ON REFLECTION OF ELECTRONS BY METALLIC CRYSTALS 895 



Obviously, in order that we shall get the solution which we require, (p{z) 

 must reduce to —ido when Vi = 0. Also (p{z) must be periodic with the 

 period TT. These conditions, together with the equations (6), suffice to deter- 

 mine (po{z), <pi{z)j <p2iz), ' • • successively and uniquely. We easily obtain the 

 results: 



e-'" 



(1 + 0,y{2 + So) J 

 (4 - 36,)/" 



+ 



I ^ -+- Ann IP. r *' 



(1 - ^o)Hi + eo)eo{2 - So) 



(Pi 



iz) 



+ 



(1 + ^o)Hl - ^o)^o(2 + ^o) (1 + eoy{2 + 0o)(3 + ^o) 



J_r (11 - 5go)e^" 



12Si L(l - 6lo)*(2 - 610)2(3 - ^o)(4 - ^o) 



4(15 - 18<?o + Sdl)e''' 2(8 - SSSl + 15^0*) 



(1 - ^o)*(l + ^0)(2 - «2(3 _ 0^>)0^ (1 _ ^2)3(4 _ ^2)^3 



4(15 + 18^0 + 56>g)g"'" 

 (1 + doYil - e,){2 + ^o)2(3 + ^o)^o 



(11 + 5^o)e"'" 



]■ 



(1 + 0o)K2 + ^c)H3 + ^o)(4+ W, 



The functions <p5{z) and ^6(2) have been computed also; but, because of 

 their complexity, they will not be exhibited here. 



It is easily found that the expression \l/3{xo)/}l/z(xo) appearing in equation 

 (4) has the value (p{Tr/4:)(a/2) in terms of our present notation^. 



In principle we now have all the information we need to calculate the re- 

 flection coefficient R. Furthermore, our experience showed that it is quite 

 easy to compute R by this method, provided that the value of E does not 

 lie too near an edge of a diffraction band. However, Brun's method proved 

 to be unsuitable for calculating values of 7? for values of E in the neighbor- 

 hood of a dijffraction band edge, and we were forced to seek another method 

 to obtain these values. After some tentative work with other methods, we 



® We must take account of the fact that the symbols ^p{x) and i/'(z), which we are using 

 for convenience, do not represent merely the same function with different arguments. 

 In fact, after the change of the independent variable from x to z we have the following 

 relation between the old and new \f/*s: [^(a;)]oid = ['^(^ + 22/a — ir/2a)]o\d = [^{z)]nt>w 



