THE STRAIN - OPTICAL CONSTANTS or SOME: : A '
SODIUM HALIDES

Thesis for the Degreé of Ph.‘ D. . 7 -
MICHIGAN STATE umvmsm
HOWARD E, PETTERSEN ~ ‘
1968

This is to certify that the

thesis entitled

THE STRAIN-OPTICAL CONSTANTS OF SOME SODIUM HALIDES

presented by

Howard E. Pettersen

has been accepted towards fulfillment
of the requirements for

PhD degree in Phy S iC S

2 . A . ‘kaLctLLM/wtm

Major professor

 

.Date In April 1968

0-169

“ m Ph- «ad

 

 

 

 

 

.Il

 

 

 

ABSTRACT

The strain-optical constants are calculated for
sodium fluoride, sodium chloride and sodium bromide from
static compensator measurements and dynamic ultrasonic
measurements. A new pulsed ultrasound-optical technique
is used to measure various ratios of the constants and a
Babinet compensator is used to measure the relative retar-
dation for light polarized parallel and perpendicular,
respectively, to the static stress. The phenomenological
theories for both types of measurement are presented. The
experimental results are compared with all previous obser-
vations and values of the strain-polarizability constants

are calculated.

THE STRAIN-OPTICAL CONSTANTS OF SOME SODIUM HALIDES

Howard E. Pettersen

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1968

 

 

 

 

 

'_ MM

ACKNOWLEDGEMENTS

The author wishes to express sincere appreciation
to Dr. E. A. Hiedemann for his sympathetic assistance, over
an extended period of time, leading finally to the completion
of this investigation. Appreciation is also due Dr. W. R. Klein
for discussions and advice both prior to, and during, the early
stages of the investigation and also, to Dr. B. D. Cook for
invaluable guidance and assistance in bringing the investiga-
tion to a conclusion. Thanks are extended to Dr. W. W. Lester
of Corning Glass Works, Corning, New York for providing the
sample of fused silica. The National Science Foundation is

to be thanked for financial support.

Howard E. Pettersen.

ii

ru-- ‘-

 

 

 

 

 

 

TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION . . . . . . . . . . . . . . . . . 1

II. PHENOMENOLOGICAL THEORY . . . . . . . . . . . it
A. General h fr
B. Static measurements 13 i
C. Dynamic measurements 21

III. EXPERIMENTAL PROCEDURE AND RESULTS . . . . . . 3O 2 x
A. Static measurements 30
B. Dynamic measurements 3h

C. Determination of the strain-optical

constants pij #1
IV. EXPERIMENTAL RESULTS OF OTHER INVESTIGATORS . 50
V. ‘MUELLER'S THEORY OF ARTIFICIAL BIREFRINGENCE . 5h

VI. CONCLUSION . . . . . . . . . . . . . . . . . 6h

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 65

 

APPENDICES . . . . . . . . . . . . . . . . . . . . . 68

iii

 

 

LIST OF TABLES

 

Table Page

1 The change in Babinet compensator reading
per kilogram change in load . . . . . . . . . . 3h

2 Ratios of the v-values . . . . . . . . . . . . H1

3 Constants used in the calculations . . . . . . . #2

h The six equations for the pij . . . . . . . . #3 {I

5 Comparison of static results . . . . . . . . . . uh

6 Comparison of dynamic results from Table 2 . . MS

7 Separate solutions of the static equations and I
of the dynamic equations . . . . . . . . . . . . #6

8 The strain-optical constants . . . . . . . . . . h?

9 Ratios and differences of the pij . . . . . . . h?

10 Comparison of observed and calculated quantities #8

11 The experimental results of other investigators . 50

12 Refractivities in cm3/mole . . . . . . . . . . . 61

13 Strain-polarizability constants . . . . . . . . . 61

1h Data sheet for static measurements . . . . . . . 68

15 Data sheet for dynamic measurements . . . . . . . 71

 

iv

 

LIST OF FIGURES

Figure Page
1 Crystal Orientation . . . . . . . . . . . . . 12
2 Theoretical light intensity curve . . . . . . 23
3 Apparatus for the static measurements . . . . 31
h Stressing apparatus . . . . . . . . . . . . . 33
5 Apparatus for the dynamic measurements . . . . 38

6 Oscilloscope display of the photomultiplier
output C C C C C C C C C C C C C C C C C C C C lIIO

7 Babinet compensator reading versus load . . . 70

 

CHAPTER I

INTRODUCTION

 

An optically isotropic transparent medium becomes
anisotropic, and an anisotropic medium has its anisotropy changed,
when it experiences a stress or a strain. If the medium is crystal-
line, the change which is produced is dependent upon the direction, 1 l
as well as the magnitude, of the stress. The strain-optical con-
stants relate the change in the optical properties to the six
components of strain and the stress-optical constants relate the
change in the optical properties to the six components of stress.

The two sets of constants are related to one another by the elastic
constants of the medium.

Prior to the discovery of the optical effect of ultra-
sound, measurements of the stress-optical constants were made
statically. The relative retardations were measured by a compen-
sator and the absolute retardations were measured interferometrically.
Dynamic measurements can replace the less accurate interferometer

measurements. (Vedam and Ramaseshan speak of the difficulties

 

involved in the interferometric measurements in Progress in Crystal
Physics, edited by Krishnanl.)

Only one investigator has studied any of the sodium
halides other than sodium chloride and he utilized interferometric
measurements. Many investigators have studied sodium chloride,

but only two of these made all of the measurements necessary for the

 

 

 

absolute determination of all of the constants. There is,
furthermore, considerable variation between the results re-

ported by the various investigators. The present investigation
determines the strain-optical constants of sodium fluoride, sodium
chloride and sodium bromide. Originally it was planned to include
sodium iodide to complete the study of the sodium halides, but
since this compound is very hygroscopic it was impossible with

the experimental arrangement to obtain any results.

The experimental work involved both dynamic and static
measurements. The dynamic measurements utilized a new pulsed
ultrasonic-optical technique to determine various ratios of the
strain-optical constants. The pulsed ultrasound alleviates the
heating problem which is encountered when a continuous sound wave
is used. In the static measurements a Babinet compensator was
used to measure the relative retardation for light polarized
parallel and perpendicular, respectively, to the static stress.
The phenomenological theory for the dynamic measurements assumes
that a longitudinal sound wave produces only longitudinal strains,
while the theory for the static measurements considers the shears
and lateral extensions produced by a compressive stress.

The crystals which were studied were in the form of
cubes 2.00 cm on an edge with surfaces oriented (lIO), (110),
(001), so that three different experimental conditions were

possible. The stress (or strain) could be applied along an axis

 

 

 

of the unit cell, and the light propagated along a face diagonal

of the unit cell. On the other hand the stress (or strain) could
be applied along a face diagonal and the light could then be pro-
pagated either along the cell axis or along the other face diagonal.
The six experimental results were then combined to obtain the three
strain-optical constants. The experimental results have been com-
pared with all previous observations and also, they were used to

calculate strain-polarizability constants.

 

For a verification of technique and apparatus, two glass
samples were investigated before the crystals were studied. One
sample, EDF-h, had been studied extensively in the ultrasonic
laboratory at Michigan State UniversityZ-S, while the other
sample was a type of fused silica which had been investigated by
Primak and Post6. (Also see Vedam, et.al.7). The dynamic measure-
ments differed by 1.8% and 0.5% for the EDP-h and the fused silica,
respectively, and the static measurements differed by 2.7% and
0.9% respectively.

The crystals were obtained from Harshaw Chemical Company

of Cleveland, Ohio.

 

 

 

__‘v-’*‘“ v.

,-,

 

 

CHAPTER II

PHENOMENOLOGICAL THEORY

A. General

 

Many crystal properties can be represented mathematically
by tensors and geometrically by surfaces. The refractive indices

of a crystal may be represented by an ellipsoid which is known as ‘

F t
the optical indicatrix or the index ellipsoid8. In a coordinate
system whose axes coincide with the principal axes of the ellip-
soid, the equation of the ellipsoid is ” x

x2 2 £3
—§-+ XE + -—§ = 1 , (1)
n n n

x y z

where nx, ny and n2 are the three principal refractive indices.

A light wave propagating through the crystal can be represented
by a vector from the center of the ellipsoid in a direction normal
to the wavefront. The plane normal to the prepagation vector
intersects the ellipsoid in an ellipse whose semi-axes represent
by their directions the directions of vibrations of the electric

displacement vectors, D, and represent by their magnitudes the

 

indices of refraction for these two polarizations. For light

propagating in the z-direction, the equation of the ellipse is

x2 2
-—2-+'L2:1 (2)
n n
x y
2 2
or allx + 822y’ = 1 . (3)

 

If the coordinate axes and the axes of the ellipsoid do not coincide,
then the equation of the ellipsoid is

2 2 2
allx + a22y + a33z + a232yz + a3122x + 8122Xy — 1, (h)

where the aij will be combinations of the principal refractive
indices and will depend upon the rotation of the axes. Again,
for light propagating in the z-direction the equation for the ellipse

will be

2 2
allx + azzy + a122xy = 1. (5)

To describe the ellipsoid in the general case above,
we see that six coefficients, aij’ must be specified. Furthermore,

if we write Eq. (h) in the form

aijxixj = 1 id = 1,2,3, (6)

9

it can easily be shown that the a are the components of a

ij
symmetrical tensor of the second rank.

For an isotropic substance the indices of refraction

are all the same and the ellipsoid becomes a sphere

o 2 o 2 o 2

allx1 + a22x2 + a33x3 = 1 (7)

where a0 = a0 = a0 = -l-. (8)
11 22 33 n2

If the isotropic solid is subjected to a longitudinal stress in
the x-direction, the sphere is deformed to an ellipsoid of

revolution

 

 

 

 

where
2An..
: 1 : 1 xL-J (10)
JJ 2 2 2 3
(n + An..) n (1 + An..) n n
JJ JJ
n
and
Aa..~a -a = ..-1—:-2—An.. - (11)
JJ JJ JJ JJ n2 n3 JJ

This relation will be used in Section B.

