Pair Production by Photons: Exact Calculation for Unscreened Atomic Field

Abstract

Exact relativistic Coulomb wave functions, in the form of partial-wave expansions, are used to obtain analytic formulas for the differential cross sections and energy spectrum for electron-positron pair production by photons. Radiative corrections, which are believed to be small (of the order of 1%), are neglected. The errors introduced by the finite nuclear size and recoil effects are quite negligible for the energy region considered here. Apart from these approximations, the present formulas give the exact cross sections for the unscreened atomic field. Numerical results are obtained for positron energy spectra for a large number of photon energies and atomic numbers in the ranges $2{m}_{e}{c}^{2}<k\ensuremath{\le}10{m}_{e}{c}^{2}$ and $1\ensuremath{\le}Z\ensuremath{\le}100$. Total cross sections are obtained by numerical integration of the spectra. A representative choice of spectra is given. The approximately 500 total cross sections calculated are given in tables and diagrams. By one- or two-way interpolation in these data the total cross section for any photon energy and nuclear charge within the given limits may be obtained with an accuracy of 1% or better. For heavy elements large Coulomb corrections to the Bethe-Heitler Born-approximation results are found. The positron energy spectra obtained are asymmetric, favoring positrons of high energy. The total cross section $\ensuremath{\sigma}$ is smaller than the Bethe-Heitler total cross section ${\ensuremath{\sigma}}_{B}$ for photon energies very close to the threshold (owing to the repulsion of the positron), while $\ensuremath{\sigma}$ is generally larger than ${\ensuremath{\sigma}}_{B}$ in the energy region $2.2{m}_{e}{c}^{2}\ensuremath{\lesssim}k\ensuremath{\lesssim}10{m}_{e}{c}^{2}$.