DETERMINATION OP THE ELASTIC CONSTANTS
OF SAPPHIRE BY DIFFERENT ULTRASONIC METHODS.

By
Walter Georg Mayer

AN ABSTRACT
Submitted to the School for Advanced Graduate Studies
Michigan State University of Agriculture and
Applied Science In partial fulfillment of
the requirements for the degree of
DOCTOR OF PHILOSOPHY
Department of Physics
1958

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DETERMINATION OF THE ELASTIC CONSTANTS OF SAPPHIRE BY
DIFFERENT ULTRASONIC METHODS.
Walter G-. Mayer

Abstract.
Three optical methods, the Schaefer-Bergmann method, the
visibility method, and the line diffraction method were used
to find the six elastic constants of synthetic single-crystal
sapphire, and a comparison of the accuracies of the methods
was made.

The accuracy of the three optical methods was fur­

ther checked by using the thickness resonance method, and It
was found that the visibility method and the thickness reson­
ance method yield results which agree within better than 0 .01 ^.
A purely electronic method, the pulse method, was used
to determine the reliability of measurements of the elastic
properties If the sapphire sample is in the shape of disks or
thin plates•
The visibility method was found to reveal structural
irregularities and misorientations in the sample, which give
rise to localized variations of ultrasonic velocities.
The elastic constants, referred to a suitably transformed
coordinate system describing the crystalline structure, were
compared with the ultrasonic velocities measured in various
directions; the results could in some cases be used to obtain
some information concerning the orientation of the samples.

Walter G. Mayer
The six elastic moduli of sapphire were found from the
elastic constants.

The numerical values of Young*s modulus

as a function of the orientation were found from the elastic
moduli for some cryatallographic directions, showing the
crystalline symmetries of the samples.

DETERMINATION OF THE ELASTIC CONSTANTS
OF SAPPHIRE BY DIFFERENT ULTRASONIC METHODS.

ByWalter Georg Mayer

A THESIS
Submitted to the School for Advanced Graduate Studies
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of
DOCTOR OF PHILOSOPHY
Department of Physics
1958

AC KNOT,^LEDGMENTS
I wish to thank Dr. E. A. Hiedemann who
suggested this investigation for his helpful
and valuable guidance throughout the course
of its development.
I also wish to express my Indebtedness
to the Linde Air Products Company for furni­
shing samples for the measurements included
in this paper and for granting financial aid.

ii

TABLE OF CONTENTS

>ag
Introduction

1

Theoretical Considerations

3

Measurement Techniques

14

Light Diffraction Methods

16

Visibility Method

28

Thickness Resonance Method

37

Pulse Method

42

Light Refraction Method

47

Transmission Through Disks

48

Computation of Elastic Constants and Moduli

55

Summary

62

Appendix

63

Bibliography

66

ill

LIST OF FIGURES

1.

_
Location of Axes and 1210 Plane

Page
7

2.

Possible Rotations of Axes

10

3.

Orientations of Samples

11

4.

Optical Arrangements

15

5* Secondary Diffraction Patterns Produced by
Light Diffracted due to Shear Wave

19

6 . Diffraction Pattern of Crossed Sound Field

21

7- Constructive and Destructive Interference of
Transverse Circles

22

8 . Line Diffraction from III B

23

9• Multiple Diffraction Produced by Light of
Three Wave Lengths

26

10.

Longitudinal and Shear Wave In IV Z

31

11.

Irregular Spacing in I Z

31

12.

Structured. Irregularities in Samples

35

13- Splitting of the Pulse due to Presence of
Sample

44

14.

Longitudinal and Shear Wave in a Plate

49

15-

Angular Dependence of Transmission

51

16.

The Wedge Method

53

17.

Angular Dependence of Young*s Modulus

60

Iv

I.

Introduction
The object of the study reported in this paper is two­

fold: the experimental determination of the elastic constants
and moduli of synthetic single crystal sapphire, and a crit­
ical study of the various methods used with respect to their
reliabilities and accuracies.
Four blocks of sapphire and a disk of known orientation
were used in the course of the experiments.

The measured

ultrasonic velocities in different crystallographic directi­
ons were related to the corresponding elastic constants.

The

determination of these velocities in the samples constituted
the experimental part of this paper.

Only in a few cases was

it possible to relate the observed ultrasonic velocities dir­
ectly to elastic constants with the aid of the general Chris­
toff el Equations.

Ordinarily these equations yield results

which, in their analysis, give rather cumbersome mathemati­
cal expressions.
matical procedure.

Thus it was decided to simplify the mathe­
By a suitable transformation of the co­

ordinate system of the sapphire, given

by four axes since the

crystal belongs to the trigonal group,

to a rectangular co­

ordinate system and a subsequent rotation of that system so
that the new Z axis is always parallel

to the direction of

sound propagation, it was possible to obtain for all cases an
expression for the square of the measured ultrasonic velocity
in terms of sums and differences of the elastic constants and
the density of the sample.

These expressions were then used

for the calculation of the elastic constants.

1

Measurements of ultrasonic velocities can be based on
optical methods or on purely electronic principles.

In the

optical methods the diffraction of light by ultrasonic waves
offers various possibilities for the measurement of their vel­
ocities.

In the electronic methods either the travel time of

an ultrasonic pulse in the sample is measured or the transmis­
sion properties of plates as a function of the angle of ultra­
sonic incidence Is used to find the velocities of propagation
in the sample.

Since the velocity in sapphire is extremely

high it was suspected that some of the standard methods might
become unreliable while other methods, which are inaccurate
in cases where the velocities are much lower, might be useful
in this special case.
The problem of the location of the coordinate systems
and various methods for finding ultrasonic velocities are
treated in the following sections.

2

II*

Theoretical Considerations•
For the purpose of this paper it is assumed that the gen­

eralized Hooke's law is valid, implying that there exists a
linear relationship between the stress and strain components.
Higher order elastic constants are neglected; they arise if
the stress components are given by polynomial functions of the
strain components.
Under adiabatic conditions the generalized Hookes law can
be expressed as

(1 )

Eq.(l) can be written
sp * spq Tq
TP

p,q = 1,2

6

(2)

°pq ^q _

if the suffixes ij and kl are replaced by p and q according to
ij or kl
porq

11 22 33 23 13 12

1 2 3 4 - 5 6 .

(3)

The quantities cpq and spq are called the elastic cons­
tants where sometimes "stiffness" refers to Cpq and "compli­
ance" to Spq.

Cpq are ordinarily called "constants" while spq

are usually "moduli", following Voigt's (1) usage.

Hearmon

(2 ) calls the cpq "parameters" and the spq "coefficients" while
the I.R.E. refers to the Mcpq constants" as "moduli of elas­
ticity" and to the Mspq constants" as "moduli of compliance".
In this paper Voigt's notation is used.

To evaluate the elastic constants from experimental data
one usually employs the so-called Christoffel equations which
were first derived by G-reen (3).

Their derivation and appli­

cation is based on the fact that with an elastic wave propa­
gated in an anisotropic infinite medium there are associated
three independent, mutually orthogonal displacement vectors D
which are functions of the direction of the wave vector k
whose direction is perpendicular to the planes of constant
phase of the propagated wave.

In general the three D belong

to three different plane waves which are propagated in the
direction of k with three different velocities.
three D are mutually orthogonal

Since the

it can be seen that In cases

where one of the D Is parallel to the direction of k one pure
longitudinal wave results which is associated with two pure
transverse waves.
Let the coordinate system of the anisotropic material be
fixed such that k has the direction cosines 1, m, and n.

The

directions of the three possible D are, in general, different
from the direction of k.

In Green's theory the equations

dealing with these directions contain a set of relations bet­
ween the phase velocity of the elastic wave, the cpq, and the
direction cosines 1, m, and n.

These general relations, ad­

apted to the crystalline structure treated here, are used in
this paper.

The three possible velocities V associated with

the propagation vector k and the direction cosines are the
roots of the cubic equation

4-

011 “ ^°vS

012

013

012

022 “ P vS

023

013

023

033 ~ >°v2

where p is the density.

The

= 0

(4)

are defined by

f^ab = lS<3XaXb * m S °2a2b + n E °3a3b + 1111:1(®2a3b + ®3a2b)

(5)

+ n l (c3alb + cla3b) + m l (°Xa2b + °2aXb)-

Using Eq.(3) the above expression can be transformed into the
2 -suffix system and contains then all possible Cpq where

p and q = 1 ,2 ,...6 .
The crystal group to which the anisotropic substance be­
longs determines the number of constants c

different from

zero, as well as the arrangement of the non-zero terms in the
appropriate matrix.

