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Electromechanical coupling factors and fundamental material constants of thickness vibrating piezoelectric plates
Electromechanical coupling factors and fundamental material constants of thickness vibrating piezoelectric plates H. F. T i e r s t e n *
The v i b r a t i o n s of p i e z o e l e c t r i c p l a t e s a r e g o v e r n e d by the l i n e a r p i e z o e l e c t r i c equations which may contain a r a t h e r l a r g e n u m b e r of e l a s t i c , p i e z o e l e c t r i c and d i e l e c t r i c constants. But under c e r t a i n v e r y r e s t r i c t i v e but p r a c t i c a l l y i m p o r t a n t c i r c u m s t a n c e s , p i e z o e l e c t r i c coupling coefficients o c c u r naturally in the solutions of the equations, a different coefficient a r i s i n g in each p a r t i c u l a r c a s e . Under m o r e g e n e r a l c i r c u m s t a n c e s many of the m a t e r i a l c o n s t a n t s r e main in the solution, and the utility or significance of any of the coupling c o e f f i c i e n t s a p p e a r s questionable. The solution
of the p r o b l e m of the t h i c k n e s s v i b r a t i o n s of an a r b i t r a r i l y a n i s o t r o p i c plate is used in d e m o n s t r a t i n g the significance and l i m i t a t i o n s of the coupling coefficient for v a r i o u s types of a n i s o t r o p y . In many of the i n s t a n c e s when the coupling coefficient t u r n s out to be useful, the f r e q u e n c y equation shows that the o v e r t o n e r e s o n a n c e s a r e not i n t e g r a l multiples of the fundamental. The d e p a r t u r e f r o m an i n t e g r a l multiple r e l a t i o n s h i p depends on the coupling coefficient only, and consequently, the coupling coefficient can be d e t e r mined from simple resonance measurements.
This is e s s e n t i a l l y the text of a talk d e l i v e r e d at the s e s s i o n on t r a n s d u c e r evaluation (Session I) of the 1968 IEEE U l t r a s o n i c s Symposium. The o t h e r f o r m a l talks at the s e s s i o n w e r e p r e s e n t e d by A. J. Bahr of the Stanford R e s e a r c h I n s t i tute, Menlo P a r k , C a l i f o r n i a and T. R. M e e k e r of Bell T e l e phone L a b o r a t o r i e s , Inc., Allentown, P e n n s y l v a n i a . Different a s p e c t s of the s u b j e c t w e r e d i s c u s s e d in the t h r e e talks. In the panel d i s c u s s i o n held a f t e r w a r d s , the t h r e e p r e s e n t a t i o n s w e r e d i s c u s s e d and c o r r e l a t e d in s o m e detail. It also s e r v e d a s a f o r u m for people to b r i n g up c e r t a i n a s p e c t s of the subject that w e r e o m i t t e d in all t h r e e l e c t u r e s and p r e s e n t a n u m b e r of d i f f e r e n t points of view. Both the panel m e m b e r s and m e m b e r s of the audience p a r t i c i p a t e d in the d i s c u s s i o n . During the c o u r s e of the d i s c u s s i o n it b e c a m e i n c r e a s i n g l y a p p a r e n t that people who attended the s e s s i o n held widely d i v e r g e n t points of view.
The s t r a i n - m e c h a n i c a l d i s p l a c e m e n t r e l a t i o n s
This p a p e r is devoted to a d i s c u s s i o n and d e m o n s t r a t i o n of the fact that in g e n e r a l a l a r g e n u m b e r of e l a s t i c , p i e z o e l e c t r i c and d i e l e c t r i c c o n s t a n t s influence the behavior of a vibr a t i n g p i e z o e l e c t r i c plate; and that, in consequence, the utility o r s i g n i f i c a n c e of any of the defined coupling c o e f f i c i e n t s a p p e a r s questionable. However, under c e r t a i n very r e s t r i c tive but p r a c t i c a l l y i m p o r t a n t c i r c u m s t a n c e s , p i e z o e l e c t r i c coupling c o e f f i c i e n t s o c c u r naturally in the solutions of the equations--a d i f f e r e n t coefficient a r i s i n g in each p a r t i c u l a r c a s e - - a n d under such c i r c u m s t a n c e s the c o e f f i c i e n t s a r e significant and useful. The solution of the p r o b l e m of the t h i c k n e s s v i b r a t i o n s of an a r b i t r a r i l y a n i s o t r o p i c p i e z o e l e c t r i c plate is u s e d in d e m o n s t r a t i n g the s i g n i f i c a n c e and l i m i t a t i o n s of the e l e c t r o m e c h a n i c a l coupling coefficient for v a r i o u s types of a n i s o tropy. The r e l a t i o n of the t h i c k n e s s v i b r a t i o n solution to the behavior of thin film t h i c k n e s s t r a n s d u c e r s is d i s c u s s e d .
S k l = ~/2 (Uk, 1 + Ul, k)
1.3
The e l e c t r i c f i e l d - e l e c t r i c potential r e l a t i o n s Ek = --~, k
1.4
The linear, p i e z o e l e c t r i c constitutive r e l a t i o n s E
Tij ~- CijklSkl -- ekijEk' s D i = eiklSkl + ~ ikEk ,
1.5
w h e r e Ti,j , u,, D~, S,,, Ej a r e the c o m p o n e n t s of s t r e s s , m e c h . j 1 ~j anical d i s p m c e m e n t , e I e c t r i c d i s p l a c e m e n t , s t r a i n and e l e c t r i c field, r e s p e c t i v e l y ; p and ~p a r e the m a s s d e n s i t y and the e l e c t r i c potential, r e s p e c t i v e l y ; and C~jkl,_ eki], ~k_ a r e the ~ _ e l a s t i c , p i e z o e l e c t r i c , and diele t r i c constants, r e s p e c t i v e l y . Throughout this p a p e r we employ C a r t e s i a n t e n s o r notation 1, the s u m m a t i o n convention for r e p e a t e d t e n s o r indices, the dot notation f o r d i f f e r e n t i a t i o n with r e s p e c t to t i m e and the convection that an index, say j, p r e c e d e d by a c o m m a denotes d i f f e r e n t i a t i o n with r e s p e c t to the s p a c e coordinate xj. The s u r f a c e s of the plate a r e t r a c t i o n f r e e and have p o r t i o n s which a r e coated with p e r f e c t l y conducting i n f i n i t e s i m a l l y thin e l e c t r o d e s with which a s t e a d y - s t a t e d r i v i n g voltage is applied, and o t h e r p o r t i o n s which a r e u n e l e c t r o d e d . D i e l e c t r i c c o n s t a n t s a r e a s s u m e d such that e l e c t r i c fields outside the u n e l e c t r o d e d portion of the plate a r e negligible. Hence the boundary conditions may be w r i t t e n Tnj : 0 o n S
1.6
~0 = ~Ooe-iwt on D n = 0 on S,
1.7
and
I.
THE EQUATIONS
OF PIEZOELECTRICITY
The small vibrations of piezoelectric plates--or any other piezoelectric body--are governed by the linear piezoelectric equations, which consist of the following: The stress equations of motion Tij, i ~ Pfij
1. 1
The c h a r g e equation of e l e c t r o s t a t i c s Di, i = 0
1.2
* P r o f e s s o r H. F. T i e r s t e n , R e n s s e l a e r P o l y t e c h n i c Institute, Troy, New York 12181
w h e r e ~0o is the s t e a d y - s t a t e d r i v i n g potential and ~ is the driving f r e q u e n c y . Note that the 22 equations in 1 . 1 - 1 . 5 can r e a d i l y be r e d u c e d to four equations in the four dependent v a r i a b l e s u, and ¢Z, by substituting 1. 3 and 1.4 in 1.5 which may then beJ substituted in 1.1 and 1.2 to yield c ~jklUk, li + ekij ~0, ki = Pfij,
1.8
s eiklUk, li -- £ ik ~p, ki = 0.
1.9
Similarly, the boundary conditions 1. 6 and 1.7 can also be e x p r e s s e d in t e r m s of uj and ~o. Equations 1.8 and 1.9 c l e a r l y indicate that the e l a s t i c conE . . stunts c i~kl, the p~ezoelectr~c c o n s t a n t s elk 1 and the d i e l e c t r i c c o n s t a n t s ~ k influence the b e h a v i o r of a vibrating ULTRASONICS J a n u a r y 1970
19
p i e z o e l e c t r i c body. If we i n t r o d u c e the c o m p r e s s e d m a t r i x notation 2 in place of the t e n s o r notation, the a r r a y s of e l a s tic, p i e z o e l e c t r i c and d i e l e c t r i c c o n s t a n t s , r e s p e c t i v e l y , for an a r b i t r a r i l y a n i s o t r o p i c m a t e r i a l may be w r i t t e n in the form
C~q =
Cll
C12
C13
C14
C15
C16
C12
C22
C23
C24
C25
C26
C13
C22
C33
C34
C35
e36
C14
C24
C34
C44
C45
C46
C15
C25
C35
C45
C55
C56
C16
C26
C36
C46
C56
C66
el2
et3
el4
e15
el6 1 ~26 / ! e36 /
-ell eip =
e21
e22
e23
e24
e25
e31
e32
e33
e34
e35
~ll
~12
~1:~
£12
(22
~23 I
£13
E23
~33J
I
~
~
Electrode.~
~.T
2h
J
2t
F i g 1 Bounded p i e z o e l e c t r i c plate
1. 10
G r e e k index is not to be s u m m e d and u i and ~) a r e not functions of t i m e b e c a u s e all dependent v a r i a b l e s have the t i m e d e p e n d e n c e e -iwt, which has been o m i t t e d in 2.4 2.7. The e l e m e n t a r y t h i c k n e s s solution s a t i s f y i n g 2 . 4 - - 2 . 7 t u r n s out to be adequate for the bounded plate in many p r a c t i c a l c i r c u m s t a n c e s . P r e c i s e d e l i n e a t i o n s of the range of a p p l i c a bility of the t h i c k n e s s solution t o g e t h e r with the r e a s o n s for the applicability have been p r o v i d e d by R. D. Mindlin~ under a v a r i e t y of c i r c u m s t a n c e s . In this p a p e r we will suppose the t h i c k n e s s solution to be adequate for the bounded p l a t e s in which we a r e i n t e r e s t e d and not d i s c u s s this point any further.
