612 BELL SYSTEM TECHNICAL JOURNAL 



factor </)(£) which itself may be a function of E the energy of the 

 particle, but is of secondary importance. Such a formula, in the 

 case of penetration from without, represented the probability of entry 

 for a single approach of the particle to the hill. This suggests that 

 we should deem it in this case as representing the probability of escape 

 for a single approach of the particle from the depths of the valley to 

 the inner side of the hill. Suppose the valley to be of breadth a, the 

 particle to be bumping back and forth in it with speed vc the number 

 of approaches of the particle per unit time to the hill will be equal to 

 Vila. If the bottom of the valley is at the same level as the axis of 

 abscissae in Fig. 16, Vi is equal to ^2Elm\ if the valley is deeper, Vi is 

 greater. We deduce for the mean sojourn of the particle within the 

 valley, which is the reciprocal of the probability-of-escape-per-unit- 

 time, the expression : 



T 



[(t;i/a)0 (£)]-' exp. [+ (47r//0 ( ^2m{NeV,n - E)dxJ (28) 



The aspect of this expression is far from encouraging to one who 

 wishes for a striking quantitative test of the theory. Its value de- 

 pends not merely on the breadth a assumed for the space within the 

 potential-hill and the height Vm of the hillcrest, but on the details of 

 the shape assumed for the potential-curve of Fig. 12 both within and 

 without the crest; and since there is little or no independent knowledge 

 of these qualities of the nucleus, they may be adjusted practically at 

 will to fit any observed value of T whatever. Furthermore it was 

 obtained by making certain crude assumptions and certain not very 

 close approximations. 



One essential test, however, can be applied to it, which it must pass; 

 and pass it does. Let values of a and Vm, and a shape for the potential- 

 hill of Fig. 12, be so chosen that for some particular radioactive ele- 

 ment, RaA for instance, equation (28) agrees with experiment; which 

 is to say, that when into the right-hand member of (28) is substituted 

 for E the observed kinetic energy of the emerging alpha-particles, the 

 value of this right-hand member becomes equal to the observed mean 

 life of the element. Now let precisely the same values of a and Vm 

 and the same shape of hill^" be assumed for some other radioactive 

 element, RaC for instance; in the right-hand member of (28), let the 

 observed kinetic energy of the (main group of) alpha-particles for 

 RaC be substituted for E; and let the value of T be computed. We 



^^ Excepting that the two hills should be expected to slope off towards infinity in 

 the manners of the two functions Zi/r and Zi/r, where Zie and Zie stand for the 

 nuclear charges of the nuclei left behind after the alpha-particle departs, and are 

 often (not always) different for two different radioactive substances. 



