ON REFLECTION OF ELECTRONS BY METALLIC CRYSTALS 89l 



observe that our V{x) is continuous, as physical considerations indicate 

 that it should be. 



We can now state the mathematical problem before us in the following 

 terms : 



V{x) being defined by (2), we are to obtain a solution }f/{x) of (1) satis- 

 fying the following conditions: 



(a) In the region x > xo the function \p{x) exp {—2iriEt/h) represents an 

 incident beam of electrons moving toward the left, and a reflected beam 

 of electrons moving toward the right. 



(b) In the region x < xq the average electron flow, if it is not zero, is direc- 

 ted toward the left. 



(c) The function \p{x) and its derivative \l/'{x) are everywhere continuous. 

 Having obtained such a solution ypix), we are then to compute the ratio of 

 the intensity of the reflected electron beam to the intensity of the incident 

 beam. In particular, we are to study the dependence of this ratio upon the 

 quantities E, Fo, and Fi. 



The paper [LAM, 1939] already referred to dealt with the special case in 

 which Fi = 0, i.e. the case in which V{x) is assumed to be constant in the 

 region x < xq. Consequently, we are now concerned chiefly with the cases 

 in which Fi > 0. 



3. Generalities Concerning the Calculation of R 

 In the region x > xq the wave equation (1) takes the form 



dx"" 



The general solution of this equation is of the form 



yp{x) = AUx) + BUoo), 



where A and B are arbitrary constants, and ^i(x) and yp2{x) are two particu- 

 lar solutions which we choose so that the functions ypi{x) exp {—liriEt/h) 

 and ^2^ exp {—liriEt/h) represent beams of electrons, of unit intensity, 

 moving to the left and right, respectively. 



In the region x < xq the wave equation takes the form 



^ + k\E + Fo - Fi sin a{x - x,)]yp = 0. (3) 



We are concerned with a solution of this equation of the form 



^P{x) = CUx\ 

 where C is a constant, and \l^z(x) is a particular solution such that the func- 



+ e[E + Q, = 0. 