This phenomenon, discovered in 1815 by Sir David
Brewsterlo, is known as artificial double refraction. An optically
isotropic transparent medium becomes anisotropic and an anisotropic
medium has its anisotropy changed when it is subjected to a stress.
Pockelsll, in 1889, studied artifical double refraction in crystals
and developed a phenomenological theory. He assumed that the

changes in the optical properties, i.e., Aaij, were linear func-

tions of the applied stresses

A3 ksl = 1,2,3 (12)

ij = qijkl Tkl
where the qijkl are the stress-optical constants and the Tk1

are the stresses. Alternatively the phenomenon could be described

in terms of strains

A811 = pijklskl (13)

where the p , are the strain-optical constants and the S
iJkl R1

are the strains. Because of the relations between stress and

strain,

 

1
RI

 

 

 

 

 

qu — cpqrssrs and Skl : SklmnTmn’

(111)

where the c are the elastic stiffnesses and the s are
quS klmn
the elastic compliances, the stress-optical constants are related

by the equations

= = . 1
qijkl pijmnsmnkl and pijmn qijklcklmn ( 5)

These are all tensors of rank four, having 81 components,
but because the stress and the strain are second rank tensors
which are symmetrical, the 81 components are reduced to 36 in-
dependent components. The crystal symmetry itself further reduces
the number of non-zero independent components. Pockels originally
worked out the scheme for the non-vanishing elements for all of
the crystal classes, but Bhagavantamlz found some errors when he
applied a group theoretical method to the phenomenon of photo-
elasticity. For an isotropic substance the number of non-vanishing
independent elements is just two, and for cubic crystals of the
class Oh, or m3m, which is the class of the sodium halides, there
are only three non-vanishing independent elements.

The tensor subscript notation, because of its symmetry,
can be simplified to a matrix notation13. This also results in a
simplification of the calculations, however, whenever a transfor-

mation is to be performed the quantity must be expressed in tensor

form. The change in notation is as follows

11-»1, 22-»2, 33+3, 23+u, 31+5, 12--6.

 

 

 

 

With this change in notation the stress can be written as

T1
T11 T12 T13 T1 T6 T5 T2
T12 T22 T23 = T6 T2 Th = T3 (16)
T13 T23 T33 T5 Tu T3 Th
T5
T6
and the strain as
S1
S11 S12 S13 S1 S6 S5 S2
S12 S22 S23 == S6 32 Sh == S3 , (17)
$13 323 S33 35 s,+ 33 SN
35
36
but Sh = 2523, S5 = 2331 and S6 = 2812. The factors 2 appear

because of the definition of strain in connection with shears
and rigid body rotationslu. The Eq.(l3) can then be written in

matrix form as

or in detail as

   

 

 

 

 

 

 

 

 

 

 

1531 1911 p12 p13 plu P15 p161 {311

A”5‘2 p21 p22 p23 p21 p25 p26 S2

A““3 p31 p32 p33 p3h p35 p36 S3 (19)
A31 = P11 912 P13 Pun P15 916 Sh .

Afls p51 p52 p53 psh p55 ps6 35

A36) \p6l P62 P63 P61 965 966} 1361

For the point group symmetry m3m, the non-vanishing

 

independent elements are p11, p1 and phh’ with

2

P11 = P22 = P33; Pun = P55 = P66, P12 = P13 = P23 (20)

and with all the other elements vanishing. The same is true for

the stress-optical constants and the elastic stiffnesses and

 

 

 

 

 

 

 

 

compliance315-17. Therefore, Eq. (19) can be written as
Aa1 p11 p12 p12 0 O O \ (31‘
A“5‘2 p12 p11 p12 0 O 0 S2
Afl3 p12 p12 p11 0 O 0 S3
bah = O O O pun O 0 Sn . (21)
Aas O O O O p1m 0 S5
(Aa6) (o O O O O pun (36/

As the various strain-optical constants and the elastic

compliances are specified in a crystallographic coordinate system,

but tflae measurements are made in a laboratory coordinate system,

 

 

10

a rotation of coordinates must be performed unless the two sets

of axes coincide. The complete program for the static measurements

is as follows

(a) the applied stress is known in the laboratory

coordinate system

Tl
(b) the stress is transformed to the crystallographic

coordinate system

T=R'R (m)
(c) the strain is determined
'95 = H (23)

(d) the change in the optical property is determined

3 (21)

(e) the change in the optical property is transformed

Eu
-oll

back to the laboratory coordinate system

——.——
— ~—_

Aa' = RAaR. (25)

Thus finally 22' is expressed as a function of ii in terms of
the pij and the skl' If the sk1 are known, and 32' and T' are
observed, then relations between the pi. can be calculated.

For the dynamic measurements we can start with the strain,
and continue as follows

(a) the applied strain is known in the laboratory

coordinate system
;.
(b) the strain is transformed to the crystallographic

coordinate system

2”

S = R3'

W

(26)

.

~r...

in;

warm

0

 

11

(c) the change in the optical prOperty is determined
A8 = P§ (27)
(d) the change in the Optical property is transformed

back to the laboratory coordinate system

N

32' = REA-f1. (28)
Again, relations between the pij can be calculated.

For the crystals used in this study, with the faces
oriented (lIO), (110), (001), (see Figure l), the rotation

operator is

 

1.1.1.01 N
15 15

== :_L_1_

R (2. 13 0 (29)
O O 1 1

 

 

Thus for a stress, T', (or a strain, S’) applied in the direction

of the x'- (laboratory) axis (a face diagonal of the unit cell),

 

the rotation transformation results in a stress (or strain) in

the crystallographic coordinate system which is given by

Tieé-o TliO o _1’:’_ 4—,;— 0
== == ===Ei
T(orS)=RT'R= TIE—$0 O 00 7% 7%?0
O O 1 O O O O O 1
1 -1 O
= 1T. (or S” ) -1 1 O (30)
211 11

 

12

 

/'
\

 

 

 

 

\
\
/

 

Figure 1. Crystal orientation.

The cube drawn with solid lines represents
the actual crystal in the primed laboratory coordinate
system. The cube drawn with dashed lines represents the
unit cell in the unprimed crystallographic coordinate

system which is rotated hSO about the z-axis from the

laboratory coordinate system.

 

 

1.- 9m E'J‘F“ z J's-5W2.” oi" .. “Tide-7“ '3' 7- _

13

Using the changes in notation (l6) and (17), this equation becomes

 

 

 

 

[11 f 1‘
1 1
O O
T=-%—T11' 0 «Sr-:— 311 O . (31)
O 0
1-11 1-21

For a stress, (or a strain), applied in the direction
of the z'—axis (an axis of the unit cell) in the laboratory the

transformation results in a stress (or a strain) given by

 

 

 

 

{01 101
O O
l l
I: T35 o «E: 33% o . (32)
O O
o 101

B. Static measurements

When a light wave is incident normally upon a plate
of material whose index of refraction is n and whose thickness
is t, the phase of the light which emerges from the far side of
the plate is related to the phase of the incident light by the
equation1

2nt(n-no)

15- a' = x (33)

 

 

1h

where O' and O are the phases of the incident and emergent light,
respectively, and no is the index of refraction of the surround-
ing medium and X is the wavelength in vacuo. If the incident
light wave is separated into two beams by the plate, i.e., an
ordinary and an extraordinary beam polarized at right angles to
each other, and with different indices of refraction, then the
relative phase retardation between the two beams would be

M = :21, - g, = 2% (“1 - n3). (31+)
The emergent light is elliptically polarized.

The Babinet compensator is a device which can be used
to either produce or to analyze elliptically polarized light19.
It is made of two thin quartz wedges with their slant surfaces
in contact and which slide relative to one another under the
action of a micrometer screw. The wedges are cut so that their
optic axes are perpendicular to one another and, also, to the
direction of propagation of light through them. Thus, what were
the ordinary and extraordinary rays, respectively, in the first
wedge are just interchanged in the second wedge. If light traverses
the compensator at a position where the two wedges are of the same
thickness, the two beams will emerge with zero relative retardation.
For any other position, one of the beams will be retarded relative
to the other one. The emergent light will be elliptically polar-
ized.

Used as a compensator, the device introduces just enough

of the opposite relative retardation to compensate for the unknown

 

   

 

 

 

15

relative retardation, and since the retardation introduced by the
compensator is known, this is then the measure of the unknown
retardation.

Starting from the position for zero relative retardation,
the relative retardation is then proportional to the change in
the reading of the micrometer screw driving the one wedge relative
to the other. We can write this as

A¢=¢1-¢j=KAB=%-TEAB (35)

where AB is the change in the reading of the Babinet compensator,
and D is the number of divisions on the compensator corresponding
to a phase change of Zn, i.e., from extinction to extinction be-
tween crossed polarizers. Combining this equation with Eq. (3h)
we see that the Babinet compensator can be used to measure the
difference in the indices of refraction for the two polarizations

of the light,

2nt 2n
T ni-nj = D—AB. (36)

Combining this with Eq. (10) and adding primes to indicate the

laboratory coordinate system, gives

12m 12m.
______ __.i.1

m
.—
m
II—
no
I
I

 

16

The quantity [ni - nj] is the quantity which is measured by the

Babinet compensator.

Case I.

Then the

mu
1

(See Eq. (36)).

We shall now consider each of the three cases in detail.