For sapphire, which belongs to the tri-

this matrix: becomes

C11

OJ
H
O

c13

c12

°1X

H
O

c13

CX3

c33

see Love

°14

0

0

"C14

0

0

0

0

0

°44

0

0

(6 )

0

0

0

0

c44

°X4

0

0

0

0

014-

c66

C14

1

H
0

0

where c66 = i(ci;L - c12).
Only 6 constants are present in Eq.(6 ) which, when sub­
stituted in Eq.(5) together with the appropriate direction

5

cosines, yield
of cpq*

Eq.(4) can now be solved for pV2 in terms

One will note that Eq.(5) contains 21

which

are reduced to 6 due to the properties of the matrix*
trigonal system the following

For the

are obtained.

^11

55clllS + c66mS + c44n8

012

= 2 c-j_4.nl 4- (c6£ + c12)ml

(8 )

<z*13

= (°44 + 013)^1

+ 2cli}.ml

(9 )

022

= c66 ie + cllm* + c44 nti " 2c ^ m n

023 = C14^1S *
^33 =

^c^mn

(7)

(1 0 )

+ ^c13 + cif4^mn

^1:L^

(la + m 2) *■ c-^n?

(1 2 )

' Before it is possible to relate observed ultrasonic velo­
cities to the elastic constants it becomes necessary to estab­
lish the direction of k with respect to the coordinate system
of the crystal, that is one has to find 1, m, and n.

A plane

in a crystal of the trigonal system can be defined in terms
of its intercepts with the 4 crystallographic axes.

It is al­

so possible to use a rectangular coordinate system for this
purpose and then relate k to it.

Mason (5) uses this proce­

dure to define the axes of quartz which belongs to the same
group as sapphire (A12 0^).

Referring to figure 1, let the

Z axis be identical with the c-axls of the hexagonal system
of axes.

The XY plane is then the plane of reflection.

There

are three positions for the X axis which are equivalent in re­
gard to the hexagonal system since there are three equivalent
i

positions for the a-^, a2 , and a^ axes.

6

a,

a-,

1210

Ax

ct.

2 MO

a,

Figure 1*

Location of Axes and 1210 Plane.

It can be seen that with a rotation of the hexagonal axes
system around 120° the same plane can be labeled 1120 or 2 1 1 0 .
With the axes fixed one can now find the direction co­
sines in terms of a rectangular coordinate system.)] Let the
ultrasonic wave be propagated with k parallel to Z and sub­
stitute the resulting direction cosines 1 - 0 , m - 0 , n » 1
Into Eqs.

(7) to (12).

All

vanish except

h i = *22

')\h
T?

7

*33

33

This gives the solution of Eq.(4) for the three values of V
- c4 4 >

>oV^ = c44»

- c3 3 *

(13)

Considering k parallel to the X or Y direction one finds
that Eq.(4) has ra/ther involved solutions.

Bhlmasenachar (6 )

solves the Christoffel equations for a few directions of k.
Cady (7) mentions the complexity of this procedure and indi­
cates the possibility of an arbitrary rotation of the entire
coordinate system.
In order to simplify the calculation of the elastic con­
st ants of sapphire It is advantageous to perform a rotation of
the coordinate system such that the Z axis of the rotated sys­
tem is In the direction of the wave propagation vector k.
Let the new Z axis be denoted by Z *.

Whatever the direction

of k may be, there will be associated with it one longitudinal
wave and two shear waves with velocities V^, Vg]_, and Vgg •
From measurement of these velocities on© will find three cor­
responding effective elastic constants which shall be written
kL» kS1> and Kg2 •

Obviously, these constants will be a com­

bination of a number of actual elastic constants Cpq*

The

combination depends on the angle of rotation between the XYZ
system and the X * Y !Z ! system.

Let the cosines of the angles

between the axes of the old system and those of the X*Y'Z!
system be

such

that the first suffix denotes the axisof

the old system and the second the axis of the new. '
Following this scheme one obtains for the trigonal system
the following values for the effective elastic constants for

8

any direction of k. (See Appendix)
c1IL(l -

2.

)"*" + c33 azz + (^c44 * 2c13^azz^1 ~ a;'ZZ

(14)

+ ^*cl4ayzazz (^axz ~ ayz ^*

(15)

axzazy) }

It will be noted that only one expression is given for Kg.
Nevertheless, two Kg can be obtained from Eq.(15)*

Consider

a k which is not parallel to the Z axis and let its direction
be given by W o angles; © being the angle between k and Z, and
0 the angle between X and the projection of k into the XY
plane.

Figure 2 shows how two possible rotations can be ob­

tained which do not violate the stipulation that Z* is to be
in the direction of k.

Using the X direction as an axis of

rotation, position B is reached from position A and k is par­
allel to Z*.

One is now at liberty to rotate the system fur­

ther with Z* as the axis of rotation to reach position C from
B.

It can be seen that the angles between the axes of the

old and the new systems will differ for positions B and C.
Regardless of which a__ „ are used in Eq.(l4), the terms in
c n will be identical in either case, i.e. there is only one
9

v/
A

C

B

Figure 2.

Possible Rotations of the Axes.

longitudinal wave velocity.

The terms in cpq In Eq.(15) will

differ, depending on which rotation is used for the calcul­
ation of Kg.

However, there are not more than two values for

every Kg. (See Appendix).
Three differently cut samples of synthetic single crys­
tal sapphire were available for the measurements of the elas­
tic constants.

These blocks were supplied by the Linde Air

Products Company (Union Carbide and Carbon Company).

The

orientation of the normals to the parallel faces was deter­
mined by Linde.

Table I lists the orientations of these nor­

mals, where the angles 0 and 9 refer to k according to figure
2.

The combination of block number and direction identifies

the face into which ultrasonic waves are transmitted.
Figure 3 shows the directions corresponding to the k !s
in Table I .

10

TABLE I
Block
I & IV
I & IV
I & IV
II
II
III
III

Figure 3*

Direction
of k
X
Y
Z
A
B
A
B

0

©

0
90
0
0
0

90
90
0
135
45
135
45

90
90

Orientations of Samples.

The basis for the evaluation of the observed ultrasonic
velocities in different directions is given in Table II*

The

effectice elastic constants Kl and Kg as obtained from E q s .
(14) and (15) are listed for the directions given in Table I
and in figure 3 .

11

TABLE II
Effective Elastic Constants for Different Directions of k.
k into
I X
I X

Effective Elastic Constant
Kl

(16)

= Oil

& < KS1 = c44
IV X
IV Y
Ksa = 4-(cn - Ci2 ) = °66
I Z
Kl
& i

IV z

(17)

(18)
(19)

= c33

(17)

Ksi = KS2 = >544
Kl

Eq.

= <t(on + 033

4c44 + 2013)

(2 0 )

II A 1
II B < Ksi = £(044 + C66)

(21)

kS2 = *(°11

(2 2 )

Kl

°33 “ 2o13)

= 4 ( c n *■ 033 + 4c44 *■ 2ci3 + ^cl4)

(23)

III A | KSi = i ( o n + 033 - 2ci3)

(22)

K32 = tr(044 + 066 - 2cl4)

(24)

Kl

= i ( o n + 033 * 4c44 + 2ci3 " ^cl4)

(25)

III B 1 K31 = i(cu ♦ C33 - 2013)

(22)

Ks2 = i(044 + 066 ♦ 2°l4)

(26)

12

It is now relatively easy to evaluate the observed vel­
ocities in different directions of the samples.

Using the

general formula

K

(27)

where V Is the observed longitudinal or shear velocity, one
obtains a numerical value for the combination of Cpq appea­
ring In the entries of Table II, corresponding to the various
directions of L
Substituting Eqs.(l6 ) and (17) into Eq.(27) is identical
with Eq.(13)*

Similar operations with other K ’s of Table II

give the same values as those obtained from the Christoffel
equations with the difference that the latter yields the equa­
tions for V in terms of Cpq with V occuring as V 2-, V4 , and V6.
The advantages gained by transforming the coordinate system
are obvious•

15

Ill.

Measurement Techniques.
A number of experimental techniques used to determine the

elastic constants of single crystal synthetic sapphire are des­
cribed in this section, and their reliability and accuracy is
discussed*

Figure 4 shows the optical arrangements employed

in the methods described.

Unless otherwise indicated, stan­

dard electronic equipment is used for the generation of ultra­
sonic energy from x-cut quartzes.

The transducers are coupled

to the samples by placing a thin film of Dow Corning Vacuum
Grease between the transducer and the block.

All frequencies

quoted are checked by a zero-beat frequency meter to an accur4
acy of 1 part in 10 , unless otherwise stated.

