The anomalous scattering of α-particles by hydrogen and helium

Abstract

1. In a previous paper, the author has shown that the anomalous scattering of α-particles in helium can be accounted for, if one postulates the validity of the wave mechanics to describe the phenomenon, and assumes a simple spherically symmetrical form for the field between the two colliding particles. The solutions obtained were exact, no approximate methods being used, but only the scattering at large angles, 34° and 45°, was considered. The purpose of this paper is threefold. Firstly, these results are extended to the case of the scattering of α-particles by hydrogen, and good agreement with experiment is obtained. Secondly, it is shown that the scattering at small angles both in hydrogen and in helium can be explained by the same field as is used to explain the scattering at large angles. It is therefore no longer necessary to assume a “plate-like” form for the α-particle, in order to explain the scattering at small angles, as formerly according to the classical mechanics. Thirdly, a discussion is given of the extent to which the explanation here advanced for the experimental results is dependent on the particular form assumed for the potential energy of the one particle in the field of the other. It is found that, provided we assume that the potential energy, V( r ), is Coulombian for distances greater than about 5 X 10 -13 cm., then whatever the form of the potential for smaller distances , the ratio, R, of the scattering to that to be expected from a purely Coulombian field is given by the formula R = | e - i / ak . logcos 2 א + iak cos 2 א (e 2iK 0 — 1)| 2 , where the scattering is in hydrogen, א is angle of scattering, and ak = hv /4πє 2 . A similar formula is found for the scattering in helium. The formula contains only one parameter, K 0 , which depends on the velocity of the incident α-particles, v , and also on the field assumed, but is independent of the angle of scattering, א. Thus we may determine K 0 from the observed scattering for a given value of v and of א, and deduce the scattering at other angles for the same value of the velocity. The agreement obtained with experiment is then quite independent of any special choice of a potential energy function. In order to calculate K 0 and its variation with v , we should have to assume a particular form for V( r ) Conversely, from the values of K 0 determined by the experimental values of R, we may calculate V( r ) as in the previous paper for the case of two α-particles. 2. The Scattering Formula for Hydrogen .—In the experiments on the scattering by hydrogen, it is found convenient to count the scintillations due to the hydrogen nuclei projected forward by the collision rather than those due to the deflected α-particles themselves, and therefore in this portion of the paper we shall denote by It the ratio of the number of protons found experimentally to be projected in a given direction, to the number predicted by the Rutherford law. The fact that R refers to the particles struck on from rest makes a considerable simplification in the collision relations, for if we consider the collision of two particles of mass m 1 and m 2 , of which m 1 is initially at rest, and if we denote by א the angle made by the final direction of motion of this particle with the incident direction of m 2 , and by θ the angle through which either particle is deflected, measured in a frame of reference moving with the velocity of the centre of gravity of the two particles, then א = ½ (π - 0), (1) independently of the ratio m 1 / m 2 . The relation between the angle θ and the angle ϕ, through which the incident particle m 2 is deflected, is on the other hand dependent on the relative masses of the particles.