . Hclv. phys. Acta 27, 309 (1954). (21) RINDERER, L.: Destruction of superconductivity by the current carried and an applied transverse magnetic field. Z. Naturforsch. lOa, 174 (1955). (22) MEISSNER, H.: Paramagnetic effect in superconductors II. Phys. Rev. 101, 31 (1956). Sect. 26. (23) FABER, T. E.: The phase transition in superconductors IV, AI. Proc. Roy. Soc. Lond., Ser. A 231, 353 (1955). (24) GALKIN, A. A., and P. A. BEZUGLYI: The kinetics of the destruction of superconductivity by a magnetic field. Zh. eksp. teor. Fiz. 28, 463 (1955).

Chap. VI. Sect. 28{3. (25) PHILLIPS, N. E. : Thermal conductivity of In-TI alloys. Phys. Rev. 100, 1719 (1955) .. Sect. 28y. (26) RENTON, C. A.: Effect of a magnetic field on the heat conductivity of a superconductor. Phil. Mag. 46, 47 (1955).

Chap. VII. Sect. 30. (27) MATTHIAS, B . T ., and E. CoRENZWIT: Superconductivity of Zr Alloys. Phys. Rev. 100, 626 (1955). (28) ZHURAVLEV, N. N., and G. S. ZHDANov: Superconducting Bi-Rh compounds. Zh. eksp. teor. Fiz. 28, 228 (1955); also ALEKSEEVSKI, N. E., G. S. ZHDANOV and N. N. ZHURAVLEV: Zh. eksp. teor. Fiz. 28, 237 (1955) . (29) GLOVER III, R.: An empirical rule for the position of superconductors in the periodic table. Z. Physik 140,494 (1 955) . (30) TEASDALE, T . S.: Permanent magnetic moments of a superconductive sphere. Phys. Rev. 99, 1248 (1 955). (31) DoiDGE, R. P .: The transition to superconductivity. Phil. Trans. Roy. Soc. Lond. 248, 553 (1956).

Chap. VIII.

Sect. 32. (32) ALBERS-SCHONBERG, H., and E. HEER: Directional correlation measurements in superconducting metals. Helv. phys. Acta 28, 389 (1955). Sect. 33. (33) McCRUM, N . G., and C. A. SHIFFMAN: The optical constants of tin below the superconducting transition t emperature. P roc. Phys. Soc. Lon d. A 67, 368 (19 54) .

Handbuch der Physik, Bd. XV.

18

Theory of Superconductivity. By

J. BARDEEN. With 20 Figures.

I. Introduction. 1. Although superconductivity falls into the domain where one would expect ordinary non-relativistic quantum mechanics to be valid, it has proved to be extremely difficult to obtain an adequate theoretical explanation of this remarkable phenomenon. In spite of the large amount of excellent experimental and theoretical work devoted to the problem, there remain major unsettled questions. However, the area in which the answers are to be found has been narrowed considerably. There are very strong indications, if not quite a proof, that superconductivity is essentially an extreme case of diamagnetism rather than a limit of infinite conductivity. The isotope effect indicates that the superconducting phase arises from interactions between electrons and lattice vibrations. That the magnetic properties come from orbital motion of electrons and not from electron spins is shown by a measurement of the g-value from the gyromagnetic effect. KrKOIN and GooBAR1 , and more recently PRY, LATHROP and HousTON 2 , from observations of the angular momentum picked up by a sphere when a magnetic field is switched on, have found that the g-value is close to unity, as expected for orbital motion. Let us first consider the nature of the electromagnetic properties: Is superconductivity, as the name implies, a limit of infinite conductivity in which the electrons are not scattered or is it a limit of perfect diamagnetism (B = 0) as is indicated by the MEISSNER effect? These two aspects are very closely related. If the conductivity is infinite, the magnetic field in the interior of a massive specimen cannot change when the external field is changed, but the field need not be zero if the specimen is cooled in the presence of an external field. If the diamagnetic aspects are assumed basic, one must show why a current flowing in a ring is metastable and does not decay. Since the discovery of the MEISSNER effect in 193 3 and the LoNDON [16] phenomenological description which followed shortly afterwards, it has generally been assumed those aspects usually associated with infinite conductivity are a consequence of the magnetic properties. The supercurrents are then always associated with and determined by the magnetic field. In other words, in the presence of a magnetic field the stable condition is that with currents, flowing near the surface, which prevent the penetration of the field. When a current, I, flows in a ring, there is a one-parameter family of solutions determined by the flux through the ring or by the current flow. The complete current distribution is determined by this parameter. The lowest state 1 I. K. KJKOIN and S. V . GooBAR: C. R. Acad. Sci. URSS. 19, 249 (1938). USSR. 3, 333 (1940). 2 R. H. PRY, A. L LATHROP and W V . HousTON: Phys. Rev. 86, 905 (1952).

J.

Phys.

Sect. 1.

Introduction.

275

corresponds to I= 0, but states with I=!= 0 are metastable and persist indefinitely. A possible explanation is that in the multiply-dimensional phase space of all of the electrons, there is only one unique path which leads to states of lower energy. Fluctuations are unlikely to lead to this path. There exists, however, no rigorous proof of the metastability of such current distributions, and some recent theories, such as those of BoRN and CHENG and of HEISENBERG and KoPPE have taken the viewpoint that persistent currents exist independently of the magnetic field, and that it is the stability or metastability of such currents which is the basic property. In a discussion of a onedimensional model, FROHLICH 1 has also suggested that currents in the absence of magnetic fields may be metastable. Opposing the view that currents are really stable is a proof of BLOCH 2 , extended by BoHM 3 to many-electron wave-functions, that in the state of lowest energy the current density must vanish. It is possible that entropy considerations may favor distributions of microscopic current loops, so that states with currents flowing are thermodynamically stable, as suggested by HEISENBERG [9). An interesting experiment which indicates that the diamagnetic property is basic is a measure of the damping and period of a superconducting sphere oscillating in a magnetic field 4 • One expects no eddy current damping from either model. However, there should be a change in period from torques introduced by undamped eddy currents if the conductivity is infinite. On the diamagnetic model, the currents are always associated with the magnetic field and stay fixed in space as the sphere rotates. There is then no additional torque and no change in period, and this is what was found experimentally. We adopt here the viewpoint that the diamagnetic aspects are basic, and show how they might follow as a consequence of quantum theory, along the lines suggested originally by F. LoNDON [14], [15]. He showed that the LONDON equation c curl Aj = -H follows if the wave functions are so rigid that they are not modified at all by a magnetic field. While many of the qualitative consequences of this equation of the LoNDON's have been confirmed, the theory has not received a really good quantitative check. PIPPARD [20] has suggested on empirical grounds a modified form of the theory in which the current density at a point depends on the integral of the vector potential over a region surrounding the point. We shall show that when first order changes of the wave functions produced by the magnetic field are taken into account, one is led to a "nonlocal" version of the theory similar to that suggested by PIPPARD. Another major question concerns the nature of the interactions responsible for the thermal transition and thermal properties. The isotope effect (SERIN 5 p. 237) shows rather conclusively that superconductivity arises from interactions between electrons and lattice vibrations, and theories based on this idea have been proposed independently by FROHLICH [4] and the author [1]. FROHLICH's theory, developed without knowledge of the isotope effect, gave a relation between critical temperature, Tc, and isotopic mass,

VMTc r-...~const H. FROHLICH: Proc. Roy. Soc. Lond., Ser. A 223, 296 (1954). z Quoted by L. BRILLOUIN: Proc. Roy. Soc. Lond., Ser. A 152, 19 (1935) . 3 D . BoHM: Phys. Rev. 75, 502 (1949). 4 W. V. HousTON and N. MuENCH: Phys. Rev. 79, 967 (1950) . Closely related experiments are those of E. U. CoNDON and E. MAXWELL: Phys. Rev . 76, 578 (1949) ; 79, 967 (1950) and FRITZ, GoNZALEZ and JoHNSON: Phys. Rev. 79, 967 (1 9 50); 80, 894 (1950). 5 Unless otherwise qualified, references to this author advert to his preceding article in this volume. 18* 1

276

J. BARDEEN: Theory of Superconductivity.

Sect . 1

which is close to that found empirically. Because of mathematical difficulties involved, neither of these theories is very satisfactory. Anything approaching a rigorous deduction of superconductivity from the basic equations of quantum theory is a truly formidable task. The energy difference between normal and superconducting phases at the absolute zero is only of the order of 10-8 ev per atom. This is far smaller than errors involved in the most exacting calculations of the energy of either phase. One must neglect terms or make approximations which introduce errors which are many orders of magnitude larger than the small energy difference one is looking for. One can only hope to isolate the physically significant factors which distinguish the two phases. For this, considerable reliance must be placed on experimental findings and the inductive approach. A great deal can be learned from the thermal and electrical properties: Specific heat, thermal conductivity, electrical conductivity observed in the penetration depth at microwave frequencies, and other properties which give information about the excited states of superconductors. A powerful method of attack in the analysis of properties of materials at low temperatures is to consider the nature of the elementary excitations; e.g. LANDAU's rotons in liquid helium II or BLOcH's spin waves in ferromagnetics 1 . The individual particle model gives a satisfactory account of excited states of electrons in normal metals. At T = 0° K, all the levels are filled up to the FERMI level, E F, and none of the higher states are occupied. Elementary excitations correspond to raising an electron from an occupied state to a higher unoccupied state. The fraction of electrons which are thermally excited at a temperature Tis of the order of kTfEF, and the average excitation energy is of the order of kT. This gives a thermal energy proportional to T2 and a specific heat proportional to T, as observed in normal metals. The thermal properties of superconductors indicate that there are excited electrons similar to those of the normal phase, but that the number is greatly reduced when T < Tc. Since the transition is of second order, the number must be substantially the same just below Tc. Thus the elementary excitations in the superconductor, as in the normal metal, are probably those of individual particles. This is essentially the basis for the various two-fluid models which have been suggested to account for the thermal properties. The "normal" component corresponds to the excited electrons. The number of low-lying excited states and thus of excited electrons is greatly reduced in a superconductor. It has been suggested that the electrons form some sort of condensed state in the superconducting phase such that a finite energy e ,....._ k Tc is required to excite an electron near T = 0° K. This has been called the "energy gap" model. This would lead to a specific heat and thermal conductivity varying as exp (- s/2 k T) at very low temperatures. While there is some evidence 2 that this is the case, the question is still open. According to the GoRTER-CASIMIR two-fluid model, which fits the data for a number of superconductors at moderate temperatures, the specific heat varies as T 3 • If this latter model is correct at T --+0 there is reduced density of low-lying states, but no true energy gap. It can be shown that the energy gap model, originally introduced to account for the thermal properties, if taken literally gives the MEISSNER effect, and in fact leads to a theory similar to PIPPARD's modification of the LONDON equation for the current density in a magnetic field. Thus the essential task of the microscopic theory is to show why there are few low-lying excited states in the superconducting phase. 1 2

GINSBURG [5] has advocated this approach to superconductivity. See SERIN, Sect. 12 and 27.

Sect. 2.

Temperature dependence of critical field.

277

Since the BLOCH individual particle approximation accounts satisfactorily for the properties of normal metals, it has been thought that superconductivity arises from one of the terms neglected in this theory. One of these is the correlation between the positions of the electrons due to CouLOMB forces, used in the HEISENBERG theory [9]. It was suggested that electrons with energies near the FERMI surface form a lattice and so tend to keep apart and reduce to the long range CouLOMB energy between electrons. Another is magnetic interactions between electrons, as suggested by WELKER 1 . A third is electron-phonon interactions, originally introduced to account for scattering of electrons and thus resistance. They also contribute to the energy of both normal and superconducting phases, and are presumably responsible for the transition. It has been shown recently by PINES and the author [2] why, as is indicated experimentally by the isotope effect, electron-phonon interactions are more important than CouLOMB interactions. The reason is essentially that the CouLOMB interaction between electrons is a screened interaction of relatively short range. The long range part of the interaction gives rise to plasma oscillations which are of such high frequency that they are not normally excited and also to coupled electron-ion oscillations which are just the lattice vibrations of long wavelength. The remaining screened interaction is sufficiently weak so that it can be treated by perturbation methods, and thus does not have a marked effect on the wave functions. On the other hand the criterion for superconductivity, as given for example in FROHLICH's theory, is essentially that the electron-phonon interaction be so large that it cannot be treated by perturbation theory. This means that the wave functions may be modified greatly by the interaction. Mathematical methods for treating such large interactions are lacking, so that we still do not have a satisfactory picture of the superconducting phase. Partly because of the difficulties involved in developing a fundamental microscopic theory, considerable effort has been devoted to the development of phenomenological theories. These include two-fluid models to describe the thermal properties, equations such as those of the LoNDON's to account for the electrodynamic properties, and theories of boundary effects to derive properties of the intermediate state and related phenomena. Most of these theories are still on a rather insecure foundation and have not been subjected to convincing quantitative checks. The only relations one can be really sure of are those based on thermodynamics. In Chap. II we give a brief outline of the thermodynamic relations, and then discuss some of the two-fluid models which have been proposed; Chap. III is devoted to the LoNDON phenomenological theory and generalizations of these equations which have been proposed by PrPPARD, Chap. IV to boundary effects and the intermediate state, including the LANDAU theory [13], and Chap. V to the microscopic theories. The latter is concerned mainly with the formulation of the problem of calculating electron-phonon interactions. An outline is given of FROHLICH's theory and other attempts, none very successful, for calculat ing the interaction energy in normal and superconducting states.

II. Thermodynamic properties and two-fluid models. a) Thermodynamic relations. 2. Temperature dependence of critical field. The critical field is determined

by thermodynamic considerations. Exclusion of flux gives an increase in magnetic energy, and when this increase more than compensates for the lower free energy 1

H.

WELKER:

Z. Physik 114, 525 (1939).

278

J. BARDEEN:

Theory of Superconductivity.

Sect. 2.

of the superconducting phase in zero field, a transition to the normal state occurs. The MEISSNER effect shows that there is a unique state of a superconductor under given conditions of temperature, pressure and external applied magnetic field, so that one should be able to derive thermodynamic relations concerning the transition parameters. As a matter of fact, thermodynamics was applied with good results prior to the discovery of the MEISSNER effect, first by KEESOM 1, and later by RUTGERS 2 and by GoRTER 3 • The reason for this success became apparent only after the discovery that the transition really is reversible. Our treatment is patterned closely after that in the excellent book of SHOENBERG [23], who has also contributed to the theory. It is most convenient to take the pressure, P, and the external field, Ha, as independent variables. We shall restrict the discussion at present to massive specimens for which we may assume that the field is zero in the interior. Boundary effects and thin films are discussed in Chap. IV. One then deals with the GIBBS free energy, G = U- T S

H.

+ P V- f

M(Ha) dHa,

(2.1)

0

where U, the internal energy, and S, the entropy, are assumed independent of Ha, and M(Ha) is component of the magnetic moment in the direction of Ha. If the free energy depends on a parameter x, the equilibrium value is such that

(~) OX T,P,H.--o .

(2.2)

The entropy, volume and magnetic moment are given by:

5 - - (-oG) -

(2.3)

v=

T,H.'

(2.4)

(:~)T,P.

(2.5)

-

oTP,H.'

(~~) oP

M =-

It will be most convenient to consider first a long rod parallel to the field for which the demagnetizing coefficient is zero. The magnetic moment is then

(2.6) If G5 (0) is the free energy in absence of an external field,

(2.7) The critical field, H,, is that for which Gs (He) becomes equal to the free energy Gn of the normal metal, assumed independent of Ha;

Gn = Gs (o)

+ 8~ V H,

2.

(2.8)

This result must be independent of the shape of the body, and it is true that the area under the magnetization curve is independent of shape: H,

- of 1 2 3

M(Ha) dHa =

1

-8 V H/, n

W. H. KEESOM: Rapp. et Disc. 4°. Congr. Phys. Solvay, p. 288. A. J. RuTGERS: Physica, Haag 1, 1055 (1934); 3, 999 (1936}. C. J. GoRTER: Arch Mus. Teyler 7, 378 (1933).

(2.9)

279

Pressure and volume variations.

Sect. 3.

Here ~ is sufficiently large to bring the specimen to the normal state. Surface effects and any small dia- or paramagnetism of the normal state are neglected. It is also assumed that the transition takes place reversibly. Relations between various thermodynamic quantities at the transition can be obtained by taking appropriate derivatives of (2.7) along the transition curve He(T). The entropy difference between normal and superconducting states is:

5

n-

VHe dHe dT.

5 -

(2.10)

.--~

The heat, Q, absorbed in going from the superconducting to normal states is

Q = T(Sn - S) 5

=-

T

He dH_c_ v4n dT ·

(2.11)

The difference rn specific heats obtained from the relation C= T(8Sf8T) is given by C _ C = TVm (H d2Hc_ + (dHe)2) (2.12) s

dT

e dT2

4n

n

'

where vm is the specific volume. In zero applied field, the transition is of second order. There is no latent heat (Q=O), but there is a discontinuity in specific heat, LJC, at the transition point: LJC =

v:: (~ir.

