Theory of Bremsstrahlung and Pair Production. I. Differential Cross Section

Abstract

The differential cross sections for bremsstrahlung and pair production are calculated without the use of the Born approximation, assuming the energy of the electron to be large compared to $m{c}^{2}$ both in initial and final state. The wave functions in initial and final state are essentially those previously proposed by Furry (Sec. II). It is proved in Sec. III that our wave functions agree with the exact ones of Darwin except for terms of relative order $\frac{{a}^{2}}{{l}^{2}}$, where $a=\frac{Z{e}^{2}}{\ensuremath{\hbar}c}$ and $l$ the angular momentum, and that this agreement holds for any energy of the electron. An independent proof is given in Sec. IX, showing that the Furry wave functions give the matrix element correctly except for terms of relative order $\frac{1}{\ensuremath{\epsilon}}$.In the matrix element for bremsstrahlung, the initial state of the electron must be represented by a plane wave plus an outgoing spherical wave, whereas the final state has an ingoing spherical wave (Sec. IV). In pair production, both electrons contain ingoing spherical waves (Sec. V). This causes essential differences between the cross sections for the two processes.The cross section for pair production is calculated in Sec. VI; the result consists of the Bethe-Heitler formula multiplied by a relatively simple factor, plus another term of similar structure. A simplified derivation is given, which is valid for the important case of small angles between electrons and quantum (Sec. VII); it provides a useful check of the cross section of Sec. VI. In Sec. VIII, the bremsstrahlung cross section is calculated and found to be the Bethe-Heitler result multiplied by a factor. This factor is different from that encountered in pair production and becomes important only for very small momentum transfer $q$. In the limit of complete screening, these small $q$ do not contribute and the cross section goes over into that of the Born approximation.The error in the cross sections calculated in this paper is estimated (Sec. X) to be of order $\frac{1}{\ensuremath{\epsilon}}$, where $\ensuremath{\epsilon}$ is the energy of the final electron in bremsstrahlung, or that of the less energetic electron in pair production, in units of $m{c}^{2}$. The total cross section for pair production by a quantum of energy $k$ may be in error by $\frac{logk}{k}$.