Theory of X-Ray Reflexion. 325 



10. Temperature Effect and Compound Crystal. 



We will next introduce an effect so far disregarded, the 

 temperature correction. This, due to the fact that the atoms 

 are at no time all in their planes, affects the reflexion from a 

 single plane, but makes no further change. Of the displace- 

 ments of the atoms, those in the plane produce no change of 

 phase, and we only have to consider displacements out of the 

 plane. The treatment here is rather different in detail from 

 that of Debye *. Let us suppose that the potential energy for 

 a displacement £ is i^f 2 . By the principle of equipartition 

 out of the M atoms per sq. cm. a number 



are displaced a distance between f and f+^f. These atoms 

 are wrong in phase to an extent expressed hy e~ ^ sin& * 

 There are a great many atoms in any region over which the 

 phases of the undisturbed atoms are sensibly constant. The 

 temperature effect can be thus expressed as a factor in the 

 value of q, and this factor is 



or exp-,- — (2&sin0) 2 or exp — - —. -, {imrf. 

 J, (T £ &a 



This is for the amplitude, and so for the intensity the tem- 

 perature vibrations introduce a factor 



exp-— 2 (2n7r) 2 (6) 



It is the same for a given order of reflexion, but diminishes 

 rapidly with the higher orders. 



We next deduce the reflexion for a crystal composed of 

 several similar interpenetrating lattices. Let N r be the 

 number of atoms per c.c. of the rth lattice, f r their scattering- 

 effect, and let the planes of this lattice be at distance u r a 

 from those of the first, let <r r give the restraining force on 

 the atoms. Then the expression 



_=*(Sbr)i 



Wfe m 



* Debye, Lev. d. Dent. PInjs. Ges. 1913, p. 671. 



t k is used in two different senses, but the difference will be clear. 



