Theory of X- Ray Reflexion. 677 



~Na, the area! density of atoms', so large that the conservation 

 of energy would be violated. In nature this is of course 

 obviated by the fact that if the atoms are too closely crowded 

 together, the wave from each will influence the others. We 

 have seen that for rocksalt q is only of the order 10 ~ 4 , so 

 that the conservation of energy is in no danger, and we 

 shall continue to use q as it stands. The direct calculation 

 of the influence of all the atoms on one of their number 

 leads to a double series of some complexity. I am informed 

 by a friend* to whom I referred the matter, that the series 

 does in general converge (which was not at all obvious at 

 first sight), but that the question is quite a difficult problem 

 in pure mathematics. I do not give the form of the series 

 as no use is to be made of it. It will be found that the 

 forces from the other atoms exert an effect like an addition 

 to the radiation term in the vibration of an electron. The 

 radiation term hardly influences the amplitude of vibration 

 of an electron under the influence of X rays (except in the 

 case of resonance), and so w T e may conclude that the mutual 

 influence of the atoms in a plane may be neglected. 



When a wave falls on a single plane of atoms, besides the 

 directly reflected wave there are others scattered. Thus 

 there will be a wave given off in any direction for which a 

 line of atoms are in phase together, while the next parallel 

 line is a phase 2tt behind. These diffracted waves are 

 destroyed by the operation of the other planes of the crystal; 

 but there remains a wave scattered in the same direction as 

 the transmitted beam. The amplitude of this wave is given 

 by —iq , where g is obtained from q by replacing f (26, k) 

 by /(0, k). As was shown in the earlier work, it is the 

 wave — iq which gives rise to the refractive index f. 



3. Combination of Planes. 



The difficulty of the problem of allowing for the mutual 

 influence of the atoms in one plane is the complete absence 

 of phase relations between the waves arriving at one atom 

 from the rest, but this is also the reason why it is justifiable 

 to neglect it. For the combination of all the planes 

 the matter is quite otherwise. Here, when the radia- 

 tion is at the angle of reflexion, all the waves reflected 

 from the successive planes are in phase together, and they 

 must be supposed to give rise to a secondary reflexion, which 



* Mr. G. H. Hardy, F.R.S., of Trinity College, Cambridge, to whom 

 I must express my thanks for his kind interest in the question. 

 t Loc.cituy. 320. 



