Table Of ContentATOMISTIC 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.
June.1950
UMI Number: EP63353
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TABLE OF CONTENTS
CHAPTER PAGE
I . INTRODUCTION .................................. 1
1.0 Statement of the problem............. 1
1.1 Importance of the p r o b l e m ........... 2
1.2 Notation............................... 5
"1-3 Organization of the thesis . . . . . . . 6
II. SECONDARY W A V E S ............................. 14
2.0 Introduction of secondary waves . . . 14
2.1 Calculation of the secondary waves . . 15
2.2 Normal incidence at a plane boundary . 17
2.3 Discussion of r e s u l t s ............... 25
2.4 Limitations of Esmarch^ work . . . . 26
III. THE EXTINCTION R U L E .............. 28
3.0 The interaction of dipoles............ 28
3.1 The exciting electric field ......... 29
3.2 The exciting magnetic field ......... 36
IV. THE RELATION BETWEEN THE EXCITING AND THE
OBSERVED FIELD ........... . . 39
4.0 Distinction between exciting and
observed field ........................... 39
4.1 A cavity in a polarized dielectric . . 40
4.2 A subtle c h a n g e ..................... 43
iii
CHAPTER PAGE
A.3 Intramolecular fields ............... 47
4.4 The magnetic f i e l d .................. 5^
V. BASIC APPLICATIONS .............. . . . . . 55
5.0 Maxwell*s equations .................. 55
5.1 Snell!s l a w ........................... 57
5.2 Incident and refracted intensities . 63
5.3 Reflection . .. V. . . . . . . . . . . 65
5.4 Further applications ......... 67
VI. RELAXATION OF CERTAIN RESTRICTIONS . . . . 68
6.0 Simplifying assumptions ............. 68
6.1 The restriction on wavelength . . . . JO
6.2 Further work of L u n d b la d ........... 75
VII. ELEMENTS OF NEUTRON OPTICS ................ 83
7.0 The problem and the basic equation . 83
7.1 Reflection and refraction by a non
absorber ............................... 86
7.2 Amplitudes of the waves............. 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
VIII.’ CONCLUSIONS AND EXTENSIONS ............... 103
8.0 The final c h a p t e r ................. 103
8.1 Electromagnetic waves ............. 103
8.2 Neutrons............................. 106
8.3 Extension........................... 107
BIBLIOGRAPHY....................... . 109
CHAPTER I
INTRODUCTION
1.0 Statement of the problem. The main problem
investigated is the derivation, 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, Qptik,
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 (1)
by Willibrord Snell in 1621. The first "theoretical"
explanation of the law was offered by Rene
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,
Maxwell’s equations for nonmagnetic dielectric media
require that the square of the index of refraction equal
the dielectric constant, if we are to assume light to be
an electromagnetic wave. The failure to predict dispersion
was one of the reasons it became apparent that Maxwell’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 Maxwell’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, 1638.
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. Within
these limitations, however, 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 Ibid., 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.