The values of the elastic constants determined with the
various methods are listed in Section IV.

14

a .- S ch aefer - Berg-mann:

b.- D iffr a c tio n

c-

CA

P5

CA

ss

CA

ss

Pattern:

V is ib ility M ethod:
sc

C
CA
F
L
M

Camera
Circular Aperture
Filter
Lens
Mercury Arc

P
Periods of In te rfe re n c e
PS P o in t Source
S
S a m p le
SC S c a le
SS S l i t Source

FigA. Optical Arrangements.
15

Light Diffraction Methods,
Depending on the geometry of the light source one obtains
either a line spectrum or the interference pattern named after
Schaefer and Bergmann.

The Schaefer-Bergmann method is exten­

sively described in a number of publications (8 ) - (12).

A

general diagram of the arrangement Is given in figure 4a.

It

differs from the standard diffraction set-up of Debye and
Sears (13) and Lucas and Biquard (14), shown in figure 4b, in
one point:

the light diffracted by ultrasonic waves is not

collimated before it reaches the sample as it is in the stan­
dard method.

In the former method the distance A between the

sample and camera is variable while in the standard method one
has a fixed distance between sample and camera.

To obtain a

rather large image size on the film, the distance A is made
big, about 2 to 4 meters.

The focal length of L2 is about

15 cm, thus the light traveling through the sample and then

coming to a focus on the film is considered sufficiently near­
parallel to be usable.

Schaefer and Bergmann (15) do not ex­

pect an accuracy better than

even under optimum conditions.

However, the accuracy with which It Is possible to meas­
ure the points on the photograph, and the fact that it Is
easier to focus the pattern on the film or plate if the image
size Is rather big, may justify the use of the method.
use of

The

in the standard diffraction pattern with a very

long focal length produces the same advantages and still makes
the use of parallel light possible.

16

The diffraction methods' possible advantage over the vi/

sibility method, treated in the next section, is that one Is
sometimes able to obtain the longitudinal and shear velocities
with one photograph, provided both waves are set up with suf­
ficient intensity at the frequency used.

But it is found that

this situation is not easily accomplished while it is not dif­
ficult at all to meet the proper conditions for the visibility
method.

Schaefer and Bergmann state that the accuracy of their

method is about the same as that of the visibility method.
general this is not the case.

In

First of all one has to deter­

mine more quantities in the S-B method than in the visibility
method, secondly, the determination of the radius of a small
circle of bright points with finite dimensions around a cen­
tral order can not yield as high an accuracy as the measure­
ments of the distance between about 20 to 30 narrow lines,
with the correct scale appearing on the photograph, as is the
case in the visibility method.
Both the

- Compare figures 5 and 10.-

S-B method and the standard diffraction method

use the following approximate formula for the calculation of
the ultrasonic velocity
n X FA
V - --un

. _.
(28)

where n = diffraction order, X = wave length of light, F =
th
frequency, Dn = distance between central Image and n
order
(or radius of the circle in the S-B method)•
The S-Bmethod yields
depend on thestructure

symmetricalfigures whose shape
of the sample.

17

Thesefigures are

used

to evaluate the elastic constants of the sample.

However, in

the case of sapphire the ultrasonic velocities are very high,
and the diffraction orders, even at frequencies above 10 Mo are
so closely spaced that the central linage of a point source in­
terferes with the first diffraction order.

Therefore it be­

comes necessary to limit the point source to a diameter of
about 25 microns.

This in turn diminishes the light intensity

necessary to photograph the above mentioned symmetrical fig­
ures.

Even with a high intensity mercury arc as light source

and overrated fast film, intermittent sound radiation with
associated interrupted exposure becomes almost a necessity.
The faint indications of symmetry in figure 5a are not partic­
ularly suited for an accurate determination of the elastic
constants.
Jona (11) points out that in some substances a bright,
grainy background is visible if the sample is transversed by
light in certain directions.

This also occurs in some samples

of sapphire and reduces the definition even further.
Ordinarily, a diffraction pattern of simultaneous longit­
udinal and shear waves consists of 0, series of points or lines.
In figure 5 the diffraction pattern due to a longitudinal wave
traveling In the Z direction is shown In the central vertical
row of orders.

The two rows of points to the right and to the

left of the principal orders are produced by multiple diffrac­
tion.

18

Figure 5*

Secondary Diffraction Patterns Produced
by Light Diffracted due to Shear Wave*

19

Two independent diffraction patterns are shown in figure
6.

They are generated by driving two transducers simultane­

ously at the same frequency and coupling them to two adjacent
faces of the sample*

A comparison of the wave lengths of the

longitudinal waves in the X and Z directions can be made un­
der Identical experimental conditions.

It can be seen that

the velocities In the two directions differ very little.

The

difference is not as great as Indicated by Bhimasenacharfs (16)
quotation of the elastic constants.

The locus of the fourth

diffraction order In the Z direction is indicated in figure 6
using Bhimasenachar*s constants.
The presence of the two rows of points on both sides of
the principal orders is in agreement with the findings of
Bergmann that light can be diffracted from every longitudinal
order by a shear wave.

The locus of the diffraction patterns

will be a. symmetrical figure indicating the dependence of shear
velocity on direction.

One will expect to see circles of rad­

ius R (corresponding to Dn In Eq.(28)) in the case of sapphire
only If the wave is traveling in the Z direction.

The two

shear waves associated with ultrasonic propagation In this
direction have the same velocity and are therefore Indistin­
guishable In that respect.

(See Eqs. (13) and (17))*

Y/herever two circles with radius R intersect one would
expect a more Intense dot since light diffracted out of two
longitudinal orders will Interfere at that point.

However,

Barnes (17) could observe in glass that due to the elliptic
polarization of light in the transverse circles with radius R

20

the light in the intersections of the circles will experience
destructive interference if their origins are the primary lon­
gitudinal orders n and n t 1 while circles with their centers
at longitudinal orders n and n t 2 will interfere constructiv­
ely.

Figure 7 Illustrates this point.

n ( O r \)

n - I

Figure 7*

Constructive and Destructive Inter­
ference of Transverse Circles.

No dots are visible at n(0 ,-l) and n(0,l) while dots with
varying intensities are observed at n(0,2), n(-l,l), etc.

The

intensities of the dots can not be expected to be equal if one
considers that the amount of light present at a point of the
secondary rows of the diffraction pattern depends, among other
things, on the amount of light present at the order out of
which it Is diffracted.

In addition, every diffraction order

due to the shear wave, e.g. n( 0 ,2 ), can now act as a central
order for a longitudinal diffraction pattern parallel to the
principal

diffraction pattern

gitudinal wave.

in the direction of the lon­

The same principle is Involved In the pres­

ence of the secondary longitudinal orders with the shear or­
ders as central orders as shown In figure 8 for a line dif22

3

Secondary Orders
with Shear Order
as Zero Order
Primary
Shear Orders
Primary Longitu­
dinal Orders

Figure 8 *

Line Diffraction from III B.

fraction pattern#

Despite the somewhat more spectacular

appearance of figure 5 , one obtains no more information con­
cerning the elastic constants

from it than from an inspec­

tion of the simple line diffraction pattern#
The almost complete disappearance of the secondary order
n(-l,l), the order opposite the central image, can now be ex­
plained partially from intensity considerations alone#

The

elliptic polarization of the light in the transverse circles
influences the observable intensities of the secondary orders
in the following manner.

It can be seen that the first longi­

tudinal order of n( 0 ,2 ) will occur at n(-l,l); so will the
first order of n(0,-2).

The light present in n(0,±2) is pol­

arized in the same sense due to its similar origin.

Thus one

would not expect a cancellation at n(-l,l) due to the light
from n(0,-2) and n(0,2)•

But at n(-l,l) there is also light

present due to the intersection of the transverse circles with
origins at n+1 and n-1.

Its polarization is such that destruc­

tive Interference takes place with light from n(0,2) and n(0,-2).
Points n(0,-2) and n(0,2) are brighter than any other
point on the secondary diffraction pattern.

They contain

light diffracted out of the strong zero order and the weak
second orders of the primary diffraction pattern.
Point n(-l,l) contains light diffracted out of the first
orders of the primary diffraction pattern and will therefore
be less Intense.
Experimental confirmation of the above explanation of the
origin of the diffraction points can be obtained by observing
24

the pautern under linearly polarized light.

In one case the

four n( 0 ,±2 ) points are visible, while the primary diffraction
pattern and the n(-lfl) points can be seen in the other case*
Traces of the inner circle

-the longitudinal circle- can

be seen in figure 5 , also longitudinal circles with n( 0 ,±2 ) as
centers can be traced out.