(2.13)

Test of RuTGERS' Relation [Eq. (2.12)]. (From SHOENBERG [23], p. 62).

Table 1.

LIC x 1o• calc. LIC x 10 obs. A comparison of observed values Tc°K (dHcfdT) T= Tc Metal cal/°K cal/°K ofthe left and right sides of (2.12), known as RuTGERs' relation, is 12.6 Pb 10 200 7.22 given in Table 1, taken from 8.2, 9 Ta 320 4.40 9.4 SHOENBERG. La 190.00 1000 4.37 13.9 2.4, 2.9 Sn 2.61 151 3.73 3. Pressure and volume variaIn 2.08 2.3 146 3.37 tions. There are small but obTl 1.48 1.47 2.38 139 Al 0.46 1.20 0.71 177 servable changes of ~ and Te with pressure. These can be related, by means of thermodynamic relations, with the small volume difference, neglected so far, between normal and superconducting phases and also with differences in thermal expansion and compressibility between the two phases. We suppose that He in (2.8) is a function of P and T, and apply (2.4) to find 1

Since

= V. _ V.(O) 5 n

VHe (oHe) + Ht(~) . oP T 8n oP T 4n

(3.1) (3.2)

we have

(av)

JT.(Hc)- JT.(O) = [jp

T

H2

8;'

(3.3)

and, from (3.1), the change in volume LJV, at the transition is LJV = V. _ V.5 (H)= n

e

(oHc) VHc oP T . 4n

(3.4)

On combining (3.4) with the thermodynamic relation

(oHe) (aT) ( oHc) oP T = - oT P BP He'

(3.5)

280

J. BARDEEN: Theory of Superconductivity.

Sect. 4.

and making use of (2.11), we find the CLAUSIUS-CLAPEYRON relation:

(:~)H, = lv·

(3.6)

The left hand side is the change in pressure required to keep the critical field at the same value, He, as the temperature is changed. Further derivatives of (3.4) with respect to T and to P give the change of the thermal expansion coefficient, Llix, and the change in compressibility, L1K, at the transition, where

(1. = --1 -·av -···· v aT·

aP K= - Vav.

(3-7)

For the transition in zero applied field, He= 0, the expressions reduce to:

L1(1. =

_1

4n

LJK = !!:_ 4n

(~He)

aT

P

(aaPHc) r ,

(8f!c)2. aP

(3.8) (3-9)

There are also relations due to EHRENFEST which apply to all second-order transitions: dT,;

dP. =

VmTcLlcx

-en _:-c:- =

LlK

K

2

Ll-;_ ·

(3 .1 0)

A much more complete discussion of these relations is given in SHOENBERG's book [23]. b) Two-fluid models. 4. GORTER-CASIMIR and related theories. Various two-fluid models have been suggested to account for the thermal properties of superconductors. They are based on two general assumptions: (1) There is a condensed state, the energy of which is characterized by some sort of order parameter; (2) all of the entropy comes from excitations of individual particles similar to those of the normal metal 1 . The number of excited electrons depends on the temperature as well as on the order parameter. An order parameter of some sort is required to give a second-order transition such that the condensation energy varies from a maximum at T = 0° K to zero at the transition temperature. The excited electrons account not only for the entropy and part of the specific heat, but also for the thermal conductivity, a.c. resistance and viscosity of electrons in the superconducting phase. The first and best-known two-fluid model is that of GoRTER and CASIMIR [8] which in the usual form leads to a specific heat varying as P. KoPPE [11] derived a particular form of the two-fluid model on the basis of the HEISENBERG theory. However, KoPPE's theory does not depend on the interaction assumed to be responsible for the condensation, and may well have more general validity. GINSBURG's theory 2 is based on the energy gap model in which it is assumed that a minimum energy e""kTc is required to excite an electron from the condensed phase. Rather general formulations which include the others as special cases have been discussed by KoPPE [11], BENDER and GoRTER 3 and MARCUS and MAXWELL 4. 1 2

3

4

The naive interpretation as two independent fluids is not justified. W. L. GINSBURG: J. exp. theor. Phys. USSR. 14, 134 ( 1946) and [5]. P . L. BENDER and C. J. GoRTER: Physica, Haag 18, 597 (1952). P.M. MARCus and E. MAXWELL: Phys. Rev. 91, 1035 (1953).

GoRTER-CASIMIR and related theories.

Sect. 4.

281

The choice of the order parameter is somewhat arbitrary. We shall follow MARCUS and MAXWELL and others and take a parameter w which varies from unity at T = 0° K to zero at T = Tc and which is such that the condensation energy relative to the normal metal is - fJ w, where fJ is a parameter, characteristic of the metal, given by (4.1) Here H0 is the critical field at T = 0° K. The HELMHOLTZ free energy of the normal phase may be expressed in the form : (4.2)

where yT is the electronic specific heat and FL is the contribution of lattice vibrations. In going to the superconducting phase, it is assumed that any change in U (0) FL is accounted for by the term - {Jw and that (~) yP is reduced by a factor K(w), so that (4-3) F.= U(O)- {Jw- ~yPK(w) +FL(T).

+

The reduction factor K may depend on T as well as on w. If superconductivity arises from interactions between electrons and lattice vibrations, the condensation energy may appear as a reduction in zero-point energy of the oscillations. If the predominant wave lengths involved are so short that the oscillations are not excited at low temperatures, as appears to be the case, the temperature dependent terms in FL (T) will not be affected by the transition. The various theories differ in what is taken for K (w). To agree with experiment, K (w) must approach zero as w -+ 1 (corresponding t o very low temperatures) and, in order to have a second order transition, must approach unity as w -+ 0 (corresponding to T = TJ. GoRTER and CASIMIR made the ad hoc assumption that K(w)

= (1 - w)",

(4.4)

which is a simple function satisfying both limiting values. They found best agreement with experiment by taking ex= ~ , the value which leads to an electronic specific heat varying as P and a parabolic critical field curve. As will be discussed in the following, MARCUS and MAXWELL find that a smaller value of ex gives a better fit to the critical field curves of several elements, so that it is probably best to leave ex as a parameter to be determined empirically. The equilibrium value of w is that which makes (4.3) a minimum: 2

w, = 1 -

(Tr)T- 0, and let H,, = H0 , Hx = Hz= 0 at the boundary x = 0. Then Hx = Hz= 0 everywhere, and 1IY depends only on x. The eq~ation for HY is:

(10.4)

The solution which vanishes as x---* oo is: (10.5)

Hy= H 0 e The currents are in the z-direction, with

(10.6)

Thus the currents are confined to a thin layer near the surface. The solution for a massive specimen in an external field is similar. The field is nearly parallel to the surface and drops off exponentially toward the interior. With different boundary conditions (10.4) may be used to obtain a solution for a slab with a field parallel to the surfaces. If the faces are at y = ±a, so that the thickness is 2a, the solution of (10.4) which gives a field H 0 at each surface is H

=H

Cos

( Y_/~).

(10.7)

° Cos(ajJ..)

The magnetic moment per unit volume is:

f (H -

I= -~ = 8 ~ a-

(10.8)

H 0) dy.

-a

With use of (10.7), this becomes I

+a

= -

Ho 4:n

(1 -

-~ Tan!!__). a

(10.9)

J..

Measurements of LocK 1 on thin films of tin and indium are in agreement with (10.9). However, the smallest values of aj). in his experiments were not much less than unity, so that a really critical test of the LoNDON penetration law was not obtained. 11. Solutions for sphere and circular cylinder. Simple solutions of LoNDON's equations which frequently are used are those for a circular cylinder with a field parallel to the axis and a sphere in a uniform external field. The equation for the field, H, which is parallel with the axis of the cylinder, is:

(11.1) where r is the radial distance from the axis. The solution which gives a field H 0 at the surface of a cylinder of radius a is: H 1

= H

Io(rjJ..)_

(11.2)

o Io(ajJ..)'

J. M. LocK: Proc. Roy. Soc. Land., Ser. A 208, 391 (1951). 19*

J. BARDEEN:

292

Theory of Superconductivity.

Sect. 11.

where 10 is the BESSEL function of imaginary argument. The magnetic moment per unit volume is: 1= - 1- 2 2n

a

f

a

0 ( (ajA)) . (H -H)rdr= - -H1 -2A - -I 1--·0 4n

a ! 0 (a/A)

(11 .))

0

To treat a sphere of radius a in a uniform external field, H 0 , we introduce spherical coordinates, r, {}, qy. Outside the sphere, the field is that of a dipole of magnetic moment M located at the center of the sphere plus the external field: H, = (H 0 + 2r~) cos{},

(11.4)

H{} = (- H 0 + 2r~) sin{}.

(11.5)

To get the field inside the sphere, we introduce the current density which is of the form: f'l' = f(r) sin{}, f, = f{}= 0. (11.6) From the equation A.2 curl curl j = j (11.7) we find the following differential equation for f (r) : 2/

d dr2+

2 df (2 1 ) ;-~- -;2+12

f=O.

(11.8)

The solution which is well-behaved at the origin is of the form cA . -r - -r Cos -") f(r) = - ( Sm 4nr2 A A A '

(11.9)

where A is a constant to be determined. From the first LONDON equation, 4:n; A.2 curlj= -cH, we obtain the following expressions for H, and H{}

l

2A2A [ Sm-. r r cos{} Hr = -Ar Cosrs A A

'

l.

A2 A [( r2 ) . r r r HI} = -:;a 1 + 12 SmT - ;;-CosT sm{},

(11.10) (11.11)

The values of M and A are determined by the requirement that H, and H{} be continuous at r=a: 3 2 3A a M=- -H 02-a ( 1---Cot-+ -3 A ) a

A

a2

'

A= _ _

3Hoa . 2Sin (ajA)

(11.12) (11.13)

When the radius of the sphere is very small, the magnetic moment is M=- Hoas. 30A2

(11.14)

SHOENBERG 1 has applied these results to analysis of measurements of the magnetic properties of mercury colloids with particle size of the order of 10-6 to 10- 5 em. Because of the range of particle sizes, it was not possible to test (11.12) 1 D. SHOENBERG: Nature, Lond. 143, 434 (1939). 49 (1940); reference [23], p . 143.

Proc. Roy. Soc. Lond., Ser. A 175,

293

Wire carrying a current.

Sect. 12.

in a quantitative way, although information about the penetration depth could be obtained from the temperature variation in the range where (11.14) is applicable. An interesting solution is that for a body of axial symmetry rotating about its axis, which was first obtained by BECKER 1 and coworkers on the basis of the acceleration theory. If the system starts from rest with no current flow, this solution is essentially that obtained from the LoNDON theory ([15], p. 78). We have noted that the LONDON theory picks out one unique solution from a variety of solutions allowed by the acceleration theory. The solution is such that nearly all of the electrons follow the motion of the positive ions, so that there is no current in the interior. Electrons within a penetration depth of the surface lag behind, to give a net current flowing near the surface. This current is such as to produce a uniform magnetic field in the interior of just such a magnitude as to give a LARMOR frequency equal to the frequency of rotation:

H

2mc e

(11.15)

=-~w.

This field, of the order of 10-4 gauss for an angular frequency of 103 sec. - 1 is probably sufficiently large to be detected in a careful experiment. In a coordinate system which rotates with the body the CoRIOLIS force just balances, to the first order in w, the force due to the magnetic field. This, of course, is the basis of the LARMOR theorem. The surface currents for the case of a sphere of radius R are given by ([15], p. 81):

j,=j,=O, j

'P

{3 r

Sm {3 R

Sin f3r) sin 1? R:;

-{3 r

~ ~-n~ e w R · .!..._ (cos f3r-

=

1-

3 ns ew e- fi(R-r)

-

{3

sin 1?,

l

(11.16)

where {3=1 /A. The magnetic moment, M, is: M

= ~ c R3w

/R Cot f3 R + pz3R2)

(1 -

R::J

-"!.j--R 3 w.

(11.17)

12. Wire carrying a current. Another simple solution is that for the current and field distribution in long straight wire of circular cross-section carrying a current parallel with the axis. We introduce cylindrical coordinates, z, r, 1?. The only nonvanishing component of current density is j.(r) and of magnetic field is H0 (r), both of which depend only on the radial distance r. Outside of the cylinder, H ___ ] _ (12.1) 0-

where ] is the total current. The equation for j. (r) is

az iz +

Tr2 The solution is of the form

.

2:n: cr '

1 djz

j. -

-.; a:r -"12- J

0

·

I 0 (rj).)

1. = 2:n:a i T,J.aj ).) '

(12.2)

{12-3)

where a is the radius of the wire, and 10 and 11 are BESSEL's functions of imaginary argument. Since I 0 ( x) approaches infinity as r" jfx, the current is confined to 1

R.

BECKER,

F.

SAUTER

and G.

HELLER:

Z. Physik 85, 772 (1933).

294

J. BARDEEN: Theory of Superconductivity.

Sect. 13.

a thin layer of the order of the penetration depth next to the surface. The field inside the wire can be obtained from jz by using the LONDON equation: H

= {I

_

4:nA.2 c

_ _I1(r/.3_~

(- ~j·) =-]

or

2nca ll(a(A.) .

(12.4)

It should be noted that (12.1) and (12.4) are equal at the boundary r =a. It is of interest to determine how the current flows from a normal conductor into a superconductor. As far as the normal conductor is concerned, the superconducting boundary at the interface is an equipotential surface. The current density is normal to the interface. Since current flow across the boundary

svpercOfldvclor

Fig. 2. Current flow from normal metal to superconductor in a rectangular bar (after LoNDON [15], p. 37).

must be continuous, this gives a boundary condition for flow in the superconductor. The mathematical problem of determining flow within the superconductor is then to find an appropriate solution of (10.2) in which the normal component of current is specified at the boundaries of normal regions and is equal to zero at a free surface. Such solutions have been given by LONDON ([15], p. 37) for flow from a normal to a superconducting region in a rectangular bar and by voN LAUE ([25], Chap. 8) for the corresponding flow in a bar of circular cross-section. Fig. 2, illustrates the flow pattern for LoNDON's case. 13. Flow in multiply connected bodies 1 • A multiply connected body is one in which there exist loops which cannot be continually deformed to a point without passing out of the body. The simplest example is a ring. While the first LONDON equation (I) gives a unique solution for a simply connected body in a static external field, it does not give a unique solution for multiply connected bodies. This allows for the possibility of persistent a b currents. The second Eq. (II) Fig. 3 a and b. Loops for integration in (a) simply, and (b) multiply connected bodies. implies that such currents are stable in time. From the diamagnetic approach, one might expect to derive an equation analogous to (I). The problem of showing that persistent currents are metastable and do not decay in time would then remain. This problem is discussed in Sect. 14. In the present section, we shall discuss the consequences of LoNDON equations (I) and (II). 1

This section is based to a large extent on reference [15], p. 73-78.

Sect. 14.

Introduction of vector potential.

295

First consider a simply connected body (Fig. 3 a) and apply STOKES' theorem to (I) for a loop entirely within the body. This gives

JJ H·dA = - c~Aj5 ·ds,

(13.1)

L

A

where dA is an element of area of the cap and ds a line element of the loop L. It follows that (13.2) H.. The electric field has only one component, Ez. Under static condition, curl E = 0, 58 that 1 E =--(JEZ curq; r · =O. (33.1) 0

Thus Ez must be independent of r . Current flow in the intermediate state is nearly parallel with the axis and normal to the boundaries of the superconducting regions. 1

Reference 2, p. 330.

Sect. 33.

Destruction of superconductivity by currents.

339

The problem is to determine how the thickness, w (r), of a discus-shaped region varies with the radial distance r. It must be such that the current density, j, is consistent with the MAXWELL equation curlH =

or

· (r)

Jz

4nj -c •

c_ ~ ~ (r H ) = !!£..£. = _4n 4n r . 'P r or

(33.2)

If a is the normal conductivity, the effective conductivity for axial flow is: w(O)

a tt(r) = ---··----- a.

03-3)

w(~-w(~

e

The current density is therefore j (r) = a ff E = _ w (0) a E_z_ = ! __1!_•_ z

and

e

w(O) -w(r)

z

w(r) _ w(o)-

1

(33.4)

4nr'

4na E,r

(33-5)

- -c~·

Thus w (r) goes to zero at a distance

R = __ :..!!__,___ _ 4na Ez

(33.6)

l

The current density between R and a is a E, so that the total current in the wire is

+ c~c_!!_ ·. = n a E z a2 + ---'-· 16naE:

]=naE,(a 2 -R2 )

c2 H2

(33-7)

The equation may be solved to express E. as a function of ]. The final result may be written in the form

t-; = ~- {1 + v~=T;n

.

