Radiative Capture of Orbital Electrons

A theory is developed of the continuous radiation spectrum which accompanies nuclear capture of atomic electrons. It is shown that quantitative predictions of the spectrum intensities must take into account the influence of the electrostatic field of the nucleus on the radiation process. This is accomplished by evaluating and making use of a particularly simple form of the Green's function for electron propagation in a Coulomb field. In a first approximation which treats the atomic electrons nonrelativistically, the spectra radiated by electrons captured from $S$-states are shown at all energies to have the form $x{\mathrm{}(1-x)\mathrm{}}^2$, where $x=\frac{E}{E_{max}}$. Radiative capture of electrons from $P$-states is shown to produce a spectrum which becomes extremely intense at low energies, where it merges continuously with the characteristic x-ray spectrum. Certain relativistic corrections to the $S$-state radiative capture probabilities are evaluated and shown to bring about an energy-dependent reduction of the intensities of the corresponding spectra. Functions are tabulated from which the spectra for allowed capture from various orbital states of any element may be determined. The calculated spectra are found to be in satisfactory agreement with those observed experimentally. In particular, the unexpectedly high intensities found at low $\gamma$-ray energies are explained by the radiatively induced capture of electrons from $P$ states.