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LIBRARY

Michigan State
University

 

ADIABATIC PIEZO-OPTIC COEFFICIENTS OF LIQUIDS

by

Ward Arthur Riley, Jr.

A Thesis

Submitted to Michigan State University
in partial fulfillment of the requirements
of the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

1966

TABLE OF CONTENTS

PAGE
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
flmmw. ... ... ... ... ... ... ... ... ... .. 3
PREVIOUSEXPERIMENTALRESULTS.................. 6
SUMMARY OF RAMAN AND NATH THEORY OF ULTRASONIC LIGHT

DIFFRACTION . . . . . . . . . . . . . . . . . . . . . . . . . 8

EXPERIMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
mmmms.... ... ... ... ... ... ... ... ... .. 17
APPENDIX: DERIVATION OF THE LORENTz-LORENz EQUATION . . . . . . 25

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

ii

Figure
Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

LIST OF FIGURES

Diagram of the Ultrasonic Diffraction Grating.
Diagram of the Optical Arrangement. . . . . . .
Diagram of the Ultrasonic Test Cell.

Diagram of the Photomultiplier Circuit. . . . .

Block Diagram of the Electronic Circuit.

Graph of (p %%)s vs Refractive Index for Liquids
Tested in this Study. . . . . . . . .

Graph of (3%)8 vs Concentration for Methanol-Water
Mixtures. .

Graph of (3%)s'vs Viscosity for Dow Corning 200

Fluid Series. . . . . . . . . . . . . . . . . . .

. Model for the Calculation of the Local Electric Field

Produced by an Electromagnetic Wave. . . . . . . .

iii

PAGE
10
12
12
16
16

21

23

2h

27

LIST OF TABLES

TABLE PAGE

I. Summary of Previous Experimental and Computed Values

5
of (p'gg s . . . . . . . . . . . . . . . . . . . . . . . . . 7

II. Summary of Experimental Results. . . . . . . . . . . . . . . 19

iv

It: n\

0 la: la NH

['11

LIST OF SYMBOLS

adiabatic elasto-optic coefficient
adiabatic piezo-optic coefficient

density, density variation
sound velocity

refractive index, refractive index variation
pressure, pressure variation
bulk modulus

particle displacement

volume, volume variation
adiabatic compressibility

light wavelength

sound wavelength

permittivity

electric displacement vector
electric intensity

magnetic induction

magnetic intensity

velocity of light in free space

electric polarization

electric susceptibility

ABSTRACT

A knowledge of the variations in the optical properties
of liquids with pressure is required in ultrasound-light inter-
action studies. One parameter used to describe these variations is
the piezo-optic coefficient, defined as the change in refractive
index produced by a unit change in the applied pressure. The
relative adiabatic piezo-optic coefficients of numerous pure liquids
and methanol-water mixtures were determined from measurements made
on light diffracted by pulsed ultrasonic waves. Absolute values
of this parameter for the liquids tested were calculated from a
consideration of experimental values previously determined. Results
were compared with the theoretical and empirical formulas presently
used to compute this parameter and a new empirical equation was

obtained.

vi

ACKNOWLEDGMENT

The author wishes to express his appreciation to
Dr. E. A. Hiedemann and Dr. W. R. Klein for the assistance
received in the formulation and solution of this problem.
Discussions with Dr. B. D. Cook throughout the study have
been very helpful. The assistance of the other members of
the Ultrasonics Group at Michigan State University is also
greatly appreciated. Financial support of this research by

the Office of Naval Research is gratefully acknowledged.

vii

INTRODUCTION

Experimental studies involving the interaction of light and
sound in a liquid medium require a knowledge of variations in the re-
fractive index produced by the acoustical pressure. Two parameters
defined here to describe these variations are the elasto-optic coeffic-
ient (p 3%) and the piezo-optic coefficient (3%» which are related,
under adiabatic conditions, by the expression

5-3) 2 (5“)

(P 5p S = P Co 5p 8 - (1)

For a liquid of refractive index n a value of (p 3%
s

can be computed from each of the following expressions:

 

 

A. Lorentz-Lorenz ’2
(pg—p) = “Z'Lgff‘a” , (2)
B. Eykman3
(”3% : in:.-1)(n+.0,1+) (3)
n + O.8n + 1
and C. Gladstone - Daleu
(p 33),) = n - 1 . (A)

Values of (3%)8 can then be calculated from Eq. (1) if the values
of p and co are known for the given medium. Other expressions from
which to evaluate these parameters can be found in the literature but

are seldom used.

