Atomic theory of index of refraction

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ATOMISTIC THEORY OF INDEX OF REFRACTION

A Thesis Presented to the Faculty of the Department of Physics The University of Southern California

In Partial Fulfillment of the Requirements for the Degree Master of Science in Physics

by Charles Bergman Shaw, Jr. J u n e .1950

UMI Number: EP63353

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T h i s thesis, w r i t t e n by ......... u n d e r the g u i d a n c e o f h.IS ... F a c u l t y C o m m i t t e e , and approved

by a l l

its

members, has been

presented to a n d accepted by the C o u n c i l on G r a d u a t e S t u d y a n d Research in p a r t i a l f u l f i l l ­ m ent o f the r eq ui re m en ts f o r the degree o f .............

....

Date.

SEP.

.1.950___

Faculty Committee

ulhairman

.....

' I **

TABLE OF CONTENTS CHAPTER

PAGE

I . INTRODUCTION ..................................

1

1.0 Statement of the p r o b l e m .............

1

1.1 Importance of the p r o b l e m ...........

2

1.2 N o t a t i o n ...............................

5

"1-3 Organization of the thesis . . . . . . .

6

II.

SECONDARY W A V E S ............................. 2.0 Introduction of secondary waves

...

2.1 Calculation of the secondary waves

IV.

14

. .

2.2 Normal incidence at a plane boundary

III.

14

.

15 17

2.3 Discussion of r e s u l t s ...............

25

2.4 Limitations of E s m a r c h ^ work

26

. . . .

THE EXTINCTION R U L E ..............

28

3.0

The interaction of d i p o l e s ............

28

3.1

The exciting electric field

.........

29

3.2

The exciting magnetic field

.........

36

THE RELATION BETWEEN THE EXCITING AND THE OBSERVED FIELD

......... ..

. .

39

4.0 Distinction between exciting and observed field ........................... 4.1 A cavity in a polarized dielectric 4.2

. .

A subtle c h a n g e .....................

39 40 43

iii CHAPTER

PAGE A.3 Intramolecular fields

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

47

4.4 The magnetic f i e l d .................. V.

. . . . .

55

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

55

5.1 S n e l l !s l a w ...........................

57

BASIC

APPLICATIONS

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

5.0 Maxwell*s equations

5.2

Incident and refracted intensities

5.3

Reflection

63 65

.........

67

. . . .

68

RELAXATION OF CERTAIN RESTRICTIONS

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

68

6.1 The restriction on wavelength . . . .

JO

6.2 Further work of L u n d b l a d ...........

75

6.0 Simplifying assumptions

VII.

.

. .. V. . . . . . . . . . .

5.4 Further applications VI.

5^

ELEMENTS OF NEUTRON OPTICS 7.0

83

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

The problem and the basic equation

.

83

7 . 1 Reflection and refraction by a non­

absorber

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

86

7.2 Amplitudes of the w a v e s .............

89

7.3 The wave equation for an absorbing m e d i u m .................................

92

7.4 Reflection and refraction by an absorber

. .

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

93

7.5 Amplitude relations ..................

96

7 . 6 Relation of oc0 to cross section

99

...

iv CHAPTER

PAGE

V I I I . ’ CONCLUSIONS AND 8.0 The final

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

103

c h a p t e r .................

103

EXTENSIONS

8.1 Electromagnetic waves

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

103

8.2 N e u t r o n s .............................

106

8.3 E x t e n s i o n ...........................

107

B I B L I O G R A P H Y .......................

.

109

CHAPTER I INTRODUCTION 1.0

Statement of the problem.

investigated is the derivation,

The main problem

from the interaction

between an electromagnetic wave and molecular scattering centers, of the index of refraction of an isotropic medium.

This is based on a review of the literature

which began, with the references cited in M. Born, Q p tik, and included a check of all later papers recorded in Physics Abstracts.

It is assumed that all light waves

obey Maxwell's equation for the vacuum in propagating through the intermolecular spaces of material media, any apparent deviation therefrom being due to interference among primary and secondary (scattered) waves.

On this

basis the Lorentz-Lorenz equation is derived in a very general form and the limitations on it indicated.

Inci­

dental to the consideration of ordinary refraction, the relaxation of certain simplifying assumptions and restric­ tions is found to account for diffusion and opalescence, the non-linearity of the Lorentz-Lorenz equation for complex molecules and, at least qualitatively,

for optical

activity and certain deviations from the Lorentz-Lorenz rule. While the concept of index of refraction has

2 generally been used for electromagnetic waves, the inci­ dence of a beam of neutrons on the plane boundary of a homogeneous medium can also be treated as an "opticalM problem, with the Schroedinger equation replacing Maxwell's.

This problem, however,

is not treated here

on a strictly atomistic basis, average effects being considered instead of the elementary scattering processes. The laws governing the direction and intensity of the reflected and of the refracted beam are found for both nonabsorbing and absorbing media, the latter case embody­ ing some original work.

Total reflection, known to be

possible at the boundary of a nonabsorbing medium,

is

found to be theoretically impossible for absorbers, but when an inequality is deduced for the minimum capture cross section necessary to produce a given destruction of total reflection,

it is found that the smallest

readily measurable deviation from total reflection re­ quires a capture cross section larger than that of most absorbing media. 1.1

Importance of the problem.

The concept of

index of refraction dates back to the earliest days of optics and the discovery of the empirical law sin i = n sin r by Willibrord Snell in 1621.