Stress applied in the direction of a face diagonal (x');

light propagated in the direction of the other face

diagonal (y').

strain is

S11 S12 S12 0
s12 s11 S12 0
O

mll
all

812 S12 s11

Shh O

From Eq. (31) we have

1
1
O
O
O
-l
O 1
O 1
O O
O O
O
811 ‘1

Putting this result into Eq. (21) yields

(38)

s11+312
s11+s12
11 12 (39)

 

17

A“1'1 (P11+P12)(311+812) + 2P12312
A"‘2 (P11+P12)(311+812) + 2p12812
— = — == - l '
A33 ‘ A” ” PS ‘ 2 T11 2p12 (311+312) + 2911512 . (ho)
Aau 0
An 0
5
A36 'Puusuu
Transforming is to the laboratory coordinate system gives
.1. :1. 0 gal A66 0 .1. .1. o
15 15 15 15
in. _‘:‘a=§_ —1—-—1 0 A86 A32 0 —'1 ——1 O
‘ ’ 15' 15 15' 15
o O 1 O 0 AB 0 O 1
Aal - Aa6 O o
O 0 Am
3

where the Aai are given by Eq. (’40).

The light is propagated in the y'-direction and is
polarized in the x'- and z'- directions. The changes in the

optical properties for these polarizations are then given by

u 1 v
Aal = Aal-Aa6 = 2 T11 [(p11+p12)(s11+812)+21>12812 + Puhshh] (“2)
. _ 1
and A83 _Aa3 2

T11 [2P12(811+512) + 2P113121' (#3)

 

 

18

These equations can then be used in Eq. (37) as follows

-2 x . . . 0' . 0' . .
—— = a_ '8. = a - a - , - a = - .
n3 tD ABad 1 J ( 1 ) (a; ) A“5‘1 A83

1 .
1 3 E T11[(P11+p12)(511+s121+2P12S12+P113111
- l T '[2 (s +s ) +2 3 ]
2 11 p12 11 12 p11 12
1 .
= 2 T11[(P11'P12)(311‘S12) + 911311]' (“4)

The superscript "zero” denotes the undeformed value and the first
and second subscripts on AB denote the direction of the stress
and the direction of the light, respectively. The "d" signifies

a face diagonal and a "c" will signify the unit cell axis.

Case II. Stress applied in the direction of a face diagonal (x');
light propagated in the direction of the cell axis (2').
The results are the same as Case I through Eq. (#1), but

in this case the light is propagated in the z'- direction and is

polarized in the x'- and y'- directions. The changes in the optical

properties for these polarizations are then given by
1
I__ I
A31 ‘ 2 T11 [(Pl1+912)(s11+312) + 21)12312 + punshu] (“5)

I 1 l _
222 = A81+A'36 = 2 T11[(P11+P12)(511+312)+2P12512 P113111- (”6)

 

19

Then in the same manner that Eq. (uh) was obtained we can get

Case III.

1 l l; I
ABdc = A“1 ' M2 = 2 T11 [21111311]. (17)

Stress applied in the direction of the cell axis (2');
light propagated in the direction of a face diagonal
(x' or y').

From Eq. (32) we have

Then the strain is

 

o
O
= 1 (18)
T = T I n o
33 O
o
0
s11 S12 S12 0 0 O O 5’12
s12 s11 S12 0 O O 0 s12
_:_ ' - I
sT _ $12 $12 311 O O O 1 T33 — T33 311 . (#9)
O o 0 51m O O o O
O o O 0 $111+ 0 O O
O O O O 0 :51+ 0 O

 

 

20

Putting this result into Eq. (21) yields

A81 (911+P12)812 + p12311

A“a2 __ __ (P11+p12)312 + p12811

Aa3 .52 = SS = T35 2p12512 + P11811 . (50)
Aah 0

225 0

226 0

_ 221 O o
21' = R 22R = o 222 O
O O Aa3 (51)

with the Aai being given by Eq. (50). The light is propagated
in the x'- (or y'-) direction (they are equivalent), and is
polarized in the y'- (or x'-) and z'- directions. The changes
in the optical properties for these polarizations are then given
by

Aag' (or Aal') =Aa = 1

2 T33 [(P11+p12)s12 + P12311

= T33' [ p11812 + p12(311+812)] (52)

3 " T33' [P11311 + 2P128121‘ (53)

 

21

Again in the same manner that Eq. (#1) was obtained we can get

'2 A I I _ l _ -
—;—3' t—DABcd = A33 - A32 — T33 [(1311 p12)(sll 312)]' (51")

The results for the three cases are then as follows

-11 MABdg
Case I -- -— = p (s -s ) -p (s -s ) +p s
Eq. (uh) n3 tD L11, 11 11 12 12 ll 12 uh Mn
(55)
AB
Case II lg AA -—E2 = Phhshh (56)
Eq. (17) n tD Lli
-2 AA cd
CaseIII —-—— =P (3 -s )-P (s '3) (57)
Eq. (SM) n3 tD L33' ll 11 12 12 11 12
where T has been replaced by
L
T=-A— (58)

where L is the load and A is the cross—sectional area. This
has been done because the quantity 9% is the quantity which

is observed experimentally.

C.. Dynamic measurements.

These measurements depend upon the optical effect of

20,21 2

ultrasound, which was discovered in 1932 Raman and Nathz
presented the theory of the diffraction phenomenon, with the
result that the intensity of the light in the zero-order of the

diffraction pattern produced by a standing wave is given by the

 

 

m l
1 -V¢=-‘
m

expres

where _
the SOL

field,

ferent
1V, to t
differe:|
Perpendi
18 the p

variatio

 

 

 

"here th.

Then the

 

 

22

expression

1 1
IO = Jo (v/2) + 2 J1h(v/2) + 2 32 (v/2) + . . . (59)

where J is the Bessel function and v in the argument is known
as the Raman-Nath parameter. This parameter is proportional to
the amplitude of the ultrasound. A plot of Eq. (59) is shown

in Figure 2. The Raman-Nath parameter is given by

2—;‘11 (so)
where u is the maximum change in refractive index produced by
the sound wave, L is the length of the light path in the sound
field, and x is the wavelength of the light.

The value of u, and therefore of v and of I0 is dif-
ferent for light polarized parallel and perpendicular, respective-
ly, to the ultrasonic wavefront. This fact corresponds to the
different refractive indices for light polarized parallel and
perpendicular, respectively, to an applied static stress and
1527

is the phenomenon which gives rise to Mueller methods and

variations of them. Analogous to Eq. (11) we can write

2 u
1
A31 = - T (61)

n
where the subscript 1 refers to the particular polarization.

Then the ratio for the two polarizations is

V
i i
R: ——T =u— =3— . (62)

L:
(.1.
La-

 

C ./ - mm View K

10

%UHmCOu:H uzwfid uCOUhOR

 

 

 

 

Percent Light Intensity

 

 

23

10 --

 

Raman-Nath parameter, v.

Figure 2. Theoretical light intensity curve.

.x-s KW“; . - .

 

 

 

Fron

so th

11188311:

and co

in the d
CUbe. A
is in th:
0r
or

where the

we Shall nO‘

Case I.
11

(y

 

21

From Eq. (18) we can write

(Aa)i (P115191
R: (221m)]. = (Pmsn)j ’ (63)

so the problem is one of finding the ratio of the v-values from
measured light intensities. This is discussed in Chapter III.

For these experiments it is assumed that only pure tensile
and compressive dynamic strains exist in the direction of propaga-
tion of the sound wave. Borgnisz3 has shown that pure longitu-
dinal waves can exist in cubic crystals in the directions, measured
from the crystallographic axes, with direction cosines (0,0,1),
(0,1,0), (1,0,0), which are the crystallographic axes and, also,
in the directions (f1, fl, 1.1) which are the diagonals of the
cube. Another solution of his equations, which he does not mention,

is in the direction of a face diagonal, i.e.,

|+

m1 = m2 , m3 = O
+
or m2 = - m3 , m1 = 0 (6h)
or 1113 = f 1111 , m2 = 0

where the 1111 are the direction cosines.

We shall now consider each of the three cases in detail.
Case I. Sound propagating in the direction of a face diagonal (x');

light propagating in the direction of the other face diagonal

(1")-

 

 

 

which,

COmpari
ments w
Pliaan
StatiC

ment St.

giVeS a2

 

 

25

From Eq. (31) we have

1
1
0
S = l s ' o (65)
2 11 ’
o
-2
which, when used in Eq. 21 yields
Aal1 P11+p12
A‘a2 p11+p12
22 — 2:2 - =3 — l s ' 2
3 — "P '2 11 P12 . (66)
A311 0
Aa o
5
A86 "21’1“:

Comparing this equation with Eq. (#0) for the static measure-
ments we see that this one does not involve the Elastic com-
pliances whereas the former one does. This is because the
static treatment starts with stresses while the dynamic treat-
ment starts immediately with strains.

Transforming Z; to the laboratory coordinate system

gives again, the same as Eq. (#1),

 

 

 

 

 

 

 

 

 

but now

polariz1

Optical

and
These eq

unknown

the equa

Where tht

Case II.

which is r

 

 

 

Aal - 226 o o
53' = o 221 +226 0 (67)
O 0 223

but now the A31 are given by Eq. (66).
The light is propagated in the y1-direction and is
polarized in the x'- and z'- directions. The changes in the

optical properties for these polarizations are then given by

A”5‘1' = A‘11 ' A‘36 = 2 S11 [P11 + P12 + zphh] (68)
and

223' = Aa = % S11 [2p12]. (69)
These equations can then be used in Eq. (63) to eliminate the
unknown sound amplitudes. If the sound amplitude were known,
the equations could be used directly. The result is

2P
%d= 'ET77§§%EEE WM

where the subscripts on R have the same meaning as they have on

AB.