Now new information can be ob­

tained from that*
In a diffraction pattern produced by a point source of
monochromatic light one obtains only a limited number of dots
from which measurements can be made*

To Increase the number

of points available on one photograph, unfiltered light from
the mercury arc is used in such a manner that the three beams
of the most intense frequencies of Hg are refracted slightly
before they reach the sapphire block.

This results in some­

what displaced central orders of the three beams of wave
lengths 5791 A yellow, 5461 A green, and 4556 % blue*

With a

standing wave present in the block one can observe a great num­
ber of dots, each system producing its own set, each displaced
slightly with respect to the others due to the varying loca­
tions of the central orders.

Due to the different wave lengths

of the light ( X ) the distances between the orders of the sets
also vary.

The three systems contained In figure 9 are used

seperately to determine the constants C33 and C4 4 .

Rather

than trying to find R directly from measurements of the dist­
ance between an* ill-defined point of the primary order and its
ccrresoonding point in the secondary orders It is more con­
venient to obtain R* from

25

<

I? i

o

•O

£

■<f?
©

%•

CD

CD

«R>
(®*
<P

o
i]
1

-

111
iH

u
*

Figure 9.

Multiple Diffraction Produced by Light
of Three Wave Lengths•

26

= [r

R

+ (d/2)]*

according to
r

♦

*■
m

%

•

*

•

j
d
-

- ■

Using Eqs .(28), (19), and (17), the three independent measure­
ments yield the following values:
_____________ V i L ____________ V s ____________ C 3 3 _________ 0 4 4

5791

11207

6856

50.2

18.7

5461
4356

11159
11129

6850
6841

49.6
49.4

18.6
18.6

The averages of above values compare favorably with those cal­
culated from other measurements using the S-B method.
Using collimated light and a 4-0" lens for L-^ in the line
diffraction grating method one finds values higher than those
given above.

27

Visibility Method.
Ultrasonic ally produced "optical11 gratings have under
many conditions grating constants which, as in the case of
ruled optical gratings, can be determined most accurately by
direct measurements using an Abbe comparator or a traveling
microscope.

A method, which is usually referred to now as the

visibility method or the method of secondary interferences,
was therefore developed by Hiedemann, first with Bachem (18)
and la.ter with other co-workers (1 9 ) - (2 2 ) to make the ultra­
sonic waves visible and to measure their grating constants.
It is found to be one of the most powerful known methods with­
in its range of applicability (transparency, etc.); it yields
very accurate values of velocities

- for measurements in
4
liquids an accuracy of better than 1 part In 10 can be ob­

tained (20) -

and It reveals some important information about

the sample which is not obtainable with the other methods de­
scribed in this paper.
The optical arrangement necessary is shown In Figure 4c.
A standing ultrasonic wave Is set up In the sample S and acts
as a diffraction grating for the monochromatic light collim­
ated by L2 •

According to Hiedemann and Schreuer (20) this

arrangement produces an interference pattern at the planes P,
P ’...

These planes are equa.lly spaced along the optical axis,

and the periodicity of the visibility of the ultrasonic gra­
tings is of the same type as that for ruled gratings which was
studied by Winkelmann (23)•
28

The interference patterns in their simplest form consist
of a series of lines whose spacings are equal to the distance
between the nodes or antinodes of the standing wave in the
sample.

If, e.g., the wave length of the sound in the sample

is 1 mm, the lines produced at P, P !... will be & mm apart.
One can observe and measure these lines either with a travel­
ling microscope focused on the Interference planes or one can
photograph directly a portion of the planes containing the
Interference lines.

A suitable photographic technique was em­

ployed to photograph from 15 to 30 lines on one frame.

Using

the suggestion of Schaefer (24) to plc.ce a calibrated trans­
parent scale in a plane eliminates a number of possible errors
in relating distances measured on an enlarged photograph to
actual distances on the film negative.*)
From this description It is seen that the only serious
error likely to be made at all is the possibility of having
the light beam not exactly collimated.

The only measurement

of a linear distance necessary with this method can be consid­
ered very reliable since the correct scale appears on the fin­
ished photograph.
The fact that the scale is not necessarily perpendicular
to the interference lines obviously has no influence on the
accuracy of the method as long as one measures along a line
One actually mounts the whole assembly C, L^, and scale
(figure 4c) so that the scale Is in focus at C, and then
moves the assembly along the optical axis until P, P f...
is also in focus.

29

which Is perpendicular to the Interference lines using the
scale visible in any direction in the plane P, P'...
The actual determination of the desired constants is now
made rather easily.

One measures the distance between a num­

ber of lines on the photograph

which yields the ultrasonic

wave length In the sample, determines the frequency, and finds
the velocity.

The velocity is that of either a longitudinal

or a shear wave*

The spacing of the lines gives an indication

of what kind of standing wave Is set up in the sample.

Shear

waves produce lines more closely spaced than those due to a
longitudinal wave.
Figure 10a is a part of a photograph of a standing lon­
gitudinal wave In the IV Z direction.

Figure 10b shows a shear

wave in the same direction, made visible by using two crossed
nicols.

The scale for both photographs Is given In figure 10b.

Measuring the distances between the lines visible in these
photographs will result in the respective values of the longi­
tudinal and shear velocities.

Regardless of what lines are

selected the results agree very well among themselves.
Figure 11 shows the situation In the center portion of
block I.

The numbers on the scale correspond to 0.1 mm.

Closer inspection of the lines reveals that they are not equal­
ly spaced, particularly the lines going through about 13*1,
13.7, and 14.4.

The spacing between the first two lines is

smaller than the average spacing of the rest of the lines;
the next two lines are suaced wider.

This leads to the as­

sumption that the velocity in this region varies and is dif30

a

b
Figure 10.

a.- Longitudinal Wave In IV Z (F - 9096 kc)
bo- Shear Wave in IV Z (F = 9055 kc)

sI ■ »!'■
* *

£

•! X
Figure 11.

Irregular Spacing in I Z

i

31

(F = 9009 kc)

ferent from the average velocity in the rest of the sample
along the same direction.

Here a specie! advantage of the

visibility method over the other methods described in this
paper becomes apparent.

It is usually assumed that the ultra­

sonic velocity in one particular direction is uniform and a
calculation of the velocity according to the results of the
other methods is actually based on that assumption, or what
amounts to the same end result, one finds an average velocity
in a certain direction.

Thus it is impossible to prove or dis­

prove the assumptIon of uniformity.

Consider what will happen

If in the single-crystal block there should happen to be a.
small layer of polycrystallinity.

Under these conditions it

will still be possible to accomodate an integral number of
half wave lengths, provided the character of the plane wave is
not completely destroyed.

However, due to the layer a minute

change in frequency will be required to make up for the regi­
onal change In ultrasonic velocity.

Observation of a standing

wave does therefore not imply that the sample has a uniform
structure throughout.

The visibility method gives an indica­

tion of the average velocity in a small slice of the sample
whose thickness is limited by the spacing between the inter­
ference lines.

By increasing the frequency it is possible to

narrow the slice, where the limit of this process is essential­
ly set by the width of the observed lines.

By rotating the

sample around the axis of ultrasonic propagation it is possible
to find also the approximate lateral dimensions of the region
of polycrystallinity•
32

Variations In the density of the sample will produce simi­
lar results, so will local drastic temperature changes.

The

phenomena shown in figure 11 are not likely to be caused by
temperature gradients since in sapphire, according to specifi­
cations published by the Linde Air Products Company (25) the
mean linear thermal expansion coefficient per degree varies
from approximately 5 x 10"^ to about 9 x 10~6 between 20° C
and 1000° C.

This Is too small an amount to be of significant

influence considering the temperature gradient experimentally
possible with this method.
Measuring the distances between the pairs of the three
lines mentioned, and using the results as basis for the evalu­
ation of the regional velocity, is somewhat inaccurate since
the lines on the photograph are not well enough defined to war­
rant such a procedure.

Averaging over too many lines adjacent

to the three lines will greatly reduce the influence of the
irregular spacing.

To find at least an approximate value for

the change in velocity the following is done:

Measuring the

distance between the 13*7 line and the 6 th line to the left of
It, and then the distance between that line and the 14-.4 line,
one obtains two different velocities.

One is the average over

six lines, including the narrow spacing, the other the average
over seven lines, including the wide spacing.

The velocities

for the two cases are 11,072 m/sec and 11,702 m/sec.
Of course, measurements over more or less lines, or over
different sets of lines will give different velocities, but one
sees that a qualitative analysis Is easily possible.
33

The over-

all average of many measurements from numerous photographs
gives an average value of 11,215 m/sec.
Other structural irregularities are seen in figure 1 2 .
The sensitivity in detecting small inhomogeneities Is increased
if If a greater number of lines per ■unit length is observed.
In order to obtain more lines without increasing the frequency
it Is possible to move the assembly C, L-^, and SC along the
optical axis.

