124 BELL SYSTEM TECH NIC A L JOURNA L 



readily shown that the stresses for 6 = 45° are given by the equations ex- 

 pressed in two index symbols 



^38 5 a 



r = 



(145) 



Tz — CizSl 4" Cl3 02 4" C33O3 



r; = Cf4 5l + //14 62 ; £1 = -/?145b + 47rLSuai'] 



Te = cf4 ^5 - /?i4 5i ; £2 = h^'x + 47r[)Su52] 



J,, ^ icn - c^2 ) _^^ . £^ ^ _^^^f^| „ 5^j ^ 4x1/333 53]. 



For a long thin longitudinally vibrating crystal all the stresses are zero 

 except the stress Ti along the length of the crystal. Hence it is more ad- 

 vantageous to use equations which express the strains in terms of the 

 stresses since all the stresses can be set equal to zero except Ti . All the 

 strains are then dependent functions of the strain Si and this only has to 

 be solved for. Furthermore, since plated cjystals are usually used to 

 determine the properties of crystals, and the field perpendicular to a plated 

 surface is zero, the only field existing in a thin crystal will be £3 if the thick- 

 ness is taken along the ^3 or Z axis. Plence the equations that express the 

 strains in terms of the stresses and fields are more advantageous for calcu- 

 lating the properties of longitudinally vibrating crj^stals. By orienting 

 such crystals with respect to the crystallographic axis, all of the elastic 

 constants except the shear elastic constants can.be determined. All of 

 the piezoelectric and dielectric constants can be determined from measure- 

 ments on oriented longitudinally vibrating crystals. 



For such measurements it is necessary to determine the appropriate 

 elastic, piezoelectric, and dielectric constants for a crystal oriented in any 

 direction with respect to the crystallographic axes. We assume that the 

 length lies along the Xi axis, the width along the .T2 axis and the thickness 

 along the Xz axis. Starting with equations of the form 



O t; ^^ Sxjlc(llcC ~l d i jmt-'m 



T (146) 



47r 



k 