Values obtained from Eqs. (Z-M) differ by as much as 15
percent for certain liquids. In addition, experimental values with
which to compare these formulas are available for only a small number
of liquids.

In this study, the relative adiabatic piezo-optic coef-
ficients of numerous pure liquids and methanol-water mixtures were
determined from measurements made on light diffracted by pulsed
ultrasonic waves. Absolute values of (%%)s and (p'%%)s for these
liquids were then calculated from a consideration of existing
experimental data and Eq. (1), and the results were compared with
values obtained from Eqs. (2-h). A new empirical equation was

obtained from which (p 3%)8 can be calculated.

THEORY

The piezo—optic coefficient, defined as the variation in
refractive index produced by a unit change in the applied pressure,
is an important parameter in ultrasonic light diffraction studies.
In general, the fraction of the incident intensity of a light beam
diffracted, upon passing through an ultrasonic grating, depends upon

the maximum value of the variation in the refractive index

where 5pmax is the maximum acoustical pressure produced by the
ultrasonic source. Since we are concerned with small but rapid
pressure variations, it is assumed that the variations produced
in the refractive index occur under adiabatic conditions.

For a plane compressional sound wave propagating in a
linear liquid medium of low viscosity, the familiar wave equation

can be derived for small particle displacements E in the form

K626 625
“‘2‘ ' "'2' = 0 (6)
P 5x Ot

where p is the density of the medium and K the bulk modulus defined

by
_ _ .EEL.
K — Ov/v ° (7)

The adiabatic compressibility BS is defined as the reciprocal of the
bulk modulus. Thus, using Eq. (7) and the principle of conservation

of mass, we find

. 5p . <8)

. , . , n
Application of the chain rule to (p 3;)3, the elasto»
optic coefficient, yields an expression relating the elasto-optic

coefficient to the piezo-optic coefficient in the form

\wn l /}n)

5 KS; S ' (9)

The general plane wave solution to Eq. (6) has the familiar

form
1 ’t t k'x
g: go 9. (“3 ) (10)
with<m‘ and k' the magnitudes of the circular frequency and prOpa-
gation vector of the disturbance, respectively. Substitution into
Eq. (6) yields for the velocity of propagation of the disturbance
l
: ...... o (11)
o Bsp
Thus Eq. (9) can be written in the useful form
5n. 2 On
(P Op)s “' P Co (5195 y (12)
since from Eq. (11) we find that
1
BS " 2 . . (13)
PCO

In general (fig) and (p é3') depend upon the light frem
Op 5 Op 3
quency used (exhibit dispersion), the magnitude of the pressure
(are non-linear) and ambient pressure and temperature. In this
study, measurements were made at atmospheric pressure, room
temperature, moderate acoustical pressures, and the mercury green

0
line (5h6l A) was used.

, On
The quantity (p 55)s can be determined from several ex-
pressions. Differentiation of the well known Lorentz-Lorenz equation
(188O)1’2, which is derived in the appendix, yields Eq. (2), an expres-

sion for the elasto-optic coefficient

2 2
egg), = In-gflwe . (M)

n

 

The Eykman formula (1895)8

g = @2-1)(n+0.h)
(P Op)s n2 + 0.8n + 1 (15)

is an empirical expression which is used frequently in experimental

h
studies. The simple empirical relation of Gladstone and Dale (1863)

(p 3%)3 = n - 1 (16)

is useful because of its great simplicity. While this expression was
originally determined from measurements on air, it yields results close
to those of Eykman for several liquids.

Equation (1h) which was obtained from the Lorentz-Lorenz
equation represents the only theoretically justified expression present-
ly being used in experimental studies. While more rigorous treatments

5-10

of the problem have been given , the results, in general, are not
easily evaluated for real dielectrics at optical frequencies. Although

differences between Eq. (1h) and experimental values are well known, a

more useful theory has not been formulated.

