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- BARSS
- WALTER MALCOMSON

ELECTRON DIFFRACTION IN GASES A Thesis Submitted to the Faculty of Purdue University by Walter Malcomson Barss in partial fulfillment of the requirements for the Degree of Doctor of Philosophy

May, 1942

PURDUE UNIVERSITY

THIS IS TO CERTIFY THAT THE THESIS PREPARED UNDER MY SUPERVISION

BY

W a l t e r Mai com son

Bnrss

ELECTRON Ll'ah'RhCTlOii

ENTITLED

IN GASES

COMPLIES WITH THE UNIVERSITY REGULATIONS ON GRADUATION THESES AND IS APPROVED

BY ME AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE DEGREE OF

Doctor of P h i l o s o p h y

\WAi-

\UrC^- W"^ June 13,

Professor In Charge of Thesis

^

3

Head of School or Department

19 42

TO THE LIBRARIAN : THIS THESIS IS NOT TO BE REGARDED AS CONFIDENTIAL.

V^rCv_4r Professor in Charge

Registrar Form 10—2-39—1M

TABLE OF CONTENTS I

II

Introduction

.

A

The Scattering of X-rays by Gases

1

B

The Scattering of Electrons by Gases

5

C

General Analysis Methods for Gas Diffraction

9

Experimental Equipment.

. . . . 13

A

Construction and Operation of Camera

13

B

Radial Intensity Compensation

16

C

A Mechanical Aid to Intensity Synthesis

19

III Practical Methods of Analysis

24

A

Measurements of the photographed diffraction pattern . . . . 24

B

Derivation of intensity curves

26

C

The Pauling and Brockway Method

3°

D

The Fourier Analysis Method

37

E

Synthesis of Intensity Curves

41

IV Application of Analysis Methods A

B

44

Comparison of Radial Distribution Methods for Carbon Tetrachloride.

V

1

..44

Determination of the Structvire of Chloropicrin CCI0NO2 . . . 45

C Partial Analysis of an Unknown Methane Derivative Conclusions

63

A

Discussion of the Calculated Structure of Chloropicrin . . . 64

B

Summary of Analysis Methods.

Bibliography

67

List of Illustrations and Tables following page Figure

1

The Electron Diffraction Camera

Figure

2

Correction for Background Intensity by a Rotating shutter. . 18

Figure

3

The

Figure

4

Radial Distribution Curves for Carbon Tetrachloride

Figure

5 Microphotometer and Intensity Curves for Chloropicrin. . . . 46

Figure

6

Intensity Curves, s^ I(s), for Chloropicrin.

46

Figure

7

Radial Distribution Curves for Chloropicrin

47

Figure

8 Use of the Radial Distribution Function in Selecting a Model 48

Figure

9 Model of the Chloropicrin Molecule, CCIoN02

aln x

Figure 10, 11, 12

13

Synthesizer

20 44

49

Comparison of Observed and Computed Intensities for Chloropicrin

.

....58

Figure 13

Intensity Curves for an Unknown Methane Derivative . . . . . 63

Figure 14

Radial Distribution Curve for an Unknown Methane Derivative. 63 following page

Plate

I

Plate

II

Plate III

The Electron Diffraction Camera and Auxiliary Equipment. . . 12 The

sin x

Synthesizer x Electron Diffraction Pattern for Chloropicrin

22 45 page

Table 1 Parameters for CCIoN02 for which Models were Calculated. . . . 57 Table 2

Correlation of best Calculated Curve vri.th the Observed Curve . 62

ABSTRACT A brief review has been given of the theory of scattering of electrons by gases. The equipment used for this research has been described in some detail, with particular reference to new or improved devices. The first of these is a continuously driven rotating shutter placed in front of the photographic plate to compensate for the rapidly increasing intensity of scattering at small angles. The second is a convenient mechanical aid to the calculation of intensity curves which facilitates the construction of the curves and the choice of model parameters. The standard methods of treatment of electron diffraction pictures have been discussed and some improvements introduced; in particular, a computation scheme adapted to the production of Radial Distribution functions for a preliminary choice of parameters. The structure of chloropicrin has been determined by the new methods and the results discussed in relation to other chemical and physical data.

