130 BELL S YSTEM TECH NIC A L JOURNA L 



In this equation the linear partial differentials have already been discussed 

 and are given by the equations 



dSij y dSij dEn dEm T 



where s^nkt are the elastic compliances of the crystal at constant displace- 

 ment, gijn the piezoelectric constants relating strain to electric displacement 

 /At, and /3l„ the dielectric "impermeability" tensor measured at constant 

 stress. We designate the partial derivatives 



dTddT,r ^'''^''■' dT,m„, dn^dT^r y'^" 



d Sij _ d'En _ ^D . d'Em __(^D 



ddnddo dTijdSo dSndSo 



(163) 



The tensors N, M, Q, and are respectively tensors of rank 6, 5, 4 and 3 

 whose interpretation is . discussed below. Introducing these definitions 

 equations (161) can be written in the form 



Em = Tkflgmkf.+ h^^ikfnTqr + Qkfmn^,] + SnlATT^mn + ^Omno^ 



Written in this form the interpretation of the second order terms is obvious. 

 N'ijkfgr represents the nonlinear changes in the elastic compliances s^jj 

 caused by the application of stress Tgr . Since the product of N nklqrTqr 

 represents a contracted fourth rank tensor, there is a correction term for 

 each elastic compliance. The tensor M'^jkfn can represent either the non- 

 linear correction to the elastic compliances due to an applied electric dis- 

 placement Dn or it can represent the correction to the piezoelectric constant 

 gijn due to the stresses Tk( . By virtue of the second equation of (162), 

 the second equivalence of (163) results. The fourth rank tensor ^Qnno 

 represents the electrostrictive effect in a crystal" for it determines the strains 

 existing in a crystal which are proportional to the square of the electric 

 displacement. Twice the value of the electrostrictive tensor ^Q^j„o , which 

 appears in the second equation of (164) can be interpreted as the change 

 in the inverse dielectric constant or "impermeability" constant. Since a 

 change in dielectric constant with applied stress causes a double refraction 

 of light through the crystal, this term is the source of the piezo-optical effect 

 in crystals. The third rank tensor Omno represents the change in the "im- 

 permeability" constant due to an electric field and hence is the source of 

 the electro-optical effect in crystals. 



These equations can also be used to discuss the changes that occur in 

 ferroelectric type crystals such as Rochelle Salt when a spontaneous polariza- 



