ional Bureau of Standards

Library, N.W. Bldg

JUN 1 7 1960

NBS MONOGRAPH 9

A Method for the Dynamic Determination

of the Elastic, Dielectric, and

Piezoelectric Constants of Quartz
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UNITED STATES DEPARTMENT OF COMMERCE . Frederick H. Mueller, Secretary NATIONAL BUREAU OF STANDARDS . A. V. Astin, Director

A Method for the Dynamic Determination

of the Elastic, Dielectric, and

Piezoelectric Constants of Quartz

Saul A. Basri

National Bureau of Standards Monograph 9

Issued June 1, 1960

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington 25, D.C. - Price 15 cents
Contents

Page

1. General considerations 1

1.1. Introduction 1 1.2. Equations of motion and boundary conditions 2

1.3. Piezoelectric effect in thin plates 2

1.4. Constants of quartz and their transformations 3

2. Longitudinal vibration of rectangular bars 5

2.1. Zeroth order solution 5

2.2. First order solution 7

2.3. Optimum orientations of bars 9

2.4. Constants of quartz at 0 = 90° 11

2.5. Concluding remarks 12

3. Thickness vibration of quartz plates 13

3.1. Thickness vibration of an infinite plate 13

3.2. Rotated Y-cut plates 15

3.3. Rotated X-cut plates 16

4. Determination of the constants of quartz 18

4.1. Program 18

4.2. C44, Cee, Ch, and Cu 18

4.3. Dielectric and piezoelectric constants 19

4.4. Ci3 and C33 21

4.5. Experimental conditions 21

4.6. Recent determinations of the constants of quartz 22

5. References 22 n
A Method for the Dynamic Determination of the Elastic, Dielectric,

and Piezoelectric Constants of Quartz

Saul A. Basri

Several dynamic determinations have been made of the constants of quartz. Most of these determinations do not take into account the piezoelectric effect; those that do, suffer from certain deficiencies which are discussed in this paper. Taking into account the piezoelectric effect, expressions for the frequency of longitudinal

\"ibration of rectangular bars and thickness shear vibration of infinite plates are derived and applied to the determination of the constants of quartz. On the basis of present theoretical knowledge, it is suggested that the best procedure is to measure the frequency of vibration of 2 particular cuts for rectangtilar bars, and 7 cuts for plates, and to measure the capaci- tance at zero frequency- of a rectangular bar. These 10 measurements provide the data for determining the 6 elastic, 2 dielectric, and 2 piezoelectric constants of quartz uniquely. For accurate measurements, it will first be necessary to determine the linear coefficients of expansion of quartz to higher accuracy than is available at present.

Conventions

The following conventions will be used throughout this report

1. The word "quartz" shall denote "a-quartz."

2. Xi, Xo, X3 are, respectively, the coordinates

along the electrical, mechanical, axid optical axes of quartz.

3. {X'l, X'2, X's} is a rectangular coordiaate sys- tem rotated \vith respect to {Xi, Xo, X3]

.

4. All imprimed tensor components are referred

to {Xi, X2, X3} and primed tensor components

to{X[,XiX3}.

5. Latin indices such as i, j, k, • take the

values 1 to 3, and Greek indices such as /x, v,

• • • take the values 1 to 6. The usual cor- respondences

11^1, 22^2, 33^^3, 23^4, 31^5, 12<-^6 are adopted here.

6. All repeated indices shall be summed over

unless otherwise indicated (summation convention).

7. All the constants of quartz discussed in the

report are assumed to be adiabatic constants

unless otherwise mentioned.

8. Electrostatic units will be used throughout

this report.

1. General Considerations

1.1. Introduction

Alpha-quartz is characterized by 6 independent

elastic constants, 2 iadependent dielectric constants, and 2 independent piezoelectric constants,

or 10 constants in all. For the complete description of quartz, it is also necessary- to know the mass density and the linear coefficients of expansion along the optical and mechanical axes. There are essentially two methods for the

measurement of the above-mentioned 10 constants

of quartz, the static method and the d>aiamic

method. The static method yields the isothermal

constants, whereas the dynamic method ^aelds the

adiaV)atic constants. The isothermal and adiabatic constants are related through the specific

heat constants. Since the specific heat constants

of quartz are not accurately known, and the main

interest is in the vibrating quartz cr^'stal, the

dAnamic method is indicated.

In order to be able to make use of the dynamic

method to determme the constants of quartz, it is necessary to have theoretical formulas that relate the frequency of vibration of the quartz cr^^stal used to the constants of interest. At present, such

formulas exist only for very simple shapes of

crA'stals such as the rectangular bar and the infinite plate. There are many simplifications and as- sumptions made in the derivation of these

formulas. Consequently, the determination of the constants of quartz with the help of these

formulas is trustworthy only if the conditions

under which these formulas are valid are met

experimentally. For this reason, a careful analysis

of the conditions of vahdity of these formulas is necessary before these formulas can be used.

Smce the dvTiamic method is chosen, aU of the constants used in the following ^vill be understood

to be the adiabatic constants.

1
1.2. Equations of Motion and Boundary

Conditions

The equations of motion of a vibrating crystal are given by

dX'j

where T'u are the stress components in an arbitrary

coordinate system {X[,X2 X3} , ^'i the components

of displacement along X'i, p the mass density and

t the time. The time variable t in (1) can be

eliminated by setting

^i=miX',, Xi X's)e'^'. (2)

Substituting (2) into (1) leads to the equation

(3)

If we write these equations explicitly, we get, with

the help of convention (5),

pw^U[=0, (4a)

dX'i

ciX{ dX', pwW,=0, (4b)

(4c)

The stress components T'tj are given by the

expression

Tij=Cij)ctSki e'kijE'k, (5)

or equivalently by

(5')

C'liv are the elastic stiffness constants, e'i^ the

stress piezoelectric constants, the strain components and E'i the ith component of the electric

field inside the crystal. S'ij are related to U't by

-''-2\dXi^^J

Written explicitly, (6) becomes

S' Q' C)U I Q, „, C)U 2 Q/ Q' _i — £11 —^ Y'' _2 —£22

^x'' —

^ —

—

(6)

dX'^

Si —2So5

3 Q'

ny 05 —^ox3

E'i consists of two terms, namely

E'i=E^'^Ef'.

(7)

(8)

Ef is due to the piezoelectric effect of a freely

vibratmg crystal and Ef are the components of; the applied field. The relation between Ef and

SJ, or T'^ will be derived in the next section.

If the surfaces of the crystal are free, the

boundary conditions that must be satisfied are

T'ijn'j=Q on all surfaces, (9) where n'j are the components of the unit normal toiis the surface imder consideration, and T'ij are given j by (5). In the case of the rectangular crystal (I shown in figure 1, (9) becomes \i r;=r;=T;=oatx;=±^

T:=r'=T'=Oat x'.= ±k. (10)

Notice that it is important that all the equations I (4) and all the conditions (10) must be satisfied, ; and not only some of them. This fact is too frequently forgotten in the literature, even by

well-known authors.

FiGUEE 1.

1.3. Piezoelectric Effect in Thin Plates

We shall now derive the expression of E'i in terms of Si_,- and T'ij for a freely vibrating crystal

plate whose thiclaiess h, is much smaller than the

lateral dimensions.

Consider a central region R (see fig. 2) whose

boundary has a distance from the edges of the

crystal which is mucli larger than thickness h. If the crystal is isolated, the total charge inside

R will be approximately zero, and the charge

inside R will be expected to vary along X'-i only.

We may thus think of the charge distribution

inside R to consist of plane sheets normal to X'z, and such that to every sheet having a uniform

surface charge density +o- there is another sheet

of uniform surface cliarge density —a. In other

words, the region R can be considered to be composed of an infinite number of parallel plate

condensers.
FiGrRE 2.

Let us consider one of these parallel condensers

with a surface charge density a (see fig. 3). Applying Gauss' law to two pill boxes about the two

plates, in a region far from the edges, we come to the conclusion that the electric displacement D

is normal to the surfaces and has the magnitude

4x0" between the two plates, but vanishes outside

them. From this it follows that the normal component of D outside the crystal vanishes at the

surfaces of the crystal over a central region such

as R.

