X-Rays from Imperfect Crystals. 819 



These equations will be true even when the scatter o£ the 

 blocks varies with the depth in the crystal, but to make 

 progress we shall suppose it constant — that is, G(u) is not a 

 function o£ z. This makes the equations linear. The end 

 conditions are that I u (z) and E„(<2r) should vanish for infi- 

 nite z. There is no need to give the solution in detail — 

 E w the value at the surface is what is required. It is * 



E tt /i=!i!!^G(„)/[M + G W 

 ' smacosu j Lr ^ v y 



+ x /{& + Gr(u)¥-[a{it)]Xl-eotfd tan 2 it)}]. . (7-2) 



The first factor represents the influence of having a crystal 

 face that is not the true reflexion plane — whether because 

 the surface is covered by a vicinal face or because it has 

 been badly ground. For the layers into which the crystal 

 was divided were drawn parallel to the actual face, and if 

 this is not the true reflexion face Gt{u) will be unsymmetrical 

 about u = 0. Now suppose that the source and point of 

 observation are interchanged — this is the same as observing 

 on the other side of the spectrometer. We must then draw 

 the x axis in the other direction, and so shall obtain a formula 

 involving —u instead of u. But if on this side we take 

 n l =—u we shall have 



E' U ./I=~H3 L ,Gt{u')/{/i+G(u , )-t etc.}. 



sin o cos u v ' ' l v ' J 



Thus if we compare together points where the u of one side 

 is the same as the u of the other, then clearly 



EVK = sin (0 + u)/sm (6-u). 



In the case of a fairly perfect vicinal face all the settings 

 which give perceptible reflexion will be not far from 

 w=«, the inclination of the face, and so the ratio of all pairs 

 of corresponding powers will -be nearly sin {6 + a) /sin (0 — a), 

 and therefore the same will be true of the integrated 

 reflexion. This factor may also be derived by simple con- 

 sideration of the area of the crystal on which a limited beam 

 would fall in the two cases. The difference of the re- 

 flexions was originally observed by Sir W. H. Bragg f, and 



* Mutatis mutandis this is substantially the solution obtained by K. W. 

 Lamson, Phys. Rev. vol. xvii. p. 624, by quite a different method. 

 If (7*1) are treated as difference equations his exact form is obtained. 



t W. H. Bragg, Phil. Mag. vol. xxvii. p. 888 (1914). At first I 

 thought my explanation was different from his; but through corre- 

 spondence it became evident that we were only regarding the matter 

 from different points of view. I wish to express my thanks to him for 

 his interest in the matter. 



3 G2 



