Theori/ of X-Ray Reflexion. 319 



finite, and complicated operations with Fresnel integrals are 

 avoided. 



It is not necessary to carry through all the details strictly 

 according to the electromagnetic theory. These can he 

 introduced later. Let us suppose that when a wave of 

 length 2irjk falls on an atom, the amplitude of the scattered 

 radiation at unit distance bears to that of the incident a 

 ratio f(^fr, &), where yfr is the angle between the direction of 

 observation and the incident beam, /is of the dimensions of 

 a length. In accordance with § § 11-13 it will be taken to be 

 real, involving no change of phase. In addition to yjr and k 

 it will depend on the direction of polarization of the incident 

 beam. We shall suppose / so small that the wave scattered 

 by one atom does not influence the amplitude of vibration 

 of the radiating system in any other. As we shall see, there 

 is an effect on the phase which can still be included. It will 

 appear that there is defini te experimental jg vidence that the 

 scattering of one n.tc>m _jToes a.ftW-t T tlmt of others, because we 

 shall find reason to believe that over a narrow range of 

 angles of incidence the reflexion is nearly perfect ; so we 

 can only regard the present process as a first approximation. 

 For simplicity we shall take a crystal composed of atoms of 

 a single element, arranged in a single lattice, but this lattice 

 may be cubic or parallelopipedal. We also neglect the 

 temperature vibrations of the atoms. These omissions are 

 very easily set right later. 



5. Reflexion from a Single Plane. 

 We first find the reflexion from one plane of atoms. Let 

 the incident beam be 0**(^~" K )/R, where R is the distance 

 from (see fig. p. 317). Taking C as origin, is the point 

 (0, 0, Ji). To find the reflexion at angle we take as 

 point of observation pcosO, 0, p smO — Ji, so that p is the 

 distance from I L . Then the point of geometrical reflexion A 

 is (Rcos #, 0, 0) where A = R sin 6. Let there be an atom at 

 Rcos# + f, n, 0. This atom will contribute a component 



where the quantities that do not vary rapidly have been 

 replaced by their values at A, and R^, r$ n are the distances 

 of f, 77 from O and P respectively. By expansion we find 

 that 