PREVIOUS EXPERIMENTAL RESULTS

Raman and Venkataramanll, using a Rayleigh interferential
refractometer, measured the adiabatic piezo-optic coefficients of six
liquids to an estimated accuracy of‘: 0.1%. These results appear to
be the most accurate measurements taken for these parameters.

Motulevich and FabelinskiilZ-lh, employing an ultrasonic
light diffraction technique similar to that used in this study, pub-
lished values of the absolute elasto-optic coefficients for seven
liquids with an estimated accuracy of i_2.5% using #358 A light.

The accuracy of this work was apparently limited by the necessity
of measuring the absolute sound pressure using the radiation pressure

15 that

method. It has been more recently pointed out by Fabelinskii
errors may have been introduced due to the effect of acoustic stream-
ing.
. . . . 16
Additional measurements by Gibson and Kincaid on benzene

17

and Waxler and Weir on several common liquids, Using similar apparatus,
yielded values for the isothermal elasto-optic coefficients. The
isothermal values have been shown to differ only slightly from the
adiabatic values for these liquids11

All of these results are summarized in Table I and com-
pared with values computed from Eqs. (lh-l6). The large deviations
existing in values obtained by Gibson and Kincaid and Waxler and
Weir from those of Raman and Venkataraman indicate variations in these
parameters with large ambient pressure. We have confined our investiga-
tion to small pressure variations accompanying ultrasonic studies. The

O
values determined by Motulevich and Fabelinskii for M358 A light may

indicate dispersion in the visible spectrum.

6

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SUMMARY OF RAMAN AND NATH THEORY OF ULTRASONIC
LIGHT DIFFRACTION

The theory of Raman and Nath18 predicts that after being
diffracted by an ultrasonic beam the light will propagate in discrete
directions described by

mR
sin Bm = I7. (17)

where k is the wavelength of the light and m is an integer. Equation
(17) is seen to be the usual grating equation except that the grating
spacing is replaced by the sound wavelength A'. The refractive index

of the medium in the region containing the sound field is described by

n(x,t) = no-} Snmax sin (k'x -¢n't) (18)

where no is the undisturbed refractive index in the absence of any
sound field and bnmax the maximum variation in the refractive index
produced by the ultrasound. The symbols k' and4n' represent the
magnitude of the propagation vector and the circular frequency of the
ultrasound, respectively.

The optical prOperties of the grating can be described

by the two parameters

 

v = k snmax L (19)
2
k' L
Q = no k (20)

where k is the magnitude of the light propagation vector in vacuum

and L the width of the ultrasonic beam.

For normal incidence of the light upon the sound field
(see Fig. l), the amplitudes of the light in the various diffraction

orders, Oh, are described by the difference-differential equation

d 2

35:2 + igfwm - gm) = i—gfivgm (21)
subject to the boundary conditions at Z = 0

so = 1

em = O m .4 O . (22)

Equation (21) has simple solutions for Q << 1. In this
case the imaginary part of Eq. (21) can be neglected and the solu-

tions at Z = L are
¢ = J (V) (23)

where Jm(V) is the mth order Bessel function of argument V. Under
these conditions the only effect the sound has upon the light beam
is to produce a modulation in phase. This ultrasonic grating is
equivalent to an idealized diffraction grating which is localized
to a plane in space and produces only a phase modulation of the
light which passes through it. At normal incidence this phase dis-
tribution is an exact reproduction of the refractive index distri-

bution of the medium within the sound wave.

10

Light

LX

' '-

 

l —* Sound

 

 

 

 

 

 

 

 

Figure 1. Diagram of the Ultrasonic Diffraction
Grating.

EXPERIMENT

In this study intensity measurements made on the central
order of the Fraunhofer diffraction pattern produced by an ultra-
sonic grating permitted determination of the relative adiabatic piezo-
optic coefficients of numerous liquids. The standard optical arrange-
ment for observing Fraunhofer diffraction shown in Fig. 2 was used.
The light source S was a General Electric 100 watt HlOO-Ath mercury
arc. A green filter F was used to select only the 5h6l: wavelength.

The light beam was collimated by lens L and limited in size by the

2
rectangular aperture A. After the light passed through the tank
containing the medium through which the sound was propagated, the

image of the source 31 was formed in the plane of the photomultiplier

l
slit so that the light in only the central diffraction order was
incident upon the slit.

