Systematics of nuclear charge distributions in Fe, Co, Ni, Cu, and Zn deduced from muonic x-ray measurements

The results of precise measurements of the energies of the $2{p}_{\frac{3}{2}}\ensuremath{\rightarrow}1{s}_{\frac{1}{2}}$ and $2{p}_{\frac{1}{2}}\ensuremath{\rightarrow}1{s}_{\frac{1}{2}}$ muonic x-ray transitions of $^{54}\mathrm{Fe}$, $^{56}\mathrm{Fe}$, $^{57}\mathrm{Fe}$, $^{58}\mathrm{Fe}$, $^{59}\mathrm{Co}$, $^{58}\mathrm{Ni}$, $^{60}\mathrm{Ni}$, $^{61}\mathrm{Ni}$, $^{62}\mathrm{Ni}$, $^{64}\mathrm{Ni}$, $^{63}\mathrm{Cu}$, $^{65}\mathrm{Cu}$, $^{64}\mathrm{Zn}$, $^{66}\mathrm{Zn}$, $^{68}\mathrm{Zn}$, and $^{70}\mathrm{Zn}$, are reported. Using a highly linear digitally stabilized Ge(Li) spectrometer system, the absolute energies and energy shifts between nuclei were measured with total errors of approximately 40-60 eV (110 eV for $^{70}\mathrm{Zn}$). The data were analyzed in terms of the Barrett moments $〈{r}^{k}{e}^{\ensuremath{-}ar}〉$ of the nuclear charge distributions from which the equivalent nuclear radii ${R}_{k}$ and the isotopic and isotonic differences $\ensuremath{\delta}{R}_{k}$ were computed. Particular attention was given to higher-order corrections of the energies of the muonic states. Appropriate quantum-electrodynamical corrections were calculated to all significant orders. Nuclear polarization corrections for multipole interactions up to and including $L=4$ were computed for each isotope. The $\ensuremath{\Delta}A=2$ isotone shifts $\ensuremath{\delta}{R}_{k}$ for even $A$ isotopes show a strong shell closure effect at $Z=28$, which is quite independent of the neutron number. The $\ensuremath{\Delta}N=2$ isotope shifts between even nuclei decrease smoothly and uniformly with increasing $N$ from $N=28$ to $N=40$ and are essentially independent of $Z$. This unexpected behavior suggests that the added neutrons interact with the entire proton core rather than with the valence protons. The $\ensuremath{\Delta}N=1$ isotope shift results show a pronounced odd-even staggering effect, which, however, is somewhat smaller than theoretical predictions. The isotone series $^{58}\mathrm{Fe}$-$^{59}\mathrm{Co}$- $^{60}\mathrm{Ni}$, which is just below the $Z=28$ shell closure, shows strong odd-even staggering, whereas the series $^{64}\mathrm{Ni}$- $^{65}\mathrm{Cu}$-$^{66}\mathrm{Zn}$ and $^{62}\mathrm{Ni}$-$^{63}\mathrm{Cu}$- $^{64}\mathrm{Zn}$ just above $Z=28$ exhibit only a very small staggering effect. A comparison of the experimental data of the rms radii ${〈{r}^{2}〉}^{\frac{1}{2}}$ with the results of spherically constrained Hartree-Fock calculations shows good agreement for all four Zn isotopes and the heavier Ni isotopes ($^{60}\mathrm{Ni}$, $^{62}\mathrm{Ni}$, $^{64}\mathrm{Ni}$), but poor agreement for the Fe isotopes and $^{58}\mathrm{Ni}$.NUCLEAR STRUCTURE, MOMENTS $^{54,56,57,58}\mathrm{Fe}$, $^{59}\mathrm{Co}$, $^{58,60,61,62,64}\mathrm{Ni}$ $^{63,65}\mathrm{Cu}$, $^{64,66,68,70}\mathrm{Zn}$; measured muonic x-ray spectra; deduced nuclear charge parameters, isotope and isotone shifts. Calculated quantum-electrodynamic and nuclear-polarization corrections. Compared charge parameters with Hartree-Fock calculations.