(33.8)

where QjQ0 is the resistance of the wire relative to that of the normal state and (33.9)

is the critical current according to SILSBE E's hypothesis. Eq. (33.8) indicates that one-half of the resistance should be restored when the critical current is reached. As shown in Fig. 18, the critical current is close to (33-9). but the jump in resistance is rather more than one-half. There is some hysteresis; on decreasing, the current drops to about 0.85 critical before the resistance disappears. It is perhaps incorrect to assume that discus-shaped regions have negligible thickness. In a more exact theory, it would be necessary to take boundary effects into account!. A rather unusual intermediate state phenomenon occurs when a longitudinal magnetic field is present along with a large current flowing in the cylinder. It has first been observed by STEINER 2 and confirmed later by others that the average flux density in the superconducting wire may be much larger than that in the applied field. This "paramagnetic" effect is not as yet completely understood, although it is almost certain that it is observed in a rather complex intermediate state phenomenon which does not involve anything basically new. H. KoPPE, Ann. Phys. Lpz. 6, 375 (1949) and C. G. KUPER, Phil. Mag. 43, 1264 (1952). K. STEINER and H. ScHOENECK: Phys. Z. 38, 887 (1937). - K. STEINER: Z. Naturforsch. 4a, 271 (1949). 1

2

22*

340

fo,

J.

BARDEEN: Theory of Superconductivity.

Sect. 34.

The paramagnetic effect is observed when the current is above a critical value, which depends linearly on the applied magnetic field:

(33.10) where ]g andy are constants for a material at afixed temperature. The maximum flux through the cylinder occurs for a current slightly larger than ] 0 • At this point of maximum flux, the total field at the outside of the cylinder is equal to ~: (33.11) H; + H: = H,2 where H"' is the field from the current, ], H

'P

=

2_1!_ . ac

(33.12)

The resistance of a cylinder in the paramagnetic state is qualitatively similar to that of the intermediate state of a long cylinder carrying a current in the absence of an external field, as discussed earlier in this section. It has been definitely established that the added flux through the cylinder comes from circular currents flowing around the cylinder 1 . The effect is destroyed if the cylinder is slotted to prevent such currents from flowing. The combination of circular plus axial flow gives flow lines following helical paths around the axis. A theory of the paramagnetic effect somwhat along the lines of LoNDON's theory has been given by H. MEISSNER 2 • The combination of H"' and H, at the surface will give flux lines which spiral around the surface. MEISSNER suggests that the superconducting domains of the intermediate state will follow more or less and be elongated along the lines of flux. The conductivity would then be highly anisotropic, with lower conductivity in a direction parallel with the field. The current would then follow helical paths and give the paramagnetic flux. While the theory is in qualitative and even semi-quantitative agreement with experiment, it does not yield a critical current (]g). Further developments will probably require a discussion of boundary energies. 34. Kinetics of phase transitions and high frequency effects. As is the case for many phase transitions, the transition between the normal and superconducting phases occurs by nucleation and growth 3 . Because of the large boundary energies involved, a relatively large nucleus must be formed before it is stable and will grow. Various aspects of the problem of nucleation and growth have been studied at a number of laboratories, and some theoretical work has been devoted to the problem. There is an excellent review of the subject by FABER and PIPPARD ([7], Chap. IX, p. 159), in which extensive references to the literature may be found. Both supercooling and superheating are observed. Actually, it is more convenient to vary the magnetic field than the temperature, so that "supercooling" refers to a metal remaining in the normal state when the magnetic field is reduced to a value lower than H, and "superheating" to a metal remaining in the superconducting state as 'the field is increased above H, . Usually supercooling is more MEISSNER, SCHME!SSNER a nd MEISSNER: Z. Physik 130, 521 (1951); 130, 529 (1951); Phys. R ev. 90, 709 (1953). Other experiments on the effect are those of T. S. TEASDALE a nd H . E. RORSCHACH jr. : Phys. Rev. 90, 709 ( 19 53) and ] . C. THOMPSON and C. F . SQUIRE: Phys. Rev. 96, 287 (1 954) . 2 H. MEISSNER: Phys. Rev. 97, 1627 (1954); 101, 31 (195 6) . 3 T. E. FABER: Proc. Roy. Soc. Land., Ser. A 214, 392 (1952). 1

132, 529 (1952). -

Sect. 34.

Kinetics of phase transitions and high frequency effects.

341

marked than superheating. This is because there usually exist local regions where the field is abnormally high at which normal nuclei may start growing. This was demonstrated by GARFUNKEL and SERIN 1 in experiments on a rod in a longitudinal field. An additional coil was placed near the center of the rod so that the field could be increased locally from below to above H,. With this geometry, which avoids a large local field near the ends of the rod, considerable superheating was observed. The velocity of propagation of the normal-superconducting boundary has been studied most extensively by FABER 2 . The studies were made for the most part by placing a number of detecting coils along a rod, so the propagation of a phase boundary along the rod could be studied. The rod was supercooled in a field a little below ~ . A superconducting nucleus could be started by suddenly decreasing the field locally by means of an auxiliary coil, and this spreads out along the rod in the order of a few seconds. The velocity of propagation is determined mainly by eddy current damping. The theory, worked out independently by PrPPARD 3 and by LrFSHITZ 4 accounts in a satisfactory way for the data. The basis of the theory is to consider the energy balance between the free energy released, when a fresh volume of normal metal is released, and the energy absorbed by the eddy currents. The latter is proportional to the velocity of propagation and the former to H,2 - H 2 . FABER 5 suggests that a superconducting nucleus spreads out in the form of a thin sheath of thickness d adjacent to the surface of the rod. According to theory, the maximum velocity of propagation of the sheath along the rod is obtained when the thickness has an optimum value given by dopt=

3 (Ll- A) H, ····· - - H,-H 4

(34.1)

The velocity for this thickness is: V

=

(H,-H) 3

C

H~ (Ll -

J.)2 '

(34.2)

where C is a constant which can be roughly estimated from the eddy current damping theory. FABER has used this result to estimate relative values of Ll- A. from his experimental data. A large amount of experimental and theoretical work has been devoted to the study of superconductors under high frequency fields. Some has involved small amplitude fields; the surface impedance is measured at microwave frequencies. A review article of PrPPARD [21] gives a summary of this work together with references to the literature. Another aspect has been the study of the kinetics of the phase transition, for which large amplitude fields in all regions of the spectrum are of interest. The theory of the destruction of superconductivity by alternating fields of large amplitude has been discussed by LIFSHITS 6 . We shall give here only a very brief summary of the theoretical aspects of work on surface impedance, because the subject is treated at greater length elsewhere in this series. M.P. GARFUNKEL and B. SERIN: Phys. Rev. 85, 834 (1952). T . E: FABER: Proc. Roy. Soc. Land., Ser. A 214, 392 (1952); 219, 75 (1953); 223, 174 (1954). 3 A. B. PIPPARD: Phil. Mag. 41, 243 (1950). 4 I. M . LIFSHITz: Z. eksper. tear. Fiz. 20, 834 (1950). 5 T. E. FABER: Proc. Roy. Soc. Land., Ser. A 223, 174 (1954); reference [7], p. 176. 6 I. M. LIFSHITs: Dokl. Akad. Nauk SSSR. 90, 363 (1953). I. M. LIFSHITS and M. I. KAGANOv: Dokl. Akad. Nauk SSSR 90, 529 (1953). 1

2

342

J.

BARDEEN: Theory of Superconductivity.

Sect. 34.

The surface impedance, Z, is defined as the ratio of the complex quantity, E 0 (w), representing the alternating electric field of frequency w at the surface

to the integrated complex current density

J (x):

Z=R+iX=

Eo(w)

00

f

(34-3)

](x) dx

0

Interpretation of data on superconductors has generally made use of the twofluid model. The electric field which comes from the time variation of the magnetic field in the penetration region, acts on the normal component and gives a loss. The problem was first considered by H. LoNDON 1 ; later PIPPARD 2 pointed out that in most experiments, the mean free path is larger than the penetration depth, and gave a semi-quantitative theory to take this into account. The mathematical theory of the "anomalous skin effect" was developed more completely by REUTER and SoNDHEIMER 3 and by MAXWELL, MARCUS and SLATER 4 • While the two-fluid model accounts in a qualitative way for the resistive part of the impedance, R, and its variation with temperature difficulties arise when a quantitative fit of the observed data is attempted. PIPPARD 5 has worked out, in part by means of dimensional analysis empirical laws which fit the observed data in different temperature ranges. At relatively low temperatures, where R in the superconducting phase is less than 5% of that in the normal phase, the data can be fitted by

(34.4) where t is the reduced temperature. The frequency dependence is contained in the factor A (w), which is found empirically to vary as w~. The usual sort of two-fluid model, such as that of the original version of the LONDON theory, predicts a variation proportional to w 2 . It appears that some other relaxation effect may be coming in to alter the frequency dependence. After consideration of various possible mechanisms, FABER and PIPPARD 6 consider the most likely one may be a relaxation process in the superconducting state with a time constant of the order of the time it takes a phonon to travel across a coherence distance, ~0 . This time is of the order of w- 9 sec., so that the relaxation would appear in the right frequency range. Another possibility is that the w~ variation is a transition range between an w 2 variation at lower frequencies and a slower variation at higher frequencies . LANDAU has suggested that "normal" electrons may be bound at low temperatures in very large orbits, and that this would give a very high dielectric constant, of the order of 109 • If this were the case, there would be appreciable displacement currents within the penetration depth at microwave frequencies. A report of a theory of ABRIKOSOV, who has extended the REUTER-SONDHEIMER theory to include displacement currents, and applications to microwave data of HAJKIN on thin films is included in SHOENBERG's review [24]. PIPPARD [21] believes that the experiments can be interpreted in other ways, and there is as yet no convincing evidence in favor of a large dielectric constant. Since there is good 1 H. LoNDON : Proc. Roy. Soc. Land., Ser. A 176, 552 (1940) . 2 A. B. PIPPARD: Proc. Roy. Soc. Land. , Ser. A 191. 385 (1947). 3 G. E. H. REuTER and E. H. SoNDHEIMER: Proc. Roy. Soc. Land., Ser. A 195, 336 (1948). 4 E. MAXWELL, P . M. MARCUS and J. C. SLATER: Phys. Rev. 76, 1332 (1949). 5 A. B. PIPPARD: Proc. Roy. Soc. Land., Ser. A 203, 195 (1950) . 6 T. E. FABER and A. B. PIPPARD : Proc. Roy. Soc. Land., Ser. A 231, 53 (1955).

Sect. 3 ).

Microscopic theories.

343

evidence that the superconducting wave functions extend over large volumes, a large dielectric constant is a possibility and should be kept in mind in analysis of microwave data.

V. Electron-phonon interactions. a) Introduction. 35. Microscopic theories. The BLOcH theory, in which it is assumed that each electron moves independently in a periodic potential determined by the ions and an averaged charge density of the valence electrons, gives a good qualitative and in some cases quantitative explanation of the electrical properties of normal metals, but fails to account for superconductivity. Most attempts to give a microscopic theory of superconductivity have made use of interactions omitted from the BLOCH theory. These include correlations between the positions of the electrons brought about by CouLOMB interactions, magnetic interactions between electrons and interactions between electrons and phonons. While all of these interactions are undoubtedly important for a complete theory, the isotope effect shows that the main one responsible for the transition is the electronphonon interaction. Prior to the discovery of the MEISSNER effect, it was thought that superconductivity was simply infinite conductivity, and that it would be necessary to show why the electrons in the superconducting state are not scattered in such a way as to give resistance. Some of the more recent theories such, as those of HEISENBERG and of BoRN and CHENG, also have attempted to account for superconductivity in terms of the stability of currents. A major stumbling block to all such theories is a theorem of BLOCH that the lowest state is one of zero current (Sect. 1). BLocH's theorem does not apply to diamagnetic currents. There can be a net current density in the lowest state in the presence of a magnetic field. LONDON's approach, which we believe to be correct, is based on the idea that all supercurrents are diamagnetic in origin. In the case of a persistent current flow in a ring, the current itself gives a magnetic field which in turn produces the supercurrents. While LoNDON has given some qualitative arguments to show why such currents should be metastable, no real proof has been given, and probably cannot be without discussion of a specific model. We shall give a brief description of HEISENBERG's theory [9] because it may contain some elements of truth, although the basic assumption that CouLOMB interactions between electrons are responsible for superconductivity is not correct. HEISENBERG1 attempted to show that electrons with energies near the FERMI surface may at low temperatures condense into electron lattices of low density moving in different directions. These electrons can be described roughly by wave packets formed from states with wave vectors within Ll k of the FERMI surface, lkl =kF. The spread of the wave packet is of order Llx=1/Lik. The kinetic energy required to localize the electron is of order 1i2 kF Ll kjm, where m is an effective mass. The gain in CouLOMB energy obtained on formation of a lattice of such wave packets was estimated to be very roughly or order - e2 Ll k log -:

k •

This will more than compensate for the increase il.. kinetic energy if Ll k is sufficiently small. A more accurate estimate of the CoULOMB energy was made later 1

W.

HEISENBERG:

Z. Naturforsch . 2a, 185 (1947); 3a, 65 (1948), also [9] and [11].

344

J. BARDEEN:

Theory of Superconductivity.

Sect. 35.

by KoPPE 1 . A difficulty in these calculations is that the screening of the fields of the individual electrons by other electrons is not taken into account. If a screened CoULOMB field of short range were used, the gain in CouLOMB energy by formation of such an electron lattice would be negligible. Since electrons near the FERMI surface are travelling in all directions, a lattice must be formed from a group of electrons in the same region of k-space, all moving in the same direction. A moving electron lattice would give a net current, which HEISENBERG argued, would be thermodynamically stable. Ordinarily, the supercurrents in different domains would be in random directions and so give no macroscopic current. The MEISSNER effect was explained by the effect of a magnetic field on the distribution of supercurrents. General theoretical objections against a theory of this sort have been given by LoNDON ([15], p. 142). Some of the detailed predictions of the theory are not in accord with observation. Perhaps the most important is a maximum current density which approaches zero as T ---*0° K. This would imply that at low temperatures there should be a marked increase in penetration depth with field, which has not been found experimentally. On the other hand, we have seen (Sect. 5) that predictions of KoPPE's two-fluid model, based rather loosely on the theory, are in at least rough agreement with observation. Another theory based on CouLOMB interactions and persistent currents is that of BoRN and CHENG 2 • It was suggested that superconductivity occurs in metals with overlapping energy bands in which the lower band is nearly full. An attempt was made to show that, below a critical temperature, the lowest free energy occurrs with an asymmetric distribution of electrons in k-space, with more electrons in some comers of the BRILLOUIN zones than others. This appears to be a violation of BLOcH's theorem that the state of lowest energy has zero current. A mathematical formulation for treating many particle wave functions in the theory of metals has been suggested by TISZA 3 , with a view toward application to the problem of superconductivity. His functions are "super" BLOCH functions which represent the coordinated motion of a group of electrons with a net momentum. The theory was not developed very far, but presumably would be useful for a theory in which persistent currents play a dominant role. We believe that the objections raised by LoNDON to all such theories are valid. Another interaction which has been suggested as being responsible for superconductivity is the magnetic interaction between electrons. Such interactions can be taken into account in the HARTREE approximation by including the magnetic fields of the electron currents in a self-consistent manner. This is of course essential when the diamagnetism is large, and has been done in the discussion of Chap. III. Electron currents are determined by the magnetic field and these currents also contribute to the field. There is no evidence, however, that it is necessary to take specific magnetic interactions between individual electrons into account. WELKER 4 once attempted to base a theory of superconductivity on magnetic exchange interactions. Another possibility, which we now believe to be correct, is that motion of the ions is involved in the transition to the superconducting state. The author 5 1 H. KOPPE: Ann. Phys., Lpz. 1, 405 (1947). Z. Naturforsch. 3 a, 1 (1948) ; 4a, 74 (1949); 6a, 284 (1951); also [11]. 2 M. BoRN and K. C. CHENG: Nature, Lond. 161, 968, 1017 (1948).- J. Phys. Radium 9, 249 (1948). - K. C. CHENG: Nature, Lond .. 163, 247 (1949). 3 L. TiszA : Phys. Rev. 80, 717 (1950). Also see J. M. LuTTINGER: Phys. Rev. 80, 727 (1 9 50). 4 H.WELKER: Z. Physik 114, 525 (1939). 5 J. BARDEEN: Phys. Rev. 59, 928 (A) (1941).

Sect. 35.

Microscopic theories.

