122 BELL SYSTEM TECH NIC A L JOURNA L 



5.4 Piezoelectric Equations for Rotated Axes 



Another application of the tensor equations for rotated axes is in deter- 

 mining the piezoelectric equations of crystals whose length, width, and thick- 

 ness do not coincide with the crystallographic axes of the crystal. Such 

 oriented cuts are useful for they sometimes give properties that cannot be 

 obtained with crystals h'ing along the crystallographic axes. Such proper- 

 ties may be higher electromechanical coupling, freedom from coupling to 

 undesired modes of motion, or low temperature coefficients of frequency. 

 Hence in order to obtain the performance of such crystals it is necessary to 

 be able to express the piezoelectric equations in a form suitable for these 

 orientations. In fact in first measuring the properties of these crystals a 

 series of oriented cuts is commonly used since by employing such cuts the 

 resulting frequencies, and impedances can be used to calculate all the pri- 

 mary constants of the crystal. 



The piezoelectric equations (111) are 



Tkl = CijkfSij — hnkC^n ; Em. = ^TTPmn^ n ~ hmijSij . (HI) 



The first equation is a tensor of the second rank, while the second equation is 

 a tensor of the first rank. If we wish to transform these equations to another 

 set of axes x', y', z', we can employ the tensor transformation equations 



, ^ dx[dx^ ^ dxldxf 

 dxk dX( dxk dx( 



[CukfSn -\~ 2Ci2k(Sl2 -\- 2Cl3t^5'l5 + C22k(S22 



+ 2c23ktS-a + C33ktS3z] - '- —-[hikth + h2k(b2 + hklh] (140) 



axk oxf 



EL = 47r p^ [/3li5i + ^':2 62 + ^isd^] - ^' 



OXm dx„, 



[hmllSl] + lllmuS 12 + 2llml3Sli + Am22«S'22 + 2(1^23^23 "f" hmSiSzs]. 



These equations express the new stresses and fields in terms of the old strains 

 and displacements. To complete the transformation we need to express 

 all quantities in terms of the new axes. For this purpose we employ the 

 tensor equations 



dXi dXj , dXn , 



where ~r~i are the direction cosines between the old and new axes. It is 



OXi 



dx ■ 3x ■ 

 obvious that -— ' = -— ^ and the relations can be written 

 OXi dx i 