At various distances between planes P, P*...

the interference of diffraction orders is such that the number
of visible lines doubles (26) •

At other locations there are

three lines visible between the lines of the simple pattern.
Figure 12a is a photograph taken under these conditions.

It

shows how the lines bend and become blurred.in some sections.
These lines correspond to block I very near the edge of the
sample.

Examination of the block under polarized light shows

that In the region in question the sample is polycrystalline.
Figures 12b and 12c are photographs of sections of block II in
which some lines are discontinuous or disappearing completely.
Some of the irregularities may well be due to changes of
density throughout the samples; the exact causes are not fur­
ther Investigated here.

It is obvious that for the determin­

ation of the ultrasonic velocities and elastic constants the
visibility method is by far superior to the other methods dis­
cussed.

In addition to that one is able to detect various

structural irregularities should they be present in the regions
over which the measurements are made.

34

35

The inability to find the same values of the

vrith

all the three optical methods compared here may have its main
cause in the inaccuracy involved in the exact determination
of the central plane of L-^, affecting the distance A, in the
case of the line diffraction patterns, and the non-parallelism
of the light in the case of the Schaefer-Bergmann method.

The

visibility method eliminates these possible sources of error.

36

Thickness Resonance Method*
If an ultrasonic wave is transmitted into a sample with
parallel faces one finds that resonance of the sample can be
achieved at various frequencies, depending on the dimensions
of the sample.

These resonance conditions, studied by Lord

Rayleigh (27), correspond to maximum ultrasonic transmission
through the sample.

From observations of a series of reson­

ances one can find ultrasonic velocities in the sample in the
following manner.
First one finds a frequency at which maximum transmission
through the sample occurs.

Then one varies the frequency un­

til another maximum is observed.

If a longitudinal wave cau­

ses the occurance of both maxima by setting up standing waves
in the sample in such a manner that the number of half wave
lengths accomodated between the parallel faces of the sample
has

changedbyexactly

one

can find the

one, due to the change In frequency,

total number of wave lengths for each maximum

provided the distance between the faces of the sample is known.
Letting the frequencies at which the respective maxima occur
be F4 and Fz , and the thickness of the sample D, one knows that
the number of half wave lengths set up in the sample are x
and x

1.

The velocity of the longitudinal wave

is con­

stant; to find VL and x one uses
F *

x/2
where

‘

^ Xu D

> Fa .

37

(xfl)/2

Fl

r

.

(29)

From this on© obtains
Fi
X

= IPWF/

(30)

The v/ave lengths ^ a n d }^2are 2D/x and 2D/(x+l), respectively.
Thus the velocity becomes

V t - - 555-

L “

x

(31)

" x+'l *

}

At frequencies around 10 Me and sample thicknesses of
about 25 mm one finds the difference of Fz and F± to be of the
order of 250 k c . According to Eq.(30) x is approximately 40.
It Is this high number of half wave lengths that makes the
application of this method to thick samples rather disadvan­
tageous.
F2 ,

Ifa smallerror is made in determining either

an x maybe found

rather than 40.

from Sq.(30) which is closer

Fz or

to 39 or 4l

Eq.(31) shows that the wrong choice of x will

give an inaccurate value of Vp,, and since the elastic constants
contain the square of the velocity, this error may become quite
significant•
Thus finding only two frequencies for maximum transmission
can not be considered sufficient, and it becomes necessary to
obtain a series of frequencies

in order to calculate the most

probable value of Vl *
The thickness resonance method is useful as a convenient
check of the velocities found with other methods.

Working

with frequencies of about 9 Me, one finds that a 2 Me trans­
ducer radiates sufficiently from about 9-5 Me to 10.5 Me if
driven near its 5th harmonic, and a 3 Me quartz from about
38

8.5 Me to 9-5 Me if driven near its 3rd harmonic.

An ade­

quate range can be covered by using these two transducers.
The following approach is useful as a check of the visi­
bility method.

Having found the average wave length in the

Z direction of block I to be 1.246 mm at 9*009 Me and knowing
the dimensions of the sample one finds x for this particular
situation
X = ££

(32)

A.

to be very close to 39*

If the value of the velocity found

with the visibility method Is correct then the x is exactly 39*
Assuming this the next maximum transmission for x + 1 and for
x - 1 must occur at frequencies
F+1- h

+ 1)VL

and

F

(x - 1)VL

2D
since in general

2D

2? =, 2D/(x + n) .

The sample is either partly Immersed in CCl^ in which a
diffraction pattern will be observed if maximum transmission
through the block occurs, or a diffraction pattern is observed
in the sample directly.

The diffraction pattern in CCI4. does

not vanish as rapidly as it does in the sample Itself•

Thus

the desired frequency can be determined more accurately If the
diffraction pattern in the block is observed.

A change by

2 kc of a frequency around 10 me is sufficient to cause the

extinction of the diffraction pattern.

Table III lists the ex­

pected and observed frequencies for various values of x.
Also listed are the velocities obtained by using Eq.(33) Into

39

which the observed frequencies are substituted while Vp is the
unknown qu ant ity .
TABLE III
Expected and Observed Frequencies.
Half Wave
Lengths

Frequenc ies (Me)
Expected
Observed

x —1 = 38

8.778

x = 39

—

Quartz
Used

Xl
(m/sec)

8.780

11215

9.009

11213

i
> 3 Me
/

x +1 - 40

9.240

9.243

11216

x +2 = 41

9.471

9.461

11201

x+3 = 42

9.702

9.716

11229

x+4 = 43

9.933

9.942

11219

► 2 Me

The average ultrasonic velocity into the Z direction is thus
11,215 m/sec - 0 .2^, and the elastic constant c-^ is then
50.2 ± 0.4^ in units of 1 0 11 dynes/cm*

The thickness resonance method was used to find the con­
stant C4.4. by using a BaTiO-^ transducer driven in the neighbor­
hood of 450 k c.

Knowing the longitudinal velocity and using

the above approach, a diffraction pattern due to the longitu­
dinal wave is expected at 462.8 kc and occurs at 462.3 kc,
corresponding to x = 2.
is 231 kc.

For x =■ 1 the corresponding frequency

A very weak diffraction pattern is observable in

CCI4. at 444.5 kc which must be due to the shear wave in the
Z direction.

However, the method is not too reliable in this

particular application.

40

The attempt, to increase the intensity of the shear wave
by using a y-cut quartz, as suggested by B. Ramachandra Rao
(28), vras not successful.

41

Pulse Method,
A great number or workers have used the pulse technique
for the determination of ultrasonic velocities in solids and
liquids (2, 29, 30).

The method described here is purely non-

optical and finds its best application in samples where the
measurable travel time of the pulse is shorter than the length
of the pulse*
Ultrasonic pulses of 10 microseconds duration with a mod­
ulation frequency of 2*5 Me are used in this experiment.

The

frequency does not enter the calculations, and no attempt is
made to measure it accurately.

BaTiO-^ transducers are used

throughout.
Essentially two methods for finding the velocity in the
sapphire sample are available: The ultrasonic pulse enters
the block directly from the transducer.and is either received
at the opposite end of the block or the pulse reflected there
is received again at the face of the block through which the
ultrasonic pulse entered.

The elapsed time of travel and the

path length give the velocity of the pulse.

In the second

method the ultrasonic pulse travels first through a liquid med­
ium with known velocity, then passes through the sample, en­
ters the liquid again and is finally received by a second,
matched transducer.

A comparison of this travel time with the

time elapsed if the sample is removed from the liquid yields
the ultrasonic velocity in the sample.
The first method is not easily applicable for the fol­
lowing reasons.

The ultrasonic velocity in sapphire is in the
42

neighborhood of

11,000

m/sec and the average thickness of the

samples available is about 25 mm.
across the blocks is about

2

Therefore the travel time

to 3 microseconds.

This time

difference is too small to be accurately recorded and evalu­
ated directly.
The second method is more successful.
show the general arrangement.

Figures 13a and b

In 13a the pulse leaves the

transducer X at the same time the sweep of the range oscillo­
scope (DuMont 256-F) is started and travels towards the matched
receiving transducer R.

A wave front is shown In three dif­

ferent locations A-A1, B-B1, and C-C 1 as it travels through
the liquid.

If t he sample is inserted in the path, as shown

In 13b, the lower portion of the ultrasonic pulse will not be
changed.

