Theory of X-Ray Reflexion. 681 



T 

 is rapidlv extinguished. Since T f = x r = (-) nr e~ r ^ 2 -» 2 > it 



follows that at a depth z in the crystal the intensity is 



onlv exp — 2~ \/q 2 — v 2 of its value at the surface, and so we 



r a 2 



may speak of an extinction coefficient 



2 V?^ 2 (7) 



Averaged across the whole region of perfect reflexion this 

 gives a coefficient 



kqjira (8) 



For the value of g which we have been using, this gives 

 about 8000, whereas the absorption coefficient, taken as 

 fi cosec $, is for the platinum rays only 300. Thus the ex- 

 tinction is complete long before the rays going in a slightly 

 different direction are appreciably absorbed. This fact is 

 important in explaining the reflexion from an ordinary 

 imperfect crystal. 



4. Spherical Wave and Effect of a Slit. 



We have so far only dealt with plane waves. A spherical 

 wave can be made by compounding together in an integral 

 a set of equal plane waves going in all different directions. 

 If we put in the reflexion factor for each of these plane 

 waves, we obtain an integral representing the diffraction 

 pattern of the reflected beam. At the distances at which 

 experiments are usually made this pattern would be of some 

 complexity. Since it would never be observed in practice 

 on account of the finite area of any actual source and the 

 imperfection of the crystal, it is unnecessary to discuss it. 

 To rind the whole intensity of reflexion we may examine the 

 effect at infinity. Here the waves are all plane, so that we 

 can apply the formula? (6) direct. If we take the intensity 

 at a point at glancing angle <£ + e, we have r = ka cos <j) . e. 

 So, making use of the abbreviation s = ^/£acos <f>, we find as 

 the intensity at a great distance p 



--. — - when e < — s 



p 2 (e- Ve 2 -s 2 ) 2 



— r. 1 when — s<€<$ )■ • • • (?) 



p- 



i 