D3 -- 0 1

2 rra + cr

D3 4 77- cr

I -2 v a

03= 0 I 2 77-a

- cr

X3 .0

Figure 3.

Since the only surface charges on the crystal are

polarization charges, it follows from the boundary

conditions on D that the normal component of D

inside the crystal ^viU also vanish at the surfaces.

Consequently, if we take a pill box inside the

crystal whose top is adjacent to the top of the

crvstal and whose bottom is insde the cr\"stal (see

fig. 4), the total electrical flux through the top

vnH be zero. Since the crystal is homogeneous,

the total flux entering the sides of the pill box will be equal to the total flux leaving the sides; in

other words, the total flux through the sides is zero. Furthermore, since all the charges inside

the piU box are polarization charges, it follows

from Gauss' law that the total flux out of the

whole box must be zero. Therefore, the flux through the bottom of the pifl box will be zero. From this we can conclude that the normal component of D vanishes everj-where within the

region R, i.e..

Figure 4.

Since the charge within R does not vary with

X'l and X'2, we conclude from symmetry that the

electric field should be along X'3 everywhere

within R, i.e., Ef'=Er8,„ (2)

where Stj is the Kronecker delta. In general,

D's=eiiEf'+47re'^,S:

=^;Er^^d^,T:, (3)

where efj and ef/ are, respectively, the dielectric constants at constant stram and constant stress, and e'iy., d'ly. are piezoelectric constants. Making

use of (1) and (2), we get from (3)

and

Therefore,

or

ef3'^3^'+4xe^.S;=0.

e3^3'^r'+4,r(^^,T;=0.

TTip/ 47r / Q

— — S' ^3iiQ.ii €33

47r „,

633

(4)

(5)

D'z=0 everywhere inside R. (1)

Either (5) or (4) can be used in (2.8) to evaluate

the piezoelectric effect in a plate whose thickness

is along the Xs-direction.

Cady [1, p. 312] obtains, instead of (4), the

result

C33

where C is a constant determined b}^ conditions on

the boundaries of the crystal. Cady finds C^^O,

whereas our derivation shows that (7=0. In his

derivation, Cady makes use of special assumptions

with regard to the charge and potential distributions inside the crystal, some of which are not

justified. In contrast, the derivation of (4) or (5)

is based on very general argiunents which are fully

stated. Our results are in agreement with those

obtained by Koga et al. [5].

1.4. Constants of Quartz and Their

Transformations

Due to the fact that quartz has 3-fold symmetry

about the Xa-axis and 2-fold symmetr^^ about the

Xi-axis, the constants of quartz are given by the

follo^ving matrices

3
Elastic Constants

<- 11 opn n n u u

0 0

0 0 0

0 C44 0 0

0 0 0 0 C44 Ci4

0 0 0 0 Ci4 Cii C12 2

^11 »S'i2 »S'l3 Su 0 0

Su 'S'13 —Su 0 0

Si3 S33 0 0 0

Sli —Su 0 Su 0 0

0 0 0 0 'S'44 2Su

0 0 0 0 <S'i4 2(*S'ii —(Sia)

Siijj=S^p, Sujk=lS^,{j9^k), Siju=\S^S^J, tc9^l).

Piezoelectric Constants

eii —^11 0 eu 0 0

0 0 0 0 —eu —Cii 0 0 0 0 0 0

du —du 0 du 0 0

0 0 0 0 ~du -2d

0 0 0 0 0 0

dijj —di^, dfjic —idifi(j ^k).

Dielectric Constants

0 0

0 €11 0

0 0 €33
Suppose we wish to transform from the crystal axes Xi to an arbitrary rectangular coordinate

system X'j. Let

Xi=a]Xj, (6) and

ajafa- • • • =ajf- ; ;

;

• (7)

Since the elastic, piezoelectric, and dielectric constants are the components of tensors of the

4th, 3d, and 2d rank, respectively, it follows that:

jkl ^mnpQ^ mnpfj

=[(ai^f^+MlD+i(«j^^2+a^^fi+ai^f^+a|^l^)]c^l

~r ("l 133 +"331 1 ~f" 0^2233 +<^332 2 ) C'l 3 +0:3333 ^-^33

_J_ {nijkl I ijkt I ijkl \ ijkl \ ijkl \ ijkl

l<a!ll23 «23 11 n «1132 T^<^3211 i^«1312 n^«132

1

"2223 "2322 <^2232 <^3222/'-^14

l«2323 +«3223 ~r«2332 +«3232

i^Q;i313 0:3113 i^«1331 "r«3131 J '^U- (8) The transformation of S,'^fci is the same as (8) except for the replacements,

Cn~^Si\, Ci2^Si2, Ciz-^Cii, C33-

' ijk p = (ain-al^-4ii-c^ en

(9)

(10)

The transformation of d'ijk is the same as

(10) except for the replacements,

€u=aiUki= (aii+aiQeu+aUez3. (12)

2. Longitudinal Vibration of Rectangular Bars

2.1. Zeroth Order Solution

Consider a quartz rectangular bar such as that

shown in figure 1. It will be assumed that

Zl»/2»^3 (1) The reason for taking li and is to satisfy the

conditions for the validity of the expressions

(sec. 1.3, eq. 4 or 5), which will make it possible

to evaluate the piezoelectric effect. The assumption Zi»/2 is made in order to minimize coupling

with other modes of vibartion and thus makes it possible to simplify the problem and get a suffi- ciently reliable solution. The effect on the

frequency of small but appreciable values of

{I2II1) will be given in the following section.

We are interested here in a solution of the form

U[=A cos kX{,

which satisfies the equation

5^^71/5:^(2=- (constant) C/;. Since

S[=W[/bX[,

this equation can be written as

5S;/dX;=-(constant)C7i. (2)

In order to relate (2) to the equations of motion

(sec. 1.2, eq 4) it is necessary to find the relation

between S^ and T,'. This, we shall do now.

In general,

(3) where S^^ are the elastic compliance coefficients,

d'i^ the strain piezoelectric constants, and E'i is given by (sec. 1.2; eq 8). For a freely vibrating

crystal,

£:f =0,and E'i=Er.

Because of (1), we can use (sec. 1.3 eq 2 and 5)

and get

E',^-{4^/e^M.T'A,.

Substituting this expression into (3), we find

If we let

we can write

SI O' 'T"

(4)

(5)

which is the desired expression. In particular, we

have

SI O' T"

Substituting this expression into (2), gives

S;;.dr;/d.Y; = - (constant)^/;.

5
In order to make this equation agree with (sec. 1.2,

eq 4a), it is necessary to assume that

and also

T[-^T', for all M 5^1.

(6)

(7) The assumption (7) is quite frequently made in

the literature. Some authors erroneously justify

(7) as follows: The boundary conditions at ±^2/2 and X3=±l3/2 are given by (sec. 1.2,

eq 10). Since lo and Z3 are taken to be small, the

stresses T^(M=trl) cannot differ appreciably from

zero in the interior and therefore can be assumed

to be zero everywhere. If this argument is correct then one may conclude, in the case of a crystal

plate whose thickness is in the X2 direction, that

T8=r2=^4—0 everywhere. This is in conflict

with the well established fact that Tg is the principle stress in the thickness-shear vibration of such

a plate. The main justification of (6) and (7), would be whether it is possible with their help,

to satisfy all of the equations of motion (sec. 1.2, eq 4) and all of the boundary conditions (sec. 1.2,

eq 10), at least approximately.

As far as the boundary conditions are concerned, all of the conditions imposed on T"^ (;u =}= 1

)

will be at least approximately satisfied because of

(7). The remaining boundary condition

T[=0 at X[=±h/2, (8) can be exactly satisfied, as will be shown later. Equation (sec. 1.2, eq 4a) becomes, with the

help of (6), ^+pw^U[=0. (9)

Furthermore, (5) and (7) yield

and in particular

SI^SnTl,

(10)

or T'-^S'-^^.

