Screening effects in radiative orbital electron capture by an atomic nucleus

Abstract

Using the results of an analytic perturbation theory for screened Coulomb wave functions, closed-form expressions are given for screening corrections to internal bremsstrahlung accompanying nuclear capture from $\mathrm{ns}$ and $2p$ states within nonrelativistic dipole approximation. The analytic perturbation theory is based on the expansion of the potential in the interior of an atom as a series of the form $V(r)=\ensuremath{-}(\frac{a}{r})[1+{V}_{1}\ensuremath{\lambda}r+{V}_{2}{(\ensuremath{\lambda}r)}^{2}+{V}_{3}{(\ensuremath{\lambda}r)}^{3}+\dots{}]$, where $a=Z\ensuremath{\alpha}, \ensuremath{\lambda}$ is a $Z$-dependent small parameter characterizing the screening, and the coefficients ${V}_{k}$ are of order unity. The nonrelativistic screened Coulomb Green's function, needed for the calculations of the matrix elements, is obtained as a series in $\ensuremath{\lambda}$ with analytic coefficients. The screening correction to radiative capture from $\mathrm{ns}$-states, in agreement with the arguments of Glauber and Martin, is simply the energy independent, multiplicative factor of the ratio of screened to unscreened bound state normalization. In radiative capture from the $2p$ state there are additional energy dependent screening corrections, largest near the resonance at an energy corresponding to characteristic x-ray radiation (due to the shift of the resonance energy position with screening), which decrease to less than 1% for photon energies several times larger than the energy of the resonance. The relation between the screening corrections to radiative capture from the $2p$ state at resonance and to the $2p\ensuremath{-}1s$ bound-bound radiative transition serves as a check of the calculation. From the screened Green's function the Fermi-Segr\`e formula, connecting the $\mathrm{ns}$-bound state energy shifts and the bound state normalizations, is rederived, together with a generalization to arbitrary angular momentum states.RADIOACTIVITY Screening corrections to internal bremsstrahlung accompanying nuclear capture of an orbital electron from $\mathrm{ns}$ and $2p$ states within nonrelativistic dipole approximation.