QUARTZ CRYSTAL APPLICATIONS 187 



to that caused by a stress along the Y axes. A detailed analysis shows 

 that the value of the electrical separation moment (dipole moment) for a 

 stress along either axis is the same value but the sign is reversed. A longi- 

 tudinal stress then can only produce a charge moment along the A^ or 

 electrical axis which is the origin of the name electrical axis. 



If, however, we introduce a different kind of stress known as a shearing 

 stress, a separation of centers of charges can occur along the mechanical or 

 Y axis of the crystal. A simple shear stress is one in which forces act normal 

 to the direction of space separation rather than along it as shown, for exam- 

 ple, by the two opposed arrows normal to the mechanical axis in Fig. 1.6B. 

 Such a shear does not occur in nature, but rather a pure shear which consists 

 of two simple shears which are directed in such a way a!s to produce no 

 rotation of the molecule as a whole about its axis. If we resolve these 

 force components along directions 45° from the crystal axes, a pure shear is 

 equivalent to an extensional stress along one 45° axis and a compressional 

 stress along the other 45° axis. Such a stress would cause the charges to be 

 displaced from their normal position, as shown in the figure. This causes 

 the center of positive charge to be displaced downward along the mechanical 

 or Y axis of the crystal while the center of negative charge is displaced up- 

 ward along the mechanical axis. 



These three relations can be written in the form 



P. = -dnX, + duYy ; P, = IdnXy (1.2) 



where Px is the polarization or charge per unit area developed on an electrode 

 surface normal to the A' axis due to the applied longitudinal stresses Xx 

 and Yy, while Py is the polarization normal to the Y axis caused by the shear- 

 ing stress Xy . dn is the piezoelectric constant and equations (1.2) show 

 that the magnitudes of all these effects are closely related. In addition 

 to these three major piezoelectric effects, quartz has two smaller effects 

 which, since they are connected with the distribution of molecules in the YZ 

 and XZ planes, cannot be demonstrated by the figures given previously. 

 The complete piezoelectric relations are then 



Px = -dnXx + dnYy - duY, ; Py = duZx + 2dnXy (1.3) 



where Fj and Z^ are respectively similar shearing stresses exerted in the YZ 

 and ZX planes respectively. The best values for the dn and du constants 

 are respectively 



dn = -6.76 X 10"' ^-:^-; du = 2.56 X 10"' ^- (1.4) 

 dyne dyne 



as obtained by recent measurements for a number of A^ cut and rotated X-cut 

 crystals discussed in appendix A. 