Equations 1.10 show that for an a r b i t r a r i l y a n i s o t r o p i c p i e z o e l e c t r i c body t h e r e a r e 21 independent e l a s t i c c o n s t a n t s , 18 independent p i e z o e l e c t r i c c o n s t a n t s , and 6 independent d i e l e c t r i c c o n s t a n t s , f o r a total of 45 independent m a t e r i a l c o n s t a n t s . This is a r a t h e r l a r g e n u m b e r of independent m a t e r i a l c o n s t a n t s which a r e r e q u i r e d to d e s c r i b e the behavior of an a r b i t r a r i l y a n i s o t r o p i c v i b r a t i n g p i e z o e l e c t r i c body--and in p a r t i c u l a r a plate. Thus it should not be s u r p r i s i n g that a single e l e c t r o m e c h a n i c a l coupling c o e f f i c i e n t cannot d e s c r i b e the r e s o n a n t b e h a v i o r of a v i b r a t i n g p i e z o e l e c t r i c plate in g e n e r a l .
The s t e a d y - s t a t e t h i c k n e s s solution to 2 . 4 - 2 . 7 has been obtained 4 and the condition for r e s o n a n c e given as
2.
is the p i e z o e l e c t r i c a l l y stiffened e l a s t i c constant and V (i) is the phase velocity of the ith (i = 1. 2, 3) wave in the r, th d i r e c t i o n and the /3k( ~)are the n o r m a l i z e d amplitude r a t i o s a s s o c i a t e d with the ith wave. In g e n e r a l , 2.8 is a 3 × 3 t r a n s c e n d e n t a l d e t e r m i n a n t , each t e r m of which is a c o m p l i c a t e d combination of m a t e r i a l c o n s t a n t s and t r i g o n o m e t r i c functions. F o r an a r b i t r a r i l y a n i s o t r o p i c plate the n u m b e r of m a t e r i a l c o n s t a n t s on which the solution depends c o n s i s t s of 6 e l a s t i c c o n s t a n t s , 6 p i e z o e l e c t r i c c o n s t a n t s , and 1 diee l e c t r i c c o n s t a n t for a total of 13 m a t e r i a l c o n s t a n t s . Clearly it would be i m p o s s i b l e to define a meaningful p i e z o e l e c t r i c coupling f a c t o r under such g e n e r a l c i r c u m s t a n c e s .
THE ARBITRARILY ANISOTROPIC P L A T E
A s c h e m a t i c d i a g r a m of a bounded p i e z o e l e c t r i c plate is shown in Fig 1. The plate is f o r c e d into s t e a d y - s t a t e o s c i l lation by an a l t e r n a t i n g voltage applied a c r o s s the s u r f a c e e l e c t r o d e s . In this d i s c u s s i o n the e l e c t r o d e s a r e s u p p o s e d i n f i n i t e s i m a l l y thin. The c o o r d i n a t e axis out of the figure is c o n s i d e r e d to be denoted by a and the extent of the plate in the a - d i r e c t i o n is 2w. The boundary conditions on the s u r f a c e s of the plate may be taken to be T~j
0 and q~=±(0Oe - i w t at x u = ±h,
2.1
Trj
0 and D r
2.2
0 at x r = ±1,
Toj = 0 and D e = 0 at xo
±w.
2.3
Since we a r e c o n c e r n e d with p l a t e s having l a r g e 1/h and w / h r a t i o s , we ignore the m i n o r s u r f a c e s and a s s o c i a t e d bound a r y conditions 2.2 and 2.3, and r e p l a c e the bounded plate shown in Fig 1 by the unbounded plate shown in Fig 2. Cl.early, a solution for the configuration shown in Fig 2 depends on the x~ c o o r d i n a t e only. Such a solution is called a t h i c k n e s s solution and s a t i s f i e s 1.8 and 1.9 with x u spatial d e p e n d e n c e and e - i w t t i m e d e p e n d e n c e only, which may then be w r i t t e n E C~kj~,Uk, ~,~, + e~,~,jCp, ~,~, 4-pw2uj = 0
2.4
s e ~ , k U k , u~ -- ~ , q ) ,
2.5
wh cob ujku' ~ - ) - cos v ( i )
eu~,je~,~, k wh~ s sin ~ {
} = 0,
~,v
2.8
where c~,jk ~,
E ci~jk ~ ~ ei,~,jel~t,k/¢~!~'
2.9
Although 2.8 is v e r y c o m p l i c a t e d in general, in many c a s e s it s i m p l i f i e s c o n s i d e r a b l y when the m a t e r i a l is even slightly l e s s a n i s o t r o p i c and then a meaningful p i e z o e l e c t r i c coupling f a c t o r can be defined. Some c a s e s in which 2.8 s i m p l i f i e s c o n s i d e r a b l y and a coupling f a c t o r can be defined and one in which it does not and a coupling f a c t o r cannot be defined a r e d i s c u s s e d in the next section. Note that when the p i e z o e l e c t r i c c o n s t a n t s vanish (or a r e ignored), 2.8 f a c t o r s and r e d u c e s to ~h cos V~~
~ O.
2. 10
This limiting c a s e c o r r e s p o n d s to Koga's 5 solution. u~
0
and the boundary conditions in 2. 1 with the s a m e d e p e n dences, which may then be w r i t t e n CujkuUk, u + eu ujc°, u = 0 at xu = ±h,
2.6
~p = ±¢oo at x u = ±h.
2.7
In 2 . 4 - - 2 . 7 we have i n t r o d u c e d the convention that a r e p e a t e d 20
I~ ~i)
ULTRASONICS J a n u a r y 1970
Note that E~jk~, defined in 2.9 is the a p p r o p r i a t e p i e z o e l e c t r i c a l l y s t i f f e n e d e l a s t i c constant and not C~jkv. When a g r e a t deal of s y m m e t r y is p r e s e n t it s o m e t i m e s t u r n s out that c~D = g. However, this is not usually the c a s e and when £D .~ ~, ~ is the a p p r o p r i a t e p i e z o e l e c t r i c a l l y stiffened e l a s tic constant. Many e x i s t i n g works on this subject, including the IRE Standards, a r e not c l e a r on this point. G. A. Coquin a has shown that the v e r y c o m p l i c a t e d t r a n s cendental f r e q u e n c y equation in 2.8 can be w r i t t e n in the
m o r e c o m p a c t and for many p u r p o s e s m o r e i l l u m i n a t i n g form ~h
wh
COS--COS--COS-V(l) V(2)
Etectrodes
I
wh V(31
2h ~
2.11
= O,
k ~ n ) - ~ -~ - t a n v - - ~ - ~ - - I
~= 1
where 2 _ (~j(n)ep pj )2 k(n) p(V(n))2 ~ ~
2. 12
2.12 shows that in g e n e r a l the k ( n ) coefficients a r e c o m p l i c a t e d functions of the p i e z o e l e c t r i c eigenwave v e l o c i t i e s for p r o p a g a t i o n in the d i r e c t i o n ~, the a s s o c i a t e d n o r m a l i z e d e i g e n v e c t o r s and s o m e of the p i e z o e l e c t r i c c o n s t a n t s .
3.
THE INFLUENCE OF MATERIAL SYMMETRY
In this s e c t i o n we c o n s i d e r the a p p l i c a t i o n of 2 . 8 [or (2.11)] to the p r i n c i p a l cuts of the p o l a r i z e d c e r a m i c s ( c r y s t a l c l a s s 6mm), a r o t a t e d Y - c u t of q u a r t z (effective c l a s s 2) and the p r i n c i p a l cuts of l i t h i u m niobate ( c l a s s 3m). The r e s u l t s show that for all of the s y m m e t r i e s c o n s i d e r e d , except for those of two of the cuts (the X - c u t and Y-cut) of l i t h i u m niobate, the t r a n s c e n d e n t a l equation, 2 . 8 [or 2.11] s i m p l i f i e s c o n s i d e r a b l y and a p i e z o e l e c t r i c coupling coefficient o c c u r s n a t u r a l l y in that t h i c k n e s s s o l u t i o n which is d r i v e n e l e c t r i cally by the s y s t e m shown in F i g 2.
Fig 2 Unbounded piezoelectric plate
equation in 3 . 2 g o v e r n s the set of p i e z o e l e c t r i c s h e a r m o d e s which a r e d r i v e n e l e c t r i c a l l y by the s y s t e m shown in F i g 2. Thus, in this s p e c i a l c a s e the p i e z o e l e c t r i c coupling f a c t o r k26 o c c u r s n a t u r a l l y in the p i e z o e l e c t r i c t h i c k n e s s solution and is s i g n i f i c a n t and useful.
3B. Polarized ceramics (class 6ram) The a r r a y s of m a t e r i a l c o e f f i c i e n t s for a f e r r o e l e c t r i c c e r a m i c poled in the x 3 - d i r e c t i o n may be w r i t t e n in the f o r m Cll
C12
C12
Cll
C13
C13
C13
C33
0
0
0
3A. Rotated Y-Cut quartz (effective class 2)
0
0
The a r r a y s of m a t e r i a l c o e f f i c i e n t s for r o t a t e d Y - c u t q u a r t z with x~ the diagonal axis may b e w r i t t e n in the f o r m
0
0
Cll
C12
C12 C~q ~- C13
eip =
C13
C14
0
0 "
C22
C23
C23
C33
C24
0
0
C34
0
0
C14
C24
C34
C44
0
0
0
0
0
0
c55
c55
0
0
0
0
C56
C66
e!l
el2 0
e13 0
e14 0
0 e25
0 1 e26
0
0
0
e35
e36 j
E22
E2
E23
£33_J
I eip =
¢.s.= ~j
wh cos ~
wh = 0, cos V-ySy = 0,
3.1
3.2
3.3
w h e r e k26 is the t h i c k n e s s - s h e a r p i e z o e l e c t r i c coupling factor for r o t a t e d Y-cut q u a r t z and is given by k~ 6 = e33/c33~:33.2 ~
0
0
0
0
0
0
0
C44
0
0
0
0
c4~
0
0
0
0
C6,
i
0 0
0 0
0 0
0 e15
e15 0
e31
e31
e33
0
0
0 e2~ 0
e~-~ 0 0
003i ~3
il
3.5
T h i s m a t e r i a l h a s c o n s i d e r a b l y m o r e s y m m e t r y than r o t a t e d Y - c u t - - o r even u n r o t a t e d for that m a t t e r - - q u a r t z , a s can b e s e e n f r o m the g r e a t e r n u m b e r of z e r o s in 3 . 5 than in 3 . 1 . When u = 3 and ~ = 1, 2 . 8 [or 2.11] f a c t o r s into the t h r e e t r a n s c e n d e n t a l equations 4
T h i s m a t e r i a l (effective c l a s s 2) is s t i l l highly a n i s o t r o p i c . However, even with this r e l a t i v e l y s m a l l amount of s y m m e t r y , for ~ = 2 and r = 1, 2 . 8 [or 2.11] f a c t o r s into the t h r e e t r a n s c e n d e n t a l equations 4 ~h wh V(1) ~(1)k~6'
0
_
cos
tan
0
C66 = 1/2 ( C l l -- C12),
0 £~j ~-
C~q =
C13
3.4
The two t r a n s c e n d e n t a l equations in 3. 3 g o v e r n two s e t s of p u r e l y e l a s t i c m o d e s which, ideally, cannot be d r i v e n e l e c t r i c a l l y by the s y s t e m shown in F i g 2. The t r a n s c e n d e n t a l
wh wh = 0 , cos - = 0, V (1) V ~2)
wh wh , tan ~ _ -v~3~ V TMk.2
3.6
3. 7
L
w h e r e k t is p r e s e n t l y called the l a t e r a l l y c l a m p e d thickness coupling f a c t o r for the p o l a r i z e d c e r a m i c and i s given by
k•
2 s e33,/~33~33 •
3.8
Obviously k t should m o r e p r o p e r l y be denoted k33 and called the t h i c k n e s s - e x t e n s i o n a l coupling f a c t o r . C l e a r l y , 3 . 6 a r e a n a l o g o u s to 3 . 3 and g o v e r n the p u r e l y e l a s t i c m o d e s and 3 . 7 is analogous to 3 . 2 and g o v e r n s the p i e z o e l e c t r i c m o d e s . The e n t i r e d i s c u s s i o n following 3 . 4 with obvious m o d i f i c a tions, n a t u r a l l y a p p l i e s in t h i s c a s e a l s o . When ~ = 1 and r = 3, 2 . 8 [or 2.11] f a c t o r s into t h r e e t r a n s c e n d e n t a l equations. Two of the t h r e e equations a r e the same ULTRASONICS J a n u a r y 19'/0
21
a s t h o s e in 3 . 6 and g o v e r n purely e l a s t i c m o d e s and the t h i r d t a k e s the f o r m wh ~_ wh tan - V (3) V (3)k~25'
3.9
w h e r e k~5 is the t h i c k n e s s - s h e a r p i e z o e l e c t r i c coupling f a c t o r f o r the c e r a m i c and i s given by 2 k~5 = e ~ 5 / c- 5 5 ~S1 .
