MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 49 



when C is the velocity of light in free space, E is the (vector) electric field, 

 j is the induction current and B is the magnetic induction. In a crystalline 

 medium the current is given by 47r/ = kE where k is the dielectric constant 

 (matrix), whence: 



CV X 5 = k^ (16.1) 



CV X E = -B (16.2) 



As the divergence of the Curl is always zero: 



VckE = Vci = 



(16.3) 



VcB = 



applying — to (16.1) and substituting (16.2) in the result: 

 oi 



-CV X V X E = kE or 



C\s7cV - VVc)£ = k'E (16.4) 



We shall try as a solution: 



E = Eoe''""''-'' (16.5) 



where Eo is the vector amplitude of the electric field, i is \/— 1, r is the 

 radius vector from the origin to any point, 9 is a constant, n is the unit 

 normal (at r) of surfaces of equal phase, and co is 27r times the frequency of E. 

 Substituting (16.5) in (16.4) we find: 



2 

 n - nEcn = ^^kE (16.6) 



Examination of (16.5) shows that - is the phase velocity along n. Writ- 



2 

 ing47r^~7' for I^ and V for — we have: 



Q 



-1 -1 V 

 k J - njck n = —j (16.7) 



This equation is independent of the absolute value of j so let us restrict j 

 to being a unit vector. 



Vck^ = = VckEoe'^'"''-'''^ = i.mV' ^"^"--"'^ 



whence jcfi = (16.8) 



That is, the current is always normal to the direction of propagation. 



