208 BELL SYSTEM TECHNICAL JOURNAL 



would be Cij or s% . In any case the difference was probably less than the 

 accuracy of measurement. 



Later measurements by Perrier and Mandrot for two of the constants ^n 

 and ^33 give the values 



^n - 127.3 X 10-"; 533 = 97 X lO-'* (A.15) 



By using the measured resonance frequencies of known modes of motion, 

 the uncertainty of the type of elastic constant can be removed, for the 

 alternations occur so fast that the leakage resistance has little effect. If a 

 crystal is lightly plated, it is shown in the next section that the resonant 

 frequency of a length vibrating bar will be determined by the zero field 

 elastic constants ^f,- . On the other hand if an unplated crystal is measured 

 in an air gap holder with a large air gap it has been shown that the fre- 

 quency measured will be determined by the zero charge elastic constants 

 s% or 6% . A careful measurement of the elastic constants of quartz has 

 recently been made by Atanasoff and Hart^. Using thickness modes for 



1 The resonances of length vibratmg crystals have been discussed by Cady, "The Piezo- 

 electric Resonator and The Effect of Electrode Spacing on Frequency," Physics, Vol. 7, 

 No. 7, July 1936, pages 237-259; and bv the writer, "Dynamic Measurement of The Con- 

 stants of Rochelle Salt," Phys. Rev., Vol. 55, pages 775-789, April 15, 1939; while the 

 resonances of thickness vibrating crystals have been discussed by Cady (above paper) and 

 Lawson "The Vibration of Piezoelectric Plates," Phys. Rev., Vol. 62, July 1, 1942, pp. 

 71-76. For a length vibrating crystal Cady shows that the resonant frequency for no air 

 gap (plated crystal) is controlled by the constant lAfi- For a crystal with a large air 

 gap, the frequency' is controlled by the constant. 



lAf, + A-,rd,l/Klsf^ = l/.f, . 



Starting with equations of the form (A. 10), the writer showed that the frequency of a bar in 

 an air gap holder would be controlled by the constant 1/xn , while the frequency of a plated 

 crystal is determined by 



K',s\ 



For a thickness vibrating crystal for which the field is applied in the direction of wave 

 propagation, Cady and Lawson find that the resonant frequency is controlled by the elastic 

 constant 



47reii2 



1 ^ 



. + ^f(f 



where D is the total separation between electrodes and t the thickness of the crystal. 

 When the separation is infinite, the controlling elastic constant is cfi + 47reii^/A'f which, 

 from equation (A. 12) is cfj. When the air gap is zero or D = t, the controlUng constant is 





r('-^) 



which, for all practical purposes, can be taken as Cif for quartz. 



2 "Dynamical Determination of the Elastic Constants and their Temperature Coeffi- 

 cients for Quartz," Phys. Rev., Vol. 59, No. 1 (85-96), Jan. 1, 1941. 