3.10
C l e a r l y the e n t i r e d i s c u s s i o n following 3 . 4 , with obvious m o d i f i c a t i o n s , a p p l i e s in t h i s c a s e a l s o . Note that V m) a p p e a r i n g in 3. "/and V (3) a p p e a r i n g in ~. 9 a r e d i f f e r e n t q u a n t i t i e s . The one in 3 . 7 is the velocity of an e x t e n s i o n a l wave propagating in the x 3 - d i r e c t i o n while the one in 3 . 9 is the velocity of an x 3 - p o l a r i z e d s h e a r wave propagating in the X l - d i r e c t i o n . Thus, again in t h e s e s p e c i a l c a s e s p i e z o e l e c t r i c coupling f a c t o r s o c c u r n a t u r a l l y in the p i e z o e l e c t r i c t h i c k n e s s solution and a r e s i g n i f i c a n t and useful.
The a r r a y s of m a t e r i a l c o e f f i c i e n t s for lithium niobate with x~ n o r m a l to the m i r r o r plane may be w r i t t e n in the f o r m -Cll
C12
C13
C[4
0
0"
C12
Cll
C13
--C14
0
0
C13
C13
C33
0
0
0
C14
--C14
0
C44
0
0
0
0
0
0
C44
C14
0
0
0
0
c~4
c6e
0 eip = --e22 e31
0 e22 e31
0 0 e33
c~),
-
0 e~5 0
el5 0 0
03h
V(2)
k~2 ) ~
3.14
V(3)
tan V (2) ~- k~3~ ~
(~3~nJ k~n ) =
+ ~3 (n) ~ e~p 2 evp 3 ~ p(V (n))2£su~
~33
3.11 Although the e l a s t i c and d i e l e c t r i c m a t r i c e s a r e identical with t h o s e of (unrotated) q u a r t z , the p i e z o e l e c t r i c m a t r i x is c o m p l e t e l y d i f f e r e n t and much m o r e c o m p l i c a t e d . As a c o n s e q u e n c e the s i m p l i f i c a t i o n s in 2 . 8 [or 2.11] that o c c u r in the c a s e of (rotated) Y - c u t q u a r t z do not o c c u r h e r e . When ~ = 1 or 2, 2 . 8 d o e s not f a c t o r c o m p l e t e l y and t a k e s the f o r m 03h wh ---- cosV(D V (~)
0
0
0
p(2)
p(3)
0
Q(~)
Q(3)
= 0,
3.12
where p(n) =
03h ~2(n)~1]221
e~2 ~
Q~) =
S
~ + ~3(n)~12321/
~(n)~
~ 2
~n)uv~zu
~2
+ f l3~ n_) e ~
sin
wh - V (n)
p(2)Q(3)
_Q(2)p(3)
= O,
wh v(n) - -
ULTRASONICS J a n u a r y 1970
~ sin ~h ~uu3/ v(n) ~
3.17
3.18
U n d e r t h e s e c i r c u m s t a n c e s the a f o r e m e n t i o n e d coupling of the two p i e z o e l e c t r i c standing w a v e s i s negligible. This s i m p l i f i c a t i o n o c c u r s b e c a u s e for high o v e r t o n e s 03 is very l a r g e and o c c u r s only in the c o e f f i c i e n t of the cosine t e r m s in e i t h e r 3.12 or 3 . 1 7 a s can be s e e n f r o m 3.13. It should a l s o be noted that if the coupling i s low, 3 . 1 5 can usually be a p p r o x i m a t e d by k~n ) v ( n ) tan 03h. ~ 1 n = 2, 3, 03h V (n)
3.19
u n l e s s the m a t e r i a l c o n s t a n t s a r e such V (2) and V (3) a r e not v e r y d i f f e r e n t or b e a r s o m e unusual r e l a t i o n s h i p to each o t h e r . In addition, for high coupling m a t e r i a l s , the m a t e r i a l c o n s t a n t s may be such that t h e r e a r e a few o r i e n t a t i o n s f o r which 3 . 1 5 can be a p p r o x i m a t e d by 3.19. However, such o r i e n t a t i o n s a r e b e s t found a n a l y t i c a l l y a f t e r the fundamental material constants are determined from appropriate measurements. In any event the k ( n ) a r e not a s i m p l e combination of the m a t e r i a l c o n s t a n t s a s is, e.g., k26 in 3 . 4 but a r e given by the c o m p l i c a t e d c o m b i n a t i o n of the m a t e r i a l c o n s t a n t s shown in 3.16. Although the e n t i r e p r e v i o u s d i s c u s s i o n in this p a r a g r a p h has b e e n for an X - c u t and a Y - c u t of m a t e r i a l s in c r y s t a l c l a s s (3m), it a p p l i e s without e s s e n t i a l change under a r b i t r a r i l y a n i s o t r o p i c c i r c u m s t a n c e s , i.e., when 2.11 holds in place of 3 . 1 5 and n = 1, 2, 3 in 3 . 1 8 and 3.19.
+ ~ (n)~ ~ wh wh ~3 u33Y]v(n ) c o s ~ v(n~
epp3 ~g, (0 ~n) cup2 + o -3( n )
22
v(n------~ c o s
3.16
At this point it should be noted that although the l o w e r m o d e s m u s t be d e t e r m i n e d analytically f r o m the r o o t s of 3 . 1 5 [or 3.17], the high o v e r - t o n e s a r e e s s e n t i a l l y e l a s t i c and may be a p p r o x i m a t e d by the r o o t s of
--el21
e11
0
3. 15
where
wh cos-=0, n=2,3. v(n) eij =
wh
t a n - - v(3) - 1,
which is obtained f r o m 3.12 along with 3.14. A s a l r e a d y noted, 3 . 1 4 g o v e r n s the u n d r i v e n purely e l a s t i c m o d e s . 3 . 1 5 [or 3.17] g o v e r n s the p i e z o e l e c t r i c t h i c k n e s s m o d e s which a r e d r i v e n e l e c t r i c a l l y by the a r r a n g e m e n t shown in Fig 2. 3 . 1 5 [or 3.17] s h o w s that the two p i e z o e l e c t r i c standing w a v e s a r e coupled at the conducting s u r f a c e s for t h e s e p r i n cipal o r i e n t a t i o n s of the c r y s t a l plate. When v = 1, both of the coupled w a v e s a r e s h e a r waves, and when ~ = 2, one of the coupled w a v e s i s a s h e a r wave and the o t h e r an e x t e n sional wave. Thus it is c l e a r that the s i m p l i f i e d p i e z o e l e c t r i c f r e q u e n c y equation 3 . 2 , 3. "/, with one p i e z o e l e c t r i c coupling constant, which o c c u r r e d in the o t h e r i n s t a n c e s , d o e s not o c c u r for the X - c u t (v = 1) or Y-cut (v = 2) of m a t e r i a l s in c r y s t a l c l a s s (3m).
=
%s = ~/'~ ( c ~
wh cos-= 0, V(~)
f r o m which it is c l e a r that 3 . 1 5 i s c o n s i d e r a b l y s i m p l e r than
3C. Lithium niobate, lithium tantalate (class 3m)
CE
Thus it is c l e a r that a p a r t f r o m the f a c t o r in the u p p e r l e f t hand c o r n e r of the d e t e r m i n a n t in 3.12, which f a c t o r g o v e r n s the u n d r i v e n purely e l a s t i c (extensional when ~ = 1 and s h e a r when ~ = 2) m o d e s , the d e t e r m i n a n t r e m a i n s e x t r e m e l y c o m plicated, As a l r e a d y noted in Sec. 2, 2.11 i s equivalent r o 2 . 8 but is m o r e c o m p a c t and m o r e r e a d i l y i n f o r m a t i v e f o r many p u r p o s e s than 2.8. Consequently, we p r e s e n t the r e s u l t s obtained f r o m 2.11 f o r the p r e s e n t s y m m e t r y in o r d e r to show the s i m p l i f i c a t i o n s that o c c u r in at l e a s t one n o n - s i m p l e i n s t a n c e . Thus, when ~ = 1 or 2, 2.11 f a c t o r s into two t r a n s c e n d e n t a l equations of the f o r m
3.13
When v = 3, 2 . 8 [or 2. 111 f a c t o r s into t h r e e t r a n s c e n d e n t a l equations. Two of the t h r e e equations a r e the s a m e a s t h o s e in 3.6 and g o v e r n u n d r i v e n purely e l a s t i c m o d e s and the
t h i r d m a y be w r i t t e n in the f o r m wh wh tan-- - - , V (3) V(3 Jk233
3.20
w h e r e k33 i s the t h i c k n e s s - e x t e n s i o n a l p i e z o e l e c t r i c coupling f a c t o r f o r a Z - c u t of a m a t e r i a l in c l a s s (3m) and is given by k33
2 S = e33/c33633"
3.21
Thus, although the g e n e r a l f r e q u e n c y equation, 2 . 8 [or 3.11], d o e s not s i m p l i f y a p p r e c i a b l y for a n X - c u t or a Y - c u t of m a t e r i a l s in c l a s s (3m), it d o e s s i m p l i f y f o r a Z - c u t and r e s u l t s in the u s u a l s i m p l i f i e d f r e q u e n c y equation, in which a w e l l - d e f i n e d e l e c t r o m e c h a n i c a l coupling f a c t o r (in t h i s ins t a n c e k33) o c c u r s n a t u r a l l y in the t h i c k n e s s solution and is s i g n i f i c a n t and useful. Note t h a t as a c o n s e q u e n c e of the f o r m of the s i m p l i f i e d f r e q u e n c y equation, o v e r t o n e r e s o n a n t f r e q u e n c i e s a r e not i n t e g r a l m u l t i p l e s of the f u n d a m e n t a l . T h i s e n a b l e s the d e t e r m i n a t i o n 8 of the p i e z o e l e c t r i c coupling f a c t o r f r o m s i m p l e m e a s u r e m e n t s of the f u n d a m e n t a l r e s o n ant f r e q u e n c y and at l e a s t one o v e r t o n e , when an equation of the f o r m of 3 . 2 0 i s valid. 4.