The molecular parameters C — Cl = 1.75^0.0/A° angle Cl — C — Cl =

110.8°df2° and C — N = 1.59±0.04 A

have been determined and the parameters

N — 0 = 1.21 A°, angle 0 — N — 0 = 127° shown to be compatible with the observed diffraction pattern.

1

I A.

INTRODUCTION

The Scattering of X-rays by Gases

According to the classical theory of the interaction of electric charges with electromagnetic radiation, whenever radiation such as x-rays passes through a region containing electrons which may be forced into oscillation by the electromagnetic field, the radiation is scattered, with an intensity distribution given by the Thomson I

= Io ( e T

formula

(1 + cos8s6 )

{1

r

where I 0 = the initial intensity R

= the distance of observation from the scatterer

given by the expression I = I e t2

(1.2)

2 where f, the atomic structure factor for x-rays, is defined as the ratio of the amplitude scattered in a given direction by the atom to that scattered in the same direction by a single "point" electron. The factor f may be calculated theoretically if the distribution of electrons in the atom is known. Although it is possible to derive an expression for f assuming that the electrons are discrete point charges the results so obtained do not agree with experimental observation. Better results are obtained using the quantum mechanical picture of smeared-out electrons, in which the orbital electrons are considered as a diffuse spherically symmetrical cloud of charge about the atomic nucleus. The form of the electron cloud is best characterized by the density function u(r) which is a function only of the distance from the nucleus. If u(r) is the true charge density in electrons per unit volume, it is convenient to define the function v(r) by the relation v(r) = 4nr2 u(r)

(1.3)

i.e. v(r) is the number of electrons in a spherical shell of unit thickness and radius r. The atomic structure factor calculated assuming a continuous charge distribution is of the form f(B) = ^

R

v(r)^£dr

(1.4)

where R = the atomic radius. This form permits the calculation of f(s) for any desired atom since v(r) may be derived theoretically. Hartree2 has calculated v(r) for the lighter elements, of atomic number less than 20, and Thomas-* and Fermi4 have calculated v(r) for the heavier elements. James and Brindley5 have compiled tables of the function f(s) based on these calculations.

In the scattering of x-rays by a monatomic gas it is possible to obtain the curve f(s) experimentally.

Application of a Fourier trans-

formation to equation (1.4) then permits calculation of v(r) by the formula v(r) = JQ

sr f(s) sin srds.

(1.5)

This provides a possible experimental means of checking on the calculations of Hartree, Thomas and Fermi. In the scattering of x-rays by groups of atoms it is necessary to consider interference between rays scattered by the individual atoms. In the case of a crystalline solid the regular arrangement of the atoms gives discrete directions of reflection corresponding to various planes of atoms in the crystal. In the case of liquids and amorphous solids of monatomic composition there are no constant separations of neighbouring atoms and no preferred directions. Zernike and Prins treated the problem of scattering by liquids by introducing a density function g(r) which is the number of atoms per unit volume at a distance r from any one atom. They obtained an expression for the fluctuating part of the intensity of the form I(s) = NI e f 2 /^4TTr 2 g 0 (r) .5i£-2£ dr

(1.6)

where gQ(r) = g(r) - g~ g = average density; atoms per unit volume N = total number of atoms irradiated f = atomic structure factor Zernike and Prins then wrote (1.6) in the form s p(s) = 4rryo

g 0 (r) r sin sr dr

(1.7)

Application of the Fourier Integral Theorem then gives T

g0(r) ~ 2$?

