X-Rays from Imperfect Crystals. 823 



The exponential and hyperbolic functions can always be 

 expanded, and if G(u) //i is not large the series will converge 

 rapidly. It will usually be right to omit the terms involving 

 sin 2 ?/, etc., even though some of these are multiplied by yu- 

 an d are being compared with others only multiplied by 

 G(u). Then 



t=G(w) sec 6 

 and 



E tt /I = g-^ , [*'^M-^ 2 G 2 (M)+f^ 3 G 3 (w)], . (8'3) 



where x' is written for x sec 6. As in § 7, if the form of 

 E„/I is found experimentally, it is possible to solve (8'3) 

 and so to obtain G(V), and consequently Q', from observation 

 of a single crystal. In this case the process will be free 

 from the objection raised in § 7 about non-uniform distribu- 

 tion of the blocks ; for, in reflecting, the surface does not 

 now receive preferential treatment over the interior. The 

 primary extinction is again untouched by the process. 



Bragg adopted a method which assumed that he could get 

 a series of plates of various thicknesses, for all of which the 

 distribution of the blocks was the same. He took the inte- 

 grated reflexion of them, and found for each set of reflecting 

 planes the thickness which gave maximum reflexion and the 

 height of the maximum. Now by (7*4) the integrated re- 

 flexion may be written as 



p « \ °°(E w /I)^ = e -^[Q^-^Q'V 2 + f^Q'V 3 ]. (8-4) 



— 00 



This has maximum at 



^=--c/ 27 +|^ T -3 , .... (8*5) 

 and the value there is 



g[l^+(W + W^| • • • (8-6) 



= Q'f [»+9&'+ (W-y*) ^}- (3-7) 



Thus Eragg's work determines <7 2 Q'? and if the distribution 

 of the blocks is the same as in the crystal used for the 

 work of § 7, it follows that his correction for secondary 

 extinction is correct to the second order, and from the 

 magnitudes of the quantities involved it is improbable that 

 the third order is sensible. 