Substituting (11) into (9) gives the equation

(11)

(12)

The solution of (12) is of the form

U[=A cos kX[-hB sin kX[, (13)

k=w(pS'u)\ (14)

where

Making use of (11) the boundary condition (8)

becomes

^=0 atX;=±^i/2.

Imposing this condition on (13) yields

-A sin k'^+B cos k^^=0, (15a)

+A sin k^^+B cos k^^=0- (15b)

The only way to get nonzero values for A and B

is to set

-sin k ~ cos k

+sin k 7^ cos k

which implies

--—2 sin k^ cosk ^==—sin kli=0,

kli=msor

k=mr/li, n = l, 2, 3, . . (16)

We can evaluate B from either (15a) or (15b)."

Using (15a), we get

B=A sin k^l cos k ~

Substituting this value of B into (13) we get

U[=(aIcos k 1^ (^cos kX[ cos k I

-fsin kX[ sin k

If we set

Ajcos k ^=a=constant,

we can write U[ in the form

U[=^a cos k(^X[--^- (17)

The frequency of vibration, /„, can be obtained

from (14) and (16).

•6
Thus

where we have from (4),

S'n =S'n-{^i^le^i){d'uY.

(18)

(19)

The solution obtained here is only a zeroth

order solution. For this reason, from now on we

shall denote U[ \iv (17) by L'?' and /„ in (18) by

fn- The 1st order solution and the two remaining

equations (sec. 1.2, eq b and c) will be discussed

in the next section.

Cady [1, p. 317] states that the piezoelectric

correction is zero for the longitudinal vibration of a quartz bar. The above result does not seem to bear out this conclusion. The origin of this dis- agreement was discussed at the end of section 1.3.

2.2. First Order Solution

We now have to find a solution which is consistent with (sec. 2.1, eqs 6 and 7) and satisfies equations (sec. 1.2, eq b and c) to terms of the order

of ihlh) or less. We have seen in section 1 that one of the consequences of (sec. 2.1, eq 7) is (sec. 2.1 eq 10), nameh^

and in particular,

Eliminating T[ between these two expressions gives

For convenience, we shall let

s^=S'^i/S'n, (1)

s;~s,s;~5,sr. (2)

and get

We shall now make use of (2) and the assumption

u[=ur+MX[,x'„x',) (3)

in order to calculate the first order solution. It should be noticed that

U'i'=a cos h (x[-^^y

Si ' =

akn sin h (^X [

dXrdX['~

539046—60 2

(4a)

(4b)

(4c)

^2=^'^ssr~^m=s,x'sr+f2(x[,x',). (5a)

Q^=f^^^s,sr-^m=.ssX',sr+ux[,x'^. m

-'~dxrdx'~^x',^dx',-^' ^^""^

^,_du[ din ^fi dx'jdxrbx',

+ssX',{-km +^~.9,,sr. (5d)

-'^dx'jdxrdx',

+s,X',{-klUr)+^^^sSr. (5e)

From (5d) and (5e) we get, respectively,

f,=s,X'Sr+iszX',VdUr-^^X',+g{X[,X',),

fi=s,x'sr+hs2X'^kiur-^^x',+h{x[,x',).

Comparing these two expressions for^i, we get

giX[,Xd=s,X'Sr +hs2X'^klUr,

KX'uXd

55X38?'

h^xmur

,

X',df^/dX[=X',df2/dX[.

The last equation yields

f2=f{X[)X's,f,=fiX[)X',.

Substituting these expressions into (5c), we find

2f{X[)=sSr ovf(X[)=hsSl'.

Gathering all of the above results, we get

U[^Ur+{s,X',+s,X^)Sr+iis,X','

+SsX'^'+s,X',X's)klUr (6a)

U',^(s,X',+is,X's)^r (6b)

mc^is,X'^+is,X',)Sr. (6c)

It remains to find the first order correction

to the frequency and satisfy (sec. 1.2, eq b and c). For this purpose, it is necessary to calculate

T'^. Making use of (2) and (3), we get

s;=m=s?'+5/i/d.Y;,s;~s,ss'(M?^i). (7)
From (sec. 1.2, eq 5) and (sec. 1.3, eq 4) we have

^33 Til us if we let.

€33 we can write

T' AT' O'

Making use of (sec. 2.1, eq 5), we get

(8)

(9)

T' ri' Q' ri' Q' T" t T'

which implies

0'^yS'y\=8i^X. (10)

Substituting the values of given in (7) into (9), and making use of (3), (6a), (10), and

(1), we get

+ Us2X','+s,X','+s,X',X',)SV], (11a)

and for T:=C',M[-{seX',+sM)UV

+i(s2z;2+,vY;2+s,x;.Y3)s?']. (iib)

If we now substitute (11) into (sec. 1.2, eq 4)

and make use of (1), (sec. 2.1, eq 14), and assume

that

^,=m{X[,X'„X',)e'"", (12)

where U'i are given by (6) and w is the corrected

frequency, we get

['^^- hHGn{S'i2X2--\- S'i3X3--\- 81^X2X3)

+ (CUS'n+ C'uSU)]m' + [C'n{SUX'2 +SUX3) —-I Oi'e (2<S'(2^2+jSuXs) — 2 '^15 (tSu^2+25'! 3X3 )] S?

=(^j {[l + h^(S[2X'2'+S[3X',

^'2 ^3

1 /o. +S'uX'2X'3)]Ur+^{SUX'2-^SUX',)S,r}, (13a)

Su

[iklCUS[2X'2'+ S[3X'3' +S'uX'2X'3) + (C[2S[,

+ G'uS'i5)]Ui' -\-lC'i^{S 'uX'2-\- S 'iiiXs) —^C'i2i2S'i2X'2

+S'uX's) -iC'u {S'uX'2 +2S[sX'3) ] Sr

=(^y'^(S[2X'2+hS'uXdS^,', (13b)

[ikoC[s{S[2X2^'\- S'i3X3^-\-S'uX2X3)-\- {C'uS'lg

+r^;3s;5)]c/r+[C';5(s;6x;+*s;5YD —hG'u{2S[2X2-\-SuX'3) —^C[i{S'iiX2-\-2S'i3X'3)]Si'

=(^)^ihS[,X2+S\3X'3)Sr- (13c)

From (4) we have

i

J-

_\'lj^^imydx[=a%/2,

J^''J^{^mX[=kla%l2,

^_\f^U\'^,'dX[=Q.

Multiplying equations (13) by

and making use of the above integrals, we get

1+ (Cl5'S'l5 +C'i6<S'i6)+ 2^ C'ii(5'i2/2+'S'l3?3)

(c;2S;,+^u<§;5)+^^;6(§iVi+syi)=o, (i4b)

(6\',s','6+c';3S;5)+ ^ ^i'5(<s;2/i+>s;3/i)=o. (hc)

According to (sec. 2.1, eq 16)

Substituting this value into (14) and setting

A-n^24 (15)

we get from (14a)

1+A (!)•
Equations (14b) and (14c) can be solved for S'l^ and S'i6, and yield

^ n' fi' c ^ ?!' 12'- 15 Ll4"-16 o' A

«J15 = ~ 'Jll'^) '-14 L i2<^13

X'l3^'l6 ^'u^'lj

(17a)

(17b)

which show that S'15 and S[e are of the order of, d or {hlhy^. In practice, since all the constants of

quartz are not known to much better than about

1 percent, it is not feasible to try to find the orientations at which (17) is satisfied. If (^2/^i)^<C0.01,

then (16) should provide a satisfactory approximation at orientations such that

Si'5~0 and Si'e^O. (18)

From this we see that the assumptions (sec. 2.1,

eq 6 and 7) are only valid at the orientations where

(18) holds, contrary to the usual belief that they

are good for any orientation. The terms in (16)

containing S[;, and S'l^, will take care of tlie fact that these coefficients may not be exactly zero. The solutions (16) will be reliable provided orientations are used for which (18) is fufilled. It should be mentioned that the type of calculation used by Davies [6] to estimate the correction

to the frequency due to the other modes of vibration is wrong, since it is assumed in this calculation that only the kinetic energy changes, due to these extra modes. An examination of the derivation here would show that the change in potential

energy is equally important and cannot be neglected. In fact, all the terms in (16) containing

C^p are due to the change in potential energy.

