Mu-Mesonic X-Rays and the Shape of the Nuclear Charge Distribution

Mu-mesonic x-rays, fast electron scattering, and electronic spectra each depend on the nuclear charge distribution $\rho \mathrm{}\left(r\right)\mathrm{}$ in different ways. Adjoining theoretical analyses of observations on these different phenomena may therefore bracket an acceptable set of functional descriptions for $\rho \mathrm{}\left(r\right)\mathrm{}$. The present paper reports the dependence of the $\mu$-mesonic x-ray spectrum of Pb on the extent and shape of the nuclear charge density. For several functional families representing conceivable forms for $\rho \mathrm{}\left(r\right)\mathrm{}$ we have obtained the resultant Coulomb potentials and have solved "exactly" by electronic computation the relativistic equations to yield the first four levels ($1S$, $2S$, and $2P$) and transition energies for mu mesons bound in the field of the lead nucleus. The next two levels ($3D$) have been found by perturbation theory. Effects omitted from the calculations have been examined and theoretical uncertainties estimated for each level. These results permit adjustment of the effective electric radius $R$, for each form of $\rho \mathrm{}\left(r\right)\mathrm{}$ to agree with known x-ray measurements of the $2P\to 1S$ transitions. Calculations completed for neighboring values of $R$ permit interpolation to improved values when more precise measurements are made and when small corrections of the raw data for electrodynamic and special coupling effects have been carried through. The doublet splitting of the $2P$ levels offers an independent measure of $R$, as well as a check on the possible existence of an anomalous muon magnetic moment. In contrast, the $3D\to 2P$ and $2S\to 2P$ transition energies are insensitivie to $R$ but sufficiently sensitive to the shape of the charge distribution, so that accurate measurements of the transition energies between the six lowest muon levels in the heavy nuclides will be able to provide information on both the electrical radius and the shape of nuclear electrification, independently of indications from the other phenomena mentioned above.