Corrections to Muonic X Rays and a Possible Proton Halo

With the aim of extracting maximum information on nuclear charge structure at large distances, we improve the theory of muonic x rays in ${\mathrm{Bi}}^{209}$ by taking into account all known effects which could change the binding energy by as much as 0.1%. We give improved estimates of vacuum-polarization and Lamb-shift effects, and incorporate Cole's recent estimate of nuclear polarization effects. Using these corrections to analyze recent experiments on muonic x rays, we obtain parameters for the charge distribution in ${\mathrm{Bi}}^{209}$ which, when compared with parameters from electron scattering, indicate a small discrepancy. We suggest that this can be explained by a proton "halo," by which we mean a small fraction of the charge (∼1%) spread over large distances (∼8 F); we show that such a "halo" is not inconsistent with electron-proton scattering and brings the theory of the Lamb shift in atomic H and D into good agreement with experiments. We find no experiment with which such a model is inconsistent.