Effect of Nuclear Charge on Internally Produced Pairs

Abstract

The effect of nuclear charge on internally converted pairs is investigated using the relativistic Sommerfeld-Maue solutions of the iterated Dirac equation. The orthonormal properties of these wave functions are studied and time-dependent perturbation theory is adapted to allow for their nonorthogonality. The integrals involved are studied using the Fourier transforms of the wave functions. The matrix elements are obtained in terms of one fundamental integral, evaluated by using an integral representation of Butler. The procedure is very simple and promises generalizations. The matrix elements are shown to be complex only through $n=\ensuremath{\mp}\frac{\mathrm{iZ}}{(137{v}_{\ifmmode\pm\else\textpm\fi{}})}$, where ${v}_{\ifmmode\pm\else\textpm\fi{}}$ is the velocity of the positron or the negatron and $Z$ is the nuclear charge. The following conclusions are drawn. Firstly, the first Born approximation results when multiplied by the well-known Sommerfeld factors of the negatron and positron are accurate to a term proportional to ${|n|}^{2}$. For 5-percent resolutions, $Z\ensuremath{\lesssim}20$, the kinetic energy must be more than 100 kev. Secondly, this result is valid for all electric and magnetic multipole transitions. Finally, this result is shown to apply to all transitions taking place between unbound states and to hold for higher order perturbation terms.