To satisfy the requirement Q << 1 as discussed in the
previous section, a one inch square 2 Mc barium titanate was used
as the ultrasonic source. This gives a value of Q::O.l. An optimum
value of V at which to work was found at V:: 1.6, where the central
order intensity J02(V) = O.hO and the slope of the intensity vs V
curve is large.

The test cell was divided into two sections by a thin
(.OOl") polyethylene membrane (see Fig. 3). The ultrasound was
then transmitted at normal incidence from section A which contained

a reference liquid, distilled water, through the membrane into section

B which contained the test liquid. The light beam was passed through

11

     

Photomultiplier

12

Sound

L3

 

 

A /’ A 81

Tank

 

       

Figure 2. Diagram of the Optical Arrangement.
Light
[0.75ch
l I
Membrane I I
A pi B
pr I pt I
—+

3-...” l'|l|'||||

I-I- - aw I'l'l'l'l'l'l'l'l'l'l

"' "" llllllllll
I! ! I '

Transducer SOHSCC—+—-4
l

 

 

 

 

Figure 3.

Diagram of the Ultrasonic Test Cell.

13

the sound field immediately upon entry into section B. Light in-
tensity measurements were then made on the central order of the
resulting diffraction pattern. This arrangement permitted the sound
source to transmit into the same immediate acoustic enviroment while
producing a diffraction pattern in liquids exhibiting various acoustic
properties. Calculations showed that the effect of absorption on the
sound field in the test liquids was negligible.

Equation (19) can be rewritten in terms of the adiabatic
piezo-optic coefficient and the maximum pressure amplitude of the

ultrasonic source by

V = kL (as) 5p . . (21)

For a given medium, an essentially linear relationship exists
between V and the rf potential applied to the transducer, Ea'
Thus, as all parameters except 5pmax are constant for a given
medium, we can express the pressure amplitude producing the dif-

fraction pattern in the test liquid by

5pmax = TAB 0 E3 (22)

where TAB is the pressure amplitude transmission coefficient through
the membrane and C is the constant of proportionality between
6p and E for water.
max a
The wavelength of the two megacycle ultrasound chosen
for this study was sufficiently greater than the thickness of the
membrane so that the effects of the membrane itself were negligible.

The transmission coefficient for the pressure amplitude through the

membrane then becomes simply

1h

TAB 2 I:?’ (23)
where
r = pBCoB (2h)
pACoA

h . .* ,
and PA’ pB’ coA’ COB are t e values of the density and sound velocity for
the liquids in sections A and B.
Initially the reference liquid, distilled water, was placed
in both sections of the test cell and the transducer voltage increased

until the central order intensity decreased to 40% of its initial value.

In this case pA COA = pB CoB’ the transmission coefficient was unity,
and 5pmax was given by Eq. (22) with TAB = 1.

Next a test liquid was placed in section B. A pulse voltage
E; which produced a relative central order light intensity of ho% was
then applied to the transducer. Since the light intensities were equal
in the two cases, the values of the parameter V for the two cases were

also equal. Since pA co was calculated using Eq. (23).

A I p3 CoB’ TAB
Equations (21 - 2h) then give

(332)]; = Ea . (25)
(5%)A TAB Ea

This expression permits the direct calculation of the relative adia-
batic piezo-optic coefficients of the liquids tested.

Clearly the small test cell, which permitted use of only
small quantities of the test liquid, would not permit the use of con-
tinuous waves due to reflections from the walls and production of
standing waves. Also, when making diffraction measurements with

continuous waves, the heating of the medium due to sound absorption

15

19

has been found to cause the diffraction grating to be unstable .
For these reasons the ultrasonic source was pulsed at a rate which
eliminated these problems.

A light beam width of 0.75 cm (about 20 sound wavelengths)
was used to insure a discrete diffraction pattern. The pulse length
was made long enough that end effects were avoided. However, the
rf pulse tended to "droop" for long pulse lengths. A pulse length
of 30 usec was found to be a reasonable compromise.