345

once suggested that there are small periodic displacements of the lattice in such a way as to produce a very large unit cell in real space and a fine-grained BRILLOUIN zone structure in k-space. The displacements were assumed to be such as to produce small energy gaps near the FERMI surface so that the energies of the occupied states are lowered. It is known that some alloys (for example, the y-phase alloys) take up a complicated structure which gives planes of discontinuity near the FERMI surface. It was supposed that the same sort of thing could occur in many metals at low temperatures, not matter how complicated the FERMI surface, if the zone structure is very fine-grained. First rough estimates indicated that the energy decrease of electrons near the FERMI surface might be sufficient to compensate for the energy required to displace the ions, but more careful estimates made later showed it too small by an order of magnitude or more. Most favorable metals are those with a large interaction between electrons and lattice and thus a large resistivity in the normal state. The diamagnetic properties were accounted for by the very small effective mass of electrons and holes with energies near the FERMI surface (see Sect. 24}. Since the best estimates seemed to indicate that transitions of this sort are not to be expected, the details of the theory were never published. Some features were retained in a later theoryl based on a dynamic interaction between electrons and lattice vibrations, which was suggested by the isotope effect. Without having prior knowledge of the isotope effect, FROHLICH [4] proposed a theory of superconductivity based on electron-phonon interactions. While such interactions had long been used to account for thermal scattering of electrons and thus the resistivity of normal metals, it had not been recognized that they would also give a contribution to the energy. FROHLICH calculated the interaction energy by use of second-order perturbation theory. H e showed that if the interaction is sufficiently large, the energy at the absolute zero would be lowered if a thin shell of electrons adjacent to the FERMI surface of the normal metal were displaced outward a small distance in k-space. He presumed that this shell distribution represents the superconducting state. There is considerable doubt about the details of the theory, because the criterion for superconductivity, the condition that the shell distribution have a lower energy then the normal one, is essentially the same as the condition that the interactions be so large that perturbation theory cannot be applied. It is believed that the basis of the theory is correct, and that the criterion gives a good indication for the occurrence of superconductivity, but that better mathematical methods are required to give a reliable picture of the nature of the supercopducting state 2 • We shall discuss FROHLICH's theory in more detail in Sect. 42. Since an adequate mathematical theory of superconductivity based on electron-phonon interactions has not been given, we shall devote most attention in this Chapter to the formulation of the problem. Both FROHLICH and the author start ed from the BLOCH theory in which it is assumed that each electron moves independently in a periodic potential field. Vibrational coordinates and interactions between electrons and vibrations were introduced exactly as is done in the theory of conductivity. The strength of the interactions was estimated empirically from the high temperature resistivity. 1 J. BARDEEN: Phys. Rev. 79, 167 (1950); 80, 567 (1950); 81, 829, 1070 (1951); 82, 978 (1951); also [J]. 2 FROHLICH himself has emphasized the need for new mathematical methods; H. FROHLICH: Physica, Haag 19, 7 55 (1953); reference [30], p. 909. There are illuminating discussions by BoHR. HEISENBERG and others following the first of these, and there are also interesting discussions following the second.

346

J. BARDEEN:

Theory of Superconductivity.

Sect. 36.

There are two objections to this formulation: First, the CouLOMB interactions between electrons should be introduced at the start; second, displacements of the electrons brought about by electron-phonon interactions have an important effect on the vibrational frequency and also on the effective matrix element for the interaction. An important part of the problem is to show how these should be determined from first principles. Starting from a formulation which includes CouLOMB interactions between electrons, we shall show that the usual BLOCH theory should be a reasonably good starting point to develop a theory of superconductivity. We also show why electron-phonon interactions have a larger influence on the wave functions than CoULOMB interactions, even though the interaction energies are much smaller. Our treatment, given in Sects. 37 to 41, follows closely one of PINES and the author [2]. 36. Importance of screening in metals. An essential point to be remembered is that CouLOMB interactions in a metal are screened interactions. This applies to the interactions between electrons and ions as well as to the interactions between electrons. This was recognized in early calculations of the electronphonon interaction by HousTON 1 and by NoRDHEIM 2 • The potential energy of an electron at a distance r from an ion was taken to be: Z e2

v(r) = - -r- e-"'',

where rJ. is a screening constant, estimated from a FERMI-THOMAS model. To calculate the change in potential resulting from a lattice vibration, it was assumed that the ions move rigidly under the displacements. In a later calculation, the author 3 determined the screening by a HARTREE self-consistent field method and applied the results to a calculation of the resistivity of monovalent metals More recently, NAKAJIMA 4 has derived nearly equivalent results by use of field theoretic methods. It is also important to take the screening into account in a calculation of the vibrational frequency. Entirely erroneous results would be obtained if the response of the electrons to the motion of the ions were not included. The HARTREE selfconsistent field method has been extended by ToYA5 to derive an expression for the vibrational frequency. Equivalent results follow from NAKAJIMA's derivation. Prior to NAKAJIMA's work, FROHLICH 6 and KITANO and NAKAN0 7 independently used similar field theoretic methods to determine the effect of electron motion on the vibrational frequency by starting from the BLOCH HAMILTONian in which CouLOMB interactions between electrons are not explicitly introduced. From a field-theoretic point of view, there is an interaction between electrons brought about by virtual emission and absorption of phonons, and it is this interaction which is believed to be responsible for superconductivity [4]. There is also a phonon self-energy, which can be quite large when the interaction is strong enough to give superconductivity. This means physically that one must take the electron motion into account in a derivation of the phonon frequency ([1], p. 264). W. V. HouSTON: Phys. Rev. 34, 279 (1929) ; 88, 1321 (1952) . L. W. NORDHEIM : Ann. Phys., Lpz. 9, 607 (1931). a J. BARDEEN: Phys. Rev. 52, 688 (1937). 4 S. NAKAJIMA: Buss. Kenkyu 65, p. 11 6 (1953); reference [30], p. 916. 5 T. TovA: Buss. Kenkyu 59, 179 (1952) . 6 H. FROHLICH: Proc. Roy. Soc. Lond., Ser. A 215, 291 (1952). 7 Y. KITANO and H. NAKANO: Progr. Theor. Phys. 9, 370 (1953).

1

2

Sect. 37.

Derivation of the HAMILTONian.

347

The reason that the BLOCH HAMILTONian is reasonably satisfactory for most problems in the theory of metals, including superconductivity, is that the CouLOMB interactions are screened out within a distance of the order of the interparticle spacing. To give an example, ABRAHAMS 1 has estimated the collision cross-section and mean free path for screened electrons in the alkali metals. He finds that the scatterings possible are so greatly restricted by the exclusion principle that the mean free path for electron-electron collisions is greater than that for electron-phonon interactions at practically all temperatures. BoHM and PINES 2 have shown that the long range part of the CouLOMB interaction leads to a coherent motion of the electrons which can be described in terms of plasma oscillations. These are of such high frequency that they are not normally excited. There remains a short range CouLOMB interaction between the individual electrons. PINES and the author [2] have extencled this theory to take the motion of the ions into account. In the combined collective motion of electrons and ions, there are high frequency modes corresponding to the plasma oscillations of an electron gas and low frequency modes corresponding to longitudinal lattice vibrations. Expressions for the electron-phonon interaction and vibrational frequency derived in this treatment are nearly equivalent to those found by the HARTREE self-consistent field method. The collective treatment is not applicable to phonons of short wave length; for these, the NAKAJIMA formulation is probably the most satisfactory. As pointed out be FROHLICH, a canonical transformation can be used to eliminate the electron-phonon interaction, from the HAMILTONian, and one is left with an interaction between electrons which corresponds to one he had derived earlier by perturbation theory methods. When the electron-phonon interaction is large, this procedure breaks down for a small number of terms with small energy denominators. These terms are not important for calculating the matrix element for the interaction and vibrational frequencies, but they are just the terms important for superconductivity. Since they cannot be treated by perturbation theory methods, they can have a large effect on the wave functions. The general plan of this Chapter is as follows. In Sect. 37, the HAMILTONian is derived in a form suitable for a field-theoretic treatment. The canonical transformation which eliminates the linear terms of the electron-phonon interaction and the NAKAJIMA method for deriving the shielded interaction and phonon frequency is given in Sect. 38, and is followed by the collective treatment in which plasma coordinates are introduced in Sect. 39. Convergence of the expansion and the criterion for superconductivity are discussed in Sect. 40. The remaining sections are concerned with attempts which have been made to calculate the electron-phonon interaction energy when the interaction is strong enough to give superconductivity. b) Formulation of the electron-phonon interaction problem. 37. Derivation of the HAMILTONian. In order to formulate properly the problem

of calculating the interactions between electrons and phonons in a metal, we derive in this section an expression for the HAMILTONian in a form sufficiently general for our purpose. CouLOMB interactions between electrons and motions of the ions are included from the start, but several approximations are made in order to simplify the equations. These amount essentially to neglect of anisotropic effects not believed to be important for the superconductivity problem. It is 1 E. ABRAHAMS : Phys. Rev. 95, 839 (1954). 2 D. BoHM and D. PINES: Phys. Rev. 82, 625 (1951); 85, 338 (1952); 92, 609 (1953). D. PINES: Phys. Rev. 92, 626 (1953).

J.

348

BARDEEN:

Theory of Superconductivity.

Sect. 3 7-

assumed that lattice waves are either longitudinal or transverse, and that electrons interact only with the longitudinal component. This is a valid approximation for waves of long wavelength, but is not correct for short waves except for certain directions of travel. We also assume, as is often done in the BLOCH theory, that the matrix elements for the electron-phonon and CouLOMB interactions depend only on the wave vector difference between initial and final states. In the calculation of CouLOMB interactions, approximations are made which amount to treating the valence electrons as a free electron gas. We assume a monatomic crystal of N atoms of valence Z, so that there are n =ZN valence electrons. Positions of the valence electrons are denoted by ri(i=1, 2, ... n) of the ions by Ri(j=1, 2, ... N). The total HAMILTONian of the crystal is the sum of four terms, the kinetic energy of the !ilectrons, the electron-ion interaction energy, the CouLOMB interaction between electrons and the HAMILTONian for the ions, including kinetic energy, CouLOMB and exchange interactions: (3 7.1)

As written, the CouLOMB interaction energies of the separate terms are very large, but these large terms tend to cancel in the sum. Included is a large negative contribution from the second term, which represents the interaction between each electron and the sum of the CouLOMB fields of all of the ions, and large positive contributions from the CouLOMB interactions between the electrons and between the ions. In order to avoid dealing with these large energies, we suppose that there is subtracted from the electron-ion interaction, the interaction of each electron with a uniform positive sea, from the electron-electron interaction, the self-energy of a uniform negative sea, and from the ion-ion interaction, the selfenergy of a uniform positive sea. Since the sum of these three terms adds to zero, the total energy is unchanged. The ion-ion interaction energy less the energy of a uniform positive sea is equivalent to the energy of the ions in a uniform negative sea, including the self-energy of the negative charge. We want to modify this HAMILTONian by introducing phonon coordinates to represmt the ion motion and by introducing occupation numbers of a set of BLOCH functions to represent the electron wave function. The transformed HAMILTONian will then contain creation and destruction operators for the electrons. The BLOCH functions, V'k (r), are a set of one-particle functions for the electrons which apply to a crystal with the ions fixed in equilibrium positions. They may be defined by a HARTREE approximation or by a HARTREE-FOCK approximation in which effects of electron exchange are included. We shall use an even simpler approximation here and assume that the density of valence electrons is uniform, so that the effective potential, V (r), in which the electrons move is that of the ions in equilibrium positions compensated by a uniform negative charge. If v (r- R7) is the potential of the ion at the equilibrium position R7, V(r)

= L.: v(r- R7)

+compensating charge.

(37.2)

1

The BLOCH equation for the one-particle functions is (37-3) The electrons are described in an extended zone scheme, so that the wave vector k is not necessarily in the first BRILLOUIN zone. The designation of the wave vector

Sect. 37.

Derivation of the HAMILTONian.

349

for a particular state presumably would be chosen so that the approximations concerning matrix elements mentioned in the preceding paragraph would be most nearly valid. This implies that an electron in state k is treated much like a free electron with the same wave vector. As usual, periodic boundary conditions are introduced to get a discrete set of k-values. We shall omit spin-orbit interactions, and where necessary indicate the spin by an index s which can take on the values ± ·~. We describe the electron wave function in second quantization by giving the occupation numbers for this set of BLOCH functions. Creation and destruction operators, ci.s, Cf x, for which collective coordinates are not introduced. The last term in this sum represents a shielded CouLOMB interaction between the individual electrons. On comparing the electron-phonon interaction with that introduced in the self-consistent field method, we see that Ek =

so that

(40.6) (40.7)

The u,. are to be determined in such a way that the new plasma and phonon variables will be uncoupled and will represent independent oscillations. It is required that there be no coupling via the subsidiary conditions as well as in the HAMILTONian. The value of u,. and the phonon frequency w,. are determined by carrying out a canonical transformation which eliminates to a given order the electron-phonon interaction terms in (40.5). It is required that to the same order there be no coupling between phonons and electrons in the transformed subsidiary conditions, and this will be the case if the phonon variables no longer appear in the conditions in this order. The canonical transformation is just that defined by (39.4) with f (k, x) and g(k, x) defined by (39-9) and (39.10). For lx I -x, by the method of Sect. 39. The transformed HAMILTONian is:

H= LEkCkCk+ L { ~ P> xc because of the difference between v~ and v,., but for these x the shielding is not very important. In (40.8) there is also a weak interaction between electrons and plasma, which may be eliminated in a manner similar to that used for electron-phonon interaction. This also leads to an interaction between electrons, actually considerably larger in magnitude than E 2 , and one which can not be treated by perturbation theory methods, implying that the wave function derived from the collective model are not accurate. Because of the large energy of the plasma quanta, this interaction probably plays no role in superconductivity. There remains in the final HAMILTONian a shielded CouLOMB interaction between electrons, represented by 2: t M; (!,. (} - ,.. This also may be treated by perturbation theory. > "' We shall discuss the convergence of the canonical transformation which ~liminates first order terms in the electron-phonon interaction and leads to the ir1teraction H 2 in the following section. We shall not discuss here the complication introduced by the subsidiary condition on the wave function, which condition transforms to (40.17) The problems introduced by this condition are similar to those in the absence of ion motion, and they have been discussed by BoHM and PINES 2 • The subsidiary condition is usually just ignored in getting an approximate wave function for the HAMILTONian, but it is doubtful whether this is really justified. 41. Convergence of the canonical transformation. The canonical transformation, S (39.4), may be thought of as introducing a new set of BLOCH functions which depend on the vibrational coordinates, and a new set of vibrational coordinates which depend on the electron coordinates. The expansion (39.2) of 1 This analysis answers objections ra ised by G. WENTZEL: Phys. Rev. 83, 168 (1951) and W. KoHN and VACHASPATI: Phys. Rev. 83, 462 (1951). For earlier discussions of the problem, seeK. HuANG: Proc. Phys. Soc. Land. A 64, 867 {1951) and also [1]. 2 D. BoHM and D. PINES: Phys. Rev. 92, 609 {1953). D. PINES: Phys. Rev. 92, 626 {1953) . See also questions raised by N . I. ADAMS jr.: Phys. Rev. 98, 1130 (1955) and subsequent comment by PINES (submitted to Physical Review).

Sect. 42.

FROHLICH's

359

shell distribution

the new HAMILTONian in a power series inS will converge rapidly if a small number of terms are omitted from S. These are the terms for which the energy denominators are small. We shall show (1.2) that the omitted terms do not contribute appreciably to the matrix element and vibrational frequency, v,. and w,. . It is, however, just these terms which are important for superconductivity. In his analysis of the problem, FROHLICH 1 suggested omitting these terms from the canonical transformation and treating them separately, and we shall follow this procedure here. The expansion coefficients for a given electron state, k, are '!i - 1 f (k, x) q,.. For the present purpose, we may omit the small term 1i2 w! in the denominator of f(k, x) [Eq. (39.10) ], and take {41.1) The expansion will converge rapidly if {41 .2) We shall omit from the expansion those x for which IEk - > 1iw..f4.

(44.11)

Except for a numerical factor, this is of the same form as FROHLICH's criterion for superconductivity (42.3). If w .. < D,., it is the vi bra tiona! mode which is pushed down by the interaction and becomes unstable. On the other hand, if D,. < w,., it is the electrons which become unstable. This latter condition corresponds to L1 E < 1i w, i.e., that the difference in energies of the electron states be less than the phonon energy. As we have stated in Eq. (42.6), it is virtual transitions of this sort which are believed to give superconductivity. Actually, of course, the electrons do not become unstable; the apparent instability simply means that the description of the electrons in terms of an equivalent set of oscillators is inadequate.

Sect. 44.

Other methods for calculation of electron-phonon interaction energies.

