Analysis of Muonic- Atom X Rays in the Lead Isotopes

We apply a radial-moment analysis to existing muonic-atom data for the nuclei ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$, ${}_{}{}^{207}{}_{}{}^{}\mathrm{Pb}$, ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$, and ${}_{}{}^{204}{}_{}{}^{}\mathrm{Pb}$, with emphasis on the directly measured isotope shifts and with special attention to charge-density functions of the forms that have been used to fit high-energy electron scattering data. The accuracy of the method is investigated and found to be sufficient for a detailed interpretation of the presently available data. When analyzed in terms of radial moments, the isotope shifts show clearly that the charge-density function must change significantly in shape (not merely in radial extent) in order to describe all of the isotopes. A like pair is formed by ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$-${}_{}{}^{207}{}_{}{}^{}\mathrm{Pb}$, which differ only in radial scale. A similar pair appears to be formed by ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$-${}_{}{}^{204}{}_{}{}^{}\mathrm{Pb}$. These last two isotopes have significantly less diffuse surfaces than ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{207}{}_{}{}^{}\mathrm{Pb}$. We find, in agreement with other analyses, that if the charge distribution is chosen to best fit those transitions which do not involve the $1s$ state, the resulting predicted binding of the $1s$ state is about 3 keV less in all four isotopes than is indicated by experiment. This difference is not experimentally significant, since it is possible to fit all transitions which are strongly affected by the charge distribution. Our analysis brings out a significant and still unexplained discrepancy between muonic-atom isotope shift measurements and electronic isotope shift measurements, the muonic shifts being typically larger by a factor of 1.5.