PIEZOELECTRIC CR VST A LS IN TENSOR FORM 95 



that these piezoelectric constants are written Z/"^^. Finally the last partial 

 derivatives of the stresses by the entropy a can be written 



dT 



'da 



") ^' = 1,^-P) Q^'^^ST^") 'iQ = -yrdQ (45) 



• /s,D 6 da /s,D 6 oa /s.d 



where dQ is the added heat. We designate 1/6 times the partial derivative 

 as — Yn and note that it determines the negative stress (compression) 

 necessary' to put on the cr>'stal to keep it from expanding when an increment 

 of heat dQ is added to the crystal. The electric displacement is held 

 constant and hence the superscripts S, and D are used. The first six equa- 

 tions then can be written in the form 



(46) 

 — h'nxhi — /U'Jo — h'na^s — y^f dQ. 



To evaluate the next three equation? involving the fields, we make use of 

 the fact that the expression for dU in equation (42) is a perfect differential. 

 As a consequence there are relations between the partial derivatives, 

 namely 



(47) 



(4.S) 



where /3 is the so called "impermeability" matrix obtained fiom '.he dielectric 

 matrix e„m by means of the equation 



&r.n = ^-^ (40) 



where A is the determinant 



(5(!) 



and a"''" the minor obtained by suppressing the wth row and ;/th column. 

 The partial derivatives of the fields by the entropy can he written 



dE^ 

 da 



A . 1 dE„\ 1 dE„,\ .s,z. ,,, .... 



/S.D U da /S.D 6 da /s n 



where q'n is a pyroelectric constant measuring the increa:£e in field required 

 to produce a zero charge on the surface when a heat /() is added to the 



