MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 69 



we assume that it can be rediagonalized to tlic matrix k' by means of the 



/I 5io bA 



transformation: k' — 8k8, where 8 = I 8-2\ 1 Sjs 1 



Vsi 532 1 / 



Since 5ii 821 + 812 822 + 8r.i 823 = we find that, to the first order of small 

 Cjuantities6,^ = —5,,. 



Expanding 5^5, to the first order of small quantities and equating the non- 

 diagonal terms to zero we find that: 



5l2 — 



A12 



^23 — 



Ao. 



and 831 = 



kn ~ ^22 ^22 — ^33 ^33 ~ ^11 



Therefore, to the first order of small quantities A,;: 



/kn + An A12 A31 \ /kn + A„ > 



8 = I A12 )^22 + A22 A23 ]8, = I y^22 + A22 



\ A 31 A23 ^33 + A33/ \ ^33 + A33/ 



where 8 = 



-A3 



^11 — ^22 ^33 — kn 



A12 . A23 



1 



-A23 

 ^^33 — kn k-a — h 



kn — k2 

 A31 



k22 



is the transformation 



.v' = X (21.18) 



The electro and piezo-optic effects of biaxial crystals can be handled by 

 these infinitesimal transformations, but uniaxial crystals and cubic crystals 

 may require finite rotations to re-diagonalize the k matrix. In the 



/k 0\ 

 case of cubic crystals we note that { ^ A J may be diagonalized by a 



\0 A kJ 



rotation of 45° about .Vi, giving 



^k > 



)^ + A 

 a) k - A/ 



(21.19) 



^k AoX / k 



^ Ai I becomes I k 



A Ai kJ VVAfTA? 



upon rotation 

 through angle 



