Coulomb Corrections in the Theory of Internal Bremsstrahlung

The internal bremsstrahlung associated with allowed $\beta$ decay is calculated for the case for which the gamma-ray energy is less than $2m{c}^2$, the kinetic energy of the electron in its final state is small compared to $m{c}^2$, and $\frac{Z{e}^2}{\hslash c}$ is small compared to 1. It is not assumed that $\left(\frac{Z{e}^2}{\hslash c}\right)mc$ is small compared to the final momentum of the electron, or to the gamma-ray momentum. Results are obtained for the gamma-ray energy spectrum and for the angular correlation between the electrons and the gamma rays. For ${\mathrm{S}}^{35}$, for which the above assumption would seem to be satisfied, the agreement between theory and experiment for the number of gamma rays per $\beta$ disintegration per $m{c}^2$ is better than that previously obtained; because of uncertainties in the experimental results, the extent of the improvement is not clear. Under the additional assumption that the final kinetic energy of the electron is small compared to the gamma-ray energy, an expression is derived for the polarization of the gamma rays.