Case II. Sound propagated in the direction of a face diagonal (x');
light propagated in the direction of the cell axis (2').
The results are the same as Case I through Eq. (67). The light

which is propagated in the z'-direction is polarized in the x'-

 

' "‘.'.. é" '

 

 

 

and y"

these p

and

The rat:

Case 11:

which Yi‘
L‘

 

 

27

and y'- directions. The changes in the optical properties for

these polarizations are then given by

I_ _ _l I
221 _ 221 226 _ 2 s11 [p11 + P12 + thu] (71)
l_ _l I -
and Aa2 ‘ Aal ma6 ’ 2 S11 [p11 + p12 Zpuh] ‘ (72)
The ratio of these changes is
R = P11 + p12 ' Bpuh . (73)

dc
p11 + P12 + zpuh

Case III. Sound propagated in the direction of the cell axis (2');
light propagated in the direction of a face diagonal
(x' or y').
From Eq. (32) we have
0

O

H

= 333 O (71)

1{fill
O

O

which yields

v 0

 

Transfor

The light

and

The ratio

 

 

 

28

A'31 p12

A82 p12

Aa3 =2; = I: 333' p11 - (75)
Aah 0

A35 0

Aa6 O

Aal O 0 p12 0 O
=l__ __ I
As _ o 222 0 _ S33 0 p12 0
o o 223 o 0 p11 . (76)

The light is propagated in the x'- (or y'-) direction and is
polarized in the y'- (or x'-) and z'- directions. The changes

in the optical properties for these polarizations are given by
I l _ = I
and M l = Aav = S I o 8

The ratio of these changes is

. (79)

 

 

as 103

Case 1
Eq.

OI'

Case I
Eq.

01'

Case II
Eq.

01‘

 

 

 

 

29

The results for the three dynamic experiments are then

as follows.

Case I
Eq. (70)

01'

Case II
Eq- (73)

or

Case III
Eq- (79)

or

R =__2112.___ (80)
dd P11 + p12 + zpuh

R _ p11 + p12 ‘ zpuu
dc p11 + p12 + zphh

(82)

O = P11(Rdc'1) + p12 (Rdc'l) + zpuu(Rdc +1)‘ (83)

_ p
Red — .12 (81)
p11

0 = p11 Rcd ' p12 ' (85)

 

 

 

 

The

dens

wave

 

CHAPTER III

EXPERIMENTAL PROCEDURE AND RESULTS

A. Static measurements.

The arrangement of the apparatus is shown in Figure 3.
The light from a mercury vapor light source passed through a con-
densing lens and a filter, which passed only the green line of
wavelength 5h61 X, and illuminated a vertical slit. A collimating
lens produced a parallel light beam which was limited in size by
a diaphragm to approximately 6 mm by 3 mm through the center of
the crystal. This was done to obtain as uniform a stress field
as possible where the light passed through the crystal. A polar-
izer in front of the crystal was set at #50 to the direction of
the applied stress so that there were equal light intensities
polarized parallel and perpendicular, respectively, to the direc-
tion of the stress. Another lens then focussed the light on the
slit of an American Instrument Company photomultiplier microphotometer.
Interposed in front of the photometer were a Babinet compensator
and an analyzer crossed with respect to the polarizer. With no
sample and no compensator in the light beam, no light arrived at
the photomultiplier because of the crossed polarizer and analyzer.
The stressed sample produced elliptically polarized light and
the compensator was adjusted to introduce just the opposite phase
changes so that again no light arrived at the photomultiplier.
The range of adjustment of the compensator permitted three dif-
ferent settings across the range, for which the light was extin-

guished. The difference in these readings corresponds to a phase

30

1:4

 

 

rims. :" '

 

 

 

 

 

31

 

 

 

 

 

 

 

 

 

 

Figure 3. Apparatus for the static measurements.

L"
ll

":1
II

U!
H

II"
II

U
II

0
ll

L _

B -

K>
II

E

mercury vapor light source
condensing lens
filter passing 5h6l R
slit

collimating lens
diaphragm

polarizer

crystal

focussing lens
Babinet compensator
crossed analyzer

photomultiplier tube

 

 

chang
In ac

times

appar.
D. Bra
at the
of 3.:
served
stress
permit!
tal to
side of
materia
tect the
With the
SUPPOIte
entire 8
Optical 1
The tube

with the

CreaSed

 

function

 

32

change of 2n, so the compensator can be calibrated on this basis.
In actual operation, for each load the compensator was set three
times at each of the three settings, in the order ABCCBAABC.

The stress was applied to the crystal by means of the
apparatus shown in Figure h. This apparatus was constructed by
D. Bradleyzu and is similar to that used by Waxler and Napolitano25
at the National Bureau of Standards. A tube with an inner diameter
of 3.2 cm and an outer diameter of 5.0 cm and a length of 2% cm
served as the guide for the cylinders which actually applied the
stress to the crystal. Ball bearings with a diameter of 0.9h cm
permitted the short cylinders which were in contact with the crys-
tal to tip so that they would not exert a greater stress on one
side of the crystal face than on another. A piece of rubber dam
material was placed in the cup of each of these cylinders to pro-
tect the surface of the crystal. A disc, which was in contact
with the uppermost cylinder by means of another ball bearing,
supported the load pan with three cables spaced at 120°. The
entire apparatus was supported rigidly, and independently of the
optical bench so that the latter would experience no deformation.
The tube had a light port on two opposite sides which were aligned
with the optical bench.

The initial load was approximately 7.5 kg and was in—
creased 1 kg at a time to a maximum added load of 11 kg. As
expected, the reading of the Babinet compensator was a linear

function of the load and the slope was determined by the method

 

 

 

Cont

Ligh

Base

Opti

 

 

 

33

 

 

 

 

 

 

1.
1

 

Guide
cylinder —--——-~

Load

transmissio
cylinder—fi"”””

 

s\\\\\\\\\\m\\\\\1
\\\\\\\\\\\\\\\\\\\Vl

 

1
-\
MO

Contact

One of 3 cables,
spaced at 120 ,
which support
the load

Crystal

 

 

cylinders <

Light I’,//”

port

 

 

 

[1

 

 

 

/1\
1

 

Baseplate

 

 

 

sample

\ Light

port

Optical
bench

 

g

3

 

Optical
bench

I’M—117

Load

 

 

 

 

Figure h. Stressing apparatus.

 

3h

of average526. The slopes, %2 , for the different samples and
orientations were used in the appropriate equation (55) or (56)

or (57). A typical data sheet and calculation are shown in
Appendix I. The results of the static measurements are summarized

in Table 1.

Table l. The change in Babinet compensator
reading per kilogram change in load.

 

 

 

NaF NaCl NaBr
Ade
L , 1.95 f 0.05 3.25 f 0.09 3.21 f 0.09
11
AB
dc 1.70 + 0.07 2.52 + 0.10 2.07 + 0.13
____T - - -
L11
as
cd 2.31 + 0.06 3.77 + 0.07 5.02 + 0.09
L ' ' ' '
33

 

 

 

B. Dynamic measurements.

Extensive use and study of Mueller'sz7 techniques have
been made in the ultrasonics laboratories at Michigan State
University. His method "B" depends upon an observation of the

plane of polarization of the diffracted light, while his method

 

 

35

”C" uses intensities of the diffracted light. Gates and Hiedemann
used both methods on glass blocks and did not get acceptable results
by method ”C" when using a photographic technique to measure light

intensities. Hagelberg3

studied the effect which different experi-
mental parameters had on the results by method "C", using a photo-
multiplier tube as a detector and concluded that the photographic
technique was the source of the earlier difficulty. Achyuthan2
and Bradleyzu made measurements on crystals of ADP and sapphire,
respectively. All of these investigations made use of a continu-
ous wave ultrasound and consequently experienced considerable
difficulty due to heating of the samples. Klein28 developed a
pulsed ultrasound technique which alleviated this difficulty. The
output of the phototube detector was displayed on an oscillosc0pe
which was triggered by the rf pulse.

Mueller pr0posed to find the ratio, R, in Eq. (63) by
considering the light intensity in the first order diffraction
line. This intensity is given by a sum of Bessel function products
similar to Eq. (59). When the Bessel functions are written in the

series form it is found that

lim :1 (v) 6 v2. (86)

V-O-O

The desired ratio for the two light polarizations is then simply

i
I}: = V+O I— . (87)

36

The low light levels in the first orders for low sound intensities,
plus the extrapolation process, reduces the accuracy of this method.
Another difficulty is that the normalization for the two polariza-
tions must be carried out in terms of the light in the zero-order
with no sound.

An alternative procedure which has been used at Michigan

State University5

is to observe the ratio of the rf voltages which
must be applied to the quartz transducer so as to produce the same
diffracted light intensities for the two polarizations. Since the
series expressions for the light intensities are identical, the
arguments must be the same if the light intensities are the same.
This technique is applicable to all diffraction orders and does
not require extrapolation to zero sound intensity.

Another procedure, the one used in the present investi-
gation, is to read the v-values corresponding to the light inten-
sities from the Raman-Nath theory. (See Figure 2.) This method,
also, is applicable to all diffraction orders and does not require
extrapolation to zero sound intensity. In addition to these two
advantages over Mueller's original suggestion, the normalization
is easily carried out in terms of the zero-order light intensity
immediately before and after the sound pulse diffracts the light,
and at the same position and angle of incidence on the phototube.

All of the methods mentioned so far will yield only the
ratios of the strain-optical constants because the sound amplitude
in Eq. (78), for example, is not known, and the ratio form permits

the elimination of this unknown. A knowledge of the sound amplitude

 

 

 

37

‘would permit the determination of the absolute values of the pij'
A number of techniques have been developed recently which attack
this problem directly. Smith and Korpel29 have immersed the test
sample in a known buffer liquid. By observing the optical effects
of the ultrasound in the liquid both before and after the sound
traverses the solid sample, they determine the sound amplitude
in the sample. Dixon and Cohen30 used a fused quartz block as
a standard for determination of the sound amplitude. They ob-
served an initial pulse of ultrasound in the quartz and then
the reflected pulses from both the front and rear surfaces of
the test sample. Another technique, used by Pedinoff and
Seguin31, involves a preliminary study of losses in the sample
and another pulse-echo type determination of the loss in the
transduction and insertion process. These authors report from
10% to 25% uncertainty in their results. When these methods
have been improved so as to obtain greater accuracy, they will
make it possible to obtain the strain-optical constants by dynamic
methods alone.