The upper portion of the pulse enters the sample

where Its velocity is considerably higher than in the liquid.
This causes a splitting of the wave front (B-B9)-

Upon lea­

ving the sample no further displacement of the two portions of
the wave front will be experienced.

A signal, schematically

represented by A-A* in 13b, will reach R and consists of two
distinguishable pulses, provided the upper portion has gained
sufficiently to have actually overtaken almost all of the
lower portion of the pulse.

Figures 13c and 13d are the re­

corded traces corresponding to cases 13a and 13b.

The dis­

placement on the time scale between the corresponding wave
trains of the two pulses in figure
velocity in the sample.

13d

is used to find the

The calculation involves the following

steps.

4-3

o

T>

JL

O
-t-

*}
lT

xL
a

c a p

44
.4"

Let
the,

the time or

arrival at R of a certain wave train of

slow pulse be Tg .

The time of arrival of the correspon­

ding wave train of the pulse which travels also through the
sample is denoted by Tf .
caused by'two facts.

The difference in Ts and Tf is

The liquid path length for the fast

pulse is shortened due to the presence of the sample.
the time thus saved be denoted by t'.

Let

It follows that this

t* = sample thickness/velocity in the liquid.

However, the

.ultrasonic pulse does spend some time t in the sample where
t* = sample thickness/velocity in the sample.
the' travel time in the sample

Therefore,

is

t =. Tf - Tg + t*
If the dimensions of

(34)

the sample are known, the ultrasonic vel­

ocity in a particualr direction in the sample is easily found
after t is determined.
Mineral oil with an ultrasonic velocity of 1449 m/sec
was used in this experiment.

Time measurements were taken

with the scale of the range oscilloscope where an estimate to
the nearest 10"® second limits the accuracy of the method.

This limitalon is rather important if one considers the three
measurements of velociy in the I Z direction which are listed
below.

The differences in the recorded times are 0.01 micro­

second in each case.

45

Measured Tf
Measured
Computed V

(psec)
(psec)
(psec)

69.29
83.92
16.78

69 •68
84.30
16.78

70.08
84.69
16.78

Computed t
Velocity

(psec)
(m/sec)

2.15
11288

2.16
11236

2.17
11184

Sample Thickness:

24.27 * 0.01 mm

These values of V l agree well with those found with the
other methods

(average 11215 m/sec).

Despite the limitations

of the time measurements, this method is found to be fast and
simple and is well suited for cases where the dimensions of
the sample and the ultrasonic velocities in it make direct
pulse measurements either impracticable or unreliable.

46

Light Refraction Method.
Lucas and Biquard (14) observed the refraction of a nar­
row light beam in a sound field and described the widening of
the slit image associated with that phenomenon.

Iyengar (31),

Loeber and Hiedemann (32), and others have used this method to
measure ultrasonic velocities in liquids and solids.
Applying this principle to measurements of ultrasonic
velocities in the sapphire blocks available yields very vague
results for longitudinal waves and no results for transverse
waves.

The rather big changes of index of refraction between

the sections of the blocks through which the light has to
travel cause variations In the light intensity received by the
photocell which are of about the same order of magnitude as
the periodic changes in light intensity produced by the pres­
ence of the ultrasonic wave in the sample.

The output of the

photomultiplier Is the sum of both effects, and the exact
detrmination of the ultrasonic wave length in the sample be­
comes impossible under those conditions.
The use of polarized light eliminates some of the effects
of the irregular index of refraction but no sufficient im­
provement can be gained.

47

Transmission Through Disks.

The transmission of1 sound through plates of isotropic
materials was investigated by Lord Rayleigh (27) whose theo­
ries were later expanded by Cremer (33), Gtitz (34), Firestone
(35), Schoch (36), and others.
In the case of perpendicular incidence of an ultrasonic
beam on the plate the conditions for maximum transmission
through the plate are given if the thickness of the plate is
n}?/29 while minimum transmission occurs if the plate thick­

ness is (2 n - l)3?/4, where n -

1 ,2

,... and k* is the ultrasonic

wave length in the plate.
If the plate Is immersed in a liquid medium and an ultra­
sonic beam impinges on the plate one finds a series of trans­
mission and reflection peaks if the angle of incidence changes.
The location of the peaks with respect to the angle of inci­
dence is a function of the ultrasonic velocity in the sur­
rounding medium, trie plate thickness, the frequency, and the
longitudinal and shear velocities in the plate.

The theory

relating the location of the peaks to the above quantities
was worked out by Schoch (36).

This theory can be used to

find elastic constants if one deals with Isotropic material.
Essentially, Schoch*s theory Is based on Snell’s law and
thus assumes that the velocities in the sample remain con­
stant regardless of the direction of propagation.

The use­

fulness of the theory is thus limited if applied to sapphire
where, due to its single-crystalline nature, the ultrasonic

48

velocities are a function of the propagation direction.

In

this case Snell*s lav/ must be changed to
v
V(/9)

sin a
sin/3

(35)

where v and oc refer to the liquid medium and V and ^3 to the
sample•
Aside from the case where f& * 90° or 0°, Schoch*s theory
can not be used in its present form for the following reasons.
Referring to figure 14 one can see that a longitudinal wave
in the plate can not reach face B if /9

90°.

In this case

it will be propagated in a direction approaching the X direc­
tion.

In Eq.(55) this corresponds to
(36)

V l = v/sin ctL
where

is the velocity of a longitudinal wave in the X dir­

ection.

Similarly, at greater oc. the shear waves will also be

prevented from reaching face B.

X

A

&
Figure 14.

Longitudinal and Shear
Wave in a Plate.

49

It can be seen that there are at least two angles, a L
and cxs, beyond which the longitudinal and shear waves respec­
tively, can not be transmitted through the plate.
mission curve, given in figure

15,

A trans­

shows two pronounced dips

in the ultrasonic intensities recorded by a transducer behind
the plate f corresponding to the angles a c and a s .
Due to the complexity of the transmission equations it
is possible to evaluate conveniently only results obtained
from sapphire disks whose orientation is such that the Z axis
is parallel to the normals to the faces of the disks.

Since

at either atL or cxg one knows that the propagation direction
in the disks is along the surface one must know the angle
between k and the XYZ system in order to find the elastic con­
stants, - or one must be certain that face A and B are paral­
lel to the XY plane because a longitudinal wave propagated in
any direction in the XY plane must have the same velocity.
(Compare Sqs.(l6 ) to (18)).
A disk

whose surfaces are parallel to the XY plane is

used to obtain the curve shown in figure 15*

Ultrasonic pul­

ses with a modulation frequency of 2.5 Me are transmitted
through the plate in the Z direction and are received behind
the plate with an amplitude corresponding to the ordinate at
cc = 0.

Then the disk is rotated in steps of about i ° and the

new amplitudes are recorded.

a u is reached at approximately

7°20 1 from the Z axis at which angle the ultrasonic pulse
travels along the XY plane.

At about 12°45 1 the shear wave

is no longer transmitted; one has total reflection.
50

< -

u o is s iu is u t u j l

51
-tf

Rcj.15. Angular Dependence of Transmission

O

With the above values of Ct substituted in Eq.(36) one
obtains the approximate velocities in either the X or Y dir­
ections
V l « 11*3 km/sec,
Vq «

6.5

km.sec.

Whether this Vg corresponds to the shear wave associated
with C4.4. or egg is questionable.

It is obvious that no great

accuracy can be expected since the dips in transmission are
not well defined and an error of minutes in ct changes the value
of sin a significantly for small angles.
The locations of the peaks and dips should be symm@tric
with respect to the line a - 0 .

This is not found to be so

for the disk examined, leading to the assumption that the faces
of the plate are not exactly parallel to the XY plane.
This assumption is further supported by the results of
measurements of longitudinal wave velocities in the direction
of the normal to the face of the disk.

The wedge method,

first described by Bhagavantara and Bhimasenachar (37), and
extensively used by Sundara Rao (38), Bhimasenachar (16),
Bhagavantam (39), and others, finds an application in this
case •
In this method a wedge-shaped quartz transducer, which can
be driven over a wide range of frequencies, is coupled to the
face of the disk, the disk Is Immersed in a liquid in such a
fashion that an ultrasonic beam transmitted through the disk
can be observed in tie liquid by means of the diffraction
oattern

3 et

up by the beam.

This arrangement is identical to
52

that saown In figure 4b , except that S has to be replaced by
the volume of the liquid adjacent to the face of the disk.
Figure 16 shows that part of the arrangement.

The direction

of the light is into the plane of the paper.

_____
a

^

{X<^cU<jL b&vel
i

Figure 16.

k

The Wedge Method.

