The passage of α- and β- particles through matter and Born's theory of collisions

In a recent paper* the present writer considered the relation between the results of observations on various phenomena connected with the passage of β-particles through matter and the requirements of the quantum theory. A close quantitative comparison of the new quantum theory with experiment was, however, not made, because the theoretical requirements were not known with sufficient accuracy. The theoretical estimates were mainly based on calculations made by Gaunt in 1927, and in his calculations the effect of close collisions with impact parameter less than atomic dimensions were not adequately dealt with. These close collisions contribute appreciably to such phenomena as the stopping-power and ionisation, and in order to allow for them in such cases somewhat arbitrary assumptions concerning their effect had to be made. The differences between the theoretical values arrived at and the experimental values were not so large that they could be definitely dissociated from these assumptions, and for this reason no fundamental significance was attached to them. A treatment of collisions more complete than that of Gaunt was given recently by Bethe on the basis of Born’s theory of collisions. Bethe’s calculations deal with the effect ofall collisions , and the formulæ obtained by him for the stopping power, primary ionisation, etc., enable us to make a closer comparison of the new quantum theory with experiment than was possible before. This is done in the first part of the present paper. It is satisfactory that the new formulæ, whilst agreeing in a general way with those previously used as a representation of the quantum theory, are in better quantitative accord with experiment. The present position is, however, not completely satisfactory. In some cases there are still large discrepancies. These discrepancies, if real, are, of course, more serious than those found in previous discussions because there is much less room for ascribing them to incompleteness or approximation in the theoretical calculations. The main assumptions made in Bethe’s calculations are that the velocity,v , of the moving particle is large compared with the Bohr-orbit velocity,u , of the atomic electrons traversed, and that it is small compared with the velocity,c , of light ; quantities of the order ofu 2 /v 2 , andv 2 /c 2 , being neglected. A third simplification which we must not overlook, especially in dealing with many-electron atoms, is the representation, in Bethe’s calculations, of the atomic electrons by hydrogen-like wave-functions. As regards the first two assumptions, ½mu 2 being roughly equal to the ionisation potential, J, the corresponding conditions of applicability of Bethe’s results may be formally writtenu 2 /v 2 ≈ J/½mv 2 ≪1, (1a)v 2 /c 2 ≪ 1. (1b) These conditions are adequately satisfied in most of the cases considered in the first part of this paper. The test of Born’s theory of collisions under these simplifying conditions is very desirable, especially in view of the considerable contemporary work which is being done on slow electrons which do not satisfy (1a).