PKOF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE. 269 



We should expect the scattering centres of the atom to be not only diffused 

 through its volume, but also to be less dense at the edges than at the centre, thus 

 producing exactly those conditions which would reduce greatly the intensities of the 

 higher orders of the spectra. 



I have attempted in what follows to make a quantitative estimate of the operation 

 of this principle, but it must be regarded as no more than approximate and provisional. 

 It is based on methods used by RAYLEIGH, SCHUSTER and A. B. PORTER. 



Let the density of the scattering centres in a stratified medium be given by 



PO+/J! sin 27T . ^, where d is the spacing. We might call this a simple harmonic 

 ct 



stratification, or for brevity a harmonic medium. 



As we know already, reflection of radiation of wave-length X cannot occur except 

 at angles given by n\ = 2d sin 0. 



Let us consider the amount of the reflection at these different angles. The amplitude 

 of the wave reflected to any point P by the stratum dx may be written as 



proportional to 



x\ f 2x sin 6\ j 



(p + Pi sin 2-n- -j ) sin ( <j> %TT . - -)dx, 



\ (*/ X / 



where account is taken both of the density of the scattering centres in the layer dx, 

 and of the loss of phase due to the depth of the stratum below the surface. 



Fig. 17. 



It seems to me that we do not contradict our principle that the intensity of the 

 reflected pencil is proportional to the mass concerned when we here take the amplitude 

 of the reflection by the layer to be proportional to the number of scattering centres or 

 the mass of the layer, although the intensity of a vibration is proportional to the 

 square of its amplitude. The amplitude, here considered, of the reflection is no doubt 

 only one of the factors which determine the intensity. The reflection is not directed 

 strictly in one direction, though it is more and more nearly so the more centres there 

 are in a layer. If we imagine the number of scattering centres iii a layer to be 

 gradually increased, the maximum amplitude is increased proportionally, but the 

 intensity of the whole reflection does not increase at the square of this rate, because 

 it is being continuously limited in its divergence from the true direction of reflection. 

 As we know from our experiments, the intensity measured, as we have measured it, is 

 proportional to the mass concerned, other things being the same. In our present 

 discussion we estimate the amplitude of the reflection disturbance at its maximum. 



VOL. CCXV. A. 2 N 