TRANSDUCER CONSIDERATIONS
The e n t i r e f o r e g o i n g d i s c u s s i o n in t h i s p a p e r a p p l i e s to t h i c k n e s s r e s o n a t o r s only and not to t h i c k n e s s t r a n s d u c e r s . However, m o s t , i m p o r t a n t q u a l i t a t i v e i n f o r m a t i o n c o n c e r n i n g the u s e of t h e s e t h i c k n e s s r e s o n a t o r s a s t r a n s d u c e r s can be d e t e r m i n e d f r o m the s i m p l e r r e s o n a t o r a n a l y s i s we have d i s c u s s e d . M o r e o v e r , r e s o n a t o r e x p e r i m e n t s a r e s i m p l e r to p e r f o r m and a r e u s u a l l y u s e d for m e a s u r i n g m a t e r i a l p r o p e r t i e s w h e n e v e r p o s s i b l e . But thin film t r a n s d u c e r s cannot s u p p o r t t h e m s e l v e s and, consequently, cannot be m e a s u r e d a s r e s o n a t o r s but m u s t b e m e a s u r e d a s t r a n s ducers. F i g 3 show a t h i n film t h i c k n e s s t r a n s d u c e r bonded to a d e l a y m e d i u m . When the p r o p e r t i e s of such a t r a n s d u c e r c o n f i g u r a t i o n a r e to b e m e a s u r e d , the a f o r e m e n t i o n e d r e s o n a t o r e i g e n s o l u t i o n d i s c u s s e d in Secs. 2 and 3 cannot be used. In o r d e r to c o r r e l a t e s u c h m e a s u r e m e n t s one m u s t u s e t h e o r e t i c a l c u r v e s obtained f o r a bonded t r a n s d u c e r 9. Howe v e r , s u c h a p r o c e d u r e i s not f e a s i b l e f o r a highly a n i s o t r o p i c t r a n s d u c e r with l a r g e p i e z o e l e c t r i c coupling, a s should b e c l e a r f r o m the d i s c u s s i o n of the highly a n i s o t r o p i c r e s o n a t o r p r e s e n t e d in Secs. 2 and 3. N e v e r t h e l e s s , when the t r a n s d u c e r a s a r e s o n a t o r h a s sufficient s y m m e t r y f o r the s i m p l i fied f r e q u e n c y equation, e.g., 3 . 2 0 to b e valid exactly, the t h e o r e t i c a l c u r v e s obtained f r o m an e l e m e n t a r y equivalent c i r c u i t t r a n s d u c e r a n a l y s i s 9 c a n b e u s e d to find the p i e z o e l e c t r i c coupling f a c t o r . When highly a n i s o t r o p i c o r i e n t a t i o n s (or m a t e r i a l s ) a r e used, even when the p i e z o e l e c t r i c coupling is s m a l l and the a p p r o x i m a t e equation, 3.19, i s valid f o r the r e s o n a t o r , the a f o r e m e n t i o n e d e l e m e n t a r y equivalent c i r c u i t t r a n s d u c e r a n a l y s i s should not b e u s e d 1° to find the coupling f a c t o r . However, for highly a n i s o t r o p i c m a t e r i a l s , e v e n with l a r g e coupling, a fe___wv e r y special--and p r a c t i c a l l y u s e f u l - - o r i e n t a t i o n s m a y e x i s t , for which the a f o r e m e n t i o n e d e q u i v a l e n t c i r c u i t t r a n s d u c e r a n a l y s i s a p p l i e s . But t h e s e o r i e n t a t i o n s c a n be found only by ignoring the coupling f a c t o r a p p r o a c h and s y s t e m a t i c a l l y finding the f u n d a m e n t a l m a t e r i a l c o n s t a n t s 7, and then d e t e r m i n i n g t h o s e few particular orien-t a t i o n s a n a l y t i c a l l y . Some of t h o s e o r i e n t a t i o n s can p o s s i b l y b e found by a c c i d e n t also. It should b e m e n t i o n e d at t h i s point that m e a s u r e m e n t s by t h e m s e l v e s , without any a n a l y t i c a l u n d e r s t a n d i n g , can s o m e t i m e s lead to confusion. F o r e x a m p l e , c o n s i d e r a n X - c u t of a c r y s t a l in c l a s s (3m) bonded to e a c h end of an i s o t r o p i c delay m e d i u m . As noted in Sec. 3C, a t h i c k n e s s d r i v i n g voltage applied to the equivalent r e s o n a t o r e x c i t e s a f u n d a m e n t a l p i e z o e l e c t r i c t h i c k n e s s - s h e a r r e s o n a n c e in which two o r t h o g o n a l s h e a r m o t i o n s with d i f f e r e n t p h a s e v e l o c i t i e s a r e coupled a t the conducting e l e c t r o d e d s u r f a c e s . T h i s m e a n s that when such a plate i s bonded to each end of a n i s o t r o p i c delay m e d i u m , the d r i v i n g t r a n s d u c e r e x c i t e s a n e l l i p t i c a l l y p o l a r i z e d s h e a r - w a v e in the i s o t r o p i c delay
F i g 3 T h i n - f i l m t r a n s d u c e r bonded to an i s o t r o p i c delay medium m e d i u m , which i s then d e t e c t e d by the o t h e r t r a n s d u c e r . However, the t r a n s d u c e r m e a s u r e m e n t s cannot b e c o r r e l a t e d with the c u r v e s obtained f r o m the e l e m e n t a r y equivalent c i r c u i t t r a n s d u c e r a n a l y s i s b e c a u s e the e l e m e n t a r y equivalent c i r c u i t e q u a t i o n s differ f r o m the g o v e r n i n g t r a n s d u c e r t h i c k n e s s equations. 5.
CONCLUSION
W h e n m a t e r i a l s p o s s e s s the a p p r o p r i a t e s y m m e t r y , it is meaningful to d i s c u s s and m e a s u r e e l e c t r o m e c h a n i c a l coupling coefficients. However, when m a t e r i a l s do not p o s s e s s the a p p r o p r i a t e s y m m e t r y , the coupling f a c t o r a p p r o a c h i s not meaningful. In m e a s u r i n g such m a t e r i a l s one m u s t p r o c e e d by f i r s t making the r e q u i s i t e m e a s u r e m e n t s to find the f u n d a m e n t a l m a t e r i a l c o n s t a n t s and then a n a l y t i c a l l y finding the p r a c t i c a l o r i e n t a t i o n s , for which the e l e m e n t a r y e q u i v a l e n t c i r c u i t t r a n s d u c e r a n a l y s i s holds.
ACKNOWLEDGEMENT The a u t h o r w i s h e s to t h a n k G. A. Coquin, A. H. M e i t z l e r , and E. K. Sittig, all of Bell Telephone L a b o r a t o r i e s , for valuable discussions. T h i s work was s u p p o r t e d in p a r t by the Office of Naval R e s e a r c h u n d e r C o n t r a c t No. N 0 0 0 1 4 - 6 7 - A - 0117-0007.
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J e f f r e y s , H. ' C a r t e s i a n t e n s o r s ' , C a m b r i d g e U n i v e r s i t y P r e s s , New York (1952)
2
T i e r s t e n , H . F . ' L i n e a r p i e z o e l e c t r i c plate v i b r a t i o n s ' , Chap 7, Sec 1, P l e n u m P r e s s , New York (1969)
3
Mindlin, R.D. ' T h i c k n e s s - S h e a r and f l e x u r a l v i b r a t i o n s of c r y s t a l p l a t e s ' , J o u r n a l of Applied P h y s i c s , Vol 22 (1951), No 316 Mindlin, R.D. ~qCaves and v i b r a t i o n s in isotropic, elastic plates', Structural Mechanics, Perg a m o n P r e s s {1960), pp 199-232 Mindlin, R. D. and G a z i s , D . C . "Strong r e s o n a n c e s of r e c t a n g u l a r A T - c u t q u a r t z p l a t e s ' , P r o c e e d i n g s 4th US National C o n g r e s s of Applied M e c h a n i c s , pp 305-310
4
T i e r s t e n , H . F . ' T h i c k n e s s v i b r a t i o n s of p i e z o e l e c t r i c p l a t e s ' , JASA, Vol 35 (1963),No 53
5
Koga, I. ' T h i c k n e s s v a r i a t i o n s of p i e z o e l e c t r i c o s c i l l a : ring c r y s t a l s ' , P h y s i c s Vol 3 (1932) No 70
6
Ref 2, Equ. (9.77)
7
W a r n e r , A . W . , O n o e , M. and Coquin, G.A. ' D e t e r m i n a tion of e l a s t i c and p i e z o e l e c t r i c c o n s t a n t s for c r y s t a l s in c l a s s (3m),JASA, Vol 42 (1967) No 1223
8
Onoe, M. T i e r s t e n , H . F . a n d M e i t z l e r , A . H . 'Shift in the location of r e s o n a n t f r e q u e n c i e s c a u s e d by l a r g e e l e c t r o m e c h a n i c a l coupling in t h i c k n e s s mode r e s o n a t o r s ' , JASA, Vol 35 (1963), No 36
9
M e i t z l e r , A . H. and Sittig, E . K . ' C h a r a c t e r i z a t i o n of p i e z o e l e c t r i c t r a n s d u c e r u s e d in u l t r a s o n i c d e v i c e s o p e r a t i n g above 0.1GHz, J o u r n a l of Applied P h y s i c s (September 1969)
10
F o s t e r , N. F. Coquin, G. A., Rosgonyi, G. A. and Vanatta, F . A . ' C a d m i u m sulphide and zinc oxide thin f i l m s t r a n s d u c e r s ' , IEEE T r a n s a c t i o n s - - S o n i c s and u l t r a s o n i c s , Vol 15 (1968), No 28 ULTRASONICS J a n u a r y 1970
23
The v i b r a t i o n s of p i e z o e l e c t r i c p l a t e s a r e g o v e r n e d by the l i n e a r p i e z o e l e c t r i c equations which may contain a r a t h e r l a r g e n u m b e r of e l a s t i c , p i e z o e l e c t r i c and d i e l e c t r i c constants. But under c e r t a i n v e r y r e s t r i c t i v e but p r a c t i c a l l y i m p o r t a n t c i r c u m s t a n c e s , p i e z o e l e c t r i c coupling coefficients o c c u r naturally in the solutions of the equations, a different coefficient a r i s i n g in each p a r t i c u l a r c a s e . Under m o r e g e n e r a l c i r c u m s t a n c e s many of the m a t e r i a l c o n s t a n t s r e main in the solution, and the utility or significance of any of the coupling c o e f f i c i e n t s a p p e a r s questionable. The solution
of the p r o b l e m of the t h i c k n e s s v i b r a t i o n s of an a r b i t r a r i l y a n i s o t r o p i c plate is used in d e m o n s t r a t i n g the significance and l i m i t a t i o n s of the coupling coefficient for v a r i o u s types of a n i s o t r o p y . In many of the i n s t a n c e s when the coupling coefficient t u r n s out to be useful, the f r e q u e n c y equation shows that the o v e r t o n e r e s o n a n c e s a r e not i n t e g r a l multiples of the fundamental. The d e p a r t u r e f r o m an i n t e g r a l multiple r e l a t i o n s h i p depends on the coupling coefficient only, and consequently, the coupling coefficient can be d e t e r mined from simple resonance measurements.