A diffraction pattern will appear only when the condi­
tions for maximum transmission perpendicular to the face of
the disk are fulfilled.

Since this method is used here to

find whether the normal to the face of the disk deviates very
much from the Z direction, one selects the number of half wave
lengths which are to be set up the sample, and from the dimen­
sions of the disk and the known longitudinal velocity in the
Z direction, one can calculate at what frequency maximum
transmission should occur.
It is found that strong transmission does occur at the
calculated frequency.

Varying the frequency by about 2%

around the calculated value shows that the transmission does
not change appreciably.

Taking in account that the thickness

of the disk is not uniform, and that the orientation of the
sample may be somewhat uncertain In various regions of the
disk, one can only find a limited frequency range at which
53

strong transmission occurs.

Considering the case where three

half wave lengths are accomodated across the disk, the 11maxi­
mum" resonance frequency and the maximum thickness of the
disk as factors yield a velocity in the "Z" direction of
11,801 m/sec while the "minimum" resonance frequency and the
minimum thickness give a velocity of 1 1 ,1 6 3 m/sec.

Under the

above conditions one can hardly speak of a most probable value
for the velocity.

If one were to calculate the elastic con­

stants from above results one would have to accept any value
between about 49 and 56 x 1 0 " dynes/cm for

033

-

Bhimasenachar

finds the value of c-^ to be 5 6 . 3 while Sundara. Rao quotes
5 0 . 6 for the same constant.

Since both authors use the wedge

method the disagreement is not surprising.

Their values fall

within the approximate experimental range given above.
However, the assumption that the normal to the face of
the disk is not exactly in the Z direction seems valid since
the range of velocities lies predominantly above the known
longitudinal velocity in the Z direction.
Considering the asymmetric location of the transmission
dips in figure 15 and their unusual flatness, together with
the uncertain results of the wedge method, one sees that
determinations of elastic constants, based on methods invol­
ving transmission through thin disks, may cause appreciable
errors•

54

JY *

Computation of* the Elastic Constants and Moduli.
After the transformation of the coordinate system is per­

formed the effective elastic constants K can he found either
by direct measurements or by computation from Eqs.(l6) to (26).
Since there is only one longitudinal velocity possible in any
direction of ultrasonic propagation there exists no doubt as
to what effective

is measured#

In the case where shear

waves can be propagated with two different velocities in one
direction one can not immediately say which shear wave is ob­
served.
The values obtatned from direct measurements of a possible
shear wave must be in accordance with the Cpq computed from
longitudinal wave velocities and other shear velocities.

Thus

it becomes possible to assign a measured Kg to the appropriate
equation, the wrong assignment leads to contradictory values
for some of the c

contained in Eqs.(l6) to (26).

The effective elastic constants are listed in Table IV.
A * following an entry Indicates that the velocity leading to
the K was directly measured and found by Bq.(27).

The rest of

the entries result from substitutions of the Cpq found from
the equations pertaining to the measured K fs into the rest of
the equations giving the unobserved K's,
are listed In Table V.

55

The resulting cpq

TABLE IV.

Equations

S-B

Vislb.

Dlffr.;

Reson.

(16)

48.6*

49.6*

50.9*

49.7*

I X (17)

18.6*

20 .6

19.6

IX

IX

(18)

19.4

19.4

'

;

i
Pulse
4 9 .4 *

1

!

20.8

i

I Z (19)

1 49.6*

50.2*

5 2 .8*

5 0 .2* | 50.3*
i

I Z (17)

| 18.6

20.6

19.6*

II A (20)

45.5* ! 47.1*
46.6
, 48.0

47.4*
49.5

II A (21)

19.2* | 20.0*

20.2*

II A (22)

21.5

20.0

46.6

47.6*
48.0

48.2* *
49.5
j

III A (23)

50.2*

51.6*

52.7*

III A (24)

15-5

16.2

III B (25)

42.8*

44.4*

46.2*

21.5

20.0

19.9*
21.9

22.8

23.8

II B (20)

20.6*

;

;
I....

i

|

*

III B (22)
III B (26)

•
1

t
< 44.2*

In units of 10u dynes/cm*-.

It will to© noted that the values for Eq.(20) differ for
II A and II B.

According to theory, the value must toe the
4

arithmetic mean of the K from Eqs.(23) and (25) in either case.
The measured values are all lower and not equal, leading to the
56

conclusion that block II Is not oriented according to the
description cited by Linde (regional polycrystallinity Is a
kind of misordentation)•

It was pointed out by Granato et al

(40) that low values of velocities (and with it low values of
elastic constants) can be expected if dislocations are present
in a sample of otherwise single-crystalline substance*

How­

ever, this consideration was not further investigated.
From Table IV and Eqs.(l6 ) to (26) the elastic constants
can now be found.
TABLE V.
Elastic Constants Found with Different Methods.
°pq

S-B

Visib.

DIffr.

Re son

Pulse

°1 1

48.6

49.6

50.9

49.7

49.4

c33

49.6

50.2

52.8

50.2

50.3

C4 4

18.6

20.6

19*6

20.6

«12

9.8

10.9

9.3

c 13

6.3

4.8

8.0

014

3.7

3-8

3.3

(c6

6

)

(19 .4)

(19 .4)

(2

0

.8 )
...

In units of 10" dynes/cm2;

57

From E q s • (16) to (26) It is obvious that every Cpn , where
p ss q, corresponds to some K which is associated with a dir­
ectly measurable velocity, while every Cpq> where p ^ q, ap­
pears only in equations for K described by sums and differences
of cpq*

Thus the c

with unequal p and q must be obtained in­

directly by substitutions and eliminations•

Small errors in

the determination of directly measurable constants may affect
the calculated values quite significantly.

This difficulty is

present whether one rotates the coordinate system or uses the
Christoffel equations.

The values given for Cqg* C13? C14>

and egg are therefore not as reliable as the values of the
other constants•

The elastic moduli Spq can be computed from the values of
c n by using the following expressions:

Ir'^A

2sll “

A ^i- C4.4 /B

2s12 " °33^A ~ c44//b

O

(37)

CVJ
H

s13 - "c13/a

s33 - (°11 +
s44 = ( O n - Ci 2 )/b

S14 = -c i 4/b

where

)

A

s

°33^°11 * c1 2 ^ “ 2 c13’

B

s

C;t;(( :11 - C12 ) - So**

}

(38)

Substituting the values from Table V into the above expressions
yields the values of the elastic moduli as shown in Table VI.

58

TABLE VI.
Elastic Moduli Found with Different Methods.
spq

S-B

Vislb.

Diffr.

811

2.22

2.18

2.10

3 33

2.07

2.02

1.97

3 44

5-59

5.04

5.24

312

-0.46

-0.50

-0.37

813
s 14

-0.22

-0.16

-0.26

-0.55

-0.49

-0.42

In units of 10"1S cm/dyne.

Young*s modulus is given by E s l/s*-^ for any direction.
The values of the S 33 in terms of s ^ are found hy using
Eq.(14) where now an s is substituted for e v e ^ c, except, due
to the slight difference in the s and c matrix, 4044 . mus^ *>e
replaced by S4 4 , and 40-^ by 2 s ^ .
pute Young *s modxilus for any k.

Now it is possible to com­

Some results are plotted In

figure 1 7 *
In figure 17a the values of 1 / 0*53 are P^O’kted as a func­
tion of © with 0 = 0 °, i.e. k Is rotated In the XZ plane.
Figure 17b is a plot of E as a function of © with 0 » 90°,
I.e. k is rotated in the YZ pla,ne.

In figure 17c the angle 0

is held constant at 60° and 0 Is changed from 0 through 2rr,
i.e. k lies on a cone around the Z axis.

The first two plots

show two-fold symmetry since the rotation of k is around a

59

VD O

Fig*. 17. Angular Dependence of Young’s Modulus.

o

60

binary axis in each case.

The third plot shows three-fold

symmetry since the rotation of k is around the three-fold
Z axis.

If k were rotated around the Z axis in the XY plane

the resulting plot of Young's modulus would be a. circle.
The radial distances in each plot are given in units of
1 0 " dynes/cm^.

The values of Sp^ as determined by the visibility method
were used for the plots of figure IT*

The values as found

with the Schaefer-Bergmann method and the line diffraction
method yield plots very similar to those given.

61

V * Summary.
Using optical and purely electronic methods the longitu­
dinal and shear velocities of ultrasonic waves in different
crystallographic directions were measured in samples of syn­
thetic sapphire, and the results were used to determine the
six elastic constants after a suitable transformation of the
coordinate system was made in order to simplify the calculation.
It was found that the visibility method is most reliable
and also offers possibilities to investigate scattered misorientations and minute irregularities in the samples.