2.3. Optimum Orientations of Bars

We have seen in the previous section that the

only orientations for which (sec. 2.2, eq 16) is reliable, are those for which (sec. 2.2, eq 18) is satisfied, namely

and S'u'^Q.

From (sec. 2.1, eq 4) and the approximately known

values of the constants of quartz, it can be verified

that

[{Ai^ieiM,di]is:.<:m.

We are thus justified in replacing the above conditions by the simpler ones

S[f^O and S[,'^Q. (1) We shall now find out the orientations at which

(1) is satisfied and for which the longitudinal

vibration can be excited piezoelectrically.

Consider the bar shown in figure 5. If we let

X'i —a.)Xj,

then we have from figure 5

where

SeC^ SeS0 Ce — (JeC\—- CeS4, Se (2) —-G^ 0 _

Se --sin d, Ce =cos Q,

Sij,-= sin </), C4 =cos 0.

The matrix (2) can be obtained as the product

of three rotation matrices representing the three

successive rotations shown in the figures below.
Thus

— o A"" u

~ rr

(J 0 1 u -Ce 0 0 1 0 0 0 1

0 0 1_ n _ — L-0 0 _i 0 0_

In the primed coordinate system we have, using

(sec. 1.4. eq 8, 9) and (2), la' 0 / ozrn o 03 03/^ ^ 1 o ( c<3/~r3 o + Se SIC4,) -\- SiziOgSeC^iS^ — CIS$84, f\)

-'rh SnidSlSlCeC^, —SS^C^Ce) + ^844

(

2SdS^ClC^-^2SeC4,ClS,j>) -\-jSqs

( 2SeClS4.-{-2SlO^S^)

O QSri Q I fit Q2\ O QS/T O I ni 02\

+-|-<S'i4*S'jC(5(70(4*S'0 — 1) —\S^^S\C4,S^{C'\ — iS'l) = ('S'n — »S'i2

\S^^ S\C\S,^{C\ — S"^

-^^,s,4S\(:\t\{^si~\).

For quartz,

S^^=2{S S\2i

.

Thus

S'x^= —ZS]^4 sin - d cos ^ cos 30, (3) from which we conclude that

S'Js^O when 0=0, 90° for all values of 0,

and 0=30, 90° for all values of Q. (4)

Similarly, it can be shown that

S[,=2S"'''-=2a\a)aWiS'^''' =sin 6>{cos (9 [2(5'33 cos- B—S^^ sin^ Q) —

{S44^2Sn) cos 2Q\~Sx4 sin 30 sin 30}. (5)

Probably the most reliable values of S^^^ are those given by Koga et al. [5] (with a correction

in sign). In units of 10-" cm-/dyne, these values

are:

;Sn = 12.8, <S33=9.7, ^44=20.0, S'lo^-l.S,

^,3= -1.2, ^^u=-4.5, *S66=29.2. (6)

Substituting these values into (5), we get

-Siec^sin Q [cos 0(19.4 cos^ 0—25.6 sin^ 0 — 17.8 cos 20) +4.5 sin 30 sin 30],

10

from which we conclude that

iSlo^O when 0=0 for all 0, 0=90 for 0 =0, 60,

0=90 for 0=70, 7r-49,

0=30 for 0=7r-7O, 49. (7)

No other values of 0 are listed in (7) since, according to (4), S\^ =^ only at 0 =30° and 90°. Examination of figure 6 wUl show that 0 =30°,

0 = 7r— 70° is the same orientation as 0 =9O° + 12O°,

0=+7O°. Since quartz has trigonal symmetry

about Xz, it follows that all the properties of

quartz at 0 = 30°, 0 = x— 70°, are the sume as those

at 0 = 90°, 0 =70°. In the same way, it can be

sliowu that the properties of quartz at 0 =30°,

0=49°, are the same as they are at 0 =90°,

0=7r— 49°. Consequently, the orientations in (7) at 0 = 30°, give no more information than those ^ at 0 =90°, and need not be considered further.

\

\

\ \

120°

\

\/90°

30° \

Figure 6.

It remains now to find out at which of the

orientations listed in (7) it is possible to excite

longitudinal vibrations piezoelectrically. Since

T\x is the stress mainly responsible in exciting

the longitudinal vibration, the driving electric

field is taken along the thickness direction of the

bar, i.e., along the X3 direction, we have from

(sec. 1.2, eq 5), T\i= ~e'zxJi'o,.

Consequently, it is possible to excite the longitudinal vibration piezoelectrically for all orientations

at which fsuJ^O.

Making use of (2), we have, with the help of

(sec. 1.4, eq 10),

e3i =sin 0 (sin 30 «ii +2 cos 0 eij.

Making use of the values of en and by Koga

et al. [5] (with correction in sign) we get in cgs units

631 ^ sin 0 (5.25 sin 0 sin 30 +2.44 cos 0)X1O*.
From this, we conclude that

e^i =0 when 6i=0° for all 0,

0=90° for 4>=0, 60°, (8) 0=25° for 0 =90°.

Comparmg (4), (7), and (8), we see that the

orientation at which S'l^^O, SJe—0, and longitudinal vibrations can be excited piezoelecti'ically, are: 0 =90°, 0=70°, 7r-49°. (9)

At 0=90°, there is the additional advantage that

514 =0 for all 6.

2.4. Constants of Quartz at 0=90°

We have seen in the previous section that all the orientations that can be used, occur at 0 =90°

(see figure 5). ^Ye shall, therefore, write down

all the constants of quartz occurring in (sec. 2.2,

eq 16) for these orientations.

Making use of (sec. 2.3, eq 2) and letting

sin n d=Sn, cos n 6=Cn,

we get

0 S (J- c s

0 0

for 0=90°. (1)

Substituting the values of a) given by (1) into

(sec. 1.4, eq 8 to 12), we get

Sn=S'Su + C'S^,+S'CK2S,^+Su)-2S'CS,,.

*S' 1 3S^S12 -\- C'Si3 -\- SCSu

SU=o.

cu=o.

r[,=2S[S'C(-a,+C^.+2Cu)

+C^(C33-Ci3-2C44)+Wl4].

d',,= -S'-dr^ +SCd^,.

d3z=dn.

d'z^ —Szdu — ^2(^14.

ezi^—Shn +S^eu- 4=0.

^33 — ^UIT T

*33 — ^llSi^ —S'ly, — (4x/ csD^siC^sji

C'i';x= C[^— (47r/e3f)e3ie3^.

It was stated at the outset that there are only 6

independent elastic constants, 2 piezoelectric

constants, and 2 dielectric constants. The

equations that relate C^^ and S^^ are given by

[2, p. 207]:

O/O ^33 I ^44 o/O ^33 ^44 ri '-'13 ^L 11 r-5-' ^(-12= 13= a p a p a

/o 'S'14 ^ 'S'n~l~'S'i2 ^, Sii S12

^- 14 — L-33 — ) L.44

)3 Cu C12 Sn

2 where

2i3'

a=5*33 ((Sn +(Si 2) —2<S'i 3

,

(2)

O O ^33 I C'n DQ C'33 C44 a p a p

rt Cl4 Q ^11 "I

"" C12 — *J33 — ! Si

C:3

7> a

S(j& =2(Sii —*S'i2) =2 -r^J where

(3)

a' = C^3(C^^+C,2)-2Ch,

0' — Cu{ Cii C12) 2 C'14. ^ The equations that relate c?,>, tfj, and e'fj are given by [2, p. 452]

611= (f11 — Cl2)c^ll~l~^^H<^14) 6i4 =2Ci4(Zii +C44(ii4.

(4)

^=r^-(2en(/ii +ei4(^i4), ^iz =4z. (5)

47r 47r

Since 5(5=0, (^35=0, ^"(5=0, and 6^5=0, it follows

that S[~0 and C'i'5=0 for 0=90° at any value of 0. Consequently, the expression (sec. 2.2, eq 16) for the frequency becomes

(1+A) (j-J=l +CA+C'nS'nA.

