898 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



In pLAM, 1939] we derived an approximate formula for R when E is 

 large for the case in which Fi = 0. The work involved in that derivation, 

 together with the equation (7), enables us to obtain the desired formula by 

 a simple calculation. The result obtained is the following: 



R= ^[l-il^l. (8) 



Vife2£3 



L 4^ J 



The range of validity of the approximate formula (8) has not been de- 

 termined. The nature of the derivation, and also the form of the result, leads 

 us to suspect that the approximation is good only so long as the ratio 

 Vi/Fo does not exceed some bound depending upon the other quantities 

 entering into the expression for R. The approximation certainly breaks 

 down when Fi/Fo reaches the value Wo/(ae^). However, this value is well 

 above any of the values with which we deal with in this work. Consequently, 

 we suspect that the formula can be used, to extrapolate our calculations of R 

 to higher values of E, without serious danger of error in the cases which we 

 consider here. 



6. The Calculated Results 



The reflection coefficient depends upon the independent variable E, and 

 upon the three parameters Vo, Vi, and a. The effects upon R of taking vari- 

 ous values of Vo and Vi seemed to be of greater interest than the effect of 

 taking various values of a; and, consequently, we confined ourselves in the 

 calculations to a single value of a, namely, a = tt X 10"^ cm~^ This value 

 of a makes the period of V{x) in the crystal equal to 2 X 10~^ cm. 



We took six values of Vo, proceeding in equal steps from 10 electron volts 

 to 20 electron volts inclusive. These values adequately cover the range 

 which is of interest in connection with actual metals. 



Including the calculations reported in [LAM, 1939], we have taken, for 

 each of the values of Vo, five values of Vi, proceeding in equal steps from 

 to 0.4 Vo. Although it is somewhat difficult to say just what value of Vi is 

 most appropriate to the case of a specific actual metal, it appears that these 

 values cover the range of values of interest adequately. 



The results of the calculations are shown in a self-explanatory form by the 

 curves given in Figs. 4 to 9 inclusive. For the sake of unity, we have included 

 the results which were previously published in [LAM, 1939]. 



7. Physical Discussion of the Results 



The results do not call for much discussion, especially in view of the dis- 

 cussion which Herring and Nichols have given in the paper already referred 

 to. However, there are a few observations which should be made. 



