OPTICAL PROPERTIES IN CRYSTALS 163 



Inserting the value of Hq. from (5), this equation takes the form 



7-v r / c ^ 1 io>\t—xiniiv\ 



Lfi = r- Leiifc WAi -c-Oi WiMiJe 



and, in general, 



V 

 Di = -[ei]k{ejkiEkfii)nk\. (9) 



V- 



Kx])an(ling the inner parenthesis, we have the components 



(£2^3 - £3^2)1; (£3«i - £1^3)2; {Eifh — E2ni)-i. (10) 



Then 



€,■^^-[(£2^3 — E-jvi); (E-iHi — Eiih); (EiUo — E2ni)]nk gives 



A = — r [(-E3W1 — Eins)m — (£1^2 — £2«i)w2] 



= [(£3^3 + £2^2 + Eini)ni - Ei{nl + «2 + th)] 



V 

 Di = —-— [{Ein-2 — Eifi^fh — (£2^3 — £3^2)^3] 



V' 



(11) 



= [(£3W3 + £2^2 + £iwi)«2 — £2(^1 + nl + nl)] 



A = — "V [(-E2W3 — Ezn2)n-2 — (E^ni — Eins)ni] 



= [(£3^3 + £2;i2 + EiHiJth — Ez{nl + lii + nl)]. 

 Xow, since n\ -\- nl + ^3 = 1 because n is a unit vector, we have 



V' V 



Di ^ —- [Ei - {Ejnjjuil or — Di - Ei - (Ejnj)ni = 0. (12) 

 v^ V- 



This equation states that Di , £, and «, are in the same plane, VLj being 

 normal to the plane as shown by Fig. 1. The energy flow vector 



Si = —- eijkEjHk (13) 



47r 



also lies in the plane since it is perpendicular to £ and H. It is at the same 

 angle 6 with n that £ is with D. The velocity of energy flow is Vcos 6. The 

 energy velocity is called the ray velocity and the energy path the ray path. 

 Next, from the relation for a material medium, that 



Di = KijEj or conversely £_,■ = ^jiDi (14) 



