ON REFLECTION OF ELECTRONS BY METALLIC CRYSTALS 893 



which an extensive theory exists. We shall recall a few of the chief facts 

 brought out in this theory, f 



Unless the constants ^o and Oi satisfy some one of certain special rela- 

 tions, the general solution of equation (3') is of the form 



lA = K.e^iiz) + K^e-'^^fi-z), 



where /x is a constant determined by do and di, f{z) is a function which is 

 periodic with the period r, and the i^'s are constants of integration. 



In certain ranges of values of the ^'s, the constant ^ is real, and in other 

 ranges it is pure imaginary. When /x is real we can obviously take it to be 

 positive; and then, in order that yl/z{x) may be bounded in the range x < Xo, 

 we must choose i/'sW to be the function e^^f^z). When /z is pure imaginary, 

 we can take it to be ^ | /x | ; and then, in order that ^3 W shall represent a 

 state in which the flow of electrons is to the left in the crystal, we must 

 choose ^sW to be the function e~>'^f{—z). 



When 11 is pure imaginary we have a non- vanishing flow of electrons to the 

 left in the crystal. Consequently, the intensity of the reflected beam must 

 be less than the intensity of the incident beam. Hence, under this condition 

 we must have R < \. On the other hand, when ju is real there is no average 

 electron flow in the crystal. Consequently, under this condition the inten- 

 sities of the incident and reflected beams must be equal, so that R = \. 

 These considerations point to the importance of discussing, first of all, the 

 conditions under which /x is real or pure imaginary. 



Figure 3 shows a well known diagram, modified slightly to suit our present 

 purposes^. Here ^0 and ^i are taken to be rectangular coordinates of a point 

 in a plane, and the plane is divided into regions of two kinds (shaded and 

 unshaded) by a system of curves. If the point (^0, ^1) is in the interior of 

 one of the shaded regions, the above /x is real; if the point is in the interior 

 of one of the unshaded regions, /x is pure imaginary. (If (^0, ^1) lies exactly 

 on the boundary of one of the regions, we have a somewhat more compli- 

 cated situation, which we do not need to consider here.) This diagram en- 

 ables us easily to determine, for any fixed values of Vq and Fi, the ranges of 

 values of E in which we have R = \. We shall call these ranges of values 

 of E the dif Taction hands. 



Now our problem has been reduced to that of computing R for values of E 

 which do not lie in diffraction bands. In treating this phase of the subject 

 we shall follow the course of the actual calculations, without any examina- 

 tion of ways in which the work might have been done more efficiently. 



t See, for instance, E. T. Whittaker and G. N. Watson, footnote 4, Chapter XIX. 

 ^ See, for instance, N. W. McLachlan, "Theory and Application of Mathieu Func- 

 tions" (p. 40), Oxford University Press, 1947. 



