Muonic Atoms and the Radial Shape of the Nuclear Charge Distribution

We find, for a rather broad set of two-, three-, and four-parameter charge-density functions, that a particular muonic-atom transition energy determines a particular moment of the nuclear charge density, $〈r^k〉$. The exponent $k$ depends on the atomic number $Z$ and on the quantum numbers of the pair of states in question, but it depends only very weakly on the mathematical form of the charge-density function. Consequently, an almost model-independent analysis of muonic-atom energies is possible. This analysis is facilitated by introduction of the equivalent radius $R_k$, defined by $R_k={\left[\frac{1}{3}\mathrm{}\right(k+3\mathrm{}\left)\mathrm{}〈r^k〉\right]}^{\frac{1}{k}}$. For $Z=82\mathrm{}$, a range of moments from $k=0.08\mathrm{}$ to $k=4.80\mathrm{}$ is provided by available data. The $2p_{\frac{1}{2}}-1\mathrm{}{s}_{\frac{1}{2}}$ transition in lead, for example, measures the $k=0.80\mathrm{}$ moment and determines $R_{0.8}\cong 7.013-1.49(E-5.788\mathrm{})\mathrm{}$ F, where $E$ is the transition energy in MeV. For this transition, as well a for several others, $k$ is approximately a linear function of $Z$.