JQ

s p(s) sin sr ds

(1.8)

Here again it is thus possible to calculate a distribution function from the fluctuating part of the observed scattered intensity. In treating the scattering of x-rays by gases the method is to calculate the intensity scattered by a molecule in a fixed orientation, then consider the effect of allowing the molecule to take all possible orientations. The intensity of the x-rays scattered by a single gas molecule is given by the Ehrenfest' formula

I = Ie I

k

i=l j=i

fi f j 5in

5

*M

(1.9)

s r^

where r^j = distance between atoms i and j f^

= atomic structure factor for atom i

In diffraction of x-rays by gases it is necessary to consider only the molecular structure and the form factors for the constituent atoms, because the large distances between molecules make the intermolecular effects negligible. In contrast, the scattering by a liquid whose molecules are polyatomic depends upon atomic, molecular and intermolecular interference with all three effects contributing to the observed pattern. It is important to note that the factor (sin sr)/sr occuring in formulas (1.4), (1.6) and (1.9) does not have the same significance in all three cases. In (1.4) it is due to consideration of simultaneous scattering by elements of charge in different parts of the atom and appears in the expression for the amplitude. In (1.6) and (1.9) the same factor occurs in the expression for the scattered intensity and is due to consideration of all possible orientations of a group of atoms with respect to the direction of the original x-ray beam. It should also be noted that only in diffraction by crystals is there a one-to-one correspondence between a diffracted ray and a specific spacing between atoms. In diffraction by liquids and gases there

is no such correspondence between rings of the pattern and specific interatomic spacings because each pair of atoms contributes a continuous intensity curve of the form (sin sr)/sr.

B.

The Scattering of Electrons by Gases g

In 1925 Elsasser

suggested that it might be possible to obtain

interactions between a beam of electrons and the regularly spaced atoms of a crystal. Two years later Davisson and Germer' succeeded in diffracting a beam of slow electrons from the face of a nickel crystal. Since then it has been found possible to treat the diffraction of electrons by considering the interference of their associated de Broglie waves scattered by the particles of an atomic or molecular scatterer. The de Broglie wavelength associated with an electron whose momentum is mv, is

X = h/mv

(1.10)

where h = Planck's constant The relativistic expression for this wavelength, in terms of the potential in volts through which the electron has fallen, is /v

I em0V/

V

1200moc2/

where m Q = rest mass of the electron. An approximate value, for lower voltages, is X

= /l50/V

x 10- 8 cm.

(1.12)

The essential form of the scattering of electrons by charged particles can be derived classically by the treatment Rutherford

used

for the scattering of alpha-particles. The effective cross section of a scatterer is given by rrb2 where b is the collision parameter. From consideration of simple mechanical laws and the Coulomb interaction between

6 charged particles one gets 2ze2 where e = electronic charge ze = charge on scatterer l/2mv2 = kinetic energy of electron 9 = half the scattering angle. The intensity, or number of electrons scattered through unit area at a distance r and angle 2.9 , is - SSI z*e** 1 1 - r3 'SWsiii4 0 T

nil* (1.14)

where Q, = number of electrons incident on scatterer N

= number of scattering particles per unit volume

t

= thickness

If the scattering particles are atoms instead of point charges ze^the factor z 2 is replaced by (z - f ) 2 where f is the atomic structure factor for x-rays. This shows that the atomic nucleus is largely responsible for the scattering of electrons and that the electrons surrounding the nucleus detract from the effectiveness of the nuclear scattering, particularly at small angles. This is to be contrasted with the x-ray case in which the effect of the nuclei was negligible. The factor (e 2 /mv 2 ) 2 when compared with the corresponding factor (e s /mc 2 ) 2 for x-rays, equation (1.1), shows that electrons are scattered more effectively than x-rays by the ratio (c/v)^. Practically, this means that scatterers must be much thinner for electrons and exposures need not be as long as for x-rays. The angular variation of intensity is given largely by the Rutherford factor, (1/sin^), which is completely different from the slowly-varying polarization factor for x-ray scattering. For electrons,

therefore, the intensity rises sharply and almost indefinitely as the scattering angle approaches zero. By using the de Broglie relation (1.10) with classical equation (1.14) we may obtain for the intensity scattered by n independent atoms, nl 0 /m(z-f)e212

1

.

where I Q = initial intensity. When we consider interference between electron waves scattered by different atoms we obtain the Wier.V

formula for scattering of elec-

trons by gaseous molecules. It has also been derived by quantum mechanical methods by Mott^2 and others.

