Does size matter? Exploring the viability of measuring the charge radius of the first excited nuclear state in muonic zirconium

Abstract

From the point of view of the electromagnetic interaction, empirical descriptions of the nucleus involve only a few parameters, one of the most important being the nuclear charge radius. This has been well measured for ground state nuclei, but it is difficult to measure for excited states, since they decay too quickly for conventional methods to be used. We study the atomic transitions in muonic ^{90}Zr and find that the nuclear charge radius of the first excited state can be inferred by measuring the gamma emissions from certain transitions. We find that with 1keV photon resolution, we can infer a difference between the charge radius of the nuclear ground state and first excited state as small as 0.13%. We will work in units where h = c = 4\pi\epsilon_0 = 1 so that e^2 = \alpha \approx 1/137 (unless otherwise specified). Mass, momentum, and energy will have units of eV, whereas distances will be given in eV^{-1}. In qualitative discussion, we will sometimes revert to discussing distances in meters due to the familiarity of typical scales (e.g. nuclear radius, Bohr radius). When working with 4-vectors in Minkowski space, we use the metric convention (+,-,-,-).