Dispersion of Diamond. 399 



In order to evaluate the vector sum appearing in formula 

 (1), fix as the i-th centre any of the lattice points, say in 

 fig. 2. Gall 1, 2, 3, 4 its four nearest neighbours. The 

 assemblage being unlimited and homogeneous, all the atomic 

 centres are in equal conditions. Disregarding the free 

 oscillations of the system, which would soon die away, let 

 us take account only of the vibrations of p; forced by the 

 external field. Then, in a homogeneous* field E, the 

 moments p; will either be all equal in size and direction, or, 

 at the utmost, split into two classes, p' and p". In order to 

 see this we can proceed as follows. Let, for instance, E be 

 parallel to and concurrent with the vector 01. Then every 

 atomic centre will be either such as 0, in which E points 

 away from it towards its next neighbour (viz. 1), or such 

 as 1, or 2, etc., where E is directed towards them, away from 

 their nearest neighbours. Let p' correspond to the first, 

 and p" to the second class of centres. Then, denoting by 12 

 the linear vector operator in formula (1), i. e. writing it, as 

 a dyadic, 



0=5^[3u.u-l], .... (2) 



the vector sum, say P, representing the total action of all 

 doublets upon the electron in will be 



p=xyp'+n"p'\ 



where O' is as in (2) with u = u / , r = r f , and with X extended 

 to the first class of points, and similarly for 11". 



Now, E being along 01, the vectors p', p" can, by reasons 

 of symmetry, have only the same or opposite directions. 

 Thus, denoting the unit vector from to 1 by i, 



p'=p'i, p" = *p% 



where 5 is a scalar (undetermined thus far), and therefore, 

 P=p / (0' + 50")i. 



But if the field E is reversed, the roles of the two classes 

 of centres will be exchanged : the moment for each member 

 of the first class will be — sp'i, and for the second class — p'i; 

 at the same time the total P will be reversed, so that (the 



* This will also hold with sufficient accuracy in the case of luminous 

 oscillations, since their wave-length is very great as compared with the 

 mutual distance of neighbour centres. The latter being of the order 

 of 10" 8 cm., a wave-length cube of visible, or of not too remote ultra- 

 violet light, will contain 10 u or 10 10 atomic doublets. 



