376 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



Applying these corrections to the frequency equations of Table II the 

 resulting compliance constants become 



smo^x) = sn = 1.271 X 10r''[l + 16.5 X 10-^(T - 50°) 



+ 58.5 X ICr^T - 50°)2 -i-SSX 10-'^(T - 50y + • • •] 



sm+zo^x) = 0.8159 X 10-i2[l + 96.4 X lQr'(T - 50) 



+ 276.5 X ia-^(r - 50)2 ^ 219 A X l(y-'\T - 50)3 + . . .] 



sm-^ox) = 1.402 X 10-i2[l + 114.4 X 10-6(r - 50) ^^^^ 



+ 178 X 10-9(r - 50)2 _ 91 6 X 10-i2(r - 50)3 + . . .] 



5?2(+6o-x) = 0.8614 X 10-i2[l + 186.4 X 10-«(r - 50) 



+ 302.2 X 10-9(r - 50)2 _^ 385 3 ^ 10-i2(r - 50)^ + • • •] 



The equation for the compliance constant 522 for an X-cut crystal at an 

 angle 6 from the Y axis has been shown to be 



fB E 4/,| Bi'4/v rt E 3/,./i 



522 = ^11 cos 6 + 533 sm 6 — 2^14 cos sm 



(22) 

 + (25f3 + sfi) sin cos ^ 



Solving for the constants in terms of the compliances for the four angles 

 measured we find 



BE E ^22 (+30°) ~" -^22 (-30°) E _ E _i_ Oo^ 



^11 = S22 (.0''X) ] Su — 1— ^ J -^33 — -^22(0°) -r -^^22 (+60°) 



4^ 2 E /r, E , E \ 10 E 2 E /'o7^ 



— ^^22 (30°) — ^^22 (-30°); ^^13 "T '^44; = " ^ ^22(0°) " ^ ^22(60°) K^o) 



_,2Se ,26e 



+ -Q-522(+30°) + -Q 522 (-30°) 



Hence adding the results we find 



5fi = 1.271 X ia-^2[i + 16 5 X 10-«(r - 50°) 



+ 58.5 X 10r\T - 50)2 + 33 x 10-i2(r - 50)^ + • • •] 

 si, = 0.971 X 10-i2[l + 134.5 X 10-«(r - 50) 



+ 144 X 10-9(r - 50)2 _j_ 570 x 10-^2(7^ _ 50)3 _|_ . . 

 su = -0.4506 X 10-i2[l + 139.5 X 10-«(r - 50) 



+ 40 X 10-9(r - 50)2 _ 54 X IQri^T - 50)3 + . . .] 

 (25 fa + si,) = 1.785 X ia-i2[i + 300 X 10-«(r - 50) 



+ 460 X 10-s(r - 50)2 _ 98 X 10-^2(2^ _ 50)3 + . . .] 



* See "Piezoelectric Crystals and Their Application to Ultrasonics," page 204, equation 

 10.26. 



(24) 



