Bremsstrahlung and the Photoelectric Effect as Inverse Processes

Abstract

The bremsstrahlung matrix element at the short-wavelength limit of the spectrum is calculated to lowest order in $\alpha \equiv \frac{Z}{137}$, for an unscreened Coulomb field. The result, valid for relativistic incoming electrons, is shown to be exactly ${\alpha }^{-1\mathrm{}}{m}^{-\frac{1}{2}}$ times the complex conjugate of Sauter's relativistic matrix element for the $K$-shell photoelectric effect. These matrix elements are the leading terms in an expansion of the exact matrix elements in powers of $\alpha$, and they are found to be derivable from the first two terms of the expansions in powers of $\alpha$ of the electron wave functions. In this sense their structure is completely analogous to that of the Bethe-Heitler bremsstrahlung matrix element.