11
Making use of (sec. 2.1, eq 18) and (sec. 2.2, eq 15), i.e.

(6) and

ll / TT \ - - ~

.

and letting;

7=(2/J)-V (7)

we get, for n=l, the equation

ySn+{y- On) ^

(0(/iS;, + /iSJa) - CuS[,= 1

.

(8) Equation (8) yields 2 independent equations, one for each of the 2 orientations given in (sec. 2.3,

eq 9). Therefore, at least 8 more equations are necessary to determine ail of the 10 constants.

2,5. Concluding Remarks

In order to be able to use (sec. 2.4, eq 16) it is necessary to use a rectangular bar for which

(see fig. 1) /2</, and /a-C/a- The bar must be fully plated on the surfaces A^3=

it/3/2. The lowest mode of vibration (^^=1)

should be used, since the correction factor A in

(sec. 2.2, eq 15) depends on n^. The mass density p occurring in (4.7) can be

obtained from the data in [4, p. 288]. An equation

of the form

can be used to obtain an exact fit of this data, and

the result valid for the range —200° r'<r< + 100°

C, is

p=2. 6510- 10-M9. 167 + 10-2r-0.167(10--V)-] (1) where t is the temperature in degrees centigrade

and p is in gm/cm^

The dimensions li, I2, h of the bar should be

measured accurately before plating at some temperature To, say room temperature. The dimensions at any other temperature can be calculated

from the curves given in [4, p. 384] for the expansivity _l{r)-m-C).

1{0°C)

.A fit to 2 or 3 significant figures (the same accuracy

to which the curves can be read) can be obtained

with an equation of the form

a={A^BT + CT')T.

The result of such fitting valid for the range

-200° (:'<r< + 100° C, is given by

= [1.282 +0. 117(10--+) +0.028(10-2t)2

+0.023(10-+^3](io-2^).

(3a)

103a3(r) = [0.687 +0.087(10-2r)+0.003(10-2r)2

+ 0.003(10-+)3](10--t),

(3b)

where ^^(t) is the expansivity associated with the

Xi direction.

From (I) we have

/(r)=Z(0)[l+air)],

Z(ro)=Z(0)[l+a(ro)].

and

Thus

/(r) l+a(r) ri 1 / Mn / m

~l-a(To)+aCr;,

from which we obtain

/(T) =/(r„ni-a(ro)+a(rj]. (4)

Let e,, e'i be the unit vectors along Xi and X'i, respectively. We can expand the U dimension

along the crystal axes as follows (not using the

summation convention)

l(ro) = /i(ro)e'i = /,(ro) 2,.a,-^^e,=2;^Z|(ro)e,-. (5a)

1 where

(5b)

and a,-

'-^ is the inverse of a). Since (ap is orthogonal, aY'^=a), and we get from C4.8j

"0 0 1

(a7i')= S -C 0 for<A=90°. (6)

_C S 0,

According to (4) we have with the help of (5b)

lKr) = l{(To)[l-aj{ro)+ajiT)]

Consequently,

1,(T) = 2,/Kr)e, =/,(r„)2,[l-a,(To)+a/r)]ar^^e, (7)

12
which shows that /i(r) is not parallel to liiT^), i.e., the orientation of the bar changes slightly with

temperature. Strictly speaking, a correction should

be made for this, but this correction is <^0.1

percent, as can be seen from (.3), and is thus of the same order of magnitude as the terms we have been

neglecting.

From (7) we have

|l.(T)i-'=l,(T)-l.(r)

1

or

^.(r)=/.(ro)^|, [l-a,(ro)+aXr)]-'(ar^O'}^ (8)

which is the desired expression for at any temperatui-e r. The frequency will be shifted due to plating,

and the frequency for zero plating can best be

obtained by measuring the frequency as a function

of plating thickness and extrapolating to zero

thickness.

3. Thickness Vibration of Quartz Plates

3.1. Thickness Vibration of an Infinite Plate where

TTe saw in the last chapter that we have to find

at least 8 more trustworthy formulas relating the

frequency of vibration to the constants of quartz.

The thickness vibration of infinite plates has been

studied extensively and thus offers a possibUit}-

for obtaining a few more reliable formiilas, if the

conditions of validity of the formulas can be met

experimentall3^ We shall see that this is possible.

Much of the theor\' of thickness vibration of

infinite plates can be found in [1]. However, for ease of reference, and in order to have a theoiy

in the form we have been developing, we shall present this theor\' in this section.

Consider a quartz plate of lateral dimensions

much larger than the thickness h. It is customary

to choose X'2 along the thickness; and we shall follow this custom here.

FlGUBE 7.

The equations of motion are given hy (sec. 1.2, eq 3J, i.e., |^^+pw2L^,=0. (1)

The expression for T<_, can be obtained from (sec. 2.2, eq 8 and 9) by replacing the index '3' by '2'. The result is Tij — CijtiSki- (2)

47r Cijkl CfjA-Z~l~ S' ^llj^ikl (3)

Making use of (sec. 1.2, eq 6) and the s^-mmetry

properties of C'iju, we can write (2) in the form

Substituting (4) into (1), we get

C' ijkl bX'dXi

(4)

(5)

If the lateral dimensions of the plate are sufficiently large compared to the thickness, the

edge effects can be neglected, and we can assume

that

Ui=U',{X',).

Thus, (5) becomes

(6)

Equation (6) constitutes 3 coupled equations of the second order. In order to solve these equations, we use the usual method of transforming

to 'normal' coordinates in which the equations are decoupled. For this pui'pose it is necessary to rem-ite (6) -with due attention to contravarient

and covarient components, and get

(6')

If /3j are the coefficients of an orthogonal trans- formation, then we have

and

(7)

(8) 13
We shall assume that /Sj is so chosen, that

•0 if k9^l

7

rO k9^lA

(9)

Substituting these expressions into (6'), and dis- continuing the summation convention for the

time being, we get

Multiplying both sides of last equality by j3i and

summing over i, we get

Thus

^Vi I 2 » n

(10)

which are the desired decoupled equations. Tiie

solution of (10) can be readily written as

where

ypi=Ai sin kiX'z-^-Bi cos ki X'2,

kl=^ip|y^ w

(11)

(12)

The problem now is to calculate /3j, yi and ki. Multiplying both sides of (8) by /3', summing over

j, and making use of (9), we get

2,5' ^2^-2^^.= ' ^/3,- ''^]yk =^k^-k''^{yk=^T''yi.

Since jSf is orthogonal, ^T^' =fi\, and we get

or (13)

i:,{C'^''^'-bhkWi=Q.

Equation (13) represents 3 homogeneous linear

equations, and thus has a nonzero solution only if

det {C'''^'—b{y,)=0, (14)

which is the well known secular equation. Expanding (14) and making use of convention (5), we get

C'es "Yk C'26 C'46

C',, C',,-y, C',, =0. (14')

^'46 ^24 ^^44 "Yk

Which is a cubic equation in y^ and has the three

real roots Ti , 72, and 73. For each of the roots 7^, two

of equations (13) and the normalization condition

2,

1

:1 (15)

can be solved unic^uely for /3|, 183. Thus with

each value of 7^ there is associated a frequency of vibration given by (12) and a direction of vibration

given by

e:=2,ftr''e;=2,)3?e;. (16)

The only thing remaining to complete the solution is to solve for ki, using the boundary conditions on the surfaces X'^^ ±.hl2, given by the

second equation of sec. 1.2, eq 10. Making use of

(4) and (7), the boundary conditions take the

form

7^/2! V ruiijk j y nnijt j ^ -^'"'^ dxr ' ^x',

-2,.5^^^-^^5^,=0atA=±|

Since these constitute homogeneous linear equations, and det(S;5'^'^-i3*)±0, it follows that

dX.

