Evidence for Higher Shape Deformations in Muonic X Rays

We have fitted the $K$ x rays and hyperfine splittings from muonic ${}_{}{}^{232}{}_{}{}^{}\mathrm{Th}$ and ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ by a model with a Fermi charge distribution with distortion terms of the type ${\beta }_{2n}{Y}_{2n,0}$, $n<~3$. We have also reanalyzed for $n<~2$ the data for rare-earth deformed muonic atoms. Our results are compared with those of inelastic proton and $\alpha$ scattering as well as Coulomb excitation. Such muonic analyses may provide the preferred determinations of quadrupole moments.