94 BELL SYSTEM TECHNICAL JOURNAL 



due to the increased piezo electric constant for this particular cut, and the 

 frequency constants are different due to the change in angle with respect 

 to the crystallographic axes. The three series of interfering modes as de- 

 scribed for the BT -c\xi have been measured for this crystal and as shown 

 in Section 6.4 are 



6.31 

 _ 254.00 



Jt't — ^7 ^^2'» 



In a manner similar to the BT case a chart has been developed of a folded 

 frequency scale showing the frequencies at which even order X„' flexure 

 modes propagated along X and odd order Zx shear modes along X as well as 

 odd order Zx shear modes along Z' will interfere with the high frequency 

 Xy' shear mode for a crystal 1 centimeter square. This is shown in Fig. 6.26. 

 Its use is the same as that described for the BT case. Insufficient experi- 

 mental work has been done to indicate the relative shift in the flexure and 

 shear modes along the X axis when they approach each other in frequency. 

 Also, most of the use of square plates and experimental work has been con- 

 fined to the ^T-cut crystals and hence no ok regions are shown for this 

 chart. 



APPENDIX B 



Equation of elastic and piezoelectric constants for rotation of axes about 

 the A^ axis. (5 = sin &; c = cos &) 

 I 



C\\ — C\\ 



C22 = C\\C + C335* + 2(2C44 + ^13)5 c -|- 4cu5C 



C33 = CvJ" + C33C 4- 2(2C44 + ^13)5 C — 4Ci45 c 



c'u = C44 + (Cll + C33 — 4C44 — 2Ci3)5V — 2ci4(c" — s') sc 



C55 = CiiC -f- C665 -r -^CuSC 



C66 == Ci\S 4" cmc — 2cusc 



Cu = C12C -\- C13S — 2cusc 



Cu = cns^ + cizc 4- 2cusc 



cn = Ch(c — s) -\- (ci2 — cn)sc 