K.) - i„(^) E [£ i**., %J?u+ *$.]

(1.16)

where s = 4TT sin 9

A

E = (z-fj/s* is the atomic structure factor for electrons f = atomic structure factor for x-rays r

ij

=

separation of axoras i and j

Sj_ = incoherent scattering function The part of this expression of interest in the study of molecular structures is the first term in the bracket. The double sum includes all pairs of atoms when the subscripts i and j are taken over all values for which i f- j. The terms with i = j, and hence r^j = 0, will contribute only a monotonically varying background to the diffraction pattern and represent the scattering by the various atoms considered independently. The terms in S take account of the inelastic scattering and also give no diffraction effects characteristic of the molecule. This function does not appear in the classical formula since it concerns cases in which the scatterer absorbs energy from the electrons. Values of the

8 function S have been tabled by Bewilogua ^ but it is usually impossible to make an exact correction for these terms, since they become indefinitely large at zero angle. The most important feature of the diffraction of both x-rays and electrons by gaseous scatterers is that each pair of atoms in a molecule contributes a continuous intensity curve of the form

sin

„, x x

whose

periodicity is characteristic of the separation of the atoms. There is thus no one-to-one correspondence between a ring in the pattern and a single atomic separation. The intensity at any part of the pattern depends on all the parameters determining atomic separations and hence the structure of the molecule. The diffraction of electrons by gases has the following distinctive features. 1. The useful fluctuations in intensity, due to interatomic interference, are in general small in amplitude compared to the general background scattered intensity, due to monotonically varying terms. In particular, the background intensity is always 3.arge enough so that even the most negative parts of the fluctuating intensity curve due to the molecular structure do not reach zero intensity. This property of molecular scattering recuires large differential sensitivity in the defecting device, e.g. high contrast in the photographic plates used to record the patterns. 2.

The background intensity varies inversely as the fourth power of the distance from the center of the pattern. This means that over most of the pattern the background is so steep that the molecular scattering appears merely as inflections in the intensity curve instead of apparent peaks and valleys. It is possible to observe such a superimposed pattern visually because the eye is sensitive to rate of

9 change of blackening, but visual measurements on rings are subject to error in the apparent positions of the peaks. These two conditions introduce experimental difficulties and seriously limit the accuracy of the calculations that can be made from electron diffraction photographs. In some cases such limitations may make the electron diffraction studies less valuable than x-ray studies in spite of the advantages of high intensities and short exposures of the electron diffraction technique.

C.

General Analysis Methods for Gas Diffraction

There are two general methods of treatment for electron diffraction pictures, either of which can be and has been used alone quite satisfactorily. However, the successive application of both methods is surest of giving accurate results and may cut down considerably the total amount of labor involved. The first is the method of synthesis, in which a reasonable model is assumed for the molecule from chemical or other data, and the intensity of electron scattering to be expected from such a model is calculated using equation (1.16). For an approximation, the simplified formula I(s) = £±

£ . ziZi

sln

sri

.i , i f j

(1.17)

is quite satisfactory for s not too small. The values of the parameters rj_j are chosen by trial until the position and shapes of the peaks on the calculated curve best match those on the observed pattern. The second is the method of harmonic analysis. The commonest form is that introduced by Pauling and Brockway ^' * and known as the Radial Distribution Method. Debye and Pirenne10 have given a complete discussion of the method and its application to the diffraction of both

10 x-rays and electrons. They conclude that the radial distribution method gives better results for electron diffraction than for x-ray diffraction, but warn that there is considerable chance for error in the positions of the peaks, particularly for the broader maxima. The electron intensity scattered by a molecule with fixed orientation may be written

-P? hkffPf*-e ik(r'{Vl"Ts)

i(s) =

avi *v-3

(1.18)

where k = 2rr/X v = O^i -