365

The excitation of the electrons can not be considered small, and ToMONAGA's theory is not applicable. A marked change in electron wave functions is indicated, but the theory cannot tell what will actually occur. Another method, discussed in [J], is to eliminate the interaction terms by means of a BLOCH-NORDSIECK transformation. The basis of the approximation is neglect of recoil of the electron during emission or absorption of a phonon. This can be expected to be valid only for interaction with phonons of long wavelength, so that the velocity of the electron is not changed very much. The interaction need not be small. The kinetic energy operator, p2f2 m is replaced by a term linear in the mon.entum, V · p, in which V represents a constant average velocity of the electron. Tht net result of the transformation is to replace the electron-phonon interaction terms by an interaction between electrons. The most important term is: U(r1 , r 2 ,

••• ,

_

rn)--

"lvxl 2 cos[x·(r;-ri)] L...- -w!- (V·~'

(44.12)

k,,,,

where v" and w" are the interaction potential and vibrational frequency as defined in Sect. 38. The diagonal terms of (44.12) give just the usual second-order perturbation theory expression for the interaction energy. This result gives some justification for use of the perturbation theory expression for large interactions, provided only that the phonon wavelengths are long. However, it is undoubtedly a poor approximation to keep only diagonal terms when the interaction is large. One should get a better solution of the ScHRODINGER equation in which the interaction between electrons is given by (44.12). The nature of the interaction (44.12) has been discussed by SINGWI 1 . Electrons near the FERMI surface move much more rapidly than the velocity of sound, 5. One may consider the emission of phonons as a CERENKOV radiation, or as a "bow wave" of a projectile in air moving faster than the velocity of sound. The disturbance is confined to a wake, the angle of which is approximately SfV ,...._,10- 3 radians. Carrying out the summation in (44.12) by taking principal parts, SINGWI finds that the interaction energy of two electrons is indeed zero except when one is in the wake of the order. The interaction is positive (repulsive) and is a maximum at the boundary of the wake, where it becomes singular. BoHM and STAVER 2 had suggested earlier that the wake nature of the interaction might be important. They proposed that chains of electrons might be formed in the superconducting state, with one electron following in the wake of another. SrNGWI also discussed this possibility. One difficulty with this picture arises from the uncertainty principle. As we have discussed earlier, there is good evidence that the wave functions of electrons in the superconducting state spread out over large regions, and it is difficult to picture them as representing localized and relatively non-interacting "chains". The nature of the emission of sound waves from an electron in a metal has been discussed by KLEIN 3 from a somewhat different point of view. He suggests that what might be important for superconductivity is the relative value of the energy uncertainty, L1 e, as calculated from the relaxation time, and the excitation energy, e, of the electron from the FERMI surface. The proposed criterion for superconductivity is L1 efe ,...._, 1 for e small, of the order of k T,, and L1 ej e should then decrease toward zero as e increases. However, the usual conductivity 1 K. S. SINGWI: Phys. Rev. 87, 1044 (1952). - K. S. SINGWI and B. M. UDGAONKAR: Phys. Rev. 94, 38 (1954). Some errors in the first of these are corrected in the second. 2 D . BoHM and T. STAVER: Phys. Rev. 84, 836 (1951) . 3 0. KLEIN: Ark. Fysik 5, 459 (1952).

366

J.

BARDEEN:

Theory of Superconductivity.

Sect. 45.

theory gives an increase in L1 s/s as s increases. KLEIN suggests that perhaps the compensation of the ionic motion is not as complete as predicted by the theory, as a result of a small part of the electron density remaining stationary as the ions move. This would give a small contribution to Vx varying as 1/x rather than as x, and would thus give a large scattering probability for x very small. This is given as a purely ad hoc assumption, with no physical basis. 45. FR6HLICH's one-dimensional model. FROHLICH 1 has investigated a onedimensional FERMI-DIRAC gas of free electrons interacting with lattice displacements. He finds that the vibration corresponding to the wave vector which connects states at the top of the FERMI distribution may become unstable so as to create a finite sinusoidal displacement of the lattice and an energy gap in the electron states at the FERMI surface. There are JUSt enough electrons to fill the states below the gap. These features are reminiscent of the author's early theory based on a three-dimensional model with energy gaps at the FERMI surface, as discussed briefly in Sect. 35. There are, however, some differences. In the one-dimensional case, an energy gap can be formed even though the interaction is fairly weak. Another difference is that FRoHLICH considers the slow motion of the lattice wave together with the electrons through the crystal 2 • This corresponds to a displacement of the entire system of electrons plus lattice wave in k-space. There is a current flow associated with a displacement of this sort . This differs from a displacement of the electrons in k-space, with the lattice distortion kept fixed. Since the band is completely filled, this latter would not lead to a new state and there would be no current flow. In contrast, a moving sinusoidal lattice wave carries the electrons along with a resultant current flow . The calculation of the energy of the system was made by an adiabatic selfconsistent field method. FROHLICH finds the following for the energy gap, W W

=

3

8EFe-2ZF- '

(45.1)

where EF is the FERMI energy, Z the number of free electrons per atom and F the interaction parameter defined in an analogous way to that for the threedimensional case (Sect. 42). In order that the approximations of the theory be valid, it is required that (45.2) E F > W > 1i Wmax , where 1i Wmax is the maximum phonon energy. If V is the velocity with which the system moves, the energy is 1

1 W2

1

)

E=n ( 3EF--8 EF-+ 2(m+mr)V2'

(45.))

where m1 is an effective mass defined by (45 .4)

Here S is the velocity of sound. Condition (45.2) implies that m1 will generally be larger than m. Since the whole configuration moves with only a single degree of freedom, FROHLICH suggests that scattering to decrease the current would be unlikely, so that the current should be stable. H . FROHLICH: Proc. Roy. Soc. Lond., Ser. A 223, 296 (1954). The general theory of coupled electron-lattice motion of this sort has been discussed by L. BRILLOUIN: Proc. Nat. Acad. Sci U.S.A. 41, 401 (1955). 1

2

Sect. 46.

Concluding remarks.

KUPER 1 has calculated the thermal properties of the one-dimensional model. He finds that the energy gap decreases with increasing temperature, and goes to zero at a critical temperature, ~. However, the approximations made in the theory are not valied unless ~ is larger than the DEBYE temperature, eD, for the model. In actual superconductors, of course, ~ is much smaller than He looked into the question of stability of currents corresponding to the disdlacements discussed above, but was unable to come to a definite conclusion. The model used is an unrealistic one, which makes it difficult to tell how much these results mean when applied to actual superconductors. The instability of the lattice for relatively weak interactions is true only for the one-dimensional case. As mentioned in Sect. 35, it appears that in three dimensions, the energy gained by the electrons when a large number of energy gaps is formed at the FERMI surface is considerably less than the energy required to deform the lattice. However, it is very likely true that there also exists for the general case a current flow in which the whole configuration of the superconducting state is displaced in k-space. It should be remembered, however, that the wave packets for the superconducting state extend over large volumes of real space, so that any such currents will also extend over large volumes. It is this feature which makes it difficult for the electrons to res1=ond to a magnetic field as do classical electrons, and which leads to a large diamagnetism. While currents of this sort probably do not play much of a role in the MEISSNER effect, they may be important for persistent currents in wires of small cross-section, where the current density does not vary much over the area.

en.

46 Concluding remarks. While there is some qualitative understanding of the nature of the superconducting state, we still do not have a good mathematical theory, or even a good physical picture of the difference between the normal and superconducting states. A superconductor is an ordered phase in which quantum effects extend over large distances in space, distances of the order of 10-4 em. in pure metals. It is this large extent of the wave packets which undoubtedly accounts for the remarkable magnetic properties. As is the case for other second-order phase transitions, a superconductor is probably characterized by some sort of an order parameter which goes to zero at the transition point. However, experimental evidence for an order parameter is inconclusive, and we do not have any understanding at all of what the order parameter represents in physical terms. The isotope effect shows that superconductivity arises from interactions between electrons and lattice vibrations, and theory indicates that, when the electron-lattice interaction is large, one can expect a marked change in the electron wave functions. Better mathematical methods for treating large interactions are required. The ToMONAGA intermediate coupling theory has been applied with success to the polaron problem 2 of an electron moving in an ionic crystal, and there is hope that some such method might be applied to electrons in a metal. One of the major difficulties is to isolate the interactions responsible for superconductivity. The energy difference between the normal and superconducting phases is only a very small part of the total electron-phonon interaction energy. Theory indicates that the interactions responsible are those for which the difference between the energies of the electron states, L1E, is less than the phonon energy, iiw. However, even if we consider only these, the energy involved in C. G. KuPER: Proc. Roy. Soc. Lond., Ser. A 227, 214 (1 955) . T. D. LEE, F . E . Low and D. PINES: P h ys. Rev. 90, 297 (195 3) . - F. E. Low and D. PINES: Phys. Rev. 91, 193 (1953). - T. D. LEE and D . PINES: Phys. Rev. 92, 883 (1953).H . FROHLICH : Adv. in Phys. 3, 325 (1954). 1

2

368

J.

BARDEEN: Theory of Superconductivity.

Sect. 46.

the phase transition is still only a small fraction of the total. It is quite possible that the significant values of L1 E are of the order of k I;, but, if so, we do not know why k I;, is so much smaller than 'liw. A possible explanation is that relatively long wavelength phonons are involved. This would be expected if the interaction energy is calculated by a self-consistent field method, such as that of NAKAJIMA, Iather than by the collective description. It will be recalled that v~, which becomes large for small x, appears in H2 (40.11) for those terms calculated by the NAKAJIMA method, while the screened interaction, v", appears for those terms treated by the collective description. Although the plasma treatment we have discussed may not be completely valid, one would expect on physical grounds that only screened interactions actually occur. Evidence that short wavelength phonons are important is that the transition temperature of thin films does not vary much with film thickness, even when the thickness is only a few atom layers [24]. It is probable that the vibrational modes are not affected very much by the phase transition. The energy difference is only a small fraction of the total zero-point energy of the modes. While it is possible that a small number of the modes might be affected to a large extent, this is not likely because one would expect that a large fraction of the modes participate in the transition. If this conclusion is correct, one should be able to treat the vibrational coordinates by perturbation theory methods, if not the electrons 1 . In this case, one could, by an appropriate canonical transformation, replace the electron-phonon interaction by an interaction between electrons. Thus one would take an interaction such as that given by ~ (40.11) seriously, and attempt to get a good description of the electron wave functions for a HAMILTONian with this interaction term. It would not be a satisfactory approximation to keep only the diagonal terms as is done in perturbation theory. In this way one would replace the electronphonon interaction problem by the still difficult problem of treating a FERMIDIRAC gas with interactions so large that they cannot be treated by perturbation theory methods. We would expect to find as a consequence of an adequate theory a justification for the energy-gap model. An essential difference between the normal and superconducting state appears to be that in the latter a finite energy, e, is required to excite an electron. The magnetic properties can be determined by perturbation theory methods, as discussed in Chap. III. A non-local theory, perhaps similar to that suggested by PIPPARD, would probably result; the LoNDON theory would represent only a limiting case not actually attained. Relaxation processes at high frequencies would depend on the details of the model. A framework for an adequate theory of superconductivity exists, but the problem is an extremely difficult one. Some radically new ideas are required, particularly to get a really good physical picture of the superconducting state and the nature of the order parameter, if one exists. The author is indebted to A. B. PIPPARD and J. M. BLATT for stimulating -:vrrespondence

about some of the controversial questions treated here and to with the manuscript.

J.

R. ScHRIEFFER for aid

General references. [J] BARDEEN, J.: Rev. Mod. Phys. 23, 261 (1951). A review of attempts which have been made to calculate electron-phonon interaction energies for application to superconductivity. [2] BARDEEN, J., and D. PINES: Phys. Rev. 99, 1140 (1955). Formulation of the electronphonon interaction problem, including effects of CouLOMB interactions. 1

See the discussion by

J.

BARDEEN, reference [30], p. 913.

General references. [3] BuRTON, E . F., and others; The Phenomenon of Superconductivity. Toronto 1934.

[4] FROHLICH, H.: Phys. Rev. 79, 845 (1950). The basic paper of FROHLICH's theory. [5] GINSBURG, W. L.: Fortschr. Phys. 1, 101 (1953). A review of theories of supercon-

ductivity, with consideration of the spectrum of elementary excitations. Good bibliography, particularly of work done in USSR. eksper. teor. Fiz. 20, 1064 (1950). An extension [6] GINSBURG, W. L., and L. D. LANDAU: of the LoNDON phenomenological theory to take into account a space variation of the order parameter. - GINSBURG, W.L.: Nuovo Cim., Ser. II 2, 1234 (1955). [7] GoRTER, C. J. (editor): Progress in Low Temperature Physics, vol. I. New York 1955. [8] GoRTER, C. J., and H. B. G. CASIMIR: Phys. Z. 35, 963 (1934). - Z. techn. Phys. 15, 539 (1934). - Physica, Haag 1, 306 (1934). Thermodynamic relations and the twofluid model. [9] HEISENBERG, W.: Two Lectures, Cambridge, 1948. One of the lectures is a review of the HEISENBERG-KOPPE theory. [10] KoPPE, H . : Fortschr. Phys. 1, 420 (1954). A review of the phenomenological theory. [11] KoPPE, H.: Ergebn. exakt. Naturw. 23, 283 (19 SO). A review, based in large part on the HEISENBERG-KOPPE theory. [12] KLEIN, 0.: Ark. Mat., Astronom. Fys. Ser. A 31, No. 12 (1944). - KLEIN 0., and J. LINDHARD: Rev. Mod. Phys. 17, 305 (1945). Calculation of diamagnetic properties of an electron gas with applications to superconductivity. [13] LANDAU, L. D.: Phys. Z. Sowjet 11, 129 (1937) . Unbranched model of intermediate state. [14] LoNDON, F.: Une conception nouvelle de la supraconductibilite. Paris 1937. Review of phenomenological theory. [15] LoNDON, F.: Superfluids, vol. I. New York 1950. Macroscopic theory of superconductivity. A basic source for the present article. [16] LONDON, H ., and F. LoNDON: Proc. Roy. Soc. Lond., Ser. A 149, 71 (1935). - Physica, Haag 2, 341 (1935). Basic papers of the LoNDON phenomenological theory. [17] MEISSNER, W.: Ergebn. exakt. Naturw. 11, 219 (1932). [18] MEISSNER, W.: Handbuch der Experimentalphysik, vol. 11 (pt. 2), 204 (1935). [19] MENDELSSOHN, K.: Rep. Progr. Phys. 12, 270 (1949). [20] PIPPARD, A. B . : Proc. Roy. Soc. Lond., Ser. A 216, 547 (1953). Basis for PrPPARD's non-local phenomenological theory. [21] PIPPARD, A. B.: Adv. Electronics a. Electron Physics 6, 1- 45 (1954). [22] SERIN, B.: Superconductivity, Experimental Part, in this volume. [23] SHOENBERG, D.: Superconductivity, 2nd Ed. Cambridge 1952. An excellent introduction to the subject, with a very complete bibliography. [24] SHOENBERG, D.: Nuovo Cim. 10, 459 (1953). A review of work on superconductivity in the USSR. [25] LAUE, M. voN : Theorie der Supraleitung, 2. Auf!. Berlin-Gottingen-Heidelberg 1949. English translation by L. MEYER and W. BAND, New York, 1952. A very complete account of the phenomenological theory.

z.

Conference Proceedings (since 1949). [26] 1949. International Conference on Low Temperatures, M.I.T., Cambridge, Mass. [27] 1951. Low Temperature Symposium, National Bureau of Standards, Circular 519,

Washington, 1952.

[28] 1951. Oxford Conference on Low Temperatures.

[29] 1953. Short [30] 1953. [31] 1953. [32] 1955. [33] 1955.

LoRENTZ-KAMERLINGH ONNES Conference, Physica, Haag 19, No.9 (Sept. 1953). papers followed by discussions. International Conference on Theoretical Physics, Kyoto and Tokyo, 1954. International Conference on Low Temperature Physics, Houston, Texas. Ninth Congress of the International Institute of Refrigeration, Paris. Conference on Low Temperature Physics and Chemistry, Baton Rouge, La. USA.

Handbuch der Physik, Bd. XV.

24

Liquid Helium. By

K.

MENDELSSOHN.

With 101 Figures.

Introduction. The phenomenon of superfluidity, like that of superconductivity occupies a unique position in the pattern of our known physical world. While at first there was a tendency to regard the unusual effects which were discovered as a limiting aspect of the properties of aggregate matter at very low temperatures, it is now quite clear that they have a more profound significance. The fact that these highly ordered states should make their appearance at temperatures which are two or three orders of magnitude smaller than the condensation of gases into the liquid and solid states may be nothing more than an accident due to the particular physical conditions obtaining on the surface of the earth. It is not at all impossible that in the universe as a whole the aggregation of matter may proceed more generally according to a pattern in which ordering of velocities takes precedence over ordering of positions. Thus, when discussing the behaviour of liquid helium, it is well to remember that the phenomena discovered so far may represent only a very limited aspect of a new pattern of assemblies of interacting particles. The peculiar analogy between superconductivity, an aggregation of charged light particles, obeying FERMI-DIRAC statistics, and liquid helium, an assembly of uncharged atoms, following BosE-EINSTEIN statistics, seems to emphasize the fundamental nature of the new state. The fact that these two rather dissimilar assemblies should follow the same pattern indicates that the pattern itself must be remarkably general. It probably has the same kind of generality as a crystal which may be brought together by a variety of forces such as ionic, VANDER WAALS or exchange interaction and may be composed of quite different atoms but nevertheless has always the same basic properties. The lack of completeness in our knowledge of superfluidity and our inability to understand the significance of its pattern make a systematic survey difficult. So far we do not know whether all essential phenomena have been observed and it is impossible to assess the relative importance of those which have been observed. In its short history the accepted ideas about liquid helium have changed profoundly on more than one occasion because new evidence had been obtained or old evidence had been regarded in a new light. In several instances the following up of some chance observation, sometimes disregarded or forgotten for a decade, has altered the picture completely. In these circumstances it would be unduly presumptuous to base an account of liquid helium on what appears essential to the author at the time of writing. The information has therefore been presented first in the form of a fairly detailed historical survey and a number of subsequent chapters in which the knowledge of the various phenomena has been brought up to date. It is hoped that in this way no observational fact which might gain particular

Sect. 1.

First liquefaction and solidification.

