324 Prof. W. L. Bragg and Messrs. James and Bosanquet : 



The following is a brief summary of the calculation, 

 treated in a slightly different manner from that in Darwin's 

 and Compton's papers, where it is worked out more com- 

 pletely. 



If an atom contains Z electrons, and the waves scattered 

 by these electrons are in phase, the amplitude of the scat- 

 tered wave will be 



A'=^.Z^ 2 (1) 



K 



me 



If the spatial distribution of the electrons is such that the 

 scattered waves are not in phase, the factor Z must be 

 replaced by a function F, which depends on the angle 

 of scattering and the positions of the electrons. F tends 

 to its maximum value Z at small angles of scattering. 



Let rays from a source S fall at a glancing-angle 9 on a 

 plane containing n atoms per unit area, and be reflected. 

 The amplitude at any point P is equal to one-half the total 

 effect due to the scattering by the atoms lying in the first 

 Fresnel zone around the corresponding point of incidence I. 

 The area of the zone is equal to 



sin 6 ' r x + r 2 ' 



where ri = SI, r 2 =TP. The number of atoms it contains 

 is therefore 



mr\ r 2 r 2 



sin G ' >'i + ?' 2 ' 



and the amplitude at P is equal to 



1 2 n-rrX rvr* . , n\ r,r 2 A -^ e~ 



A = . n . — ■ — . — . r . . 



2 ' 7T sin 6 ' i\ + r 2 ' sin 6 ' r Y -f i 



2 '2 



me 



If i\ is great compared with r 2 , so that the incident rays 

 may be considered as a parallel beam, we get the relation 



Amplitude of reflected beam _ D' n\ ^ e 2 ._ 



Amplitude of incident beam ~~ D ~~ sin#" ' me 2 ' 



Considering now a thin slip of crystal consisting of 

 p planes at a distance d apart, the reflexion will be most 

 intense when 



???A, = 2</sin 6. 