It was

also found that the determination of ultrasonic velocities in
thin pla tes of sapphire must be considered rather unreliable
if even small variations in thickness and orientation are
present.
From the elastic constants the six elastic moduli were
computed which were then used to find the angular dependence
of Young's modulus.

62

VI * Ann end Ix .
Arbitrary Rotation of* Coordinate System*
If the entire cartesian coordinate system describing the
trigonal system is rotated such that the Z axis coincides with
the direction of k, where the new system shall be called the
X'Y'Z* system, then the direction cosines relating the two
systems can be expressed by the scheme
X?
X,

*3

X3

axx

x£
aXy

axz

ayx

ayy

ayz

azx

azy

azz

(Al)

An elastic wave traveling in the direction of k has assoc­
iated with it one longitudinal and two shear velocities.

By

conventional definition -see Sokolnikoff (41)- the constant
c ^ refers to the longitudinal wave while the C44 belong to
the shear waves.

These constants, which In this paper are

called Kl and Kg , will be combinations of various unprimed cpq
To find the combination one can use Voigt's (Ref. 1, pg 579
and 58 9 - 5 9 2 ) derivation for the transformation of spq and
adapt it to the

pq , or one can use the tensor transformation

/
‘rstu =

6 x£»

dxs 6x-£, 6xu ^
6x± 6x j 6xk 6xx
1Jkl

where i,j,k,l = 1,2,3 and r,s,t,u = 1,2,3.
The ^x^/dx-j, are the direction cosines of (Al).

63

Since one has to find only the effective elastic con­
stants c-53 and c ^

one substitutes the direction cosines cor­

responding to rstu = 3333 and then rstu = 2323 from (Al) into
(A2) together with the

from Eq.(6 ).

It is important to

treat the elements of the matrix E q .(6 ) according to their
positions In the array and not according to their suffixes
which are a result of incidental symmetry properties referred
to the Cartesian system#

Summing over all the terms thus ob­

tained and replacing the ijkl by their corresponding qp as
they actually occur as suffixes in the trigonal matrix, one
finally obtains the expressions for the effective elastic
constants as given In Eqs.(l4) and (15)*
It is quite obvious that the computation of Kg Is more
involved than that of K^.
of K l I s Indicated here.

Therefore, only the computation
In what follows, the

have

been replaced by their corresponding cpq where the pq merely
gives the position of the element in the matrix*
E q . (6 ) has 18 non-zero elements but due to symmetry all
coq w ^ ere P / ^

occur twice (that is Cpq » O q p ) •

there are 12 C3333 terms to be summed.

cll^axz)

Therefore,

These terms are:

2c24^ayzazz^

2c1 2 ^ x z ay z)

°33(a*z)

2c^-j(ax z a z z )

c ^ ( a y Zaz z )

(A3)
2 cl 4 (axzayzaz z )

°5 5 ^ax z azz^
2 c5 6 ^axzayzazz^

c2 2 ^ayz^
2c2 3 (ayzazz>

c6 6 (ax z ayz>

64

Nov; one replaces the suffixes by their corresponding
suffixes in Eq.(6 ), also noting that 055 » i(cil " C1 2 )> anci
then arranges the a's according to the 6 remaining cpq*

After

summing all the terms and rearranging them one obtains Eq.(l4).
A similar procedure results in Eq.(15)*
in 4 different ways in terms of
2233, 2323, 3322, 3232.

Kg can be vrritten

since the rstu can be

This results In slightly different

expressions for Kg because there are two sets of equivalent
arrangements of the rstu.

Rather than finding two expressions

for Kg only one is given in Eq.(15)> the other values of Kg
are found when needed for various orientations of the samples
by rotating the X ^ ’Z 1 system around Z f.
It was pointed out that only one Kp was possible for a
given orientation*
from (A3).

This can be seen from either Eq.(l4) or

All direction cosines occurlng contain a suffix z,

thus there Is no dependence on the angles between the X and Y
axes and the X* and Y* axes.

This Implies that a rotation

around Z* does not change the value of Eq.(l4)*

65

VII.

Blbl1o&raphy.

1. Voigt,V/. : "Lehrbuch der Kristallphysik", Enl.ed., Teubner,
Leipzig 1928.
2. Hearmon,R.F.S.: Rev.Mod.Phys. 18, 409 (1946) and
P b i l .M a g •Suppl• £, 323 (1956).
3. G-reen,G-.: Trans .Cambr.Phil.Soc. £, 121 (1839).
4. Love,A.E.H.: "Mathematical Theory of Elasticity", Dover,
New York 1944.
5. Mason,W.P.: "Piezoelectric Crystals", 1 st ed. Van Norstrand, New York 1950.
6. Bhimasenachar,J.: Proc.Ind.Acad.Sci. 22, 199 (1945).
7. C a d y , W .0.: "Piezoelectricity", McGraw-Hill, New York 1946.
8. Bergmann,L. : "Der Ultraschall", 6 th ed., Hirzel,
Zurich 1954.
9. Schaefer,C.? Bergmann,L., G-oehlich,H.J.: G-lastechn.Ber.
15, 449 (1937).
10. Schaefer,C., Bergmann,L.: Naturw. 22, 685 (1934).
11. Jona,P. : Helv.Phys.Acta 2£, 795 (1950).
12. Jona,F., Scherrer,P. : Helv.Phys.Acta 2f>, 35 (1952).
13. Debye,P., Sears,F.W.: Proc.Nat.Acad.Sci. 18, 409 (1932).
14. Lucas,R., Biquard,P.: C.R.(Paris) 194-» 2132 (1932).
15. Schaefer,C., Bergmann,L. : Ber.Preuss .Akad .Vf. 13 (1934) .
16. Bhimasenachar,J . : Proc.Nat.Inst.Sci.India 16, 241 (1950).
17. Barnes,J . M •: "The Determination of the Elastic Constants
of Optical Glasses by an Ultrasonic Method", Thesis,
Michigan State College (1955).
18* Bachem,C., Hiedemann,E., Asbach,H.R.: Z.Phys. 8 7 , 734 (1934) .
19. Hiedemann,E., Bachem,C.: Z.Phys. £1, 418 (1934).
20. Hiedemann,E., Schreuer,E. : Z.Phys. 1 0 7 . 463 (1937).
21. Hiedemann,E., Hoesch,K.H.: Naturw. 2^, 705 (1935).
22. Hoesch,K.H•: Z.Phys. 109, 606 (1938).

66

23. Wlnkelmann,A. : Ann.Phys. 2J, 905 (1908).
24. Scha.fer,J.: "Eine nene Methods zur Messuns der Ultraschallgeschwindigkeit In Festk8rpern , Thesis,
Universit&t Strassburg (1942).
25* K©bler,R.W.: "Optical Properties of Synthetic Sapphire**,
Linde Air Products Company, New York, No Date.
26. Blenck,A. : **Experiment ell e Untersuchungen tiber die Periodizit&t der Abbildung stehender Ultraschallwellen
und dber die dabei beobachteten Feinstrukturen",
Thesis, Universit&t Strassburg (1944).
27. Lord Rayleigh: **The Theory of Sound II1*, Dover, New York,
Reprint 1944.
28. Ramachandra Rao,B. : Curr.Scl. 1£, 148 (1950) .
29. Galt,J.K.: M.I.T. RLE Report 45 (1947).
30. Teeter,C.E.: J.Acoust.Soc.Am. 18, 488 (1947).
31. Iyengar,K.S .: Proc.Ind.Acad.Sci# 4 1 A . 25 (1955).
32. Loeber,A.P*, Hiederaann,E.A.: J.Acoust.Soc.Am. 28, 27 (1956).
33. Cremer,L.: Akust.Z. J, 81 (1942).
34. G8tz,J.: Akust.Z. 8, 145 (1943).
35. Firestone,F .A.: Non-Dest.Test. X, Nr.2 (1948).
36. Schoch,A.: Acustlea 2, 1 (1952).
37• Bhagavantam,S ., Bhimasenaehar,J .: Curr.Sci. 1 J , 229 (1944).
38. Sundara Rao,R.V.G.: Proc.Ind.Acad.Sci. 2 9 A , 352 (1949).
39• Bhagavantam,S .: P r o c • 33rd Ind.S c i .C o n g •, Bangalore 1946.
40. Granato,A., DeKlerk,J., Truell,R. : Phys.Rev. 1 0 8 . 895 (1957).
41. Sokolnlkoff,I .S .: "Mathematical Theory of Elasticity**,
2nd ed., McGraw-Hill, New York 1956.

67