371

importance in future work has been omitted. Even so, the reader is warned that for serious work on the subject he should have recourse to the considerable number of detailed summary articles and to original papers. The literature on liquid helium is very large and cannot be easily subdivided into important and less important contributions. For the reasons mentioned above it is always possible that some early and obscure paper may contain information of considerable value which has been disregarded by later workers. In order to make the present article manageable, references have been restricted to work mentioned in the text. For the remainder, reference should be made to summaries with exhaustive literature index which have been listed at the end of this article.

A. Historical survey.

The lines of the helium spectrum were first seen by a number of observers of the sun's atmosphere in 1868, and in the following years they were ascribed to a new element which had not yet been found on earth. The first terrestrial occurrence of the element was discovered by RAMSAY who in 1895 separated a small quantity of the gas from uranium bearing minerals. Five years later, he and TRAVERS showed that helium failed to be condensed in liquid hydrogen and therefore had a lower boiling point than the latter. From a number of experiments in which samples of helium were compressed and expanded at low temperatures and from measurements of the gas isotherms the boiling point was estimated by various authors to lie below 6 °K. In only one observation, made by KAMERLINGH ONNES, was there reliable evidence of a mist of liquid drops. 1. First liquefaction and solidification. One obstacle to large scale liquefaction was the scarcity of the new element. The abundance of helium in the earth's atmosphere is about 0.0005 volume percent and its separation from the air requires considerable quantities of liquid hydrogen. Monazite, from which the gas for the first liquefaction was obtained contains about 1 to 2 cm3 of gas per gm. It was only after the large scale extraction of helium from certain well gases that it became generally available. In the first successful liquefaction of helium in 1908, KAMERLINGH ONNES used the conventional type of LINDE-HAMPSON cycle based on the JoULE-KELVIN effect. For experiments on a smaller scale this method was supplemented by helium liquefaction due to desorption cooling and by single adiabatic expansion, both these methods being due to SIMON. In 1934 KAPITZA liquefied helium by cooling the gas in a reciprocating expansion engine and this method has more recently been adapted by CoLLINS to a form of helium liquefier which is com. mercially available 1 . On the same day on which KAMERLINGH ONNES titst liquefied helium [J], he tried, by reducing the vapour pressure above the liquid, to reach the triple point. This and subsequent attempts of the same kind failed, and it became clear that helium under its own saturation pressure will remain liquid at all temperatures below the critical point. The main object of this work was to reach as low a temperature as possible and to determine the vapour pressure curve over the region investigated. The actual temperatures reached by pumping off the vapour are subject to a small degree of uncertainty, considering the temperature scale used by the authors. The early results were re-calculated by KEESOM on the basis of the "1932 scale" and yield the following picture. In his first liquefaction, on July 10th, 1908, KAMERLINGH ONNES reached a temperature of 1.72 °K. In this and the following three attempts in 1909, 1910 and 1919 mechanical 1

For more details cf. Vol. XIV, articles by DAUNT and CoLLINS, this Encyclopedia. 24*

372

K.

MENDELSSOHN:

Liquid Helium.

Sect. 2.

pumps were used and the temperatures attained were 1.38, 1.04 and 1.00 °K respectively. Using diffusion pumps he reached in 1922 a vapour pressure of 0.013 mm Hg, corresponding to a temperature of 0.83 °K, and ten years later, KEESOM succeeded in pumping helium down to 0.71 °K. The success of the magnetic cooling method in the following year, 19)), diminished interest in the attainment of very low temperatures with helium, but the pumping off experiments had clearly demonstrated that down to less than a seventh of its critical temperature, helium retained its liquid state of aggregation. This did not, of course, preclude the possibility of a triple point below that temperature and for a satisfactory solution of the problem, the melting curve had to be investigated. In 1926 KEESOM [2] used a cryostat whose temperature could be changed and into which a strong-walled capillary containing helium under pressure was immersed. He found that at the boiling temperature the capillary was blocked at a pressure of 128 kgfcm2 but free at 126 kgfcm2 and c'oncluded that the melting pressure of helium must lie between these values. Since the melting pressures in the temperature range where the liquid is stable are not excessively high, the same author constructed a cryostat embodying a pressure container made from glass which permitted visual observation of the melting process [3]. In it an iron stirrer could be moved by an external magnet and it was observed that, as the melting curve was passed, the stirrer lost its mobility. This was the only visual indication that solidification had taken place since solid helium turned out to be perfectly transparent. 2. The diagram of state. The melting curve above 2.5 °K was found to rise rapidly with temperature and extrapolation of this section to lower temperSO atures would not preclude the possible exist~ atm ence of a triple point. However, below this 40 temperature, the measured values of the meltsolid ing curve showed a surprising deviation to higher pressures from any such extrapolation (Fig. 1). Indeed, TAMMANN [4] showed that the 1\ relation between the melting pressure (in atmospheres) and the absolute temperature can be Hel[ HeI _l expressed as : 10 T - 1 = log (p - 24.0) {2.1) ~

I

v

1/ 1\

0

/.0

1.8

\

z.o

J.#.

r - #.2

5.0"1

thereby indicating that even at absolute zero a pressure of 24 atmospheres would be required to force helium into the solid state. The realisation that helium has a diagram of state which differs essentially from that of any other substance by the fact that solid and vapour phase cannot co-exist provided the first indication of the unique position of this substance. F ormerly it had been supposed that owing to the symmetry of the constituent particles, helium would serve as an ideal model substance for investigations of the solid, and particularly of the liquid, state. The unusual shape of the melting curve and the obvious absence of a triple point showed clearly that helium was not a very representative liquid and that some new factor, which was not operative in "normal" substances, had to be taken into account. An indication of this new factor was actually apparent in KAMERLINGH ONNES' first experiment of July 10th, 1908. He had then made a rough determination of the liquid density and found it to be about 0.15, an astonishingly low value. The great number of new facts discovered then and shortly afterwards seem to have, however, overshadowed the smallness of the absolute value and Fig. 1. Phase diagram of liquid helium.

Sect. 3.

373

The lambda-phenomenon.

an explanation of the unusual equilibrium between kinetic energy and interaction forces had to wait until 192). At that time BENNEWITZ and SIMON discussed the deviation from TROUTON's rule in hydrogen with reference to the zero point energy. SIMON [5] now applied these considerations to the case of helium in which he postulated the zero point energy to be so high that it would prevent solidification. Expressing the extension of TROUTON's rule as

(L

+ E )/T = const

(2.2)

0

where Lis the latent heat of evaporation and E 0 the zero point energy, he deduced for the latter in the case of helium a value of 64 caljmol. His postulate that the high zero point energy of the substance prevents its solidification under saturation pressure emphasized the unique position of liquid helium. It also explained that this unique position is directly due to the influence of the quantum principle. Six years later, WoHL [6] correlated the measured density of the liquid with the very much higher estimate of its density from the gas kinetic diameter of the helium atom and the application of the law of corresponding states. The introduction into this law of a quantum parameter such as was introduced into the extension of TROUTON's rule proved of great success in DE BOER's prediction of the vapour pressure curve of the light helium isotope. It is as well to separate this general evidence for the fact that liquid helium is a "quantum fluid" from the anomalous behaviour which has attracted so much attention. Such evidence as exists at present suggests that failure to crystallize under its own vapour pressure owing to the influence of the zero point energy may be a phenomenon which is shared by both isotopes. The anomalous transport properties, on the other hand, may, possibly for quantum statistical reasons, be confined to the 0148 heavier isotope. --0.144

·-- \ '\

3. The lambda-phenomenon. The discovery of the lambda-phenomenon, as the anomalous be- 0.140 haviour of He4 in its liquid state has been called, has been a gradual process. This is not surprising if O.!Jo one realizes that the observed phenomena were totally unexpected and have no counterpart in the OIJ2 behaviour of any other liquid. The way in which the liquid density varies with temperature was 0.128 first investigated by KAMERLINGH 0NNES in 1\ 1911. Measurements were carried out between 0.12'1-(} 4 °K 5 2 J I 1.5 and 4.3 °K which yielded the somewhat surTprising result of a density maximum near 2.2 °K. Fig. 2. Density of liquid helium. The experiments were repeated with greater ac. curacy in 1924 in which this maximum was well defined [7] (Fig. 2). While the care takento eliminate errors due to a possible anomaly in the expansion of the glass vessel indicate the importance which KAMERLINGH ONNES attached to the result, its significance was not yet understood at the time. The analogy with the density maximum in water was clearly tempting enough to exclude other explanations. The first realisation that the density maximum might indicate a more profound change in the liquid arose from a determination of the latent heat of vaporisation by DANA and KAMERLINGH ONNES in 1926 [8]. They found that this quantity which between 1.5 and ) .5 °K is in first approximation independent of temperature showed a slight minimum and that this minimum co-incided with

t

\

1

374

K.

MENDELSSOHN:

Sect. 4.

Liquid Helium,

the anomaly in the density curve (Fig. 3). The authors suggested that the two anomalies were possibly a sign of some discontinuous change taking place in the liquid at this temperature. Thus, in the last year of his life, KAMERLINGH 0NNES reported the suspected existence of two states of liquid helium. At the same time DANA and KAMERLINGH 0NNES had carried out determinations of the specific heat of the liquid but their publication only lists values above 2.6 °K. It appears that work was also carried out by them at a somewhat lower temperature. Since, however, they had reason to believe that some of their results were falsified by secondary causes, they did not include in their report very high values of the specific heat obtained in this region. A full investigation of the problem and a clear demonstration of the two states of the liquid was left to KEESOM and his co-workers. First the dielectric constant of the liquid was measured and a change similar to that in the density 0

_...

cal/gm 5

3.0

~ \

3

\

\

I

2.0

\ \

I /.0

2

1\

II I

I

0

col/gm,(Jeg

I

z

3

T-

4

I 5 °K

Fig. 3. Latent heat of evaporation.

/ (j

/

v t6

I

/ 2.0

T-

2.4

2.8

0

K

Fig. 4. Specific heat of liquid helium.

was found [9]. These observations were followed by the most important investigation on the static properties, the determination of the specific heat between 1. 3 and 4.1 °K by KEESOM and CLusrus [10]. It revealed a large anomaly, somewhat resembling in shape the inverted Greek letter lambda from which the phenomenon has derived its name. Particular attention was devoted to the peak of the anomaly and it was found that, while a discontinuity occurs in the specific heat at about 2.19 °K, there is no latent heat connected with this transition (Fig. 4). The sharpness of the transition was estimated by later experiments of KEESOM and Miss KEESOM [11] to be within a few thousandths of a degree. 4. Helium I and Helium II. Using the same apparatus, KEESOM and CLusrus also investigated the position of the transition, called the lambda-point, as the pressure is changed. By taking cooling curves of the calorimeter filled with helium under more than saturation pressure, they observed that the lambdapoint moved with rising pressure to progressively lower temperatures. The melting curve is reached at a temperature of 1.75 °K where the melting pressure is about 30 atmospheres. It is significant that this is the region in which the melting curve looses its downward slope and becomes t emperature independent. The liquid region in the diagram of state of helium is therefore divided by the lambda-line into two completely separated regions. Following an early suggestion by KEESOM and WoLFKE, the two forms of the liquid above and below the lambdaline have been named liquid helium I and liquid helium II (Fig. 1).

Sect. 5.

Super heat conduction.

375

Further work, especially by KEESOM and Miss KEESOM, to which we shall refer later has shown that all along the lambda-line the same condition holds as at saturation pressure, namely that the two liquids are not separated by a latent heat. This means that under equilibrium conditions liquid helium I and II can never be co-existent. Occasional reports by later observers who claimed to have seen a liquid-liquid boundary are probably erroneous. The discovery of the lambdy-transition in liquid helium led EHRENFEST [12] to consider this type of transformation in more general terms. He proposed a distinction between different types of transformation according to the discontinuities in the derivatives of the thermal potential. He defined the order of a transformation by whether discontinuities will occur in the first, second or higher derivatives of the potential. Thus a change involving a latent heat, such as melting, has to be considered as of first order whereas the lambda-transformation is of the second order since no discontinuity exists in the thermal energy but only in the specific heat. For the change of the lambda-point with pressure this then leads to dp) _ Llcp (dT;.(4.1) Tv Ll ap

and

(4.2) where L1 cp and L1 ap are the discontinuities in the specific heat and the coefficient of expansion at the transition between helium I and II. Since then, a number of careful measurements leading to the entropy diagram and the diagram of state of liquid helium have been carried out which will be discussed in detail later. This work did not lead to the discovery of any salient new facts, but it emphasized the curious position of the phase equilibrium bet ween liquid and solid helium at low temperatures. According to the third law of thermodynamics the entropy of the liquid as well as of the solid phase must become zero at absolute zero. The lambda-anomaly in the specific heat of the liquid now indicates a rapid loss of entropy within a few tenths of a degree below the lambda-point. Quite apart from the interesting question about the way in which order is established in the liquid in this region, the entropy loss must make itself felt in the shape of the melting curve. The variation of the melting pressure with temperature, which according to the CLAUSIUS-CLAPEYRON equation is the ratio of entropy change to volume change, will become zero as the entropy difference between solid and liquid phase disappears. Therefore, as SIMON [13] has pointed out, the change in slope of the melting curve is closely bound up with the lambda-phenomenon since at these temperatures the entropy of the liquid drops to a value which is not far from that of the solid entropy. At absolute zero, and effectively in the last 1.5° above it, the melting process of helium bears no similarity to that observed normally. Solidification will not take place on cooling but solely on the application of external pressure. The melting heat disappears and the phase change becomes a purely mechanical process at which no thermal change takes place. 5. Super heat conduction. What has been said about the discovery of the lambda-phenomenon is a fortiori true for the discovery of the anomalous transport effects. The appearance of a transformation in the liquid phase was unexpect ed and surprising, but the thermal effects showed at least some resemblance to transitions observed in the solid state. The transport effects, however, have no counterpart at all in physical observations, if we except the equally enigmatic phenomenon of superconductivity. It may appear astonishing that

376

K.

MENDELSSOHN:

Liquid Helium.

Sect. 5.

it took more than 25 years after the first liquefaction of helium for these striking phenomena to be recognized. However, as we shall see, curious facts were observed and recorded and other features of the experiments, which appear striking in retrospect, were passed over without comment. It is simply that the obvious conclusions to be drawn from these observations would make no sense, just as the now established effects still do not fit into the known pattern of the behaviour of aggregate matter. A very conspicuous change takes place in the liquid when it is cooled through the lambda-point by pumping off the vapour. It must have been seen by a great number of observers many times and was, in fact, recorded in 1932 by McLENNAN, SMITH and WILHELM [14] who write: " ... the liquid was watched closely as the triple point was approached. When a pressure of 38 mm was reached, the appearance of the liquid undenvent a marked change, and the rapid ebullition ceased instantly. The liquid became very quiet and the curvature at the edge of the meniscus appeared to be almost negligible." This sudden cessation of boiling is indeed quite a dramatic effect and has since been used generally to demonstrate the lambda-point to a large audience. Neither McLENNAN and his co-workers nor anyone else tried to interpret the effect, and it was not until the enormous increase in the heat conduction was found directly that the obvious connection between the two phenomena was realised. This failure to perceive the true reason for the change in the aspect of the liquid is clearly due to the fact that no mechanism was known by which the heat conductivity in a dielectric liquid can suddenly increase up to a million times. The idea of a large heat conduction only occurred when the conclusion had become quite inescapable. In their calorimetric experiments KEESOM and Miss KEESOM [11] set out to determine the sharpness of the lambda-point from a plot of the actual observations in which the temperature rise of the calorimeter on heating was recorded. It was then found that at the lambda-point not only the rate at which heat is taken up changed but also the way in which it was taken up by the liquid (Fig. 5). Below the lambda-point the calorimeter temperature became instantly steady as soon as the heating current was switched off. Above the lambda-point over-heating was very evident. Since the shape of such records is a standard test in calorimetry for the equalisation of temperature, it now became clear that there existed a rapid change in heat conduction at the lambdapoint. The magnitude of the change was, however, not yet appreciated as is evident from the fact that the first apparatus designed for a determination of the heat conduction proved completely unsuitable. In this first attempt KEESOM and Miss KEESOM [15] tried to measure the temperature difference at the faces of a flat disc of liquid. This method worked with helium I but when the apparatus was filled with helium II, the temperature difference proved unmeasurably small. Only when a narrow capillary had been substituted for the disc shaped chamber, could measurements be taken. Determinations at 1.4 and 1.75 °K yielded a value of approximately 190 calfdegree · em which by far exceeds the heat conductivity of any other substance. It is about 200 times larger than that of copper at room temperature and three million times that of helium I. Although the extreme magnitude of the heat conduction as discovered in these experiments in 193 5 and 1936 only constitutes part of the heat flow phenomenon, it provided the stimulus for research into the transport phenomena. An important additional fact which had escaped notice in these first experiments was observed in the following year by ALLEN, PEIERLS and UDDIN in Cambridge [16]. They, too, measured the heat conduction of helium II enclosed in a capillary

377

The thermo-mechanical effects.

Sect. 6.

but found that, besides being very large, it also depended on the temperature gradient. A little later the authors themselves called this result in doubt and possibly caused by a complicating effect. However, later work established that the value of the heat flow is not only influenced by the temperature gradient but also by the dimensions of the apparatus in which it is measured. The concept of "heat conductivity" in the accepted sense as a constant ratio of heat current density to temperature gradient has thus lost its usefulness when dealing with liquid helium II. Limiting the disooo; "K cussion to one capillary size and

I

2

0

Time-

Fig. S. Fig.

s.

