PIEZOELECTRIC CR VST A LS IN TENSOR FORM 127 



termined, the measurement of the lowest mode of the face shear crystal 

 gives one more relation and hence the values of 5i2 and S6& can be uniquely 

 determined. 



Similar measurements with crystals cut normal to Xi and width along Xs 

 and with crystals cut normal to X2 and width along Xi determine the constants 

 SAi , 523 and 555 , Siz respectively. The equivalent constants are obtained 

 by adding 1 to each subscript 1, 2, 3 or 4, 5, 6 for the iirst crystal with the 

 understanding that 3+1 = 1 and 6+1 =4. For the second crystal 2 

 is added to each subscript. 



Finally the remaining three constants can be determined by measuring 

 the face shear mode of three crystals that have their lengths along one of 

 the crystallographic axes and their width (frequency determining. axis) 

 45° from the other two axes. 



Any symmetry existing in the crystal will cut down on the number of 

 constants and hence on the number of orientations to determine the funda- 

 mental constants. 



6. Temperature Effects in Crystals 



In section 2 a general expression was developed for the effects of tempera" 

 ture and entropy on the constants of a crystal. Two methods were given, 

 one which considers the stresses, field, and temperature differentials as the 

 independent variables, and the second which considers the strains, displace- 

 ments and entropy as the independent variables. In tensor form the 10 

 equations for the first method take the form 



Em= — hm i jS i J + ■iir^m'n 5 n " qll dQ (155) 



The piezoelectric relations have already been discussed for adiabatic condi- 

 tions assuming that no increments of heat dQ have been added to the 

 crystal. 



If now an increment of heat dQ is suddenly added to any element of the 

 crystal, the first equation shows that a sudden expansive stress is generated 



S.D 



proportional to the constant X;t^ which has to be balanced by a negative 

 stress (a compression) in order that no strain or electric displacement shall 

 be generated. This effect can be called the stress caloric effect. The 

 second equation of (155) shows that if an increment of heat dQ is added to 

 the crystal an inverse field Em has to be added if the strain and surface 

 charge are to remain unchanged. This effect may be called the field caloric 