The arrangement of the apparatus is shown in Figure 5.
The polarizer, analyzer and Babinet compensator of the static
apparatus have been replaced by a Wollaston double-image prism
after the sample. The Wollaston prism was oriented so that the

two images were polarized parallel and perpendicular, respectively,

to the sound wavefront and were displaced laterally relative to

 

 

 

 

 

 

 

38

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rf
rf 7 g, I Gated m Oscilloscope
generator c.w. amplifier
rf
pulse Signal fi
Q
‘ l
._1
"‘ { \1 1>
'45‘ _
._A|
’ _______..
Hg L1 F S L2 D C W 3 PM
Figure 5. Apparatus for the dynamic measurements.
Hg = mercury vapor light source
L1 = condensing lens
F = filter passing 5h61 X
S = slit
L2 = collimating lens
D = diaphragm
C 2 crystal
Q = quartz crystal
W = WOllaston prism
L3 = focussing lens
PM.= photomultiplier tube

 

 

 

39

one another. The photomultiplier was mounted on a micrometer
screw so that its slit could receive either of the two images.
Instead of the static stressing apparatus, an x-cut
quartz crystal produced an ultrasonic wave in the crystal sample.
The quartz was bonded to the sample by a silicone high vacuum
grease. It was driven at approximately 8.3 mc, the exact fre-
quency being chosen so as to produce the strongest standing wave
in the sample crystal. The rf voltage was supplied by a General
Radio Standard Signal Generator whose output was gated and ampli-
fied by an Arenberg Pulsed Oscillator. The standing ultrasonic
wave acted as an optical phase grating, thus producing a dif-
fraction pattern at the focal plane of the lens L3. The output
of the photomultiplier tube was displayed, as shown in Figure 6,
on a Tektronix 511A dual-trace oscilloscope. The percentage of
light remaining in the zero-order was observed for each polari-
zation and for a number of different values of the rf voltage.
The v-values corresponding to the percentage of light were read
from the Raman-Nath theoretical curve (Eq. (59),. Figure 2). For
a given rf voltage the ratio of the v-values for the two polar-
izations was determined. These ratios, R, were then used in the
appropriate equation (81) or (83) or (85), depending upon the
orientation. A typical data sheet and calculation are shown in
Appendix II. The resultS* of the dynamic measurements are sum-

marized in Table 2.

 

'rvavm~-

 

“-m‘

 

 

 

 

The upper trace is the rf voltage which is applied to the
quartz crystal. The lower trace is the photomultiplier

output. The broken line represents zero light, with down
being the positive direction. 'Measurements were made with

the time-base expanded as shown below.

 

Figure 6. Oscillosc0pe display of the photomultiplier
output.

n.- *7
a“ ‘m
.4. -
m .

 

 

#1

Table 2. Ratios of the v-values.

 

 

 

NaF NaCl NaBr
Rdd 2.21 f 0.02 1.21 f 0.01 1.08 f 0.01
Rdc 1.73 f 0.01 1.17 f 0.01 1.00 f 0.01
Rcd 3.98 t 0.01 1.39 f 0.01 1.29 f 0.01

 

0. Determination of the strain-optical constants pij'

The static Eqs. (55)-(57) and the dynamic Eqs. (81),
(83), (85) constitute six equations in the three unknowns pij'
The numerical values used in the equations are given in Tables 1,
2 and 3. The six equations, with the numerical values inserted,
are given in Table A. The best method of solving these equations
is to solve the static equations for the two unknowns (pll-plz)
and phh’ and solve the dynamic equations for the two unknown
ratios plz/p11 and pun/p11, and then to solve these four equations

for the individual pij values.

 

 

 

12

Table 3. Constants used in the calculations.*

 

A = cross-sectional area of crystal = h.00 f 0.005 cm

t = path length in crystal = 2.00 f 0.005 cm

2

 

 

 

D = change in Babinet reading for a retardation of
2n radians = 1287 f l
X = wavelength in vacuum = 5161 f l X
NaF NaCl NaBr
n° 1.336 f 0.001 1.511 f 0.001 1.611 t 0.001
s11 11.5 x 10'13cm2 22.79 x 10’13cm2 28.50 x 10'13cm2
dyne dyne dyne
-l3 -l3 -13
312 -2.36 x 10 -u.691 x 10 -6.31 x 10
-l3 -13 ~13
Shh 35.71 x 10 78.31 x 10 101.0 x 10

 

* The elastic compliances are from Aleksandrov and

32

and the indices of refraction are from

33.

Ryzhova

the Handbook of Chemistry and Physics

 

13

 

 

 

 

Table 1. The six equations for the pij.

NaF -2.832 = p11 x 13.86 - p12 x 13.86 + p1m x 35.71 (A) (88)
-1.235 = pun x 35-71 (B) (89)
-1.678 = p11 x 13.86 - p12 x 13.86 (C) (90)

0 = p11 x 2.21 + p12 x 0.2M + phh x h 48 (D) (91)
O = p11 x 0'73 + p12 x 0.73 + p1m x 5 16 (E) (92)
o = p11 x 3.98 - p12 (F) (93)

NaCl -3.058 = p11 x 27-18 - p12 x 27-18 + Pun x 78.31 (A) (91)
-1.186 = P11 x 78 31 (B) (95)
-1.771 = p11 x 27.18 - p12 x 27.18 (C) (96)

0 = p11 x 1.21 - p12 x 0.76 + pML x 2 18 (D) (97)
0 = p11 x 0.17 + p12 x 0.17 + pun x 1 31 (E) (98)
o = p11 x 1.39 - p12 (F) (99)

NaBr -2.516 = p11 x 3h.8h - p12 x 3h.8& + phh x101.0 (A) (100)
-0.8112= pML xlOl.O (B) (101)
-1.967 = p11 x 31.81 - p12 x 31.84 (C) (102)

0 = p11 x 1.08 - p12 x 0.92 + pML x 2.16 (D) (103)
0 = pun x 1 (E) (101)
o = p11 x 1.29 - 212 (F) (105)

 

 

 

 

11

The static equations are solved for (p11 - ) from

p12
equation (C) directly and also, by substituting equation (B)

into equation (A) and finally taking a weighted average. The
value for phh is obtained from equation (B) directly and also,

by substituting equation (G) into equation (A) and finally

taking a weighted average. The results are listed in Table 7.

It is interesting to note the extent of agreement between the
three static equations in a straightforward fashion. From the
equations in Table h, we see that the right-hand side of equation
(A) is equal to the sum of the right-hand sides of equations

(B) and (0). Therefore, the same should be true of the left-

hand sides. This comparison is shown in Table 5.

Table 5. Comparison of static results.

 

Left-hand side of equations % difference
taken from Table h
Eq- (A) EQs. (B)+(C)
NaF 2.832 2.913 2.8%
NaCl 3.058 2.960 3.3%
NaBr 2.516 2.778 9.9%

 

 

 

The dynamic equations (D)-(F), after dividing through
by p11, are solved for p12/p11 from equation (F) directly and
also, by solving equations (D) and (E) simultaneously. These

values are combined to find a weighted average. The value for

 

 

“5

phh/pll is found by solving equations (D) and (E) simultaneously
and also, by substituting the average value for pl2/Pll into
each of them separately. An average value is then obtained. The
results are listed in Table 7. The agreement between the dynamic
equations cannot be demonstrated in quite as straightforward a
manner as was done for the static equations, however, the Eqs.

(80)-(85) can be combined to yield
Rdd (Red + 1) = RCd (Rdc + 1). (106)

This comparison is shown in Table 6.

Table 6. Comparison of dynamic results from Table 2.

 

Rdd ( Red + 1 ) Rcd ( Rdc + 1 ) % difference
NaF 11.2 10.9 2.7%
NaCl 2.96 3.02 2.0%
NaBr 2.17 2.58 1.3%

 

 

 

 

 

 

 

 

 

A6

Table 7. Separate solutions of the static

equations and of the dynamic

 

 

 

equations.

Compound Static equations Dynamic equations

NaF pll-p12 =-O.119 plz/p11 = h.Ol

Pun =-0.0338 Pun/p11 = -0.710

NaC1 pll-plz =-OOO6S6 plz/pll = 1037
pML =-0.0156 pun/p11 = -0.0881

NaBr pll-plz =-0.05h5 p12/p11 = 1.25
*

p1m =-0.00723 pun/p11 = 0.016

 

 

 

The values for the individual pij are obtained by
solving the equations given in Table 7. These values are listed
in Table 8. Various ratios and differences also have been calcu-
lated for use in Chapter IV. These are shown in Table 9. In
each of these last three tables we note a definite trend in the
values as we move from.the fluoride to the chloride to the

bromide.

*Because of considerable uncertainty in this number, it was not

used in further calculations.

 

”7

Table 8. The strain-optical constants.

 

p11 p12 Pun
NaF 0.0h36 0.169 -0.032u
NaCl 0.177 0.213 -0.0156
NaBr 0.218 0.272 -0.0072

 

 

 

Table 9. Ratios and differences of the pi

 

 

J
p _p 312 Pen

11 12 p11 p11 + p12
NaF -0.125 3.88 -0.152
NaCl -0.066 1.37 -0.037
NaBr -0.0Sh 1.25 -0.0lS

 

 

 

A final check on the values listed in Table 8 is made
by using these values to calculate the quantities which were
observed experimentally, i.e., the left-hand sides of Eqs. (55)-
(57) for the static measurements and the ratios, R, of Eqs. (80),
(82) and (8h) for the dynamic measurements. These comparisons

are exhibited in Table 10. It would appear that the values of

 

 

 

 

 

 

#8

 

 

 

 

 

Table 10. Comparison of observed and calculated quantities.
Observed Calculated % difference
value value
NaF -2.832 -2.889 2.0%
-1.235 -1.157 6.6%
-1.678 -1.732 3.2%
2.21 2.29 3.6%
1.73 1.88 8.3%
3.98 3.88 2.5%
Average—m
NaCl -3.058 -3.036 0.7%
-1.186 -1.222 3.0%
-1.77h -1.81h 2.2%
1.2h 1.25 0.8%
1.17 1.16 0.9%
1-39 1.37 _1;£§_
Average 1.5
NaBr ~2.516 -2.611 3.7%
-0.811 -0.730 10.5%
-1.967 -1.881 h.5%
1.08 1.13 h.5%
1.00 1.03 3-0%
1.29 1

.25 .1
Average .9

 

 

 

1‘9

the strain-optical constants which were obtained in this
investigation have a precision of better than 2% for the sodium
chloride and better than 5% for the sodium fluoride and the

sodium bromide.

