# Radial-moment analysis for muonic x-rays

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The crucial question to be answered in studying the radial shapes of spherical nuclei is: Ã©xactly what parameter of the nuclear charge distribution is determined by each transition energy in muonic atoms? The measurement of the 2p , ->-ls. . energy for a particular muonic atom determines, of course, only a single parameter of the nuclear charge distribution. Since the earliest days of analyzing muonic x-ray energies, workers have recognized that different transitions determine different parameters, so that, in principle, energy measurements for a number of different transitions in a given atom might provide information about the radial shape of the nuclear charge density. T ^Y radial shape, we mean simply the form of the charge density p(r) "] . However, it has not been clear exactly what nuclear parameter is determined by a particular transition, except in the low-Z limit, where non-relativistic perturbation theory shows that level shifts are proportional to even moments of the charge density. In this limit, for instance, the 2p->-ls transitions measure the mean square radius . One occasionally hears or reads that the same is true for heavy atoms. This is not then case. Even for Z as low as 6, the 2p-^ls transitions do not determine exactly the mean square radius.