(17)

Because the crystal is driven by electrodes of

opposite polarity, only that part of the solution

(11) which is antisymmetric in X'2 is of interest,

and thus

\pi=Ai sin kiX'2. (18)

Applying (17) to (18), we get

cos^"jA/2=0,

from which it follows that

h

or

^^=(2^+1)2' w=0, 1,2,

kin=n ijjh), n=l, 3, 5, . (19)

Substituting (19) into (12), we get finally for the frequency of vibration

where

i=l, 2, 3, n=l, 3, 5, (20b)

In order to find out whether any of the normal

modes described by (18) and (20) can be excited

piezoelectrically, we have to calculate the piezoelectric coefficients pertaining to these modes.

14
Making use of (1) and (7) we get

2y

from which we get

Thus the effective stress components that excite

the mode are not T^-j but 2i/3^T-2. Making

use of (sec. 1.2, eq 5) and recaUing that the applied

field is along the X'^ direction, we get

and

T;-2= —el-j2£"fc= —esta-E" (21)

where E is the magnitude of the applied field, and

e'i =&\en2 (22)

is the desired piezoelectric coefiicient. Making

use of convention (5), we can write (22) in the

form

e;=j8l4+/3^4+^34. (23)

3.2. Rotated Y-Cut platesThe secular equation (sec. 3.1, eq 14') is very

complicated in general, and it is deskable to find

special cases in which (sec. 3.1, eq 14') gets simplified. A well known special case in which this occurs is the case of the so-called "rotated Y-cut

plates" or "Y'-cut plates" shown in figure 8. For

such plates, we have from figure 7.

where

X'i =a)Xj

"1 0

0"

0 c -s

0 s c

•I , ''I Figure 8.

and

(S'=sin e, C=cos 6.

Furthermore, according to [3, p. 247, 248],

C;5=C;6=0 for M= 1,2,3,4 (la)

621=622=623=^24=0 (lb)

It thus follows from (1) and (sec. 3.1, eq 3) that

^26 =0, 645 =0, 622= C22, C2i=C2i, Cii=Cii

and (sec 3.1, eq 14') becomes

^6-7. 0 0

0 6^22 —7ft = 0.

0 C24 ^^44-7.-

Expanding this secular equation, we get

(C^;6-Tft)[7|-(t';2+f'44)7ft+(a2C':4-C24)^] =0

whose roots are

^C22'\~Ci4\ I r"/C22 ^24

and

(2a)

;b) 72.3=(^—^—

j±L(^—^;+t^24j • (2

Equations (sec 3.1, eq 13 and 15) can be

written as

(<5;6-7ft) /3f=o

(O^2-7ft)^f+C^4ft*=0

C'^4i8|+(<^:4-7ft)ft'=0

(|Sf)^+(^l)'+(/3|)-^=l.

When k=l, we notice that

(^"22^71) C21

C2i (C44 I'l)

(3a)

(3b)

(3c)

(3d)

0

and thus

(4a)

On the other hand when k^2 and [3, C66?^72,3

and thus = We then get from (3b, c, d),

after a little algebra.

/3!=0, /3i=

^^24 m+{C22-y2y^' [C'2l+C'22-l2y]^'

(4b)

15

(?;,-72
and eq 5) we have

C22——Ts

(4c)

Since the direction of vibration is given by (sec.

3.1, eq 16), we see from (4) that

el=0\e[, eI=/3^e;+^^e3(for k=2,3),

and thus \f/i represents thickness-shear vibration

in the XJ^Yg-plane and \p2, 'As represent vibrations

in^ mutually perpendicular directions in the

XjXs-plane.

Making use of (lb) the piezoelectric excitation

coefficient given in (sec. 3.1, eq 23) becomes

«*=/3lC26-

It thus follows from (4) that

^1 =626, 62=63=0. (5) Consequently \l/2 and rp^ cannot be excited piezoelectrically, i.e., the only mode which can be

excited piezoelectrically is the thickness-shear

mode

77/ TT ,;':„=^sinyA';,(w=l,3,5, . . .) (6)

which has a frequency of vibration

(7)

where we have made use of (sec. 3.1, eq 18 to 20),

and (2a).

From [3, p. 247 and 248] we have

G^5=S^0n-\-C^C^Q—2SC Cii, (8<i)

e'2,= CiSeu~Ceu), (8b)

e2i=C'ef,+ S'ei,. (8c)

We thus get from (8), (7) and (2a)

^66= S'^C44-[- C^Cge

2SC Ci4

=p(2hfnln)\ (9) From (9) we see that by cutting Y'-cut plates at

7 different orientations, we can evaluate all of

the dielectric constants, all of the piezoelectric

coefficients and three of the elastic constants. The question now is what are the orientations

that give minimum coupling with other modes

of vibration. Let us first find out what stresses can be excited piezoelectrically. From (sec. 1.2,

or

Making use of (lb), we see that T[, T'^, T^, and

T'i cannot be excited piezoelectrically, and we only

need to worry about T'^ and T'^. Furthermore,

from (sec. 3.1, eq 2) and (1) we get

5 — '^55*-'5~r'^56*~>6»

-^6 — 56^5~r "-^ 66*->6-

(10a)

(10b)

The only strain component which is involved in

the thickness-shear mode in the X'l Xj-plane, is S'^. It thus follows from (10) that the interfering mode

is determined by S5, which is coupled to S'^ by

means of C^g. We can thus reduce coupling to other modes to a minimum by choosing such

orientations as to make

&55=t^;6+(47rA;f)44=0.

Since the second term is less than 1 percent of C56 and the constants of quartz are not known to better

than 1 percent, we can neglect the second term in

the preceeding equation and get [3, p. 247].

C',,= (C- S') Cu- SC(Cu- CJc^O. (11)

This is exactly the condition that leads to the AC

and BC cuts [1, p. 454] . Making use of the values

of C^, given by Koga et al, [5] (after a correction

in sign),

(7i4=18.06X 101° dynes/cm^, C44-C66 = 18.39X101° dynes/cm^, we find that (11) is satisfied for

0= +31.5°,-58.5°.

However, since these values depend upon temperature, we shall use, for convenience, the angles

?=+30°,-60' (12)

It will be shown in (4.2) that this choice will make

it much easier to obtain a solution for the elastic constants in terms of the frequencies of vibration.

3.3. Rotated X-Cut Plates

For the rotated A"-cut plates shown in figure 9, we have

S 0 c

{a))= C 0 -S , (1)

0 1 0

16
Figure 9. where S=sm 6 and C=cos 6, as before. Making

use of (1), we obtain, for the constants occurring

I in (sec. 3.1, eq 14)

^22 — C^^n,

e26=(ySen,

As in the previous section,

/^i I 47r / /

*22 (2)

From these expressions, it can be seen that, in

general, none of the C^p's are zero; which implies

that the equation for is cubic. Thus, the

formula for the frequency will be very complicated, and it becomes difficult to solve for the constants of quartz.

_ At 0=90° (C=0, S=l), the situation becomes

simpler; but e22=62'4=e26=0 and, according to

(sec. 3.1, eq 23), thickness vibration cannot be

excited piezoelectrically.

The X-cut plate 0=0, 5=0, C= 1) is more promising, and, in fact, is the only simple case among

the X'-cut plates which can be excited piezoelectrically. If we set S=0, (7=1 in the above equations, we get for an X-cut.

6*22 = Ci1 , C44 = Cei

^24 =0, C26 =0,

622=611, 624=0, 626=0

,'S_ s

*22 —

Consequently, (sec. 3.1, eq 14') becomes

C44-7*

0

C14

0 Cu

C'22-jk 0

0 Cee-Tft

= 0, (3)

where

^^22 — C'li —g ell, til and (sec 3.1, eq 23) becomes

6l=i3|6ii.

Solving (3) for j/., we get:

y,=pi2hfJny=C:,2=Cn- eh' en

(4)

(5^

In exactly the same way as'in (sec. 3.2, eq 3 and

4), it can be shown that:

^1=131=0, 131=1; 131=^1=0.

