Precision Muonic-Atom Measurements of Nuclear Quadrupole Moments and the Sternheimer Effect in Rare-Earth Atoms

The ground-state quadrupole moments of ${}_{}{}^{151}{}_{}{}^{}\mathrm{Eu}$, ${}_{}{}^{153}{}_{}{}^{}\mathrm{Eu}$, ${}_{}{}^{155}{}_{}{}^{}\mathrm{Gd}$, ${}_{}{}^{157}{}_{}{}^{}\mathrm{Gd}$, ${}_{}{}^{159}{}_{}{}^{}\mathrm{Tb}$, ${}_{}{}^{163}{}_{}{}^{}\mathrm{Dy}$, ${}_{}{}^{167}{}_{}{}^{}\mathrm{Er}$, ${}_{}{}^{177}{}_{}{}^{}\mathrm{Hf}$, ${}_{}{}^{179}{}_{}{}^{}\mathrm{Hf}$, ${}_{}{}^{191}{}_{}{}^{}\mathrm{Ir}$, and ${}_{}{}^{193}{}_{}{}^{}\mathrm{Ir}$ were determined with an uncertainty of less than one percent by measuring the quadrupole hyperfine-splitting energies of muonic $M$ x rays. The results are used to determine experimentally Sternheimer shielding factors for the $4f$, $5d$, and $6p$ electronic states of the respective atoms. The deduced shielding factors for the $5d$ electronic states were found to vary considerably among these elements, presumably as a result of configuration mixing.