This is e s s e n t i a l l y the text of a talk d e l i v e r e d at the s e s s i o n on t r a n s d u c e r evaluation (Session I) of the 1968 IEEE U l t r a s o n i c s Symposium. The o t h e r f o r m a l talks at the s e s s i o n w e r e p r e s e n t e d by A. J. Bahr of the Stanford R e s e a r c h I n s t i tute, Menlo P a r k , C a l i f o r n i a and T. R. M e e k e r of Bell T e l e phone L a b o r a t o r i e s , Inc., Allentown, P e n n s y l v a n i a . Different a s p e c t s of the s u b j e c t w e r e d i s c u s s e d in the t h r e e talks. In the panel d i s c u s s i o n held a f t e r w a r d s , the t h r e e p r e s e n t a t i o n s w e r e d i s c u s s e d and c o r r e l a t e d in s o m e detail. It also s e r v e d a s a f o r u m for people to b r i n g up c e r t a i n a s p e c t s of the subject that w e r e o m i t t e d in all t h r e e l e c t u r e s and p r e s e n t a n u m b e r of d i f f e r e n t points of view. Both the panel m e m b e r s and m e m b e r s of the audience p a r t i c i p a t e d in the d i s c u s s i o n . During the c o u r s e of the d i s c u s s i o n it b e c a m e i n c r e a s i n g l y a p p a r e n t that people who attended the s e s s i o n held widely d i v e r g e n t points of view.
The s t r a i n - m e c h a n i c a l d i s p l a c e m e n t r e l a t i o n s
This p a p e r is devoted to a d i s c u s s i o n and d e m o n s t r a t i o n of the fact that in g e n e r a l a l a r g e n u m b e r of e l a s t i c , p i e z o e l e c t r i c and d i e l e c t r i c c o n s t a n t s influence the behavior of a vibr a t i n g p i e z o e l e c t r i c plate; and that, in consequence, the utility o r s i g n i f i c a n c e of any of the defined coupling c o e f f i c i e n t s a p p e a r s questionable. However, under c e r t a i n very r e s t r i c tive but p r a c t i c a l l y i m p o r t a n t c i r c u m s t a n c e s , p i e z o e l e c t r i c coupling c o e f f i c i e n t s o c c u r naturally in the solutions of the equations--a d i f f e r e n t coefficient a r i s i n g in each p a r t i c u l a r c a s e - - a n d under such c i r c u m s t a n c e s the c o e f f i c i e n t s a r e significant and useful. The solution of the p r o b l e m of the t h i c k n e s s v i b r a t i o n s of an a r b i t r a r i l y a n i s o t r o p i c p i e z o e l e c t r i c plate is u s e d in d e m o n s t r a t i n g the s i g n i f i c a n c e and l i m i t a t i o n s of the e l e c t r o m e c h a n i c a l coupling coefficient for v a r i o u s types of a n i s o tropy. The r e l a t i o n of the t h i c k n e s s v i b r a t i o n solution to the behavior of thin film t h i c k n e s s t r a n s d u c e r s is d i s c u s s e d .
S k l = ~/2 (Uk, 1 + Ul, k)
1.3
The e l e c t r i c f i e l d - e l e c t r i c potential r e l a t i o n s Ek = --~, k
1.4
The linear, p i e z o e l e c t r i c constitutive r e l a t i o n s E
Tij ~- CijklSkl -- ekijEk' s D i = eiklSkl + ~ ikEk ,
1.5
w h e r e Ti,j , u,, D~, S,,, Ej a r e the c o m p o n e n t s of s t r e s s , m e c h . j 1 ~j anical d i s p m c e m e n t , e I e c t r i c d i s p l a c e m e n t , s t r a i n and e l e c t r i c field, r e s p e c t i v e l y ; p and ~p a r e the m a s s d e n s i t y and the e l e c t r i c potential, r e s p e c t i v e l y ; and C~jkl,_ eki], ~k_ a r e the ~ _ e l a s t i c , p i e z o e l e c t r i c , and diele t r i c constants, r e s p e c t i v e l y . Throughout this p a p e r we employ C a r t e s i a n t e n s o r notation 1, the s u m m a t i o n convention for r e p e a t e d t e n s o r indices, the dot notation f o r d i f f e r e n t i a t i o n with r e s p e c t to t i m e and the convection that an index, say j, p r e c e d e d by a c o m m a denotes d i f f e r e n t i a t i o n with r e s p e c t to the s p a c e coordinate xj. The s u r f a c e s of the plate a r e t r a c t i o n f r e e and have p o r t i o n s which a r e coated with p e r f e c t l y conducting i n f i n i t e s i m a l l y thin e l e c t r o d e s with which a s t e a d y - s t a t e d r i v i n g voltage is applied, and o t h e r p o r t i o n s which a r e u n e l e c t r o d e d . D i e l e c t r i c c o n s t a n t s a r e a s s u m e d such that e l e c t r i c fields outside the u n e l e c t r o d e d portion of the plate a r e negligible. Hence the boundary conditions may be w r i t t e n Tnj : 0 o n S
1.6
~0 = ~Ooe-iwt on D n = 0 on S,
1.7
and
I.
THE EQUATIONS
OF PIEZOELECTRICITY
The small vibrations of piezoelectric plates--or any other piezoelectric body--are governed by the linear piezoelectric equations, which consist of the following: The stress equations of motion Tij, i ~ Pfij
1. 1
The c h a r g e equation of e l e c t r o s t a t i c s Di, i = 0
1.2
* P r o f e s s o r H. F. T i e r s t e n , R e n s s e l a e r P o l y t e c h n i c Institute, Troy, New York 12181
w h e r e ~0o is the s t e a d y - s t a t e d r i v i n g potential and ~ is the driving f r e q u e n c y . Note that the 22 equations in 1 . 1 - 1 . 5 can r e a d i l y be r e d u c e d to four equations in the four dependent v a r i a b l e s u, and ¢Z, by substituting 1. 3 and 1.4 in 1.5 which may then beJ substituted in 1.1 and 1.2 to yield c ~jklUk, li + ekij ~0, ki = Pfij,
1.8
s eiklUk, li -- £ ik ~p, ki = 0.
1.9
Similarly, the boundary conditions 1. 6 and 1.7 can also be e x p r e s s e d in t e r m s of uj and ~o. Equations 1.8 and 1.9 c l e a r l y indicate that the e l a s t i c conE . . stunts c i~kl, the p~ezoelectr~c c o n s t a n t s elk 1 and the d i e l e c t r i c c o n s t a n t s ~ k influence the b e h a v i o r of a vibrating ULTRASONICS J a n u a r y 1970
19
p i e z o e l e c t r i c body. If we i n t r o d u c e the c o m p r e s s e d m a t r i x notation 2 in place of the t e n s o r notation, the a r r a y s of e l a s tic, p i e z o e l e c t r i c and d i e l e c t r i c c o n s t a n t s , r e s p e c t i v e l y , for an a r b i t r a r i l y a n i s o t r o p i c m a t e r i a l may be w r i t t e n in the form
C~q =
Cll
C12
C13
C14
C15
C16
C12
C22
C23
C24
C25
C26
C13
C22
C33
C34
C35
e36
C14
C24
C34
C44
C45
C46
C15
C25
C35
C45
C55
C56
C16
C26
C36
C46
C56
C66
el2
et3
el4
e15
el6 1 ~26 / ! e36 /
-ell eip =
e21
e22
e23
e24
e25
e31
e32
e33
e34
e35
~ll
~12
~1:~
£12
(22
~23 I
£13
E23
~33J
I
~
~
Electrode.~
~.T
2h
J
2t
F i g 1 Bounded p i e z o e l e c t r i c plate
1. 10
G r e e k index is not to be s u m m e d and u i and ~) a r e not functions of t i m e b e c a u s e all dependent v a r i a b l e s have the t i m e d e p e n d e n c e e -iwt, which has been o m i t t e d in 2.4 2.7. The e l e m e n t a r y t h i c k n e s s solution s a t i s f y i n g 2 . 4 - - 2 . 7 t u r n s out to be adequate for the bounded plate in many p r a c t i c a l c i r c u m s t a n c e s . P r e c i s e d e l i n e a t i o n s of the range of a p p l i c a bility of the t h i c k n e s s solution t o g e t h e r with the r e a s o n s for the applicability have been p r o v i d e d by R. D. Mindlin~ under a v a r i e t y of c i r c u m s t a n c e s . In this p a p e r we will suppose the t h i c k n e s s solution to be adequate for the bounded p l a t e s in which we a r e i n t e r e s t e d and not d i s c u s s this point any further.
Equations 1.10 show that for an a r b i t r a r i l y a n i s o t r o p i c p i e z o e l e c t r i c body t h e r e a r e 21 independent e l a s t i c c o n s t a n t s , 18 independent p i e z o e l e c t r i c c o n s t a n t s , and 6 independent d i e l e c t r i c c o n s t a n t s , f o r a total of 45 independent m a t e r i a l c o n s t a n t s . This is a r a t h e r l a r g e n u m b e r of independent m a t e r i a l c o n s t a n t s which a r e r e q u i r e d to d e s c r i b e the behavior of an a r b i t r a r i l y a n i s o t r o p i c v i b r a t i n g p i e z o e l e c t r i c body--and in p a r t i c u l a r a plate. Thus it should not be s u r p r i s i n g that a single e l e c t r o m e c h a n i c a l coupling c o e f f i c i e n t cannot d e s c r i b e the r e s o n a n t b e h a v i o r of a v i b r a t i n g p i e z o e l e c t r i c plate in g e n e r a l .