Therefore, it follows from (4) that only the mode

for k=2 can be excited piezoelectrically. The fre- quency of this mode is given by (5) and the

direction of vibration by:

62 =^ i0iBi =G2-

Consequently this mode represents thickness exten- sional vibration.
4. Determination of the Constants of Quartz

4.1. Program

In the last two chapters, we found 4 different

crystal cuts for which we have reliable formulas

relating the frequency of vibration to the 10

elastic, dielectric, and piezoelectric constants of

quartz. They were rectangular bars cut at (see

sec. 2.3, eq 9, and fig. 8)

i=90°, 6=70°, 131°, (1)

and Y'-cut plates oriented at (see 3.2, eq 12, and

fig. 8)

d=+30, -60°. (2)

Solutions for other modes of vibration, such as

flexure and face shear, are not as reliable as the ones we have obtained so far, and will therefore

not be considered.

Since 4 out of 6 of the stress components (T'l, T2, T'z, T'i) are zero for Y'-cut plates, and the

expression for the frequency of vibration of these

plates is relatively simple (see sec 3.2, eq 9) it is advisable to make as much use of Y'-cut plates as

is possible.

From (sec. 3.2, eq 9), i.e., — »S'"'C44-1- C^Cgs

ISC Cli

-Pi2hfjny, (3)

we see that 7 constants are involved, namely;

C44, Cee, Cu, efi, efa, Cu, and eu. Thus, all the

information that can be obtained from such plates,

should be possible to obtain from 7 different

orientations. We already have 2 orientations in

(2), and we need to choose 5 more. For simplicity

and convenience, we shall take the following values

of 6:

0°, 45°, 60°,-30°,-45= (4)

However, because the dielectric constants occur

in the denominator of the last term of (3), we shall see in the following section that this will make it possible to determine all the constants in (3) except efi. The elastic constant Cu can be determined from

the formula for the extensional-thickness vibration

of an X-cut plate (sec. 3.3, eq 5), namely

C',,= Cu-^eh=p{2hfn/ny, (5)

€11

in conjunction^with the information obtained

from (3). In order to determine efi, a method used by

Mason [2, p. 65] is probably as good as any.

This method will be described in section 3.

The only constants that remain to be determined''

are C13 and C33. These two constants can be obtained from the formula for the longitudinal vibra- 1 j tion of a rectangular bar given by (sec. 2.4, eq 7!i and 8), as will be shown in section 4.

4.2. C44, Cee. Ci4, and Cn

For convenience, let

efi/47r=€i, e|i/47r=e3. (1)

)

Evaluating (sec. 4.1, eq 3) at the angles in (sec. 4.1,

eq 1 and 4), we get

^1

^66(45) = I (C44+ C,,-2Cu) + h

^66(-45) = KC4.+ ^6+2^14) +i ^^^4^

(2)

cum=HCu+sc,,-2./3Cu)+m 3ei + €3

g-,(-30) = i(6^44+3<:^56+2V3(7i4) +3/4^%+^'"^'

cum= i (3(744+ c,,-2^cu) + 1 ^^^"'r;'"^'

C';6(-60) = K3C44+C66+2V3C7h) + |

From these equations, we have

A=UCU-i5)-CU45)]^C,,^

^11^14 (3)

A,=^[C',,i-SO)-C',,m]=Ci,+P^ (4)

3€i + e3

(5)

B,= \CU-^5) +CU'^5)] =C,,+Ce,+'^^ (6)

B,=2[cu(-m+oum=Cu+3Ce,+3 3ei+£3

(7)

g3=2[(7;e(-60) +6';e(60)] =3C44+^a6+ '^To («)

18
Equations (3 to 5) can be rewritten as

Aiei-\-Aie3 =€iCii+e3Cu+eueii, (9)

3^2ei+^2€3=3eiC'i4 +e3Cu+3enei4, (10)

^361 +3^363 =ei Cu+3 €3^14 +611^14. (11)

Multiplying (9) by 3, and subtracting it from (10), we get

3(yl2-vli)£i+(^2-3^i)e3=-263Ci4. (12)

Moreover, if sve subtract (9) from (11), we obtain

(yl3-^i)€i+ (3^3-^1)63= +2e3(7i4. (13)

Adding (12) and (13), we find the following

relation between ei and €3: (3^2+^3-4^1)61+ (A2+3^3-4^i)e3=0,

or

where

63=— aei.

4^1 —3^2—^3

'4^1-^2-3^3

(14)

(15)

We can now obtain Cu by substituting the

value of €3 given by (14) into (13). Thus

Ci4=K3^3-^i)+^ (A-A). (16)

From (2), we have

^0—<-^6l fl (17)

Furthermore, solving (3) for enCu, we get with the

help of (14), —=(l-a)(^i-a4).

*i (18)

Squaring both sides of (18), and making use of

(17), we obtain

^1 Aq— Cee (19)

Consequently, if ei is determined, €3, Cu, and eu can be calculated from (14), (17), and (18),

respectively. We shall show how ei=en/4ir can be

determined in the following section.

If we substitute the value of €3 given by (14)

into (6) to (8), and then the value of en/ei given

by (17) into these equations, we get after a little algebra,

5i(l-a)-A=(l-a)C44-«a6+-> (20)

€1

(l-^)-3vlo=(l-^) C,,-aC,,+'f\ (21)

Bz A=(l-Sa)Cu-aCe,+'f- (22)

C44 can now be obtained, by subtracting (20)

from (21) and then solving for C44. Thus

<^44=^ [B,(3-a)-3B,{l-a)-6Ao]. (23)

Substituting this value of C44, and the value of

ef4/ei, given by (19) into (20), we get after a httle

algebraic manipulation,

aC'i6+[(l-a)(5i-C44)-(l+a)A]a6

+ [{'i--c,y{A,-C,,y-(l-oc)A,(B,-C,d+Al]=0.

The solution to this quadratic equation can be

written in the form

•^ee^^ [(l+a)Ao~(l-a){B,-Cu)

±(l-a)^{B,-C,,-A,y-MA-Cuy]. (24)

Only one of the signs in (24) will give the correct answer. Cu can be obtained by eliminating

efi/ei from (sec. 4.1, eq 5) and (2). Thus

Cii —C22 Aq C^^. (25)

From the values of 6*14, C44, C^s, and Cu obtained

in this section, we can calculate the following constants (see sec. 2.4, eq 2 and 3):

C'i2=C'n 2CgS) (26)

Si,= -Cu/P', Su=2aS'> S,,=2C,S' (27)

where

/3 /2C(i^Cn C14. (28)

4.3. Dielectric and Piezoelectric Constants

Since the plated rectangular bar can be considered as a parallel plate condenser, it should be

possible to determine the dielectric constant of the

bar by measuring the capacitance between the two

electrodes at zero frequency. This method was

already used by Mason [2. p. 65] ; but he employed

a roundabout method of obtaining the relation

between the capacitance C and the dielectric constant. We shall give here a more straight-forward

derivation of the same result. As usual, the capacitance C is defined by

(1)

19
where Q is the total electric charge on one elec- trode, V the potential difference between the two

electrodes, a the surface real charge density on the

electrodes, and E is the static electric field inside

the crystal. The problem now is to evaluate ajE.

Since the situation in figure 3 is applicable here,

we have for the normal component of the electric displacement inside the crystal,

where

In our case,

Thus,

Z>3=47ro-, or (t^DzIAtt, (2)

(3a)

(3b)

(4)

Since T'l is the only important stress,

^\=S'ii,TlArd'nE'i = S'nT[-^dziE,

or

(5)

The equation of motion is given by

dT[ 1 dS[ 1 d'U[

dX[ s[,dX[ s[,dX?'

The solution to this equation is U[=A cos kX[-^B sin kX[

where

as was shown in (sec. 2.1, eq 13 and 14). However, in this case, the boundary conditions are

1 T[=^ iS[-dnE) =0 at X[=±h/2, Cff —

Oil or dU[ S[=^=ki-A sin kX[+B cos kX[)

=d'siE&t X[=±^^,

instead of S'i = 0, as before. The solution obtained

here is the particular integral, whereas the solution

(sec. 2.1, eq 17) was the complementary function.

Writing down the boundary conditions explicitly, we have

k(^-A sin k 1+5 cos k ^^=d'uE,

k(^Asmk^^+B cos k ^-^=duE.

