Radiative corrections to atomic photoeffect and tip bremsstrahlung. III

Abstract

The radiative corrections to the photoeffect from the $K$ shell were evaluated recently by McEnnan and Gavrila assuming a hydrogenlike atomic model. The result was expressed in the form of a corrective factor [$1+(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}})\ensuremath{\delta}$] multiplying the basic photoeffect differential cross section. An expression for $\ensuremath{\delta}$, correct to lowest order in $\ensuremath{\alpha}$ and $\ensuremath{\alpha}Z$, was derived in analytic form in terms of a large number of single and double integrals over Feynman parameters, requiring numerical integration. $\ensuremath{\delta}$ depends on the photon energy $\ensuremath{\omega}$, on the electron-ejection angle $\ensuremath{\theta}$, and on the energy threshold $\ensuremath{\Delta}E$ below which we allow the additional (soft) photon, emitted together with the ejected electron to go undetected. The $\ensuremath{\Delta}E$ dependence of $\ensuremath{\delta}$ reflects the inseparable connection between the $K$-shell photoeffect and the Compton effect from the $K$ shell with (soft) photons emitted in the range $\ensuremath{\Delta}E$. It was shown that the same expression for $\ensuremath{\delta}$ with an appropriate redefinition of the variables, also radiatively corrects the bremsstrahlung spectrum at its high-energy tip. In the present work we have carried out the computation of $\ensuremath{\delta}$. It covers the $\ensuremath{\omega}$ energy range from 0.5 keV to 50 MeV at all relevant angles. The relative error was kept below 0.001, except at high energies where it was only 0.01. We find that ($\ensuremath{-}\frac{\ensuremath{\alpha}\ensuremath{\delta}}{\ensuremath{\pi}}$) is always positive and increases with $\ensuremath{\theta}$ and $\ensuremath{\omega}$. At energies of about 500 keV, ($\frac{\ensuremath{\alpha}\ensuremath{\delta}}{\ensuremath{\pi}}$) becomes of the order of 1% and at 5 MeV it has grown to about 5%, when taking $\ensuremath{\Delta}E=0.01$${\mathit{mc}}^{2}$. By integrating over the angles we have derived the quantity $\ensuremath{\Delta}$ which radiatively corrects the total photoelectric cross section, in the form of a factor [$1+(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}})\ensuremath{\Delta}$]. We also infer by exponentiation the correct form of $\ensuremath{\delta}$ and $\ensuremath{\Delta}$ in the limit $\ensuremath{\Delta}E\ensuremath{\rightarrow}0$. Finally, we derive the expression for the cross section describing the energy spectrum of the electrons emitted in the vicinity of the photoelectric peak, which is the basic information coming from coincidence experiments.