The s t e a d y - s t a t e t h i c k n e s s solution to 2 . 4 - 2 . 7 has been obtained 4 and the condition for r e s o n a n c e given as
2.
is the p i e z o e l e c t r i c a l l y stiffened e l a s t i c constant and V (i) is the phase velocity of the ith (i = 1. 2, 3) wave in the r, th d i r e c t i o n and the /3k( ~)are the n o r m a l i z e d amplitude r a t i o s a s s o c i a t e d with the ith wave. In g e n e r a l , 2.8 is a 3 × 3 t r a n s c e n d e n t a l d e t e r m i n a n t , each t e r m of which is a c o m p l i c a t e d combination of m a t e r i a l c o n s t a n t s and t r i g o n o m e t r i c functions. F o r an a r b i t r a r i l y a n i s o t r o p i c plate the n u m b e r of m a t e r i a l c o n s t a n t s on which the solution depends c o n s i s t s of 6 e l a s t i c c o n s t a n t s , 6 p i e z o e l e c t r i c c o n s t a n t s , and 1 diee l e c t r i c c o n s t a n t for a total of 13 m a t e r i a l c o n s t a n t s . Clearly it would be i m p o s s i b l e to define a meaningful p i e z o e l e c t r i c coupling f a c t o r under such g e n e r a l c i r c u m s t a n c e s .
THE ARBITRARILY ANISOTROPIC P L A T E
A s c h e m a t i c d i a g r a m of a bounded p i e z o e l e c t r i c plate is shown in Fig 1. The plate is f o r c e d into s t e a d y - s t a t e o s c i l lation by an a l t e r n a t i n g voltage applied a c r o s s the s u r f a c e e l e c t r o d e s . In this d i s c u s s i o n the e l e c t r o d e s a r e s u p p o s e d i n f i n i t e s i m a l l y thin. The c o o r d i n a t e axis out of the figure is c o n s i d e r e d to be denoted by a and the extent of the plate in the a - d i r e c t i o n is 2w. The boundary conditions on the s u r f a c e s of the plate may be taken to be T~j
0 and q~=±(0Oe - i w t at x u = ±h,
2.1
Trj
0 and D r
2.2
0 at x r = ±1,
Toj = 0 and D e = 0 at xo
±w.
2.3
Since we a r e c o n c e r n e d with p l a t e s having l a r g e 1/h and w / h r a t i o s , we ignore the m i n o r s u r f a c e s and a s s o c i a t e d bound a r y conditions 2.2 and 2.3, and r e p l a c e the bounded plate shown in Fig 1 by the unbounded plate shown in Fig 2. Cl.early, a solution for the configuration shown in Fig 2 depends on the x~ c o o r d i n a t e only. Such a solution is called a t h i c k n e s s solution and s a t i s f i e s 1.8 and 1.9 with x u spatial d e p e n d e n c e and e - i w t t i m e d e p e n d e n c e only, which may then be w r i t t e n E C~kj~,Uk, ~,~, + e~,~,jCp, ~,~, 4-pw2uj = 0
2.4
s e ~ , k U k , u~ -- ~ , q ) ,
2.5
wh cob ujku' ~ - ) - cos v ( i )
eu~,je~,~, k wh~ s sin ~ {
} = 0,
~,v
2.8
where c~,jk ~,
E ci~jk ~ ~ ei,~,jel~t,k/¢~!~'
2.9
Although 2.8 is v e r y c o m p l i c a t e d in general, in many c a s e s it s i m p l i f i e s c o n s i d e r a b l y when the m a t e r i a l is even slightly l e s s a n i s o t r o p i c and then a meaningful p i e z o e l e c t r i c coupling f a c t o r can be defined. Some c a s e s in which 2.8 s i m p l i f i e s c o n s i d e r a b l y and a coupling f a c t o r can be defined and one in which it does not and a coupling f a c t o r cannot be defined a r e d i s c u s s e d in the next section. Note that when the p i e z o e l e c t r i c c o n s t a n t s vanish (or a r e ignored), 2.8 f a c t o r s and r e d u c e s to ~h cos V~~
~ O.
2. 10
This limiting c a s e c o r r e s p o n d s to Koga's 5 solution. u~
0
and the boundary conditions in 2. 1 with the s a m e d e p e n dences, which may then be w r i t t e n CujkuUk, u + eu ujc°, u = 0 at xu = ±h,
2.6
~p = ±¢oo at x u = ±h.
2.7
In 2 . 4 - - 2 . 7 we have i n t r o d u c e d the convention that a r e p e a t e d 20
I~ ~i)
ULTRASONICS J a n u a r y 1970
Note that E~jk~, defined in 2.9 is the a p p r o p r i a t e p i e z o e l e c t r i c a l l y s t i f f e n e d e l a s t i c constant and not C~jkv. When a g r e a t deal of s y m m e t r y is p r e s e n t it s o m e t i m e s t u r n s out that c~D = g. However, this is not usually the c a s e and when £D .~ ~, ~ is the a p p r o p r i a t e p i e z o e l e c t r i c a l l y stiffened e l a s tic constant. Many e x i s t i n g works on this subject, including the IRE Standards, a r e not c l e a r on this point. G. A. Coquin a has shown that the v e r y c o m p l i c a t e d t r a n s cendental f r e q u e n c y equation in 2.8 can be w r i t t e n in the
m o r e c o m p a c t and for many p u r p o s e s m o r e i l l u m i n a t i n g form ~h
wh
COS--COS--COS-V(l) V(2)
Etectrodes
I
wh V(31
2h ~
2.11
= O,
k ~ n ) - ~ -~ - t a n v - - ~ - ~ - - I
~= 1
where 2 _ (~j(n)ep pj )2 k(n) p(V(n))2 ~ ~
2. 12
2.12 shows that in g e n e r a l the k ( n ) coefficients a r e c o m p l i c a t e d functions of the p i e z o e l e c t r i c eigenwave v e l o c i t i e s for p r o p a g a t i o n in the d i r e c t i o n ~, the a s s o c i a t e d n o r m a l i z e d e i g e n v e c t o r s and s o m e of the p i e z o e l e c t r i c c o n s t a n t s .
3.
THE INFLUENCE OF MATERIAL SYMMETRY
In this s e c t i o n we c o n s i d e r the a p p l i c a t i o n of 2 . 8 [or (2.11)] to the p r i n c i p a l cuts of the p o l a r i z e d c e r a m i c s ( c r y s t a l c l a s s 6mm), a r o t a t e d Y - c u t of q u a r t z (effective c l a s s 2) and the p r i n c i p a l cuts of l i t h i u m niobate ( c l a s s 3m). The r e s u l t s show that for all of the s y m m e t r i e s c o n s i d e r e d , except for those of two of the cuts (the X - c u t and Y-cut) of l i t h i u m niobate, the t r a n s c e n d e n t a l equation, 2 . 8 [or 2.11] s i m p l i f i e s c o n s i d e r a b l y and a p i e z o e l e c t r i c coupling coefficient o c c u r s n a t u r a l l y in that t h i c k n e s s s o l u t i o n which is d r i v e n e l e c t r i cally by the s y s t e m shown in F i g 2.
Fig 2 Unbounded piezoelectric plate
equation in 3 . 2 g o v e r n s the set of p i e z o e l e c t r i c s h e a r m o d e s which a r e d r i v e n e l e c t r i c a l l y by the s y s t e m shown in F i g 2. Thus, in this s p e c i a l c a s e the p i e z o e l e c t r i c coupling f a c t o r k26 o c c u r s n a t u r a l l y in the p i e z o e l e c t r i c t h i c k n e s s solution and is s i g n i f i c a n t and useful.
3B. Polarized ceramics (class 6ram) The a r r a y s of m a t e r i a l c o e f f i c i e n t s for a f e r r o e l e c t r i c c e r a m i c poled in the x 3 - d i r e c t i o n may be w r i t t e n in the f o r m Cll
C12
C12
Cll
C13
C13
C13
C33
0
0
0
3A. Rotated Y-Cut quartz (effective class 2)
0
0
The a r r a y s of m a t e r i a l c o e f f i c i e n t s for r o t a t e d Y - c u t q u a r t z with x~ the diagonal axis may b e w r i t t e n in the f o r m
0
0
Cll
C12
C12 C~q ~- C13
eip =
C13
C14
0
0 "
C22
C23
C23
C33
C24
0
0
C34
0
0
C14
C24
C34
C44
0
0
0
0
0
0
c55
c55
0
0
0
0
C56
C66
e!l
el2 0
e13 0
e14 0
0 e25
0 1 e26
0
0
0
e35
e36 j
E22
E2
E23
£33_J
I eip =
¢.s.= ~j
wh cos ~
wh = 0, cos V-ySy = 0,
3.1
3.2
3.3
w h e r e k26 is the t h i c k n e s s - s h e a r p i e z o e l e c t r i c coupling factor for r o t a t e d Y-cut q u a r t z and is given by k~ 6 = e33/c33~:33.2 ~
0
0
0
0
0
0
0
C44
0
0
0
0
c4~
0
0
0
0
C6,
i
0 0
0 0
0 0
0 e15
e15 0
e31
e31
e33
0
0
0 e2~ 0
e~-~ 0 0
003i ~3
il
3.5
T h i s m a t e r i a l h a s c o n s i d e r a b l y m o r e s y m m e t r y than r o t a t e d Y - c u t - - o r even u n r o t a t e d for that m a t t e r - - q u a r t z , a s can b e s e e n f r o m the g r e a t e r n u m b e r of z e r o s in 3 . 5 than in 3 . 1 . When u = 3 and ~ = 1, 2 . 8 [or 2.11] f a c t o r s into the t h r e e t r a n s c e n d e n t a l equations 4
T h i s m a t e r i a l (effective c l a s s 2) is s t i l l highly a n i s o t r o p i c . However, even with this r e l a t i v e l y s m a l l amount of s y m m e t r y , for ~ = 2 and r = 1, 2 . 8 [or 2.11] f a c t o r s into the t h r e e t r a n s c e n d e n t a l equations 4 ~h wh V(1) ~(1)k~6'
0
_
cos
tan
0
C66 = 1/2 ( C l l -- C12),
0 £~j ~-
C~q =
C13
3.4
The two t r a n s c e n d e n t a l equations in 3. 3 g o v e r n two s e t s of p u r e l y e l a s t i c m o d e s which, ideally, cannot be d r i v e n e l e c t r i c a l l y by the s y s t e m shown in F i g 2. The t r a n s c e n d e n t a l
wh wh = 0 , cos - = 0, V (1) V ~2)
wh wh , tan ~ _ -v~3~ V TMk.2
3.6
3. 7
L
w h e r e k t is p r e s e n t l y called the l a t e r a l l y c l a m p e d thickness coupling f a c t o r for the p o l a r i z e d c e r a m i c and i s given by
k•
2 s e33,/~33~33 •
3.8
Obviously k t should m o r e p r o p e r l y be denoted k33 and called the t h i c k n e s s - e x t e n s i o n a l coupling f a c t o r . C l e a r l y , 3 . 6 a r e a n a l o g o u s to 3 . 3 and g o v e r n the p u r e l y e l a s t i c m o d e s and 3 . 7 is analogous to 3 . 2 and g o v e r n s the p i e z o e l e c t r i c m o d e s . The e n t i r e d i s c u s s i o n following 3 . 4 with obvious m o d i f i c a tions, n a t u r a l l y a p p l i e s in t h i s c a s e a l s o . When ~ = 1 and r = 3, 2 . 8 [or 2.11] f a c t o r s into t h r e e t r a n s c e n d e n t a l equations. Two of the t h r e e equations a r e the same ULTRASONICS J a n u a r y 19'/0
21
a s t h o s e in 3 . 6 and g o v e r n purely e l a s t i c m o d e s and the t h i r d t a k e s the f o r m wh ~_ wh tan - V (3) V (3)k~25'
3.9
w h e r e k~5 is the t h i c k n e s s - s h e a r p i e z o e l e c t r i c coupling f a c t o r f o r the c e r a m i c and i s given by 2 k~5 = e ~ 5 / c- 5 5 ~S1 .