20

Thus

and

yl=0, kB=dnE/cosk^^,

'^i=kB cos kX[=dziE cos kX'ijcoB k ^ .1

At w=n, k=0, ^i=dziE and, according to (5), T[ =Q. Thus, we have from (4), 0-=— E,. 4t

Substituting this value of cr into (1), we get

(7=-

or

1 f-U _f3_ A-K III', (6)

in agreement with the result obtained by Mason i [2, p. 65]. The same result would have been obtained, had we used (3b) instead of (3a).

The relation between efi and efi is given by (sec. 2.4, eq 5) to be

Cu fa — (2eiic?ii+eH(^i4). (7)

47r 47r Making use of the relations

din —^ivSvfjtj and the symmetry properties of quartz, we get

dn = £n(Sn —S12) -\- euSu='^euS^s-\- CuSu, (8a)

'^14=eil('S'i4 »S'24) +ei4*S'44 =^en*S'i4 +ei4*S'44. (8b)

We thus have, with the help of (sec. 4.2, eq 17,

18, and 19)

2eiiC?n +614^14= A^eeeii +4S'i4eiiei4

+^44e?4=/3efi/4x, (9) where

An—Cr

From (7), (9), and (6), we have finally,

_efi_ ^3 C

4x 1 +^

(10)

(11)

Now that efi and efi can be calculated from

(11) and (6), e33 = ef3 can be obtained from

(Sec. 4.2, eq 14). Furthermore, en, can be cal- culated from (sec. 4.2, eq 17 and 19) and dn, du

from (8). Thus, all the dielectric and piezoelectric

constants can be determined.
4.4. Ci3 and Q where

The only remammg mdependent constants to

, be determined are C13 and C33. As was abeady

pomted out m section 1, these constants can be

obtained from (sec. 2.4, eq 8), evaluated at the

two angles in (sec. 4.1, eq 1). We notice from (sec. 2.4) that we can write

^"'~(Cn+C,,) C33-2Cf3"^"" ^

^ and

Ci=E,,C^s+F,,Css+G,„ (2)

where aU the constants Ai^ • are known.

In view of (1) and (2), (sec 2.4, eq 8) has the

following form

A.1C13 -|- A2C33 -\- A3C13 -|- A4C13C33 -(- A5C33

-\-A.sCi3C33-\-A.-!Cis=As. (3)

It is difficult to see how to solve simultaneously

l|| two equations of the form (3) for the two unkno%vns

ji Ci3 and C33. For this reason, we shaU use the

If following iteration method for solving (Sec. 2.4,

eq 8) : First of aU, we use the best known values

of Ci3, C33 only in (2), i.e., in the correction terms

containing C{i and Ci'g. This will make (sec. 2.4, eq

8) take the simple form

has+b2Q3+hCh=h. ' (4) Two equations of this form can be easily solved

for Ci3 and C33 by eliminating C33 first. The solution obtained in this wslJ, can then be used in (2) and the process repeated. This method is expected

to lead quickly to the correct values for C13 and

, C33, since the terms containing Cu and C'iq in

' (sec. 2.4, eq 8) are about 1 percent of the other

! terms, and the error made by using the best known

I values of C13 and C33 in (2) is probably no greater

i than 1 percent. Thus the error in the first solution

obtained wiU be about 0.01 percent.

Probably the best values available for Cn and

633, are those given by Koga et al. [5], namely

C,3= 1.193 X 10" djmes/cm^, (5)

(733=10.59X10" dynes/cm^.

After C13 and C33 are evaluated, Su, Su, S13, S33 can be evaluated from (sec. 2.4, eq 3), i.e.,

a' ={a^+ 0,2)0,3-20,3, &' =2{0,,0,,-Cu). (10)

With this, aU the elastic, dielectric, and piezoelectric constants of quartz can be determined. It should be recalled that the elastic constants are the adiabatic ones at constant electric field.

4.5. Experimental Conditions

The following conditions must be fulfilled in using Y'-cut plates: (1) The plates must be flat on both sides. (2) The ratio of lateral dimensions

to thickness should be as large as possible (perhaps

about 20:1); (3) The plates must be excited at a harmonic high enough so that/„/n. is a constant. One should measure experimentallj' , and find

out the value of n at which this occurs. (4) The

thickness of the plates before plating must be

measured accurately. (5) The electrodes should

cover the fuU surfaces of the plates. (6) The

frequency for zero plating must be determined by

extrapolation of the curve of frequency versus

plating thickness. (7) Both chcular and square

plates should be tried. Theoretically, if the

harmonic is high enough, the shape of the plate

should not make any difference.

The conditions for longitudinal vibration or rectangular bars are as follows: (1) Zi/Z2>10 and

h/ky^O (see fig. 10). The bars should be fully

plated at the top and bottom smiaces. (3) The

fundamental frequency of vibration must be used.

(4) The recommended values of d, 70° and 131°

are only rough, and a little experimentation with

the value of 6 about these values, will determine

the best cut for minimum coupling with other

modes of vibration. (5) AU dimensions should

be measm^ed accurately before plating. (6) As

condition 6 above.

Figure 10.

C33 . C44 It is imderstood that all the crystals are made to =—r+-;^' (t»; sions at r can be calculated from the measm-ed

vibrate at the same temperature r. The dimenSi2=Su — iSt,i, (7) dimensions at room temperatm-e tq by means of

(sec. 2.5, eq 8). 5-53= — Cu/a', (8) If the variation of the constants of quartz with

temperature is desii-ed, then all the constants

S33={On-'rOi'^)la', (9) must be determined at different temperatures, and

21
a plot for the temperature variation of each constant can be obtained. The temperature coeffi- cients can then be calculated from the definition,

4.6. Recent Determinations of the Constants

of Quartz

Several authors have tried to determine dynamically the constants of quartz, e.g., Anatasoff and

Hart [7] and Mason [8] and other authors mentioned in Cady's book [1]. Most of these authors

have not taken the piezoelectric effect into account

and have used expressions for the frequency of

vibration which were not too reliable for the

experimental conditions they had.

Recently, Koga et al., [5] have evaluated the

constants of quartz taking the piezoelectric effect into account. However, they used the value 4.50

for the dielectric constant in all their formulas;

whereas it is clear from the formulas of section

4.2 that the dielectric constant €22 is not the same

for aU the plates, but depends upon the angle of

cut. Furthermore, Koga et al., only determined

the constants of quartz in the neighborhood of 20° C; and there is much need for the evaluation

of these constants at low temperatures, where the

aging characteristics of quartz improve considerably.

The particular choice of crystal cuts recom- mended in this report should give a closer estimate |{ of the actual values of the constants of quartz

than those used by other authors; at least for the

cutsin (sec. 4.1, eq 1 and 2). It thus seems that

the determination of the constants of quartz by

the method proposed here, in the temperature

range -200° C to +100° C will be a valuable

contribution to our knowledge of quartz.

It should be mentioned here that the linear

coefficients of expansion of quartz, given by (sec. 2.5, eq 3), are only known to three significant

figures, and there does not seem to be any data on

the subject beyond that given in Sosman's book

[4] since 1927. Thus, for highly accurate determinations of the constants of quartz, it seems that more accurate data on the linear coefficients of

expansion of quartz is necessary.

5. References

[1] W. G. Cadv, Piezoelectricity (McGraw-Hill Book

Co., Inc., New York, N.Y., 1946).

[2] W. P. Mason, Piezoelectric crystals and their appli- cation to ultrasonics (D. Van Nostrand Co., Inc., New York, N.Y., 1950).

[3] R. A. Heising, Quartz crystals for electrical circuits (D. Van Nostrand Co., Inc., New York, N.Y.,

1946).

[4] R. B. Sosman, The properties of silica (Reinhold

Publishing Corp., New York, N.Y., 1927).

[5] Koga et al, Phys. Rev., 109, 1467 (1958).

[6] R. M. Davies, Phil. Mag., 16, 104-109 (1933).

[7] Anatasoff and Hart; Phys. Rev., 59, 838 (1941).

[8] W. P. Mason, B.S.T.J., 32, 178 (1943)

Boulder, Colo., July 23, 1959.

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