The first "theoretical"

explanation of the law was offered by Rene

(1)

3 Descartes^ on the basis of his metaphysical ideas, and every succeeding theory of optics has-had to derive it as one of the basic tests of validity.

Thus, for example,

M a x w e l l ’s equations for nonmagnetic dielectric media require that the square of the index of refraction equal the dielectric constant, an electromagnetic wave.

if we are to assume light to be The failure to predict dispersion

was one of the reasons it became apparent that M a x w e l l ’s equations would have to be modified.

Now in earlier days,

before the electromagnetic theory of light, various physicists

(such as Sellmeier and Helmholtz) had explained

dispersion on the assumption that ponderable bodies con­ tain small particles that are set vibrating by the inci­ dent light and that have a certain mass.

If light repre­

sents an electric field, what then was more natural than to consider these particles charged?

(This in fact was

one of the ideas leading to the theory of electrons.2 ) Instead of merely using M a x w e l l ’s equations for dielectric media and saying the polarization appearing therein satisfies a certain vibration equation, one could then study the fields of these electrons under the influence

1 R. Descartes, Dioptriques, Meteores, Leyden, 1 6 3 8 . 2 H.A. Lorentz, Problems of Modern Physics, p. 170.

4 of light.

Assuming electronic equations of motion of in­

creasing complexity,

there were obtained increasingly good

approximations to the curves representing the actual dis­ persion and absorption.3

in this way the actual behavior

and nature of the electrons could be studied. In making more detailed, more general, and more careful attempts to predict the optical properties of matter from its molecular constitution, much is learned about the limitations of the classical theories of the electromagnetic field and of the nature of matter. these limitations, however,

Within

it is possible to correlate

index of refraction and molecular structure in such a way as to serve many practical purposes, particularly in the fields of organic and colloid chemistry.^

The desirability

of putting on a sound theoretical basis, especially as regards its limitations, anything of great practical value seems obvious, and the practicality of index of refraction needs no greater evidence than the number of papers

3 I b i d ., pp. 169-174. 4 For a few significant examples, see: A.S. Chakravarti, Current Science, 15:105, April, 1946. W. Heller, Physical Review, 68:5* July 1 and 15* 19^5. E. Kordes, Zeitschrift M r physikalische Chemie, B44:249 and B44:327* 1939.

5 published each year under the title,

“A New Refractometer.H

To the physicist, however, the new information obtained on the picture of light and matter must be the primary source of interest in a study of the theory of index of refraction. Where the reflection and refraction of neutrons is con­ cerned, the opportunity to verify and clarify present concepts of m a t t e r - i n c l u d i n g the neutrons themselves-is even more obvious.

Even the brief treatment in Chapter

VII, which certainly could not go very deeply into basic questions,

leads to results which are rather interesting.

1.2

Nota t ion.

Throughout the work on electromag­

netic waves the electric and magnetic fields are represented by complex quantities so that the convenient exponential form may be used.

It is understood that only the real part of the

quantity in question is to be considered. E - A e 1 ^*' = (a + ib)

Thus

(cosw/t + i s i n ^ t )

is understood to mean actually E = a cosrJ,

and the integral on the right is readily transformed. On

r>

r' j

by G r e e n ’s theorem

... where the last integral extends over the external surface P of the medium and over S(1T, r 0 ), and N is the outwarddirected normal to the surface element dS-a, at R ! . About this point Oseen assumed that the functions f ; satisfy the equations

.

(g )

As a new condition this overdetermines the unknowns E ! and f;,

32 but it can be shown to be satisfied for plane waves.^ Assuming then that (6 ) can be justified, it follows

/-*s Now we may take curl” curl” back inside the transformed integral, and let R lf= R.

Thus by (5 ) and (7)*

' - - r / e . ) -j

u, c -r/d) A_

-

rr ----------- J

(8)

This is O s e e n fs result, obtained without his compensating errors in the interchanging of integration and differenti­ ation. From the vector identities curl (aB) = a curlB + grada * B, c u r l ( A * B ) = A divB - B divA + (B * v)k - (A * v)B, where by definition

and the fact that the fj (R1) are constant vectors so far as the operator curl is concerned, it is readily verified that curl a u r i

2 Cf. post. Chap. VI, 6.0.

~ fj

33 where' ; n' ■>.' v/r)

K V* ~

Thus

e * ~ ‘* - r/c)

*•.= . j_£

r

T —

J r

on S(R,r« ), where (d/dN) = -(d/dr), c u r / c U r l L j % f] = ~ ( r-

~

and

r & ( Jr ) ( ^ f e - "r)j

_

Curl c u r l C S $ $ ] = [ ( £ • ? ) ~[(~6

’jfvV) c)ra d f

-

J*

— -fj 3 * ^ ^

- a ^ r f

- ( § - $ ) * & ( £ ) ] .

Writing A r-

- I ~~

c.

f ; ( R'J — f } ( F() -r ( r .y) f j (R.)

= --jrfp.7) f/CR;

, _ ..

C 3

c. 1

f

2 (

3 ~fj ( R)

‘

•+ j r i r - v ^ & C R ) - ■ ■■

Except for quantities which vanish with r 0 , the result is

K

.

G (ft + i

(Ki

^

7

(X1

}

Adding the contribution of the integral over F, and summing over j, we obtain finally j= t

Z.

r»r,

cvr)

c. u r I /C O *

=

( f t ; ■£- r/,£ ) 1 ±s

C t/r) m

"V':;

/-

*-2) j?* £ T?

Jl X : d

>-r

t

'

j

_

tr £•

+•