3

min

~

/..0

-~ /.2

/

/./;

I

I

I

!.()

T-

I

-- \

v

1.8

\

20 "K 22

Fig. 6.

The change in heat conduction on passing the lambda-point as shown by thermometer readings (in arbitrary units) in a measurement of the specific heat. The arrow marks the position of the lambda-point. Fig. 6. Temperature dependence of the heat flow in liquid helium II under constant temperature gradient.

constant temperature gradient, it was found that on cooling below the lambdapoint the heat conduction rises rapidly to a maximum at about 2 °K and then falls off again to lower temperatures (Fig. 6). 6. The thermo-mechanical effects. The complicating feature which made the interpretation of these results doubtful and which was traced shortly afterwards by ALLEN and JoNES [17] turned out to be another unexpected effect which again is characteristic for helium II only. The heat conduction apparatus used in Cambridge consisted of a thermally insulated glass reservoir embodying a heater which communicated with the helium bath through a glass capillary (Fig. 7). The heat flow through the capillary was measured by applying a heating current and comparing the height of the meniscus in the reservoir with the level of the bath. Since the heated end of the liquid had a higher vapour pressure than the bath, the level in the reservoir was depressed and the degree of separation of the menisci therefore acted as a Fig. 7. Arrangement for meassensitive differential thermometer. uring the heat conduction of heliumll (diagrammaliquid At small heat currents the separation of the levels tical) used by ALLEN, PEIERLS was very small and sometimes almost non-existent. This and UDDIN. could be understood in terms of very large heat flow under small temperature gradients. However, under certain conditions of temperature gradient and absolute temperature, there also appeared to be cases in which the level in the reservoir showed a definite rise above the bath level. In terms of vapour pressure differences this would have meant that on supplying

378

K.

MENDELSSOHN:

Sect. 6.

Liquid Helium.

heat to the helium in the reservoir its temperature fell, and this was clearly nonsensical. The experiment was therefore varied by opening the top of the reservoir so that no difference in vapour pressure was possible. Repeating the heat flow experiment under these conditions (Fig. 8), yielded the astonishing result that on supplying heat, the meniscus in the reservoir rose above the bath level. The authors were able to enhance the effect very much by shining a light on a tube closely packed with emery powder which carried a fine nozzle projecting above the level of the helium bath. A free jet of liquid helium was then seen to rise as high as 30 em into the vapour space above the bath level. This striking demon ..

Fig. 8 a und b.

Fig.9.

Fig. 8 a and b. The thermo-mechanical ("fountain") effect. (a) First observation, (b) fountain produced in a tube filled with fine powder (P) and closed by a cotton wool plug (C). Fig. 9. The mechano-caloric effect.

strati on earned the phenomenon the name "fountain-effect" which is still frequently used. However, the conditions in this demonstration experiments are somewhat complex and tend to obscure the true nature of the phenomenon which is presented in a much clearer form by the first experiment. We therefore prefer the descriptive term "thermo-mechanical effect" which now has come into general use. The meaning of these observations is that as heat is supplied to the reservoir, a flow of liquid towards this supply of heat will take place. As the column of liquid at the heated end of the capillary rises, its weight will push helium back through the capillary into the bath and a dynamic equilibrium is established in which the thermo-mechanical flow is balanced by the return flow. The reason why a so much greater height of liquid column could be established in the fountain experiment, was evidently due to the return flow being largely inhibited by the flow resistance of the powder filled tube. Further experiments with a powder filled tube in which the height of the column of liquid was carefully measured showed that for constant temperature difference the reaction force showed a dependence on absolute temperature which was not unlike that of the heat conduction. The discovery of the thermo-mechanical effect immediately suggested the possible existence of another phenomenon which would be its thermo-dynamical

Sect. 7.

379

Superfluidity.

counterpart. The thermo-mechanical effect shows that in liquid helium II the establishment of a temperature difference will cause the appearance of a difference in fluid pressure. The question therefore arose whether the establishment of a pressure difference will set up a corresponding difference in temperature. A search for such a "mechano-caloric effect" was made in the following year by DAUNT and MENDELSSOHN in Oxford [18] (Fig. 9) who observed that flow of helium II from a higher to a lower level is indeed accompanied by a temperature gradient. Their experiment was carried out with a small Dewar vessel which was completely closed except for a small orifice at the bottom. The lower part of the vessel was filled with closely packed jeweller's rouge which formed a plug, P, with many fine channels and above this plug a resistance thermometer, T, was fitted. When the vessel was partly immersed in the bath of helium II the meniscus inside it adjusted itself eventually to the same height as the bath level and the temperature inside the vessels was the same as b a that of the helium bath. On withdrawing the Dewar vessel from the bath, liquid helium was seen to run out through the orifice at the bottom and it was observed that LJP at the same time the t emperature inside the vessel rose slightly. Conversely, if, starting from the equilibrium position, the vessel was lowered further into the bath so that liquid flowed in through the Fig.10 a and b. The (a) mechano·caloric and (b) thermo-mechani· plug a fall in temperature was cal effects in two volumes of liquid helium II connected by a capillary link C. noted. This showed that the heat content of the fluid which had passed through the plug was lower than under starting conditions and that the heat content of the fluid which was left behind rose accordingly. The powder plug was thus found to act as an "entropy filter" and rough estimate from the first experiment indicated that the entropy of the liquid which had passed through the filter was very small and possibly zero. 7. Superfluidity. While the high heat flow in helium II was the first indication of the existence of anomalous transport phenomena, the discovery of the thermo-mechanical effects was preceded by that of another unexpected phenomenon. On various occasions it had been noted that small leaks became noticeable in evacuated containers when these were immersed in helium II and it had, in fact, been suggested by KEESOM and KEESOM in 1932 that the viscosity of the liquid might decrease below the lambda-point. The first measurements of the viscosity were carried out three years lat er by WILHELM, MISENER and CLARK [19] in Toronto who made observations of the damping of an oscillating cylinder in liquid helium at four temperatures above and one just below the lambda-point. In their experiments the damping increased very appreciably from 4.2 to 2.4 °K and was then found to be lower again at 2.2 ( ?) K. BURTON [20] deduced from these data viscosity values of 110 micropoise at 4.2 °K, rising to 270 micropoise just above and then falling to 33 micropoise just below the lambda-point. This interpretation of the data has been called in doubt by a number of authors who drew attention to the likely occurrence of turbulence, but it seemed clear that the change at the lambda-point wa real. In 1938 KEESOM and MAGWOOD [21] repeated the Toronto experiments, using an oscillating disc method which permitted somewhat easier interpretation. They found a gradual drop with falling 0

)80

K.

MENDELSSOHN:

Liquid Helium.

Sect. 8.

temperature in helium I to be followed by a discontinuity at the lambda-point with a higher value for the viscosity in helium II and a subsequent rapid decrease as the temperature was lowered further. More recent measurements which will be discussed below have cleared up the discrepancies between these results in helium I and at the lambda-puint. The observations of KEESOM and MAcWooD confirm, however, the drop in viscosity below the lambda-point found by the Canadian workers. Early in 1938 two short papers appeared in the same issue of "Nature", by KAPITZA [22] and by ALLEN and MISENER [23] respectively, in which the flow of helium II through narrow channels was described. In both cases the liquid was flowing from a raised glass reservoir under gravity back into the helium bath. The link between reservoir and bath used by KAPITZA was the gap between two optically flat plates whereas ALLEN and MISENER studied the flow through capillaries. The width of the flow channel was varied in the former experiment by inserting spacers in the gap and in the latter two different capillaries were employed. It was in these observations that a new striking phenomenon of liquid helium II was revealed which has become known as "superfluidity ", a term suggested by KAPITZA in this first paper. He found that when in his arrangement the glass plates were opposed without an intervening spacer so that the gap, as determined from the optical fringes, was of the order of 5 X 10- 5 em, the flow of helium I could just be detected over several minutes. In the helium II region, however, the whole reservoir was emptied within a few seconds. The drop in viscosity on passing the lambda-point was estimated to be at least 1 500 times. A similar result was obtained by ALLEN and MISENER who observed moreover in these and subsequent experiments that the flow velocity was greatly independent of the pressure difference and of the diameter of the capillary. Indeed, for the finer capillaries it was found that this velocity increased with ·decreasing capillary diameter. 8. Film transfer. In the same volume of "Nature" in which the discovery of superfluidity was reported, and again in the same issue two letters dealing with still another strange phenomenon in liquid helium II were published. The authors, DAUNT and MENDELSSOHN [24] in Oxford and KIKOIN and LASAREW [25] in Kharkov respectively described observations on the helium film. The first indication of a peculiar transport effect in liquid helium was observed even before the lambdapoint was discovered. In 1922, when KAMERLINGH ONNES [26] made an attempt to reduce further the temperature to which liquid helium could be cooled by pumping off the vapour, he. employed an arrangement in which two concentric Dewar vessels were used (Fig. 11). The object of the experiment was to shield the liquid in the inner vessel thermally from the influx of radiation by the liquid in the outer vessel. He therefore expected the liquid surrounding the inner vessel to evaporate more rapidly and the meniscus in the inner vessel to remain higher. He found, however, that the liquid levels in both vessels fell at the same rate as vapour was drawn off. Moreover, when by shining a lamp on the cryostat the surrounding liquid was evaporated more rapidly and the outer meniscus fell below the inner one, the two levels re-adjusted themselves again to the same height once the lamp was removed. A similar adjustment in the opposite direction was observed after the outer level had been raised by scooping liquid from the inner into the outer vessel. KAMERLINGH ONNES believed this effect to be a distillation phenomenon and the experiment was not repeated for another 16 years. In 1932 Cwss and MENDELSSOHN [27] reported msome detail on a disturbing effect in calorimetric measurements at helium temperatures which they traced to

)81

Film transfer.

Sect. 8.

the evaporation of a layer of helium from the surface of the calorimeter. They noted that the effect only occurred when the calorimeter had been cooled to below ,._,z 0 K. Another disturbing effect was observed by ROLLIN and SIMON [28], also in attempts at pumping down to a low temperature a cryostat filled with liquid helium II. They found that the rate of evaporation was much higher than was to be expected when allowance was made for the known sources of heat influx into the cryostat. From a series of experiments in which this effect was investigated they came to the conclusion that the inner wall of the tube connecting the cryostat with the pumping line was covered with a film of liquid helium. This work was carried out in 1936 in conjunction with experiments revealing the anomalously high heat conduction of liquid helium II. It was therefore only natural that these authors at first ascribed the high rate of evaporation from their cryostat to a high heat conduction of the helium film on the walls of the ~rmomeror

~ofer

Fig. t I.

observation of liquid helium transport between concentric vessels.

KAMERL!NGH ONNES'

Fig. 12.

KIKOIN

and LASAREW's experiment on the helium film.

connecting tube. However, this process of heat influx into cryostats containing helium II is quite different as was demonstrated by the experiments of DAUNT and MENDELSSOHN as well as by the subsequent work of RoLLIN and SIMON [29]. The observation of KIKOIN and LASAREW followed a pattern somewhat similar to those of RoLLIN and SIMON. They used a glass tube whose lower end dipped into a bath of liquid helium II while at its upper end a heating coil and a thermometer were attached (Fig. 12). In the helium II region the upper end of the tube had always the same temperatures as the lower one when no heating took place. The same was true for small heating currents but at a critical value of the current, which increased with falling temperature, the temperature at the upper end of the tube rose rapidly. This observation corroborated the idea of a surface film of helium II and the critical currents were regarded as the heat input necessary to evaporate the film completely. However, these authors, too, explained their observations as due to a very high heat conduction of the film and thus failed to recognize the true nature of the film phenomenon. In fact, they regarded their results as a refutation of KAPITzA's idea of convection currents in the liquid, pointing out that the film was far too thin to allow for such a convection process. The experiments of DAUNT and MENDELSSOHN, on the other hand, were designed in the first place to repeat KAMERLINGH 0NNEs' observation of a readjustment of levels in liquid helium which had never been reported again in

382

K.

MENDELSSOHN:

Sect. 8.

Liquid Helium.

the intervening sixteen years. Their first apparatus [30] consisted of two small glass vessels on top of each other which were joined by a co-axial glass tube. Some liquid was introduced into each vessel and the variation with time of the menisci

mm aJ

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5

I

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i'".

I

I

/(}

:

5

I I

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Fig. 13.

DAUNT

and

MENDELSSOHN's

/(}

first observation of transfer of liquid helium II through the film.

in the two vessels was watched. The result was disappointing since no rapid re-adjustment, such as had been seen by KAMERLINGH 0NNES, did take place. The effect was found again, but it was so small that it almost escaped detection. In the course of 30 min the upper level dropped by 7 mm while the lower one rose, but only by 3 mm (Fig. 13). The gain of liquid in the lower vessel could either be accounted for by distillation as suggested by KAMERLINGH 0NNES or by :-:.-:.~-=::.-==.~~ transport of liquid along :::::-:....:::=-:.--=:::--:--..:: the surface of the connect-------~_;::._;:-::==.::_==.::_~ ing glass tube. In order - - --- - to decide this, the authors Fig. 14. Transfer into and out of beakers. increased the solid surface connecting the two volumes of liquid helium by inserting a number of wires into the apparatus. An increased loss from the upper vessel coupled with a proportionate gain in the lower vessel which lasted as long as the wires dipped into the liquid demonstrated that the flow took place in a film along the solid surface. For a better study of the phenomenon, named" transfer effect" by the authors, DAUNT and MENDELSSOHN used the simple device of a small cylindrical glass beaker which was suspended on a fine glass fibre and could be lowered and raised with respect to the helium bath (Fig. 14). This has proved to be a very convenient

Sect. 8.

Film transfer.

)8)

type of experiment and has been used since in a great number of investigations on transport along the film. On lowering the empty beaker partly into the liquid, it was observed to fill up gradually until the meniscus of helium inside was at the same height as the level of the bath. After raising the beaker slightly the process was reversed, helium now passing from the beaker into the bath until again equalisation of the levels had taken place. Finally the beaker was lifted completely out of the bath, and it was then seen that drops of helium formed at the em 6 E E

r--...

/.

..._r-

i-

---

r=- z ~-=~ :01--F: :::____ ---t=,0

r- ...._r--

1--

r- f--. ...._ 1-

10

Z/J

._

t- ........

I I

JO

4.0mm

Fig. 15. Film transfer of helium II from a beaker plotted against time. The change of outer level at minute 33 did not affect the rate of transfer.

bottom of the outside surface of the beaker which grew and fell back into the bath at regular intervals. On timing the rate at which liquid helium was transferred back into the bath from a filled and raised beaker, DAUNT and MENDELSSOHN found that it did not change appreciably throughout the process of emptying. In their first as in all subsequent experiments it was noted that the transfer is slightly higher within a few millimetres of the rim but then settles down to a steady rate which is uninfluenced even by sudden changes in the relative heights of the menisci. The beaker experiment thus showed that the transfer of helium along the film is independent of the potential difference, of the length of the path and of the height of the 3/ min ~ 10 intervening barrier since these all _ _ _ _ change in the COUrse of the ex- Fig. 16. The limiting effect of a constriction above the liquid level on the film transfer. periment (Fig. 15). The experimental conditions were then slightly varied by introducing a constriction of the walls inside the beaker. As the empty beaker was lowered partly into the bath, liquid began to collect inside at the same rate as in the previous experiment. However, in this case the vessel holding the bath itself was also fairly narrow so that its level fell noticeably as the meniscus inside the beaker rose (Fig. 16). In agreement with the previous experiment the transfer rate remained constant until the bath level had fallen to the height of the constriction inside the beaker. From then onward the transfer proceeded at a reduced rate, the ratio between this and the original rate being the same as the diameter of the constriction to the inside diameter of the unconstricted beaker. The last observation seemed to indicate that the transfer is limited by the minimum width of the connecting surface above the upper level. Another

384

K.

MENDELSSOHN:

Sect. 8.

Liquid Helium.

experiment was, however, carried out to demonstrate that the result obtaines had not been simulated by thermal effects. A beaker in the form of an unsilvered Dewar vessel was constructed, the inner section of which was made up of a wider cylindrical vessel on top and a narrower one at the bottom. Heat exchange in this arrangement could only take place by evaporation through the surface and, if the transfer was limited by this, it should have been proportional to the squares of the upper and lower diameters of the inner vessel. The result of the experiment showed, however, that the rates of transfer were strictly proportional to the diameters and not to their squares (Fig. 17}. While these and other observations showed that at any given temperature the rate of transfer per centimetre width of the connecting surface was completely mm

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~

i""-~ocm; 11= 7.9x10-6 cm; III~s.oxw-•cm: IV~4 . 38xw-•cm.

v

v

0 0

j..

y b

,.;2

/

--

I-

~

.,.,, .

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w

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8

e-

8.

10

0

x!O'cal/sec

Fig. 48. Rate of flow V and temperature difference .dT in thermo·mechanical transport through a narrow channel as function of the heat input Q.

