Bremsstrahlung and the Photoelectric Effect as Inverse Processes

The bremsstrahlung matrix element at the short-wavelength limit of the spectrum is calculated to lowest order in α≡Z/137, for an unscreened Coulomb field. The result, valid for relativistic incoming electrons, is shown to be exactly α^-1m^-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 α, and they are found to be derivable from the first two terms of the expansions in powers of α of the electron wave functions. In this sense their structure is completely analogous to that of the Bethe-Heitler bremsstrahlung matrix element.This simple relation between the matrix elements derives from an approximate equality (through first order in α) between the Coulomb wave functions for bound and zero-momentum continuum states, which can be understood as due to the neglect of the Coulomb binding energy, a second-order quantity in α.Finally, the range of validity of Sauter's approximation is examined in detail. The lower bound of this (energy) range is found to be simply related to the radius of convergence of the expansion of the photoeffect matrix element in powers of α.