Theory of X-Ray Reflexion. 329 



their number. Consider the effect of an atom in a direction 

 inclined to the incident beam at angle 20. Of the v electrons,, 

 all in any plane at angle give waves in phase together. 

 Let p s be the distance of an electron from some plane drawn 

 in this direction. The whole resultant amplitude is then pro- 

 portional to Sexp — ik . 2p s sin 0. To find the corresponding 

 intensity we multiply by the conjugate imaginary and get 



v+ 2 22 cos U sin 6(p s -p t ) .... (9) 



This expression accounts for the phenomenon of the " excess 

 radiation " which is observed in the scattering at small angles 

 by an amorphous substance*. For k is of the order 10 9 for 

 medium rays and ps—pt is at most 10~ 8 , so if is less than 

 5° every single electron in the atom contributes to the excess. 

 At broader angles a few of the outermost electrons fail to 

 help, but there will be still a number of contributors. Finally,, 



rrr 



when = ~ - only those contribute which are less than a 



quarter wave apart. Moreover, the same excess will be 

 exhibited at a broader angle for the softer raysf. 



When we come to consider the reflexion of a crystal we 

 get the formula 



v+2 22cos-^-(p,-/> t ). 



We may probably assume that the interiors of the atoms are 

 oriented in all directions, since any forces which were strong 

 enough to turn them into one direction, would probably 

 show some optical effect, and this would mean that a cubic 

 crystal should show double refraction. So we may suppose 

 that the average of p s —pt is independent of 6. Then the 

 excess radiation contributes a constant factor to the reflexion 

 in each order, but that factor is greater for the lower than 

 for the higher orders. 



Using the value of / as above, we arrive at the folio wing- 

 formula for the reflexion of white radiation from a crystal 

 composed of one lattice. 



* Barkla, Phil.' Mag. vol. xxi. p. 270 (1911); Crowtker, Proc. Roy. 

 Soc. A. vol. lxxxv. p. 29 (1911). 



t Various hypotheses have been put forward to account for the excess 

 radiation, by J. A. Crowtker (Proc. Camb. Phil. Soc. xvi. p. 534, 1913) 

 and by D. L. Webster (Phil. Mag. xxv. p. 234, 1913). Tke last assumes 

 it due to the cooperation between electrons. He does not distinguish 

 between electrons in tke same and in different atoms. Except in the 

 region of the diffraction pattern of the primary beam, the latter are as 

 likely to contribute a negative as a positive term to (9). 