The pulse repetition rate was chosen to be 60 sec.1 in
order to synchronize the rf pulse with the intensity variation of
the mercury light source. Consequently the duty cycle, the fraction
of the time which the sound was actually introducing energy into the
medium, was only .0018. Also, these values for the pulse repetition
rate and pulse length minimized errors due to the limited size and
geometry of the tank. The ultrasonic pulse generator, an Arenberg
PG 650, was triggered by a unit pulser, General Radio type 1217-A,
and adjusted so that the sound pulse crossed the light beam at maxi-
mum intensity.

A photomultiplier circuit capable of responding during the
time interval in which the sound pulse crossed the light beam is shown
in Fig. A. An RCA 931A photomultiplier was used. Use of variable
resistor R0 permitted convenient adjustment of the output sensitivity
of the photomultiplier circuit. To assure linearity, the voltage across
the final dynode was biased as shown in Fig. h. A complete block
diagram of the electronic circuit used in making measurements is shown
in Fig. 5. The pulse voltages and light intensities were measured

using a Tektronix type 515A oscilloscope.

16

Q } Anode

 

To Oscilloscope

A\A A
v

V’Ro Dynodes
f

.__.. m .3
~100v
i

 

 

Figure h. Diagram of the Photomultiplier Circuit.

Oscilloscope 60 cycle sync

O

    
     
 

 
  

   

Unit
Pulser

  

rf
Pulser

Light

Photomultiplier Transducer

Figure 5. Block diagram of the Electronic Circuit.

17

All measurements were taken at atmospheric pressure and

I o + o
at a temperature of 2M - 2 C.

RESULTS

The relative adiabatic piezo-optic coefficients of 18 pure
liquids and 6 methanol-water mixtures were measured (relative to
distilled water) using the technique described in the previous section.
The liquids chosen for this study exhibited a useful range of refrac-
tive indices and adiabatic compressibilities.

Values determined for the relative and absolute piezo-
optic coefficients should be accurate to f h%. The main source of
error probably arose from the inhomogeneous sound field existing in
the Fresnel region of the transducer. Since the sound diffraction
pattern in the Fresnel region depends upon co for the particular
medium, one would expect differences in the sound field upon trans-
mission into different test liquids. The voltage measurements were
probably accurate to f 2%.

Raman and Venkataraman11 have measured the absolute
adiabatic piezo-optic coefficients for six liquids including water,
methanol and benzene which were also treated in this study. Using
the method of least squares, an absolute value of (3%)8 for water
was computed from a comparison with the results of Raman and
Venkataraman for the three liquids common to the two studies.
Absolute values for all liquids were obtained once a most probable

value was determined for water. Using the relation

(p —>. = (30: (3% S (26)

18

the absolute elasto-optic coefficients were calculated using
published density and sound velocity measurementsZO-ZS. Experi-
mental values for the parameters of interest are presented in

Table II. Values obtained in this work are given in Columns (5 - 7).
In Fig. 6, Eqs. (lh-l6) are compared to the experimental values
determined from this study and Raman and Venkataraman's earlier data.

Using the method of least squares an empirical relation

of the form
an 2
(p $8 = 63% - .395n - .263 (27)

was determined from the experimental values. This expression is
also shown in Fig. 6. Nearly all liquids tested were found to
have values for (p 3%)8 which lie within experimental error of this
curve. The Eykman formula, which was determined only for liquids
having high carbon content, agrees well with values obtained from
this study in the lower range of refractive indices. One is aware
that a relatively simple expression accurately predicting the elasto-
optic coefficients of all liquids is probably not obtainable. However,
this new empirical formula appears to give accurate results over a
wide range of refractive indices.

Figure 7 shows the absolute values of (3%)8 as a function
of concentration for methanol-water mixtures. Upon calculation of
the elasto-optic coefficients from experimental values of (3%)8, one

finds (p 3%)8 to be a constant within experimental error. Thus, we

see that (3% s varies directly as the adiabatic compressibility.

19

 

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__ A
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D: n - l

 

l l J l l l
1.31 1.36 1.38 1.10 1.h2 1.uh 1.h6 1.u8 1.50 1.52

l L l I
Refractive Index n

Figure 6. Graph of (p 3%)8 vs Refractive Index for Liquids

Tested in this Study.
21

[‘0
N

The average elasto-optic coefficient for these mixtures was found to be

0.3h9. Thus from the expression

A
O/l Q/
“U :3

) = O 3M9 BS (28)

s
we can determine the piezo-optic coefficients of any given methanol-
water mixture. Equation (28) is compared to experimental values in
Fig. 7.

