LOW TEMPERATURE COEFFICIENT QUARTZ CRYSTALS 79 



cut while Fig. 4 shows the frequency constant of the crystal; i.e., the 

 kilocycles for one millimeter thickness plotted as a function of the 

 angular cut. The AT crystal which occurs at an orientation of 

 + 35° — 20' has a frequency constant of 1662 kilocycles for one 

 millimeter thickness while the BT cut which occurs at — 49° has a 

 frequency constant of 2465 kilocycles for one millimeter thickness. 



The frequency curve of Fig. 4 agrees very closely with the frequency 

 calculated from the elastic constants used in the formula for the 

 velocity of propagation of an aeolotropic medium given by E. B. 

 Christofel.'^ Christofel showed that for any direction of propagation 

 in an elastic solid, there were three different waves whose velocity of 

 propagation could be obtained from the determinant 



Xll — pC', Xi2, Xi3 



Xi2, X22 — PC'i X23 



Xi3, X23, X33 — pC^ 



= 0. (1) 



In this equation p is the density, c the velocity of propagation, and 

 X's are related to the elastic constants of the crystal by the formulae 



Xii = CiiP + Ceem"^ + c-o^n'^ + Ic^emn + 2c\^nl -\- 2c\^lm, 



X12 = Cie/^ + ci^m^ + CaoU^ + (c46 + C2b)fnn 



+ (ci4 + Co6)nl + (ci2 + Ceejlm, 



Xi3 ^ CnP + C46nP + C35w2 + (f45 + C36)fnn 



+ (Cl3 + C55)nl + (fl4 + C56)lm, 



X23 = Cs,6P + CanP + CsiU^ + {cu + C2z)inn 



+ (C36 + Cii)nl + (C25 + Ci6)lm, 



X22 = c&^P + co^ni' + Cun^ + 2c24mn + 2c46W^ + Ici^m, 



X33 = c-^-Jr + di^yp + c^zn^ + Iczimn + 2czi>nl + 2cii,lm, (2) 



where /, w, and n are respectively the direction cosines between the 

 direction of propagation and the x, y, and 2 axes. For quartz 



("22 = Ci\', C2A = — ^h! ^55 = Cii', Cs6 = Cu', ("66 — ('"ll Cu) /2 



and 



Cn = Ci6 = C25 = C26 = C34 = C35 = ^36 = C45 = f46 = 0. (v3) 



For a rotation about the x axis for which a positive angle is measured 

 in a counter clockwise rotation for a left handed crystal and a clockwise 

 direction for a right handed crystal when an electrically positive face 

 ' See Love's "Theory of Elasticity," page 298, fourth edition. 



