MODES OF MOTION IN QUARTZ CRYSTALS 89 



6.52 BT Type Crystals 



As discussed in Section 6.4 the modes showing the greatest coupling to the 

 high frequency thickness shear are of two types: high orders of Xy> flexure 

 propagated along the X axis and high order Z, shears along the X and Z' 

 axes independently. Complex orders of the flexure and plate shear as illus- 

 trated in Fig. 6.2 and Fig. 6.4 do cause considerable difliculty and their 

 analysis calls for special treatment and is not within the scope of this text. 

 For the case of the ^T-cut the three primary interfering series of modes are 

 given by 



/*/ = —3^ tixf kilocycles 



fx» = — ^ — «z« kilocycles 6.30 



fz; = ' w«', kilocycles 



where X and Z' are given in centimeters and Jxj is the frequency at which 

 integral orders of flexure modes along the X axis would coincide with the 

 high frequency thickness shear mode. In a similar manner fxs and fz's 

 relate the same conditions for integral orders of the plate shear modes. 

 These equations are true only in the case where the thickness is of such a 

 value as to place the high frequency thickness shear mode at the same fre- 

 quency as the computed interfering mode. In most practical cases for oscil- 

 lator use the electric field is applied to the crystal by means of a flat electrode 

 on each side of the crystal plate. Under this condition only odd order Xy- 

 shear modes along the X axis are excited and hence the strongest couplings 

 to the Xy> flexure modes will be only for even order values of nxf in equation 

 6.30. In a similar manner the greatest interference between the Xy' shear 

 mode and high orders of the Z, shear modes along both X and Z' will occur 

 for odd orders. Therefore the strongest interference from these modes will 

 occur only for odd integers of fixs and w^'j in equation 6.30. These assump- 

 tions of only even flexures and odd shears showing appreciable coupling 

 are based upon a crystal plate cut precisely along its proper axis and of 

 uniform contour assembled in a holder using electrodes of uniform air gap. 

 Deviations from these conditions will of course alter the ideal results de- 

 pendent upon the amount and type of deviation. 



The relationships shown in equation 6.30 may be more clearly seen when 

 plotted graphically. Assuming a BT-oxi crystal plate 1 centimeter square 

 we may determine the frequencies at which an interfering mode will coincide 

 with the high frequency Xy> shear by assigning even integers to «,/ and odd 