3.10
C l e a r l y the e n t i r e d i s c u s s i o n following 3 . 4 , with obvious m o d i f i c a t i o n s , a p p l i e s in t h i s c a s e a l s o . Note that V m) a p p e a r i n g in 3. "/and V (3) a p p e a r i n g in ~. 9 a r e d i f f e r e n t q u a n t i t i e s . The one in 3 . 7 is the velocity of an e x t e n s i o n a l wave propagating in the x 3 - d i r e c t i o n while the one in 3 . 9 is the velocity of an x 3 - p o l a r i z e d s h e a r wave propagating in the X l - d i r e c t i o n . Thus, again in t h e s e s p e c i a l c a s e s p i e z o e l e c t r i c coupling f a c t o r s o c c u r n a t u r a l l y in the p i e z o e l e c t r i c t h i c k n e s s solution and a r e s i g n i f i c a n t and useful.
The a r r a y s of m a t e r i a l c o e f f i c i e n t s for lithium niobate with x~ n o r m a l to the m i r r o r plane may be w r i t t e n in the f o r m -Cll
C12
C13
C[4
0
0"
C12
Cll
C13
--C14
0
0
C13
C13
C33
0
0
0
C14
--C14
0
C44
0
0
0
0
0
0
C44
C14
0
0
0
0
c~4
c6e
0 eip = --e22 e31
0 e22 e31
0 0 e33
c~),
-
0 e~5 0
el5 0 0
03h
V(2)
k~2 ) ~
3.14
V(3)
tan V (2) ~- k~3~ ~
(~3~nJ k~n ) =
+ ~3 (n) ~ e~p 2 evp 3 ~ p(V (n))2£su~
~33
3.11 Although the e l a s t i c and d i e l e c t r i c m a t r i c e s a r e identical with t h o s e of (unrotated) q u a r t z , the p i e z o e l e c t r i c m a t r i x is c o m p l e t e l y d i f f e r e n t and much m o r e c o m p l i c a t e d . As a c o n s e q u e n c e the s i m p l i f i c a t i o n s in 2 . 8 [or 2.11] that o c c u r in the c a s e of (rotated) Y - c u t q u a r t z do not o c c u r h e r e . When ~ = 1 or 2, 2 . 8 d o e s not f a c t o r c o m p l e t e l y and t a k e s the f o r m 03h wh ---- cosV(D V (~)
0
0
0
p(2)
p(3)
0
Q(~)
Q(3)
= 0,
3.12
where p(n) =
03h ~2(n)~1]221
e~2 ~
Q~) =
S
~ + ~3(n)~12321/
~(n)~
~ 2
~n)uv~zu
~2
+ f l3~ n_) e ~
sin
wh - V (n)
p(2)Q(3)
_Q(2)p(3)
= O,
wh v(n) - -
ULTRASONICS J a n u a r y 1970
~ sin ~h ~uu3/ v(n) ~
3.17
3.18
U n d e r t h e s e c i r c u m s t a n c e s the a f o r e m e n t i o n e d coupling of the two p i e z o e l e c t r i c standing w a v e s i s negligible. This s i m p l i f i c a t i o n o c c u r s b e c a u s e for high o v e r t o n e s 03 is very l a r g e and o c c u r s only in the c o e f f i c i e n t of the cosine t e r m s in e i t h e r 3.12 or 3 . 1 7 a s can be s e e n f r o m 3.13. It should a l s o be noted that if the coupling i s low, 3 . 1 5 can usually be a p p r o x i m a t e d by k~n ) v ( n ) tan 03h. ~ 1 n = 2, 3, 03h V (n)
3.19
u n l e s s the m a t e r i a l c o n s t a n t s a r e such V (2) and V (3) a r e not v e r y d i f f e r e n t or b e a r s o m e unusual r e l a t i o n s h i p to each o t h e r . In addition, for high coupling m a t e r i a l s , the m a t e r i a l c o n s t a n t s may be such that t h e r e a r e a few o r i e n t a t i o n s f o r which 3 . 1 5 can be a p p r o x i m a t e d by 3.19. However, such o r i e n t a t i o n s a r e b e s t found a n a l y t i c a l l y a f t e r the fundamental material constants are determined from appropriate measurements. In any event the k ( n ) a r e not a s i m p l e combination of the m a t e r i a l c o n s t a n t s a s is, e.g., k26 in 3 . 4 but a r e given by the c o m p l i c a t e d c o m b i n a t i o n of the m a t e r i a l c o n s t a n t s shown in 3.16. Although the e n t i r e p r e v i o u s d i s c u s s i o n in this p a r a g r a p h has b e e n for an X - c u t and a Y - c u t of m a t e r i a l s in c r y s t a l c l a s s (3m), it a p p l i e s without e s s e n t i a l change under a r b i t r a r i l y a n i s o t r o p i c c i r c u m s t a n c e s , i.e., when 2.11 holds in place of 3 . 1 5 and n = 1, 2, 3 in 3 . 1 8 and 3.19.
+ ~ (n)~ ~ wh wh ~3 u33Y]v(n ) c o s ~ v(n~
epp3 ~g, (0 ~n) cup2 + o -3( n )
22
v(n------~ c o s
3.16
At this point it should be noted that although the l o w e r m o d e s m u s t be d e t e r m i n e d analytically f r o m the r o o t s of 3 . 1 5 [or 3.17], the high o v e r - t o n e s a r e e s s e n t i a l l y e l a s t i c and may be a p p r o x i m a t e d by the r o o t s of
--el21
e11
0
3. 15
where
wh cos-=0, n=2,3. v(n) eij =
wh
t a n - - v(3) - 1,
which is obtained f r o m 3.12 along with 3.14. A s a l r e a d y noted, 3 . 1 4 g o v e r n s the u n d r i v e n purely e l a s t i c m o d e s . 3 . 1 5 [or 3.17] g o v e r n s the p i e z o e l e c t r i c t h i c k n e s s m o d e s which a r e d r i v e n e l e c t r i c a l l y by the a r r a n g e m e n t shown in Fig 2. 3 . 1 5 [or 3.17] s h o w s that the two p i e z o e l e c t r i c standing w a v e s a r e coupled at the conducting s u r f a c e s for t h e s e p r i n cipal o r i e n t a t i o n s of the c r y s t a l plate. When v = 1, both of the coupled w a v e s a r e s h e a r waves, and when ~ = 2, one of the coupled w a v e s i s a s h e a r wave and the o t h e r an e x t e n sional wave. Thus it is c l e a r that the s i m p l i f i e d p i e z o e l e c t r i c f r e q u e n c y equation 3 . 2 , 3. "/, with one p i e z o e l e c t r i c coupling constant, which o c c u r r e d in the o t h e r i n s t a n c e s , d o e s not o c c u r for the X - c u t (v = 1) or Y-cut (v = 2) of m a t e r i a l s in c r y s t a l c l a s s (3m).
=
%s = ~/'~ ( c ~
wh cos-= 0, V(~)
f r o m which it is c l e a r that 3 . 1 5 i s c o n s i d e r a b l y s i m p l e r than
3C. Lithium niobate, lithium tantalate (class 3m)
CE
Thus it is c l e a r that a p a r t f r o m the f a c t o r in the u p p e r l e f t hand c o r n e r of the d e t e r m i n a n t in 3.12, which f a c t o r g o v e r n s the u n d r i v e n purely e l a s t i c (extensional when ~ = 1 and s h e a r when ~ = 2) m o d e s , the d e t e r m i n a n t r e m a i n s e x t r e m e l y c o m plicated, As a l r e a d y noted in Sec. 2, 2.11 i s equivalent r o 2 . 8 but is m o r e c o m p a c t and m o r e r e a d i l y i n f o r m a t i v e f o r many p u r p o s e s than 2.8. Consequently, we p r e s e n t the r e s u l t s obtained f r o m 2.11 f o r the p r e s e n t s y m m e t r y in o r d e r to show the s i m p l i f i c a t i o n s that o c c u r in at l e a s t one n o n - s i m p l e i n s t a n c e . Thus, when ~ = 1 or 2, 2.11 f a c t o r s into two t r a n s c e n d e n t a l equations of the f o r m
3.13
When v = 3, 2 . 8 [or 2. 111 f a c t o r s into t h r e e t r a n s c e n d e n t a l equations. Two of the t h r e e equations a r e the s a m e a s t h o s e in 3.6 and g o v e r n u n d r i v e n purely e l a s t i c m o d e s and the
t h i r d m a y be w r i t t e n in the f o r m wh wh tan-- - - , V (3) V(3 Jk233
3.20
w h e r e k33 i s the t h i c k n e s s - e x t e n s i o n a l p i e z o e l e c t r i c coupling f a c t o r f o r a Z - c u t of a m a t e r i a l in c l a s s (3m) and is given by k33
2 S = e33/c33633"
3.21
Thus, although the g e n e r a l f r e q u e n c y equation, 2 . 8 [or 3.11], d o e s not s i m p l i f y a p p r e c i a b l y for a n X - c u t or a Y - c u t of m a t e r i a l s in c l a s s (3m), it d o e s s i m p l i f y f o r a Z - c u t and r e s u l t s in the u s u a l s i m p l i f i e d f r e q u e n c y equation, in which a w e l l - d e f i n e d e l e c t r o m e c h a n i c a l coupling f a c t o r (in t h i s ins t a n c e k33) o c c u r s n a t u r a l l y in the t h i c k n e s s solution and is s i g n i f i c a n t and useful. Note t h a t as a c o n s e q u e n c e of the f o r m of the s i m p l i f i e d f r e q u e n c y equation, o v e r t o n e r e s o n a n t f r e q u e n c i e s a r e not i n t e g r a l m u l t i p l e s of the f u n d a m e n t a l . T h i s e n a b l e s the d e t e r m i n a t i o n 8 of the p i e z o e l e c t r i c coupling f a c t o r f r o m s i m p l e m e a s u r e m e n t s of the f u n d a m e n t a l r e s o n ant f r e q u e n c y and at l e a s t one o v e r t o n e , when an equation of the f o r m of 3 . 2 0 i s valid. 4.
