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, 〈rᵏ〉. 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 Rₖ, defined by Rₖ=[1/3(k+3)〈rᵏ〉]1/k. For $Z=82$, a range of moments from $k=0.08$ to $k=4.80$ is provided by available data. The 2p1/2-1s1/2 transition in lead, for example, measures the $k=0.80$ moment and determines R0.87.013-1.49(E-5.788) 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.