The creation of electron pairs by fast charged particles

Abstract

We shall discuss in this paper the creation of electron-pairs by the collision of fast charged particles. This calculation goes farther than other calculations on this subject in considering the effect of screening, and in investigating the probability of the creation of a pair as a function of impact parameter, i . e ., the least distance of approach between the two colliding particles. We shall also treat certain other cases which have not been considered before, among them the creation of very slow pairs such that the kinetic energies of the electron and positron of the pair are small compared to their rest energy. When the energy of one of the colliding particles is large compared with its rest mass, we shall also show that to a certain approximation most of the formulae given by the direct calculation can be obtained quite simply by a method similar to that given by Weizsäcker for calculating the emission of radiation by fast electrons on colliding with nuclei. The procedure consists in calculating the probability of the transition of an electron from its initial state of negative energy to a final state of positive energy under the perturbing influence of the two colliding particles, the electron and resulting hole then appearing as the electron and positron of the created air. We shall throughout use the Born approximation, in which the interaction between the particles is treated as a perturbation. The transition from the initial to the final state of the system can then happen in two ways. The electron in the negative energy state may either interact with one of the colliding particles and jump at once to its final state, the colliding particle going over into an inter-mediate state. This particle can then interact with the other colliding particle and both jump to their final states. Or, the electron in the negative energy state may interact with one of the colliding particles and jump to an intermediate state, after which its interaction with the other colliding particle causes it to jump to its final state. Both processes are strictly of the second order, but for brevity we shall call the former process a “first-order process,” only in the sense that it involves just one matrix element of the interaction of the electron of the created pair with the colliding particles. The second process involves two matrix elements of the interaction of the electron of the created pair with the colliding particles, and we shall call it a “second-order process.”