The chief difficulty in establishing and testing a relation of the above kind is the small range of sizes available for observation. In the neighbourhood of 10-5 em the accurate determination of channel width becomes very difficult and thus does not permit reliable evaluation of results. On the other hand, in channels much larger than 10- 4 em the flow becomes very pressure dependent, as can be seen from Fig. 46 and consequently the determination of a critical velocity, or even its existence, becomes problematic. A decision as to the correct form of the dependence of velocity on channel size will thus have to wait until a method has been found which allows to define and measure critical velocities in wide channels. The recent work on the determination of the thermal resistance in wide capillaries mentioned in part F (Sect. 32) may possibly offer such an opportunity. Corroberative evidence for the existence of a critical velocity in channels of 1o- 5 to 10- 4 em width was furnished by KAPITZA [ 42] who measured the flow rate under a thermo-mechanical pressure gradient. He used the arrangement shown in Fig. 22. On supplying heat to the inside of the bulb, liquid helium was drawn into through the slit between the optically flat discs. The width of this could be changed by the insertion of spacers and was measured with optical fringes. The rate of volume flow was measured for different rates of heat input, and a typical result at a constant ambient temperature is shown in Fig. 48. For the lowest heat inputs there is a roughly linear rise of volume flow which, however, at a certain value of Q begins to lag behind the impressed heat input. This behaviour suggests strongly the existence of a critical velccity up to which the volume flow will without friction follow the heat flow. KAPITZA also measured

Sect. 26.

413

Narrow channels.

the temperature difference Ll T between the inside of the bulb and the helium bath. He noted no increase in temperature inside the bulb up to the critical heat input (corresponding to the critical velccity just defined) when a gradual rise of the temperature in the bulb over that of the bath was observed. Considering H. LONDON's equation according to which any change of liquid level in the bulb must be accompanied by a change in temperature, the last mentioned result of KAPITZA's must appear strange. However, the temperature changes due to Z4r-.----,- ---,---.---- -,---,-, LoNDON's equation were, under the conditions of his experiment probably em He too small to be detected. The rise in LIT at the critical heat input should /8t-t---hl---t----+---..,._-_·--t-.::.:·-~-..~~ -====p.=.:-.:-_ -- ;{. therefore be considered as a surge in / r---=-= ____ ---- ::::....,. temperature far above the value of L1 T -----or-__ _ corresponding to the thermo-mecha- t 12 beexisting nical pressure difference ---r----o.- ---- __ ~ tween bulb and bath. ---- ---- __,., The existence of these surges was 8 t--H/._'---_-_-+-_-_-~--:_+;-=_==-_ ..._-+f>---+----1 -l _ subsequently demonstrated by MEYER 1 and MELLINK [82] in Leiden who used a similar arrangement buth with a 0 0 more sensitive thermometer. They 3 I 2 .tiTfound, as is shown in Fig. 49, relaxa- Fig. 49. Relaxation effect observed in the establishment of a thermo·mechanical pressure difference. tion effects which were particularly noticeable in narrow slits and close to the lambda-point. The full line represents the true variation of pressure with temperature difference under equilibrium conditions which is in agreement with the LoNDON equation. However, the way in which these equilibrium conditions could be realised depended on the rate and manner of heat supply. Only for

/-1----

r ---- ______:;:., ., c_-...--

---------------Time-Fig. SO. Outflow through • slit (1 micron) tmder a hydrostatic pressure difference with intermediate pressure measured in a static tulJe. Level in reservoir C, in static tube Sand in bath B.

small rates of heat input was the full line followed without deviation. On larger (supercritical) rates of heat input a rapid rise in temperature, indicated by the broken lines, was noted. On switching off the heat, the temperature in the bulb decreased again until a point on the equilibrium line was reached. The existence of critical velocities was clearly shown in experiments in Oxford in which in the same flow channel the flow under hydrostatic and thermo-mechanical pressure differences was investigated. Of a number of devices used that employed by BowERS and MENDELSSOHN [89] gave the most satisfactory results. The object of their experiments was not only to measure the value and pressure dependence of the flow rate but also the intermediate pressure at an arbitrary place along the flow channel. Their apparatus which is shown in Fig. 50 is similar

414

K.

MENDELSSOHN:

Liquid Helium.

Sect. 26.

to that used by KAPITZA in the discovery of superfluidity. It was a cylindrical reservoir to whose lower end a flat glass disc was attached. This disc was opposed by another optically flat glass plate so that helium could flow in and out of the reservoir through an annular channel of about 1 micron width. An annular groove was cut into the upper plate, about half way along the flow channel, to which a narrow "static tube" for pressure measurement at this point in the flow channel was attached. As the reservoir was filled and withdrawn from the bath, its level C was dropping at an almost constant rate, indicating superflow with a critical velocity. The level S in the static tube dropped immediately to the height of the bath level B and stayed there throughout the experiment, the small level difference with the bath being due to surface tension in the narrow tube. This showed that there was no pressure gradient inside the flow channel and, since the static tube was placed at an arbitrary position, one can only conclude that the whole pressure drop in a channel carrying superflow must occur at the end. Flow under a thermo-mechanical pressure difference was produced by closing the top of the reservoir and supplying energy to a heating coil inside the reservoir. As heat was supplied, the level inside the reservoir rose, showing that liquid was drawn in under a thermo-mechanical pressure difference. The level in the static tube, however, again remained at the height of the bath level. This demonstrated that the flow channel failed to maintain a thermo-mechanical as well as a hydrostatic pressure difference. Since in the last experiment the pressure difference between bath and bulb is accompanied by a temperature difference, the behaviour of the static tube suggests that in thermo-mechanical superflow, too, there is no temperature gradient in the flow channel and that the temperature difference is concentrated at one end of the channel. In these experiments, too, critical flow rates were found. Three different criteria for the critical rates were observed which agreed with each other within the experimental error. The first criterion was similar to KAPITzA's shown in Fig. 48. There was a linear rise of the flow rate with Q and a fairly sharp departure of the curve from linearity at the critical heat input. Secondly, a phenomenon akin to the relaxation effect by MEYER and MELLINK was observed. When the rise of level in the reservoir with heat input was observed under increasing values of Q, a change of behaviour occurred at the critical rate. Below this rate the rise stopped immediately when the heating was switched off, whereas above it the level continued to rise for a while after heating was discontinued. This indicated that beyond a critical rate superfluid could not pass through the flow channel sufficiently fast to keep pace with the heat supply and overheating took place. The third criterion was observed in the static tube. Its level, which remained with the meniscus of the bath for small flow rates, fell below the bath level when a certain flow rate was exceeded. The explanation for this behaviour is possibly that above the critical rat e the helium column in the reservoir lags behind the thermo-mechanical equilibrium pressure and that the helium in the flow channel now flows to an effectively lower pressure. Measured values for these three criteria are given in Fig. 51 together with the flow rates under three hydrostatic pressure differences. Since the pressure dependence of the flow is weak, the three curves differ little and the critical values obtained under thermo-mechanical pressure are all well within this spread . Using essentially the same apparatus SwiM and RoRSCHACH [90] at the Rice Institute have accurately measured the pressure dependence for gaps between the plates of 2.4 and 4.3 microns. They employed a tall reservoir which allowed them

415

Flow through packed powder.

Sect. 27.

to go up to pressure heads of 2 X 103 dynejcm 2 and a set of curves for the narrower gap is shown in Fig. 52. Expressing the volume flow rate in terms of (Ll p)n, they found that n varied between 0.33 and 0-36 from 1.4 to 2.1 °K in the case of the wider slit and between 0.27 and 0.28 between 1.4 and 1.77 °K for the narrower one. They also confirmed the behaviour of the level in the static tube observed by BoWERS and MENDELSSOHN. WINKEL [91] and others in Leiden also have determined a few critical velocities in slits between 0.43 and 3.1 microns. They used various criteria, but readings were only taken at three different temperatures. These suggest a falling off of the velocities with decreasing temperature between 1.9 and 1.5 °K. BoWERS and WHITE [92] in Oxford investigated the flow of helium II through etched copper membranes with an average channel size of 1 micron. Nothing was known concerning the actual shape of these 0.01 4 em- 1-- ,........ 30 The work was limited to flow channels. emf sec ~"'-.. em/sec 2.5cmunder hydrostatic pressures since the good ~ .,.,_ r;;.. !'-" 1-L em heat conduction of the membranes did not r~"'-..

t o.oz

~~

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Q:;

15 TFig. 51.

20

~

z

t

I~

10

~

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-

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z

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d=Z.Mo- 4cm 56 78

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/0121416 20 •102 dynefcm 2

Fig. 52.

Fig. 51. CritiCal rates determined by three different criteria in thermo-mechanical flow and flow rates under hydrostatic pressure heads as function of temperature for a 1 micron slit. (V average flow velocity.) Fig. 52. Flow rates under hydrostatic pressure through 2.4 micron slit (a) 1.40 ' K, (b) 1.64 'K and (c) 1.77 'K

permit the setting up of a temperature difference. The pressure dependence of the flow was found to be similar to that of slits between discs, n having a value of 0.2 between 1.2 and 1.9 °K, rising to almost 0.3 at 2.1 °K. The work with membranes permitted an extension of the measurement of the intermediat e pressure. In the experiments with slits the ratio of the normal flow resistances of the two parts of the annular gap amounted to 4:1 with the total pressure drop in superflow being concentrated in the part with the higher normal resistance. Geometry did not allow in this arrangement to make the normal resistances more nearly equal. However, in the work with membranes, two of these could be selected which had a ratio of 4: 3 in the normal flow resistances. Here the full pressure drop under superflow was again t aken up by the membrane with the slightly higher flow resistance, irrespective of whether it was placed between reservoir and static tube or between the bath and static tube. 27. Flow through packed powder. In their fundamental experiments on superflow, ALLEN and MISENER [88] also used a tube which was closely packed with jeweller's rouge, i.e. finely powdered Fe 20 3 , the grain size of which was estimated as being of the order of 10-s em. The size of the individual flow channels between the grains was thus about 1o-6 em which is smaller than the finest wire tubes used by the same authors and approximating the thickness of the film. On the other hand, it is clear that the geometry of the channels in such a powder must be complex.

416

K.

MENDELSSOHN:

Liquid Helium.

Sect. 27

Flow observations were made between the lambda-point and 1.1 °K, and it was found that for constant pressure head the flow rate changed with temperature in much the same way as in wire tubes or in the film. The pressure dependence of the flow, however, was marked and but for a temperature very close to the lambda-point the flow rate was always proportional to the square root of the pressure. This is characteristic of turbulent flow which evidently, as the lambdatemperature is approached, changes into laminar flow. For the latter region the coefficient of viscosity was estimated as being of the order of 1o-s poise. In view of the striking difference in behaviour between the tubes filled with wires and those filled with powder, the latter were further investigated by CHANDRASEKHAR and MENDELSSOHN [93] who used a static tube and also extended the work a

b

b

____ /}!_____ Po

T/me-

Fig. 53 a and b. Superflow under (l) hydrostatic, and (2) thermo·mechanical pressure in (a) slits, and (b) powder packed tubes. (Pz level in static tube.)

to flow under thermo-mechanical pressure differences. Their observations under hydrostatic pressure completely confirmed the results of ALLEN and MISENER. On the other hand, it was noted that when liquid helium was drawn into the reservoir by the fl.pplication of a heat current, the level in the static tube remained at the height of the bath meniscus. Thus under a thermo-mechanical pressure difference the behaviour of the powder tube was identical with that of a fine slit, suggesting no dissipation inside the flow channels and the existence of a thermomechanical pressure difference at the entrance of the powder plug. Care was taken that in flow under hydrostatic as well as under thermo-mechanical pressures the same velocity range was investigated. The results on slits and powder tubes, taken together, thus reveal an inconsistency in the flow mechanism which has as yet not been resolved. A schematic diagram of this inconsistency is shown in Fig. 53 in which (a) and (b) refer to slits and powder tube respectively while (1) and (2) denote hydrostatic and thermo-mechanical pressures. In thermo-mechanical flow through the powder tube the same criteria for critical velocities as in slits were observed. The sudden onset of friction when a critical flow rate is exceeded is even more pronounced than in the observation on slits. Fig. 54 shows a plot of the rate of volume flow against heat input in which the sharp departure from linearity is clearly marked. When comparing

417

Wide capillaries.

Sect. 28.

the critical rates obtained in experiments with thermo-mechanical flow with the flow rates under varying hydrostatic pressure heads it was found that they corresponded to a pressure difference of "-'1200 dynejcm 2 at all temperatures (Fig. 55). Since it was noted that in each case the pressure drop occurred at that end of the powder tube which was connected to the reservoir, experiments have recently been carried out in which heat could be supplied either to the reservoir or to the bath. It was found that the pressure drop now always took place at the warm end of the powder tube. In the work of CHANDRASEKHAR and MENDELSSOHN, too, a comparison between the results and the expected viscosity in helium II has been made. The power law by which the hydrostatic pressure is related to the flow velocity 8 oriNlrilry 11111ls

8

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/(

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0

I

j

--

30

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20

t m1n 7fl .ffl lfl JYJ JrJ lfl ll KINS [154] who suggests a relation between variation -- - - - Fig. 85. Formation of bulk liquid from the film. Drops formed between of rate with the change of minutes 30 and 45. film thickness with height. However, it has to be kept in mind that his value of z in Eq. (3 7.1) is anomalously high and that the usual thickness determinations would demand a very much increased variation of transfer rate over the length of the beaker. Re-examining the question of height dependence of rate, MENDELSSOHN and WHITE [149] could not detect a definite variation in observations on carefully shielded beakers except for a drop of about 10 % when the levels approached to within 3 mm. This effect was subsequently explained by EsELSON and LASAREW [156] who showed that it is due to insufficient heat contact between the inside of the beaker and the bath which results in thermo-mechan ical pressure difference. Under more isothermal conditions the effect was not observed. The appearance of liquid drops at the bottom of a test tube filled with helium II shows that at some stage in the transfer process bulk liquid will form out of the film. DAUNT and MENDELSSOHN [30], using a beaker with a funnel shaped inset as shown in Fig. 85, observed that drops of liquid would fall off the tip of the funnel if the latter was below the height of the bath level. From this they concluded that bulk liquid can be formed at constrictions below the higher level. The question was further investigated by (HANDRASEKHAR and MENDELSSOHN [86] who employed a beaker with staggered diameters below which were arranged glass skirts of different width. The film could thus originate at three different ·perimeters AA, BE, CC and the maximum diameters were arranged in the sequence D D > A A > E E > B B > C C (see Fig. 86) . The three st ages of the experiment show that liquid drops will always form when the film has t o flow over a perimeter which is narrower than the original one. It is also evident that the film will re-form from the bulk liquid even below the higher meniscus

-- -- -

446

K.

MENDELSSOHN:

Liquid Helium.

Sect. 38.

since no drops were observed on DD when BB was the perimeter at which the film originated. The nature of the bulk liquid so formed has been demonstrated in a striking observation by HAM and JACKSON [157]. They employed a conical beaker of stainless steel in which the inner diameter at A was the same as the outer diameter at B and the inner diameter at B A equal to the outer diameter at C as shown in Fig. 87. Using the optical method for the determination of film thickness, they observed that when the filled

A

A

8

Fig. 86. Experiments on the formation of bulk liquid from the film.

Fig. 87. Steel beaker for the observation of liquid helium drops forming out of the film.

beaker was raised out of the helium bath drops, which appeared as bright spots, could be seen running down the outer surface. The number of drops increased towards the lower end of the beaker because of the diminishing outer perimeter. No drops were observed between A and B at which stage the outer circumference of the beaker was equal to the originating perimeter. As the experiment proceeded and the inner level in the beaker fell, fewer drops were observed until finally they ceased completely when the inner level had reached B. The authors, comparing their observations with FRENKEL's theory for drops on a wetted surface found in agreement with it that the smallest drops did not run but appeared stationary. Bulk liquid can form from the film above the meniscus of the helium bath in cavities of such a size and at such height that due to normal surface tension the liquid will be stable. Although KELVIN already pointed out that a fine capillary suspended in the vapour Fig. 88 _ Formation of heabove the level of a liquid should fill up with liquid lium II under surface tension under equilibrium Conditions, the process is not observed in a capillary. as distillation under isothermal conditions is far too slow. In helium II the conditions are different as the film can transfer liquid along the walls of the vessel and over the suspension into the capillary. LANE and DYBA [158] showed that a capillary to whose lower end a wide tube was attached which in turn dipped into a bath of liquid helium II would fill up with liquid from the bath (Fig. 88). The height of the liquid corresponded to half the height to which a freely suspended capillary of equal diameter could be filled up with helium I, allowing for the slight change of surface tension with temperature. The reason is that in the case of helium I

=------=---=----=-=--=

447

Transfer rates.

Sect. 38.

the liquid column in the capillary is held up by a "skin" at the top and at the bottom whereas in helium II the lower skin is "pierced" by the film. Another observation by the same authors could be explained equally well. They noticed that the gap between two plates, separated by spacers, only one of which dipped into the bath, would also fill up. Here the film had evidently to ferm bulk liquid

•

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-+--------+