 

 

 

 

 

CHAPTER IV

EXPERIMENTAL RESULTS OF OTHER INVESTIGATORS

While there is only one report in the literature of

investigations on sodium fluoride and on sodium bromide, there

are several articles reporting work on sodium chloride.

The

various values reported will be presented in this chapter along

with some comments concerning them. The values of the pij have

been recalculated for this presentation on the basis of the up-

to-date elastic constants given in Table 3. The results are

summarized in Table 11. With the exception of the present study,

they are presented in chronological order. Notes concerning

some of the results can be found following the table.

Table 11. The experimental results of other
investigators.

 

‘2 _____"M+

Observer pu-p12 phh p11 p11+ p12

p11
NaF

Ersaent -0 125 -0.032h 0.0h36
1:5u61 A

Leibssle3h -O.100 -0.02h plus absolute=$=0.077
A=5893 A retardations

p12

0.169

0.177

 

 

NaBr

Present
A: 5h61 A -0.05h -o.0072

Leibssle3u -0.03h7 -0 0036
.A=5893 A

 

50

 

 

51

Table 11 (continued).

 

 

_ p p
Observer p11 p12 phh -£§ -——:—E&——— P11 p12
p11 p11 p12
NaCl
Present -0.o66 -0.0156 1.37 -0.037 0.177 0.2h3
A=5h61 A
Wertheim35 (a)
-0.052h
Pockels36 (b)
1:5890 A -0.0h39 -o 0109 plus absolute ==€=> 0.1h0 0.18h
retardations
Maris37 (c)
A;5M61 A -0.0h22 increasing load
-0.056O decreasing load
Banerjee38
A:589O A -0.0h8h -0.0120
Kidani39 (d)
A=589O A -0.0591 "elastic region", first loading
-0.0553 ”hardened state", subsequent loadings
Galtho 1.35 -0.0u2
hl
Burstein and Smith (e)
-0.0h3 -0.010 0.11
Bhagavantam and Krishna Murtyhz
AF589O A -0.0h1h -0.0lO7
Bansigir and Iyengarh3 (f)
Az5890 A -0.0h26 -0.0117
-0.095O -0.0199 (these values calculated in the
present study)
Srinivasan1m (g)
Aeh800 A -0.oh07
1:5800 A -0.0375
Leibssle31L
1:5893 A -o.0h77 -0.0107 plus absolute) ==g> 0.128 0.176
retardations
Krishna Rao and Krishna Murtyh5
A:589OA -0.0h1 -0.0120 1.31 0.132 0.172

 

 

 

52

Notes

(a) Coker and Filonu6 converted Wertheim's value to
the present day stress-optical coefficient expressed in brewsters.
It is this value which I used to calculate (pll-plz).

(b) Pockels' samples for the relative retardation
measurements were not sufficiently parallel for the absolute
retardation measurements, so different samples were used for the
latter measurements.

(c) Maris did not calculate the slopes of the retarda-
tions versus stresses properly, and also included data for mini-
mum load and maximum load which did not exhibit a linear relation.
The values in the table are those which I have calculated from
the linear portions of his data.

(d) Kidani worked at stresses under 2.7 kg/cmz. A
first loading up to 1.32 kg/cmz yields different results than
subsequent loadings up to 2.68 kg/cmz. He reports a quadratic
relation between relative retardation and stress. The values
listed here are calculated on the basis of the slopes of his
quadratic equations at 3.00 kg/cmz, where the data of Maris first
exhibits a linear relation.

(e) Burstein and Smith do not indicate how the value
for p11 was obtained.

(f) Bansigir and Iyengar also report values for KCl
and KBr. They present their data in the form of graphs. From

measurements of the slopes I was unable to calculate the results

 

 

 

 

 

 

53

which they reported. Further, attempting to compare their six
values with the six values which I calculated, there was no
relationship whatsoever.

(g) Srinivasan says his values are good only to

10-20%.
With the exception of the values for p11 and p12 for
P
sodium fluoride and -—-E&-— all of the values from the present
p11+ p12

study are higher than previously reported values. It is interest-
ing to note that there is better agreement among the values obtain-
ed dynamically, the ratios, than among the values obtained statical-

ly, i.e., Pll-plz and Phh'

 

._ _'E" . .

 

 

are

$8

 

 

 

CHAPTER V

MUELLER'S THEORY OF ARTIFICIAL BIREFRINGENCE

Various theories of artificial birefrigence have been
presentequ-hg, none of which are completely satisfactory because
of invalid assumptions, neglected effects, or because the theory
depends upon quantities to be determined experimentally and which
are not known with sufficient accuracy. It is the purpose of this
section to consider the theory proposed by Mueller.

The interaction of light and matter can be represented
as a charged particle oscillator in the electromagnetic field of
the light wave. The response of the oscillator depends upon the
effective electric field at the position of the oscillator and
this field is due to (1) the applied field, (2) the Coulomb fields
due to the other ions and (3) the Lorentz-Lorenz field due to the
fact that the medium is polarized by the applied field. The k-t-h

ionic oscillator will then have an induced dipole moment

“k = 91 Eeff (107)

where 0k is the polarizability of the ion. From electromagnetic
theory we can write

D = e E = n2E = E + hn P (108)
and then

(n2-1)E=th=1+yr§N,u,=1+st§N.a.E

J J J J eff' ’ (109)

where N is the number of ions of type j. In this study we are

3

interested in the change in the index of refraction which is

5h

 

Fwy-1"" ' .

 

 

 

 

 

 

 

 

 

wit

 

 

 

55

produced by a strain, i.e., we must differentiate equation (109)

with respect to strain. Doing this yields the result

dn an dN d1. dE

.__ _-__ __di __J_ eff
2“ dS E § dS “j Eeff + Nj dS Eeff+ Njaj dS ' (110)

 

Earlier theories (Banerjee; Herzfeld and Lee) considered the
change in the density of the oscillators and the change in the
effective field due to a strain, but Mueller has also included
the change in polarizability and has assumed that the polari-
zability is a linear function of the strain.

To relate Eq. (110) to the phenomenological theory,

we combine Eqs. (11) and (18) to obtain

-2 .
‘33'Anl = Dal e pijsj 1,3 = 1,...6 (111)
n

in the matrix notation. Then differentiating, we can write

_ ;_Q &i _ _2 5(n1-no) _2 ani
pij ‘ no3 EEE' “ 03 as 03 as. . (112)
n J

 

n 3
We see that the right-hand side will be obtained from Eq. (110).

Mueller introduced some additional constants

 

 

h h
o 0 5n
px = g 2 p12 = 2n 2 -23 SS1 ’ (p12: p13) (113)
(n -1) (n -1) no ” 3
h h
P 5%;2 p11 =__..2_2:_2 5% 5: ’ (911: P ) (111*)
Z (n -1) (n -1> n° 3 33
h h
and p ,= 2n° p = 2n° -2 anu . (115)
x 2 2 “A 2 2 as
(n -1) (n -1) n03 1.

 

 

.3— - _&Y1L"..""' _ ‘

 

w

 

 

 

56

Each of these constants, as shown by Eq. (110), will depend
upon the strain-induced changes in the oscillator density, the
polarizability and the effective field. These terms contributing

to px are written as

L c A
px — p0 + px + px + px (116)

with similar equations for p2 and Px" The symbol po represents
the contribution from the change in oscillator density, pr
represents the contribution from the change in the Lorentz-
Lorenz part of the effective field, pxC represents the contribu-
tion from the change in the Coulomb part of the effective field,
and pr represents the contribution from the change in polariz-
ability. Because this last contribution is the new aspect in
Mueller's theory, it is the only term in Eq. (116) which will

be considered in further detail before writing down Mueller's
final result.