TRANSDUCER CONSIDERATIONS
The e n t i r e f o r e g o i n g d i s c u s s i o n in t h i s p a p e r a p p l i e s to t h i c k n e s s r e s o n a t o r s only and not to t h i c k n e s s t r a n s d u c e r s . However, m o s t , i m p o r t a n t q u a l i t a t i v e i n f o r m a t i o n c o n c e r n i n g the u s e of t h e s e t h i c k n e s s r e s o n a t o r s a s t r a n s d u c e r s can be d e t e r m i n e d f r o m the s i m p l e r r e s o n a t o r a n a l y s i s we have d i s c u s s e d . M o r e o v e r , r e s o n a t o r e x p e r i m e n t s a r e s i m p l e r to p e r f o r m and a r e u s u a l l y u s e d for m e a s u r i n g m a t e r i a l p r o p e r t i e s w h e n e v e r p o s s i b l e . But thin film t r a n s d u c e r s cannot s u p p o r t t h e m s e l v e s and, consequently, cannot be m e a s u r e d a s r e s o n a t o r s but m u s t b e m e a s u r e d a s t r a n s ducers. F i g 3 show a t h i n film t h i c k n e s s t r a n s d u c e r bonded to a d e l a y m e d i u m . When the p r o p e r t i e s of such a t r a n s d u c e r c o n f i g u r a t i o n a r e to b e m e a s u r e d , the a f o r e m e n t i o n e d r e s o n a t o r e i g e n s o l u t i o n d i s c u s s e d in Secs. 2 and 3 cannot be used. In o r d e r to c o r r e l a t e s u c h m e a s u r e m e n t s one m u s t u s e t h e o r e t i c a l c u r v e s obtained f o r a bonded t r a n s d u c e r 9. Howe v e r , s u c h a p r o c e d u r e i s not f e a s i b l e f o r a highly a n i s o t r o p i c t r a n s d u c e r with l a r g e p i e z o e l e c t r i c coupling, a s should b e c l e a r f r o m the d i s c u s s i o n of the highly a n i s o t r o p i c r e s o n a t o r p r e s e n t e d in Secs. 2 and 3. N e v e r t h e l e s s , when the t r a n s d u c e r a s a r e s o n a t o r h a s sufficient s y m m e t r y f o r the s i m p l i fied f r e q u e n c y equation, e.g., 3 . 2 0 to b e valid exactly, the t h e o r e t i c a l c u r v e s obtained f r o m an e l e m e n t a r y equivalent c i r c u i t t r a n s d u c e r a n a l y s i s 9 c a n b e u s e d to find the p i e z o e l e c t r i c coupling f a c t o r . When highly a n i s o t r o p i c o r i e n t a t i o n s (or m a t e r i a l s ) a r e used, even when the p i e z o e l e c t r i c coupling is s m a l l and the a p p r o x i m a t e equation, 3.19, i s valid f o r the r e s o n a t o r , the a f o r e m e n t i o n e d e l e m e n t a r y equivalent c i r c u i t t r a n s d u c e r a n a l y s i s should not b e u s e d 1° to find the coupling f a c t o r . However, for highly a n i s o t r o p i c m a t e r i a l s , e v e n with l a r g e coupling, a fe___wv e r y special--and p r a c t i c a l l y u s e f u l - - o r i e n t a t i o n s m a y e x i s t , for which the a f o r e m e n t i o n e d e q u i v a l e n t c i r c u i t t r a n s d u c e r a n a l y s i s a p p l i e s . But t h e s e o r i e n t a t i o n s c a n be found only by ignoring the coupling f a c t o r a p p r o a c h and s y s t e m a t i c a l l y finding the f u n d a m e n t a l m a t e r i a l c o n s t a n t s 7, and then d e t e r m i n i n g t h o s e few particular orien-t a t i o n s a n a l y t i c a l l y . Some of t h o s e o r i e n t a t i o n s can p o s s i b l y b e found by a c c i d e n t also. It should b e m e n t i o n e d at t h i s point that m e a s u r e m e n t s by t h e m s e l v e s , without any a n a l y t i c a l u n d e r s t a n d i n g , can s o m e t i m e s lead to confusion. F o r e x a m p l e , c o n s i d e r a n X - c u t of a c r y s t a l in c l a s s (3m) bonded to e a c h end of an i s o t r o p i c delay m e d i u m . As noted in Sec. 3C, a t h i c k n e s s d r i v i n g voltage applied to the equivalent r e s o n a t o r e x c i t e s a f u n d a m e n t a l p i e z o e l e c t r i c t h i c k n e s s - s h e a r r e s o n a n c e in which two o r t h o g o n a l s h e a r m o t i o n s with d i f f e r e n t p h a s e v e l o c i t i e s a r e coupled a t the conducting e l e c t r o d e d s u r f a c e s . T h i s m e a n s that when such a plate i s bonded to each end of a n i s o t r o p i c delay m e d i u m , the d r i v i n g t r a n s d u c e r e x c i t e s a n e l l i p t i c a l l y p o l a r i z e d s h e a r - w a v e in the i s o t r o p i c delay
F i g 3 T h i n - f i l m t r a n s d u c e r bonded to an i s o t r o p i c delay medium m e d i u m , which i s then d e t e c t e d by the o t h e r t r a n s d u c e r . However, the t r a n s d u c e r m e a s u r e m e n t s cannot b e c o r r e l a t e d with the c u r v e s obtained f r o m the e l e m e n t a r y equivalent c i r c u i t t r a n s d u c e r a n a l y s i s b e c a u s e the e l e m e n t a r y equivalent c i r c u i t e q u a t i o n s differ f r o m the g o v e r n i n g t r a n s d u c e r t h i c k n e s s equations. 5.
CONCLUSION
W h e n m a t e r i a l s p o s s e s s the a p p r o p r i a t e s y m m e t r y , it is meaningful to d i s c u s s and m e a s u r e e l e c t r o m e c h a n i c a l coupling coefficients. However, when m a t e r i a l s do not p o s s e s s the a p p r o p r i a t e s y m m e t r y , the coupling f a c t o r a p p r o a c h i s not meaningful. In m e a s u r i n g such m a t e r i a l s one m u s t p r o c e e d by f i r s t making the r e q u i s i t e m e a s u r e m e n t s to find the f u n d a m e n t a l m a t e r i a l c o n s t a n t s and then a n a l y t i c a l l y finding the p r a c t i c a l o r i e n t a t i o n s , for which the e l e m e n t a r y e q u i v a l e n t c i r c u i t t r a n s d u c e r a n a l y s i s holds.
ACKNOWLEDGEMENT The a u t h o r w i s h e s to t h a n k G. A. Coquin, A. H. M e i t z l e r , and E. K. Sittig, all of Bell Telephone L a b o r a t o r i e s , for valuable discussions. T h i s work was s u p p o r t e d in p a r t by the Office of Naval R e s e a r c h u n d e r C o n t r a c t No. N 0 0 0 1 4 - 6 7 - A - 0117-0007.
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J e f f r e y s , H. ' C a r t e s i a n t e n s o r s ' , C a m b r i d g e U n i v e r s i t y P r e s s , New York (1952)
2
T i e r s t e n , H . F . ' L i n e a r p i e z o e l e c t r i c plate v i b r a t i o n s ' , Chap 7, Sec 1, P l e n u m P r e s s , New York (1969)
3
Mindlin, R.D. ' T h i c k n e s s - S h e a r and f l e x u r a l v i b r a t i o n s of c r y s t a l p l a t e s ' , J o u r n a l of Applied P h y s i c s , Vol 22 (1951), No 316 Mindlin, R.D. ~qCaves and v i b r a t i o n s in isotropic, elastic plates', Structural Mechanics, Perg a m o n P r e s s {1960), pp 199-232 Mindlin, R. D. and G a z i s , D . C . "Strong r e s o n a n c e s of r e c t a n g u l a r A T - c u t q u a r t z p l a t e s ' , P r o c e e d i n g s 4th US National C o n g r e s s of Applied M e c h a n i c s , pp 305-310
4
T i e r s t e n , H . F . ' T h i c k n e s s v i b r a t i o n s of p i e z o e l e c t r i c p l a t e s ' , JASA, Vol 35 (1963),No 53
5
Koga, I. ' T h i c k n e s s v a r i a t i o n s of p i e z o e l e c t r i c o s c i l l a : ring c r y s t a l s ' , P h y s i c s Vol 3 (1932) No 70
6
Ref 2, Equ. (9.77)
7
W a r n e r , A . W . , O n o e , M. and Coquin, G.A. ' D e t e r m i n a tion of e l a s t i c and p i e z o e l e c t r i c c o n s t a n t s for c r y s t a l s in c l a s s (3m),JASA, Vol 42 (1967) No 1223
8
Onoe, M. T i e r s t e n , H . F . a n d M e i t z l e r , A . H . 'Shift in the location of r e s o n a n t f r e q u e n c i e s c a u s e d by l a r g e e l e c t r o m e c h a n i c a l coupling in t h i c k n e s s mode r e s o n a t o r s ' , JASA, Vol 35 (1963), No 36
9
M e i t z l e r , A . H. and Sittig, E . K . ' C h a r a c t e r i z a t i o n of p i e z o e l e c t r i c t r a n s d u c e r u s e d in u l t r a s o n i c d e v i c e s o p e r a t i n g above 0.1GHz, J o u r n a l of Applied P h y s i c s (September 1969)
10
F o s t e r , N. F. Coquin, G. A., Rosgonyi, G. A. and Vanatta, F . A . ' C a d m i u m sulphide and zinc oxide thin f i l m s t r a n s d u c e r s ' , IEEE T r a n s a c t i o n s - - S o n i c s and u l t r a s o n i c s , Vol 15 (1968), No 28 ULTRASONICS J a n u a r y 1970
23