In Fig. 8 values of (3% s for the Dow Corning 200 fluid
series are plotted as a function of the viscosity specified by the
manufacturer. One also finds an essentially constant value of 0.h20
for the elasto-optic coefficients for this series. The useful

expression
an
(5;)8 = 0.h20 BS (29)

permits calculation of the piezo-optic coefficients of liquids in
this series which exhibit a refractive index range of 1.375 to
l.h00. Equation (29) is compared to experimental values in Fig. 8.
The lower viscosity liquids in this series, as a result of their
relatively large adiabatic compressibilities, were found to have
adiabatic piezo-optic coefficients up to five times that of water and
approximately twice that of the other liquids used in this study.
Their relatively large adiabatic compressibilities enable one to
produce a much greater variation in the refractive index for a given
applied pressure. For this reason these liquids have been found use-
ful in light modulation studieslg.
Measurements made on distilled and tap water from several

sources yielded insignificant differences in their relative piezo-

Optic coefficients.

(.

a» 8

12

x 10,

Gm
dyne

38

36

31+

32

30

2h

22

20

18

16

 

23

(yp) = .3119 BS .

O

I I I I I, I I

I
0.1 0.2 0.3 0.11 0.5 0.6 0.7 0.8 0.9 1.0

Concentration (mole fraction)
Figure 7. Graph of (8%)3 vs Concentration for Methanol-Water

Mixtures.

2h

3; = .1120 as

(53):;

x 1012 _

 

dyne ‘ .

55 "' .

 

IIIII J I llllIlJ I
l 2 5 10 ' 20

Viscosity (centistokes)
Figure 8. Graph of (8%)3 vs Viscosity for Dow Corning 200

Fluid Series.

APPENDIX
Derivation of the Lorentz-Lorenz Equation.

A derivation of the Lorentz-Lorenz equation can be found in
numerous standard references discussing electromagnetic wave propaga—
tion in dielectric media. This equation is the alternating electric
field analogy to the well-known Clausius-Mossotti:equation. BaSically,
the difference arises from the fact that both the dielectric constant
and the polarizability of the molecules must in general be considered
to be dependent upon the frequency of the electric field. Even though
one assumes the molecules to be isotropic in this derivation Hans Mueller
showed that only small deviations from this equation should exist even
with molecules exhibiting considerable anisotropy. Several other theo-

retical paperss’7-10

compare their results to those of Lorentz and
Lorenz either by making a suitable correction to the final form of
this equation or by discussing basic assumptions made in the deriva-
tion. Consequently, because of the great importance of this expression
to experimental studies and its frequent appearance in theoretical papers,
it is derived here for the reader's convenience.
For a plane transverse electromagnetic wave propagating in a

uniform isotropic non-permeable medium characterized by permittivity
e and the relations

2= es (1.)

and

i = E (ea)

25

26

we can describe the behavior of the associated electric and magnetic

fields by Maxwell's equations which take the form, in the absence of

sources,
2-§=0 (3a)
1 BB
YX§=-;-a-'§- (he)
52. §,=<D (5a)
6 0E
2x2=337 we

Upon combining the two curl equations and making use of the vanish-
ing divergences, we find that the cartesian components of the field

vectors E_and B must satisfy the wave equations

6'
V6 Ei - 02 E1 = 0 (78)
i = X, y, z
e'
VZBi-C—ZBi=O (8a)

These are well known wave equations and imply that the velocity of

propagation of these waves u is

C

61/2

 

(98)

11:

Equations (7a) and (8a) have identical general solutions which
satisfy similar boundary conditions. The optical properties of
the medium are clearly determined from Eq. (7a).

Within the medium, the effective electric field E' acting
on a particular molecule differs from the observed macroscopic field
go because of the spacing between the molecules and the net polariza-

tion of the surrounding medium. This polarization is expressed in

27

 

x
Figure la. Model for the Calculation of the
Local Electric Field Produced by
an Electromagnetic wave.
terms of macroscopic quantities by
3=Xe§o (10a)

where X5, the electric susceptibility, is related to the permittivity
by

Xe = 55:71 (11a)
Consider a single molecule and imagine it to be surrounded by a small
aphere whose radius is still large compared with the molecular dimen-
sions. The effective field at the site of this particular molecule can

be expressed as

I _
E' - Eo + Ein + 1E-out (12a)
where};in is the contribution due to the matter inside the sphere
and gout the contribution due to the matter outside. For simplicity

we shall consider a linearly polarized light wave travelling in the

28

x direction and having a component of electric field in the y direction
only.