The Lorentz-Lorenz equation relating the atomistic
quantity, the polarizability, to the phenomenological quantity,
the index of refraction, is
a: 3— %—11 (117)

hnN (n + 2)

where N is the number of molecules per unit volumeso. The

molar refractivity is given by

2
R: ER a=L4Ln_2;_11 (118)
3 o P (n + 2)

where No is Avogadro's number, M is the molecular weight and

 

 

 

if

t1‘

‘7

 

57

p is the density. If we consider first a hydrostatic pressure,
so that the symmetry of the crystal is not disturbed, then the
change in the index of refraction must be due to the change in

density. Differentiating Eq. (118) with respect to density yields

dnggz _ (n2 - 12(n2 + 2) (11

if R is considered to be independent of density. If, however,

the refractivity also depends upon the density then we must

 

write
dn an an dR
—= — + — — (120)
dp 8p R 8R P dp
where Eq. (119) has become just the first term of this new
equation. Mueller assumes that the refractivity is a linear
function of the strain
AN A
R = R0 (1 + 10 v—) = R0 (1 - 10 $9), (121)
so that
$3 _ -Roxb (122)
dp p

where R.o is the refractivity of the unstrained crystal and

Ab is a phenomenological constant which Mueller calls the
strain-polarizability constant, with the subscript zero in-
dicating the case of hydrostatic pressure. Re-writing Eq. (120),

it then becomes

 

58

 

(n2 - 1)(n2 + 2) (n2 - 1)(..2 + 2) -beb
6np + 6nR ( p )

O

Dela.
'D 23

n2 - l n2 + 2 8n
= (l - 10) up = (1 - 1b) (35% . (123)

The observed values of p(§E) are indeed less than the values
predicted by Eq. (119)51 and would be more properly given by
Eq. (123). Now turning to a consideration of the strains in—
volved in the present investigation, for a normal strain in the
z-direction, and light polarized in the x- and z-directions, the

equations similar to Eq. (121) would be

ij

Rj (1 + 83x53) (12h)

Rjz = R1 (1 + 13253) (125)

where the subscript j refers to the type of ion.
For a strain in the x'-direction (a face diagonal),

and light polarized in the x'- and y'- directions, we write

R = Rj (1 + A x.36) (126)

jX'

Rjy, = Rj (1 + xjy,s6) . (127)

The anisotropy of the strain polarizability is inter-
preted as due to the deformation of the electron cloud surround-

ing the ion when the crystal is strained. According to Burstein

 

 

59

51

and Smith , this deformation is related to the amount of homo-

polar binding and the amount of overlap of the wave functions

of the electrons associated with one ion with those of a neighbor-

 

ing ion.
Combining all of the effects, Mueller's final results
are
2 2
h R R
“ (pll'plz) _ l 0.09(Rf+R:) + 582111112 + 2.82z(f—1-f—2)
2 2 _ 2 1 2
(n -1) (R1+R2)
2
- 33 5——)-n +2 (128)
3 (n2 -1)
and
2nlL phh 1 2 2 2 R2
—— = —— O.6O(R1+R) - 2.52R R - 1.862(R1 - 2)
2 2 2 2 1 2 '- '-
(n -1) (R1+R2) f1 f2
2
- 2:3. Liz—+31 (129)
3 (n -1)

where R is the refractivity and f is the oscillator strength.
The subscripts 1 and 2 refer to the negative and positive ions,
respectively; z is the valence of the ions, which in the case of
the alkali halides is unity. The shear strain polarizability,

A', is defined by

k. = i (*jx' ' A'jy') Nle

2 i NjRj

and A is the normal strain polarizability, defined by

(130)

 

 

 

6O

2. .__ x - 1 = i "JzNJRJ " i xJxNJRJ

 

° (131)

The A0 of Eq. (121) is related to Az and A.x by the equation

1
10 = -3- (A2 + 2xx). (132)

The first two terms in Eqs. (128) and (129) come from the
Lorentz-Lorenz contribution to the effective field and the
third term is the Coulomb contribution to the effective field.
The last term is due to the optical anisotropy of the ions due
to the strain.

For a verification of Mueller's theory it is necessary

to know the values for R f1, A and A'. If certain values are

i,

assumed for R1 and f1, then the experimental values of (pll-plz)

and of phh permit the calculation of A and A'. If we combine

Eq. (123) with the equation51

3
d2 __ “ (P11 + 21.12) , (133)
dp - (p

 

then Ab can be calculated. If we know A.o and A, then Eqs. (131)

and (132) can be used to calculate Az and Ax. This has been

done and the values which were obtained are presented in Table 13.
The refractivities used are from a paper by Pirenne and

KartheuserSz. There is reason to doubt that the refractivities of

the ions are strictly additive and are independent of the other

ion in the crystal. Pirenne and Kartheuser have attempted to modify

 

 

’-"'~';.— . o
“-#W£m-J" .1 “if

 

 

 

 

61

Table 12. Refractivities in cm3/mole.

 

 

 

 

 

 

 

 

Lorentz-Lorenz Seitz Pirenne and Kartheuser (PK)
Na 0.5 0-73
F 2.5 2.22
NaF 3.12 3.0 2.95
Na 0-5 0-73
C1 9.0 7.5M
NaCl 8.52 9.5 8.27
Na 005 0'73
Br 12.67 10.hh
NaBr 11.59 13.2 11.17
Table 13. Strain-polarizability constants.
NaF NaCl NaBr
A.o 0.590 0.379 0.305
A 1.h86 1.177 1.158
Az 1.581 1.16h 1.077
Xx 0.095 -00013 ‘00081
N. -00096 0.058 0.087
Constants used in calculating the A's:
fun = 2.37 fF = 2.37 fCl = h.58 fBr = 1.90
The R1 are the (PK) values from Table 12. The pij are
from Table 8.

 

 

 

 

 

 

gi

it

 

 

 

 

 

62

the additivity relation to take into account some interaction
between the ions. In Table 12 are listed various values of the

53

refractivities. The values quoted by Seitz are actually for

the gaseous phase and are given by Fajans and JoosSu. A compar-

ison of the values shows agreement with the fact that the polar-

izability of the negative ion is less and that of the positive

ion is greater in the solid phase than in the gaseous phasess.
The values for the oscillator strengths are those

56

given by Seitz for the corresponding inert gas, i.e., the

inert gas which has the same electron configuration as the

57

particular ion. Mayer reports that the "electron number" of
the Cl ion is the same in NaCl as in KCl, but is less than that

for the corresponding inert gas, Ar.

fC1 = 3.25 < fAr = 11.61. (131+)

He reports that the same is true for the I ion.
fl = h.OO < er = 5.61. (135)

It would appear to be incorrect to assume that the oscillator
strength is dependent only on the electronic configuration.

From the values listed in Table 13, one can see the
regular trends in the A's as the compound changes from the fluoride
to the chloride to the bromide.

It is obvious that the values of the A's, with the
exception of Ab, will depend upon the values used for the refrac-

tivities and oscillator strengths. While the situation is

 

 

63

considerably better than when Mueller published his theory,

there is still much work that needs to be done on these quanti-
ties. Before his theory can be completely verified, some in-
dependent means of determining the strain-polarizability constants

must be devised.

 

Chapter VI

CONCLUSION

Using both static and dynamic methods, the strain-
optical constants of sodium fluoride, sodium chloride and
sodium bromide have been determined. Also, their strain-
polarizability constants have been calculated. The dynamic
method utilized a new pulsed ultrasound optical technique.
There was better agreement between the dynamic measurements and
earlier dynamic measurements, than between the static measure-
ments and earlier static measurements.

Vedam and Ramaseshan1 comment that the experimental
errors cannot account for the discrepancies between directly

observed values for (p11 ) and for phh as reported by

' p12
various investigators. They suggest four possible explanations:
(1) non-uniform stress distribution in the sample, (2) conditions
during growth and annealing, (3) previous history, both thermal
and elastic, of the specimen and (h) any imperfections in the

crystal, i.e., impurities, defects and dislocations.

6h

BIBLIOGRAPHY

10.

11.

12.
13.
1h.
15.
16.
17.

18.

 

BIBLIOGRAPHY

R. S. Krishnan, Progress in gEystal Physics (Interscience
Publishers, New York, 1958), p. 111.

K. Achyuthan, "Ultrasonic Methods and Photoelastic Constants",
Ph.D. Thesis, Michigan State University, 1962.

M. P. Hagelberg, "The Diffraction of Linearly Polarized Light
by Ultrasonic Waves in Transparent Solids", Ph.D. Thesis,
Michigan State University, 1961.

H. F. Gates and E. A. Hiedemann, J. Acoust. Soc. Am. 28,
1222 (1956).

H. E. Pettersen, Ultrasonics Lab Memo No. 26, Feb. 196M,
Michigan State University.

w. Primak and D. Post, J. Appl. Phys. 39, 779 (1959).

K. Vedam, E. Schmidt and R. Roy, J. Am. Ceram. Soc. £2,
531 (1966).

M. Born and E. Wolf, Princi les 2: thics (Macmillan,
New York, 196M), 2nd ed., p. 73.

J. F. Nye, Physical Pro erties of ggystals (Oxford University
Press, London, 1957), p.16.

D. Brewster, Phil. Trans. (1815) p. 60, (1816) p. 156,
Trans. Roy. Soc. (Edinburgh) 8, 353 (1818).

F. Pockels, Ann. Phys. und Chemie 31, 1AA, 269, 372 (1889),
32, 11110 (1890).

S. Bhagavantam, Proc. Indian Acad. Sci. A16, 359 (19h2).
J. F. Nye, loc. cit. p. 13%.

Ibid. p. 97.

Ibid. pp. 1&0, 251.

R. S. Krishnan, loc. cit. pp. 76, 125.

W. Mason, Physical Acoustics Vol. I; Part B (Academic
Press, New York, 19 5 , p. 355.

 

 

L. Filon, A_Manua1 of Photo-elasticity fgr En ineers
(Cambridge University Press, Cambridge, 1936), p. 19.

65

19.

20.

21.
22.
23.
2h.

25.

26.

27.
28.
29.

30.
31.

32.

33-

3h.
35-

 

BIBLIOGRAPHY (continued)

G. Monk, Li ght tPrinci les and Experiments (McGraw-Hill,
New York, 1937), p. 228.

P. Debye and F. Sears, Proc. Natl. Acad. Sci. U. S. 18,
1+09 (1932)-

R. Lucas and P. Bicquard, J. phys. radium 3, h6h (1932).
0. Raman and N. Nath, Proc. Indian Acad. Sci. g, 1106 (1935).
F. Borgnis, Phys. Rev. 28, 1000 (1955).

D. Bradley, "Photoelastic Constants of Synthetic Sapphire“,
M. S. Thesis, Michigan State University, 1963.

R. Waxler and A. Napolitano, J. Research Natl. Bur. Standards
22, 121 (1957)-

D. Davis, Em irical Equations and Nomography (McGraw-Hill,
New York, 19E3), p. .

H. Mueller, z. Krist. 99, 122 (1938).
W. R. Klein, private communication.

T. Smith and A. Korpel, IEEE J Quantum Electronics QE- 1,
283 (1965)

R. Dixon and M. Cohen, Appl. Phys. Letters 8, 205 (1966).

M. Pedinoff and H. Seguin, IEEE J Quantum Electronics QE-3,
31. (1967)

K. Aleksandrov and T. Ryzhova, Sov. Phys. - Crystallogr.
g; 228 (1961).

Handbook 2: Chemistr and Physics (Chemical Rubber Co.,
Cleveland, 19 5 , th ed., pp. B-221-223.

