﻿Distribution of Electrons in Na and CI Atoms. 439 



crystal we can suppose the atoms all to lie in a series of 

 planes, successive planes being separated by a distance d. 

 We get the nth order spectrum formed at a glancing angle 

 by the reflexion from such a set of planes if 



2d sin = n\. 



This spectrum represents the radiation diffracted by the 

 atoms in a direction making an angle 26 with the incident 

 beam, and it is formed because in this particular direction 

 the radiation scattered by any pair of atoms lying in suc- 

 cessive planes differs in phase by 2mr. Thus the amplitude 

 of the beam scattered in this direction is the sum of the 

 amplitudes scattered by all the neighbouring atoms taking 

 part in the reflexion. 



Let us consider the contribution to the reflected beam of 

 a group of atoms lying in a reflecting plane. To obtain the 

 amplitudes of the reflected wave, we sum up the amplitudes 

 contributed by the electrons in all the atoms, taking due 

 account of the fact that the electrons do not in general lie 

 exactly in the reflecting plane and so contribute waves 

 which are not in phase with the resultant reflected wave. 

 By symmetry, the phase of the resultant wave will be the 

 same as that reflected by electrons lying exactly in the 

 geometrical plane passing through the mean positions of all 

 these atomic centres. The phase of the wave scattered in a 

 direction 6 by an electron at a distance x from the plane 

 differs from that of the resultant wave by an amount 



— -.2?sin 6. 

 A, 



We will suppose that there is in every atom an electron 

 which is at a distance a from the centre, and that all direc- 

 tions of the radius joining the electron to the atomic centre 

 are equally likely to occur in the crystal. In finding the 

 effect of these electrons for all atoms (M in number) of the 

 group, we may take it as equivalent to that of M electrons 

 distributed equally over a sphere of radius a. It can easily 

 be shown that, if x is the distance of an electron from the 

 plane, all values of x between +a and —a are equally likely 

 for both cases. Such a shell scatters a wave which is less 

 than that scattered by M electrons in the plane in the ratio 



sin 6 , 

 — — -, where 



* 4tt 



(/)= --a sin 0. 