The field E. is very difficult to determine rigorously26.

in
Calculations have been made for media exhibiting rigid symmetrical
molecular distributions and media containing completely random
distributions. The electric field E1“ was found to be zero in both

instances. Thus, we shall assume in this derivation that};in = 0.
The problem is then reduced to that of finding the contribution of
the electric field at the center of an evacuated sphere due to the
polarization induced on the surface of discontinuity (see Fig. 9).
This charge density is - §'- 2' where n is the outward

normal from the spherical surfaces. Upon integrating over the

spherical surface, one finds the resulting field at the center to be

hr
Eout = 3—W3 ° (138)

Substitution for P from Eqs. (10a) and (11a) yields

 

é'-1
gout — 3 ‘Eo ° (lha)
Finally we obtain
€E+2
I _
E _ 3 E0 . (15a)

The polarization of the medium can also be defined in
terms of molecular parameters by

g=N<oz>§' (16a)

where N is the number of molecules per unit volume and < a2> the
mean polarizability of the molecules. Comparing Eqs. (10a) and

(16a) and using Eq. (11a) we find

 

39

 

This can be written in terms of the density of the medium p,

Avogadro's number N and the molar polarizability ah as

 

A
733 _ 65-1
3 NA at P ‘ ea+2 °

The refractive index of the medium is defined by

n .2
I! u .
Since from Eq. (9a) u = -£I72 we find the familiar Maxwell's
é?
formula
[12:6

 

(17a)

(18a)

(19a)

(20a)

(21a)

L-"E. in w ...-

 

10.

11.

12.

13.

1h.

15.

16.

17.

18.

19.

BIBLIOGRAPHY

H. A. Lorentz, Ann. Physik 9, 6hl (1880).

L. Lorenz, Ann. Physik ll, 70 (1880).

M. J. F. Eykman, Rec. des Trav. Chim. 15, 185 (1895).

J. H. Gladstone and T. P. Dale, Phil. Trans. 153, 337 (1863).

V. Raman and K. S. Krishnan, Proc. Roy. Soc.(London) A111,
589 (1928); Phil. Mag. '5, A98 (1928).

H. Mueller, Phys. Rev. 29, 5H7 (1936).
J. G. Kirkwood, J. Chem. Phys. E, 592 (1936).
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C. J. F. BSttcher, Theory of Electric Polarisation (Elsevier,
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A. Isihara, J. Chem. Phys. 38, 2&37 (1963).

V. Raman and K. S. Venkataraman, Proc. Roy. Soc. (London)

111. 137 (1939).

G. P. Motulevich and I. L. Fabelinskii, Bull. Acad. Sci.
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Advances in Physical Sciences (State Publishing House for

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vol. 63., p. 500.

R. E. Gibson and J. F. Kincaid, J. Am. Chem. Soc. 60, 511
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R. M. Waxler and C. E. Weir, J. Research Natl. Bur. Standards
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C. V. Raman and N. S. Nagendra Nath, Proc. Indian Acad. Sci.
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W. R. Klein, Ph.D. Thesis, Michigan State University (196%).

30

20.

21.

22.

23.
2h.

25.

26.

31

W. Schaaffs, Molekularakustik (Springer-Verlag, Berlin,
1963)-

Raymond M. Wilmette, Inc. Ultrasonic Spectrum Analyzer
Report No. 1h. Final Report to Rome Air Development
Center, Dec. 20, 1955. AD 88698.

Handbook of Chemistry and Physics (Chemical Rubber Publishing
Co., Cleveland, 1958)739th ed.

Dow Corning Bulletin 05-061 (1963).
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L. Bergmann, Der Ultraschall (S. Hirzel Verlag,
Stuttgart, l95h).

M. Born and E. Wolf, Principles of Optics (Pergamon Press,
New York, 1959).

_'.-.
a 1"