On the Reflection of Electrons by Metallic Crystals 



By L. A. Mac COLL 



This paper gives the results of some calculations of the reflection coefficient 

 for electrons incident normally on a plane face of a metallic crystal. The physical 

 situation is treated as being one-dimensional; and it is assumed that the potential 

 energy of an electron is a sinusoidal function of distance inside the crystal, and 

 obeys the classical image force law outside the crystal. The reflection coefficient 

 is computed as a function of the energy of the incident electrons, over the range 

 from to 20 electron volts, for a variety of values of the parameters which define 

 • the model of the crystal. 



1. Foreword 



^TT^HE work which is presented in this paper was undertaken as a result 

 -*- of conversations had with Dr. C. J. Davisson at various times during 

 the years 1938 and 1939, when he was investigating the reflection of elec- 

 trons impinging on the surface of a metal Uc crystal. The results for a simple 

 special case of the general problem were pubhshed in 1939^ Thereafter the 

 work on the general problem continued intermittently, and it was almost 

 completed by the early part of 1942, when it was brought to a halt by the 

 onset of wartime activities. Since then nothing has been done on the prob- 

 lem, and the results already obtained have never been published in extenso. 

 However, C. Herring and M. H. Nichols have included an illuminating dis- 

 cussion of some of the more significant of the results in their recent mono- 

 graph on thermionic emission^. 



Although the intervening years, by bringing new problems in physics to 

 the fore, have caused this work to lose some of the interest which it possessed 

 at the time it was being done, it still seems to be worth while to put the full 

 results upon record. The present occasion, when his friends and former col- 

 leagues are celebrating Dr. Davisson's seventieth birthday, is an especially 

 appropriate one for this purpose. 



2. Formulation of the Problem 



We consider electrons moving with energy E and impinging on a plane 

 face of a metallic crystal. (Fig. 1.) According to quantum mechanics there 

 is certain probability R, generally neither nor 1, that an electron will be 

 reflected by the crystal, and caused to move backwards toward the source; 

 and there is the complementary probability 1 — R that the electron will 



^ Physical Review, v. 56, pp. 699-702. This paper will be referred to henceforth as 

 (LAM, 1939]. 



2 Reviews of Modern Physics, v. 21, pp. 185-270 (1949). 



888 



