370 



THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



Since the density is the inverse of the volume, the square of the frequency 

 constant for a crystal whose dimension is measured at 50°C must be multi- 

 plied by the factor 



/, _ h 



in order to correct for the effect of temperature expansion on the elastic 

 constant. This correction is shown by the third column of Table I. The 

 fourth column is then the value of C44 for the various temperatures. The 

 fifth column shows the values of the fli , ^2 and a^ constants for the tem- 

 perature variation of C44 • 



Table I evaluates one of the elastic constants of the frequency equation 

 (4). To evaluate the other two constants, use is made of the frequency 



Table I 



constants for the AT and BT cuts given by equation (3). Over a tempera- 

 ture range the thickness / is given by the equation 



t = to(ll cos' e + ll sin' d) (8) 



where ly and Iz are the values of unit lengths along the Y and Z axis ex- 

 pressed as a function of temperature. Inserting the values of (6) and (7) 

 in equation (3), the elastic shear constants for the AT and BT cuts become 



cif (AT) = 2.924 X 10ii[l - 12 X 10r\T - 50) 



+ 12.8 X ia-^(r - 50)2 _|_ 172 X io-i2(r - soy + • • •] 



fE. 



(9) 



cif (BT) = 6.877 X lO^^l - 20 X 10-\T - 50) 



- 176 X 10-»(r - 50)2 _ 238 X 10-^2(7 - 50)^ + • 



From equation (4), the frequency equation, we have 



cd(BT) = 0.4258^6^6 + 0.5742 cu + 0.9890 cu 



cif(AT) = 0.6661 cSi + 0.3339 cu - 0.934 cu 



Since cu is already known, tke two equations can be solved for Ca and 

 Cu and we find 



(10) 



