604 BELL SYSTEM TECHNICAL JOURNAL 



bility of the particle being at that point. Let us hold, however, to the 

 picture of a dense stream of corpuscles approaching the potential-hill — 

 say that of Fig. 14 — from the right, and a much weaker stream receding 

 from it on the left. If there are 5 times as many corpuscles per-unit- 

 volume on the right as on the left, then there cannot be a steady flow 

 unless only one out of 5 incident particles traverses the hill. The re- 

 ciprocal of 5 is the fraction of particles getting through the hill, or the 

 probability of a single particle getting through; and it is also the 

 ratio ("^^*)trans./("^^*)inc.. This statement, however, is too narrow, 

 being valid for the case where the speeds v and Vr of the particles 

 on the two sides of the hill are the same. In the general case, we have : 



Probability of transmission or penetration 



= (V^)(^**)trans./(^^*)inc. 



= {vrlv)-4>{E)-exp.l- {lirlh) p dx^2m{E- neVm)^. (27) 



In Gamow's approximation the product of the first two factors has the 

 pleasantly simple constant value of 4. In the approximations of 

 Gurney and Condon and of Fowler and Nordheim for the cases of 

 Figs. 14 and 15, the product is some function of E which the reader 

 can construct from the foregoing equations. In all these cases, how- 

 ever, it is the exponential factor which dominates the trend of either 

 member of (27) considered as function of E. 



Now immediately one sees, that if transmutation is due to the 

 penetration of a charged particle through a potential-hill or potential- 

 barrier surrounding a nucleus — and if this penetration is governed by 

 laws of quantum mechanics as illustrated in the one-dimensional cases 

 — then when the number of observed transmutations is plotted against 

 the kinetic energy K of the impinging particles, the curve should be 

 expected to rise with a gradual smooth upward curvature from the 

 axis of K', and there should be no critical minimum value of K for the 

 advent of the phenomenon, but rather the beginning of perceptible 

 transmutation should be observed at progressively lower and lower 

 energy-values, as the sensitiveness of the detecting-apparatus is im- 

 proved ; and it may well be that transmutation can be detected when 

 K is still so low, that the quotient of K by the charge of the particle 

 is far smaller than any reasonable guess that can be made of the 

 height of the barrier. All these are features of such curves as those 

 of Fig. 17, or Figs. 16 and 17 of the Second Part. The adoption of 

 quantum mechanics permits us to accept these features without ascrib- 

 ing them to the hypothetical functions denoted by/i and/2 in equation 



