X-Rays from Imperfect Crystals. 807 



values of a, ft, 7 and «', ft', <y', and as all pairs of molecules 

 occur in the sum this is the same as taking the mean of/ 

 and squaring its modulus. This quantity will be denoted 

 by/ 2 ; it is different from the mean intensity scattered by a 

 single molecule*. 



Suppose that the crystal is set so that f is near 0, where 

 is given by the equation ka sin = nir, which determines 

 reflexion in the nth order. For angles far from this the 

 reflexion is insignificant, so we may put 



and treat u, v, yfr as small. This approximation excludes 

 all the other reflexions from consideration. The reflected 

 intensity is then 



(J/r 2 )/ 2 2) X cxnik{(3O-x'){u-v)sin0 + {y-y')^cos0 



- (z - z')(u + v) cos 0}. . . (4-4) 



From this a factor exp — 2ik(z — z) sin has been rejected, 

 as it is equal to unity. Now on account of the smallness of 

 u, v, yjr the exponential terms only vary slowly with a, ft, y, 

 and so it will be legitimate to replace the summations by 

 integrations. The number of terms contained in a volume 

 dY is NdfV and so the intensity becomes 



(J/r 2 )N 2 / 2 r (6 W^Y'exp^{F(^-^) + G(^-z/')+H(c--^)} 5 

 J ... (4-5) 



where F=(u — v)sin0, G = ^cos#, H= — (u + v) cos 0, and 

 the volume integrations are each taken over the whole 

 crystal. 



We shall now suppose that the crystal is put through one 

 of Bragg'' s experiments. An instrument is placed so that it 

 can catch all the reflected radiation. The element of area 

 at r is r 2 cos x &X ^ty or r 2 cos dvdyjr, and so (4*5) must be 

 multiplied by this and integrated over all values of v and ^. 

 Further, the crystal is made to rotate with angular velocity to 

 about the y axis, and the total energy E received by the 

 instrument is measured. This is equivalent to an integra- 

 tion \ du/co taken over all values of u. Then 



E = i du/(o 1 ( r 2 cos dv df x (4-5). 



—00 — » 



* See Bragg-, James, and Bosanquet, Zeitschrift fur Physik, vol. viii. 

 p. 77 (1921). 



