Relativistic Radiative Transitions

We have calculated the oscillator strength for electric dipole transitions with retardation between the $1s$ state and higher discrete states for a single Dirac electron in a Coulomb field. Numerical values for atomic number 82 show that the retarded relativistic oscillator strength for each shell is about 0.8 of the nonrelativistic value. We give numerical values for the relative intensity of the different $K$ x-ray lines for lead, neglecting screening: the $\frac{K{\alpha }_1}{K{\alpha }_2}$ doublet intensity ratio is 1.93 as compared to the NR value of 2.0. We introduce an average "oscillator density" for transitions for discrete states, and extrapolate to the series limit to find the photoelectric cross section at threshold. Our numerical value for lead of 740 barns is 23% larger than the value given by Hulme et al. We also calculate the summed oscillator strength, using a cutoff at an arbitrary high energy to give convergence. The numerical results for lead are insensitive to the numerical value of the cutoff used: using 10 Mev, we find a summed oscillator strength of 0.85, with an increase of 0.01 for each factor of two increase in the cutoff energy. Our value of the summed oscillator strength is less than that calculated nonrelativistically by Thomas, Reiche, and Kuhn; or relativistically by Gell-Mann, Goldberger, and Thirring.