892 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



tion \l/-i{x) exp {—2wiEt/h) represents a state in which the average flow of 

 electrons in the crystal either vanishes or is directed toward the left. 



The actual forms of the functions \l/i(x), 1^2 W, 1^3 W will be discussed 

 presently. 



Now the continuity of the functions \f/(x) and \l/'ix) gives us the system 

 of equations 



A\l/i{xo) + B\l/2(xo) = C4/z(xq) 

 Axf^i'ixo) + B^P2'(xo) = Ch'M, 



from which we can calculate the ratio B/A in terms of the \l/t{xo), ^/(xo). 

 Our required reflection coefficient R is | B/A |^ and so we obtain the for- 

 mula 



R = 



1^3(^0) 



^3(Xo) 



(4) 



It was shown in [LAM, 1939] that the functions i/'i(x) and \p2{x) are given 

 by the formiilae 



^iW = w,,m, Moo) = Tr_xj(-^), 



where 



and the symbols W\,{{^), W-\,\{—^) denote the usual functions occurring 

 in the theory of the confluent hypergeometric functions'*. The earher work 

 gives us all the information concerning ^i(x) and ^2(x) that we shall require. 

 Hence, in order to calculate R, we have, in effect, only to identify a suitable 

 f.olution \J/z(x) of equation (3), and then to calculate \l/z{xo)/\l/z{xo). 



4. The Solution of Equation (3) 

 In order to facilitate the use of known results, it is convenient to write 



a2 



Then equation (3) takes the form 



2'+ (^2 + 2^1 cos 22)^^ = 0. (30 



This is one of the canonical forms of Mathieu's differential equation, for 



* E. T. Whittaker and G. N. Watson, "Modern Analysis" (Chapter XVI), Cambridge 

 Univ. Press, 4th Ed., 1927. 



