Nuclear Polarization in Muonic Atoms of Spherical Nuclei

The nuclear polarization effect in muonic atoms is calculated for ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$. The nuclear intermediate states of ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ are collective particle-hole states calculated from realistic nucleon-nucleon forces. The total effect is an increase of binding of 6.0±0.6 keV for the $1s$ level, 1.9±0.2 keV for the $2p$ level, and 1.2±0.2 keV for the $2s$ level. We are able to set the uncertainties of the calculation at the relatively small values given above for the following reasons: The muonic intermediate states are treated exactly with the reference-spectrum method; the energies of nuclear intermediate states of $l>\mathrm{}2\mathrm{}$ turn out to be small compared with muonic intermediate state energies; the nuclear dipole states have been well studied experimentally and theoretically; and the nuclear monopole polarization calculated from the particle-hole states has been checked with a Thomas-Fermi theory which gives the observed surface thickness and the surface energy. Similar calculations for ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ give the correction to the $1s$ level as 0.2 keV. Henley and Cooper's closure approximation, which leads to a large correction of 58 keV to the ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ $1s$ level, is repeated with two of the following improvements: The intermediate muonic states in the continuum are treated properly, and the two-nucleon cross terms in the ground-state nucleus are included. The correction then comes down to 6 keV.