66 BELL SYSTEM TECHNICAL JOURNAL 



m, n and p represent reversals of phase along the three directions and may be 

 termed overtones. The values of k and ki are inserted to correct for the 

 change in shear velocity resulting from a change in Young's modulus in the 

 three directions. For most work with oscillator crystals where the length 

 and width are large compared to the thickness, the following simplification 

 of equation 6.9 is most useful. 



f-l^ 6.10 



When high frecjuency shear type crystals are used in connection with selec- 

 tive networks, it is necessary to make use of equation 6.9 to determine where 

 the next possible pass regions will occur. 



6.34 Effects of Rotation About the CKystallographic Axes on, the Resonant 

 Frequencies and Coupling between Modes of Motion 



Several of the elastic constants have been used in equations expressing 

 the resonant frequencies. Since most of the crystal cuts now in use are 

 rotated at some particular angle about the -Y crystallographic axis, it is of 

 int:rest to know the effect of this rotation upon the elastic constants since 

 they determine the resonant frequencies and the coupling between certain 

 of the modes of motion. The general stress-strain equations for an aeolo- 

 tropic body are given in equation A.l of Appendix A together with their 

 definitions. In the case of quartz where the axes of the finished plate are 

 aligned with the crystallographic axes the constants reduce to 7 and are 

 shown in equation A. 8. Examination of these equations shows that there 

 are extensional and shearing strains resulting from dissimilar extensional 

 and shearing stresses through the elastic constants Sij and Cj,-. This results 

 in coupling between modes of motion where a so-called cross strain exists. 

 These couplings may be made zero or small by proper orientation of the 

 crystal plate about the X crystallographic axis. The mathematics of this 

 operation is simplified by the use of matrix algebra . Upon performing 

 this ojieration a new set of elastic constants are obtained and are plotted 

 graphically together with the piezoelectric constants on Fig. 6.7. From 

 this figure we may see that the coupling resulting from the S2i constant will 

 be zero if the crystal plate is orientated by — 18.5° about X with respect to 

 the crystallographic axis. This constant determines the coupling between 

 the extensional mode along the length (I^' dimension) and the face shear 

 mode (F'Y' dimensions). This analysis resulted in the use of the —18.5° 

 cut in the channel filters of the coaxial system. Two other crystal cuts 

 resulting in low coupling between different modes of motion are the AC and 



5 "The Mathematics of the Physical Properties of Crystals," W. L. Bond, B.S.T.J., 

 Jan. 1943. 



