96 BELL SYSTEM TECH NIC A L JOURNA L 



crystal. Since the voltage will be of opposite sign to the charge generated 

 on the surface of the crystal in the absence of this counter voltage a nega- 



• • • , S,D 



tive sign is given to g „ . 



Finally the last partial derivative 



6e\ , 1 ae\ _ , i ae\ ._ dQ 



aa/s.D U OCT /s.D U da /s,d pC„ 



represents the ratio of the increase in temperature due to the added amount 

 of heat dQ when the strains and electric displacements are held constant. 

 It is therefore the inverse of the specific heat at constant volume and constant 

 electric displacement per gram of material times the density p. Hence 

 the ten equations of equation (44) can be written in the generalized forms 



— h'nlh — llnlh — il'nzh " In dQ 

 Em, = —h\mSl — him^l ~ I'Sm'^S ~ ^UmSi — ll^mS^ — IlimSt ~\~ -iTrfSml^l 



+ ^Tr&^ + -iw^^ - qlf dQ (53) 



Je=— e[7i ^1 + 72 02 + 73 03 + 74 04 + 75 OS + 76 OeJ 



—Q[qi 5i + 92 ^2 + 93 53] + -ttd • 



11= 1 to 6; m = 1 to 3 



If, as is usually the case with vibrating crystals the vibration occurs 

 with no interchange of heat between adjacent elements dQ — and the 

 ten equations reduce to the usual nine given by the general forms 



Tn = CnlSl + Cn^Si + CnsSi + CniSi + CnbSf, + C ntS e 



— hni5i — hnih — hnzh 



Em = —JllmSl — IhviSi — IhmSz — /74m'S'4 — IhmSb — /'Cm'S'e 



+ 47r/3mi5i + 4T/3I262 + 47ri3'l3 53. 



(54) 



In these equations the superscript a has been dropj^ed since the ordinary 

 constants are adiabatic. The tenth equation of {S3>) determines the increase 

 in temperature caused by the strains and displacements in the absence of 

 any flow of heat. 



If we introduce the e.xpression of equations (53) into equation (42) the 

 total energy of the crystal is per unit volume. 