H. Leibssle, z. Krist. 11h, #57 (1960).

G. Wertheim, Ann. Phys. und Chemie (Poggendorf) 86 321,

325 (1852), which appears to be translations of Compt. rend.
32, 289 (1851), 33. 576 (1851)

F. Pockels, Ann. Phys. und Chemie 32, hho (1890).

H. Maris, J. Opt. Soc. Am. $2, 19h (1927).

66

39-

A1.
A2.

A3.

AA.
A5.
A6.

A7.
A8.
A9.
50.

51.

52.
53-

5A.
55-
56.
57-

 

BIBLIOGRAPHY (continued)

K. Banerjee, Indian J. Phys. 2, 195 (1927).

Y. Kidani, Proc. Phys.-Math. Soc. Japan, 3rd Series 21, A57
(1939)-

J. Galt, Phys. Rev. 13, 1A6O (19A8).
E. Burstein and P. Smith, Phys. Rev. 12, 229 (19A8).

S. Bhagavantam and Y. Krishna Murty, Proc. Indian Acad.
Sci- 5525 399 (1957)-

K. Bansigir and K. Iyengar, Proc. Phys. Soc. (London) 11,
225 (1958).

R. Srinivasan, Z. Physik 122, 281 (1959).
K. Krishna Rao and V. Krishna Murty, Nature 129, A29 (1961).

E. Coker and L. Filon, Photo-elasticity (Cambridge University
Press, London, 1931), p. 209.

K. Banerjee, Indian J. Phys. fig 195 (1927).

K. Herzfeld and R. Lee, Phys. Rev. 28, 625 (1933).
H. Mueller, Phys. Rev. 21, 9A7 (1935).

M. Born and E. Wolf. loc. cit. p. 87.

Burstein and P. Smith, Proc. Indian Acad. Sci. A28,
377 (19A8)

J. Pirenne and E. Kartheuser, Physica 32, 2005 (196A).

F. Seitz, The Modern meogy 21 Solids (McGraw-Hill, New
York, 19AO), p. 661.

K. Fajans and G. Joos, Z. Physik 23, 1 (192A).
J. Mayer and M. Mayer, Phys. Rev. 23, 605 (1933).
F. Seitz, loc. cit. p. 6A5.

J. Mayer, J. Chem. Phys. 1, 270 (1933).

67

   

APPENDICES

 

 

4‘

Appendix I.

 

 

 

 

 

 

Data sheet for static measurements.

 

 

Table 1A.
NaBr Static T Date 6/10/67-1
In
Light beam: 6 mm x 3 mm T along cell axis
PM detector B I L along face
.1 diagonal
2 / F
'5
Added Babinet readings for extinction
108: A Ave. B Ave. C Ave.
n g a.d. a.d. a.d.
P-e- P-e- p.e.
0 289A.6 2893.6 A181.0 A180.7 5A69.2 5A69.2
2893.0 0.70 A180.5 0.23 5A68.A 0.50
2893.1 0.A A180.5 0.1 5A69.9 0.3
1 2886.8 2888.3 A175.1 A17A.A 5A6A.0 5A63.6
2889.3 1.00 A172.1 1.57 5A63.2 0.27
2888.7 0.6 A176.1 0.9 5A63.6 0.2
2 2883.9 288A.9 A170.5 A170.6 5A57.8 5A56.A
2885.5 0.63 A170.2 0.33 5A56.3 0.90
2885.2 0.A A171.1 0.2 5A55.2 0.5
3 2877.3 2877.1 A16A.2 A163.7 5A52.0 5A52.3
2876.3 0.53 A163.7 0.37 5A52.0 0.37
2877.7 0.3 Al63.1 0.2 5A52.8 0.2
A 2871.9 2872.2 A158.9 A158.9 5AA6.0 5AA5.A
2870.5 1.30 Al59.5 0.37 5AA6.2 0.93
287A.1 0.8 A158.A 0.2 5AAA.o 0.6
5 2866.5 2868.5 Al53.7 A153.9 5AA1.A 5AA1.7
2870.8 1.53 A152.9 0.77 5AA2.2 0.37
2868.2 0.9 A155.0 0.5 5AA1.A 0.2
6 2863.3 2863.3 A151.3 A150.3 5A36.6 5A38.l
2862.5 0.57 A1A7.2 2.07 5A38.3 1.03
286A.2 0.3 A152.A 1.2 5A39.5 0.6
7 2856.3 2856.6 A1A2.A A1A3.5 5A3l.1 5A31.2
2857.3 0.A3 A1A1.9 1.77 5A31.A 0.17
2856.3 0.3 A1A6.1 1.1 5A3l.0 0.1
8 285A.l 2853.A A138.0 A139.2 5A26.8 5A26.5
2853.1 0.50 A139.6 0.83 5A26.7 0.33
2852.9 0.3 A1A0.1 0.5 5A26.0 0.2

 

 

68

 

 

 

Full 1*. air:-

 

‘ll‘lfld‘lls .

Appendix I continued

 

Added Babinet readings for extinction
:38: A Ave. B Ave. C Ave.
g a.d. a.d. a.d.
p.e. p.e. p.e.

9 28A7.0 28A7. 9 A135.1 A13.
28A6.11.77 A133. 9
2850.5 1.1 A133. 2

5A21.8 5A21.3
5A20.6 0.A7
5A21.5 0.3

c>c>;-
«are
C)

n
10 28A5.1 28A5. A A128. 2 A12 29 6 5A17.2 5A18.2
28AA.6 0.70 A130.1 0.97 5A17.6 1.1
28A6.A 0.A A130.6 0.6 5A19.9 0.7

A

0

0

11 2836.3 2838. A A12A. 3 A12
2839.7 1.A3 A123. 3
2839.3 0.9 A125.0

2 5A12.A 5A12.2
.60 5A12.3 0.20
A 5A11.9 0.1

 

 

 

 

 

 

 

 

g A.99 1- 0.06 5.0A f 0.06 5.03 :- 0.0A
AL kg kg k8
Eq. (36)
p _ p _ 4251 1 ABcd
11 ‘ ,
12 tD (311- H12>n 3 L33

2 x A.00 cm2 x 5A6l x 10..8 cm

— 2.00 cm x 1287 [0.2850 -(-0.063A)] x 10-11‘293

dyne

f§.OZ
9.8 x 105dynes

- 0.0565'

69

 

Appendix I continued

 

H o m
/

0/ 0.34m O/O OmH: /°
AVIIIIII/nv ONJW IIIIIIIIOFIIIIIII OOON
/ O
U mphflv Av
/ 0:: /
0/ 0/0
0.
9% /o ow: /o

[I’ll/IMNIIIIIII
8%

03m

0 mwcwuuom m mwsfiuuom

a omeaouom

.wmoH msmuo> mdfipuou HOuomcomaou uocfinmm .P shaman
.ws ea coca cocoa

S 2 m m s m m a m m H o

my . w u u u n A u 7v. u u 1?
/ Ofimwm if
o 8 m ..

< o>u=u IIIIIIIAu w
nu
/ ONHJ 8®N 4r
nYllIIIII AU’IIIIII

o 82 ..

 

21 31q93 mozg

°uornou73xa Jo; sBurpsax Jonssuedmoo naurqsq aBeraAv

7O

 

—-._ -. _-.,J‘*" '
I“ _. Fwd...“- .
e .07.; .15. 13“" ‘ 7‘ -

 

 

Appendix II.

 

 

 

 

 

 

Data sheet for dynamic measurements.

 

Table 15.
Date 2/8/67-2
IN C
S along cell axis
NaCl
H L along face diagonal
I ‘___
L
S
Frequency GR 7.9-77.A mc Output = 0.2 v
Pulser 9.7A mc Pulse length = 3.6 cm
Power supply to phototube = 1025 v Trace flat for 2.5 cm

Oscilloscope on line sweep
Pulser triggered from scope
Sweep at 50 usec/cm

100% light = 0.2 v/cm x A cm

Reading at 3.6 cm
Wollaston after sample
Right image polarized.i.wave

 

 

rf voltage P-P Light in
0 order
v/cm cm v
Left Right
image image
cm cm
5 x 10 6.A0 3.15 3.55
6.65 3.00 3.A5
7.75 2.80 3.30
10x10 A.A0 2.70 3.25
A.90 A90 2.60 3.20
5.A5 2.50 3.10
5.95 595 2-A0 3-05
6.60 2.30 2.95
7.10 710 2.20 2.85
20x10 3.85 770 2.05 2.75
A.20 8A0 1.95 2.70

front
Left image polarized llwave
front
% light Theoretical vL Dev.
in 0 order curve --
vR
Left Right vL vR
image image
78.75 88.75 0.98 0.70 1.A0 0.01
75.00 86.25 1.07 0.78 1.37 0.02
70.00 82.50 1.20 0.88 1.36 0.03
67.50 81.25 1.27 0.91 1.A0 0.01
65.00 80.00 1.32 0.9A 1.A0 0.01
62.50 77.50 1.38 1.01 1.37 0.02
60.00 76.25 1.A6 1.0A 1.A0 0.01
57.50 73.75 1.5A 1.11 1.39 0.00
55.00 71.25 1.61 1.17 1.38 0.01
51.25 68.75 1.72 1.2A 1.39 0.00
A8.75 67 50 1.79 1.27 1.A1 0.02
Average 1.39 0.013
p.e.0.01

 

71

 

 

 

   

 

 

 

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(1711711)"S

3174 4778

   

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