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THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 





k^ [E - Vix)U = 0. 



(1) 



(Here k^ — WmlW-, where h is Planck's constant, and m is the mass of an 

 electron.) This assumption deals in a summary way with various compli- 

 cated processes involving electrons in crystals. Discussions of the vaUdity 

 of the assumption are to be found in various works on the electron theory 

 of metals. 



Fig. 2 — Assumed potential energy as a function of the coordinate x. 

 Assumption III. Specifically, the function V(x) is given by the formulae 



V{x) = — Fo + Fi sin a(x — Xo), x < Xq 



= — €V(4x), X > Xq 



xo = eV(4Fo), 



(2) 



where e is the absolute value of the electronic charge, and Fo, Vi and a 

 are suitable non-negative constants. (A graph of this function V(x) is 

 shown in Fig. 2.) According to this assumption, an electron in the region 

 X > ^0 is subjected to the classical image force. This is known to be in good 

 agreement with the facts, at least if x is not too small.* Also, according to 

 the assumption, the potential energy of an electron in the depths of the 

 crystal is a periodic point function with a negative mean value. This part 

 of the assumption is as correct as any assumption can be which attempts to 

 account for the complicated actual processes in terms of a potential energy 

 function. However, our particular choice of a periodic function is based 

 largely upon mere considerations of mathematical convenience. Finally, we 



* See Herring and Nichols, footnote 2, p. 245 et seq. 



