Nuclear charge distributions in1f72-shell nuclei from muonic x-ray measurements

Abstract

Precise $2p\ensuremath{-}1s$ muonic x-ray transition energies and energy shifts were measured for the $1{f}_{\frac{7}{2}}$-shell nuclei $^{39,41}\mathrm{K}$, $^{40,42,43,44,46,48}\mathrm{Ca}$, $^{45}\mathrm{Sc}$, $^{46,47,48,49,50}\mathrm{Ti}$, $^{51}\mathrm{V}$, $^{50,52,53,54}\mathrm{Cr}$, $^{55}\mathrm{Mn}$, and $^{56}\mathrm{Fe}$. The data were analyzed in terms of the Barrett moments $〈{r}^{k}{e}^{\ensuremath{-}ar}〉$ of the nuclear charge distribution, from which equivalent nuclear radii ${R}_{k}$ and isotopic and isotonic differences $\ensuremath{\Delta}{R}_{k}$ were computed. For those nuclei for which electron-scattering data were available, model-independent rms radii and radius differences were deduced from a combined analysis of the present data and elastic electron-scattering data. The $\ensuremath{\Delta}N=2$ isotope shifts between even-$A$ nuclei decrease nearly linearly with increasing $N$ and become negative in the second half $24\ensuremath{\le}N\ensuremath{\le}28$ of the $1{f}_{\frac{7}{2}}$ shell; a strong shell-closure effect is evident at $N=28$. The isotope shifts are independent of $Z$, suggesting that the added neutrons interact with the entire proton core rather than with the valence protons. The $\ensuremath{\Delta}Z=2$ isotone shifts between even-$A$ nuclei decrease smoothly and uniformly with increasing $Z$ for $20\ensuremath{\le}Z\ensuremath{\le}28$ and are essentially independent of $N$. Like the isotope shifts, the isotone shifts display a sudden increase at $Z=28$. Both the $\ensuremath{\Delta}N=1$ isotope shifts and the $\ensuremath{\Delta}Z=1$ isotone shifts show pronounced odd-even staggering. A strong correlation is found between nuclear deformation, as revealed by experimental $B(E2)$ values, and the measured isotope and isotone shifts. The measured isotope and isotone shifts, including those for odd-$A$ nuclei, are satisfactorily described by a simple empirical formula. The experimental results were compared with Hartree-Fock calculations; some recent calculations that include ground-state correlations display improved agreement with experiment through the inclusion of these deformation-related effects.NUCLEAR STRUCTURE $^{39,41}\mathrm{K}$, $^{40,42,43,44,46,48}\mathrm{Ca}$, $^{45}\mathrm{Sc}$, $^{46,47,48,49,50}\mathrm{Ti}$, $^{51}\mathrm{V}$, $^{50,52,53,54}\mathrm{Cr}$, $^{55}\mathrm{Mn}$, $^{56}\mathrm{Fe}$; measured muonic x-ray spectra; deduced nuclear charge radii, isotope and isotone shifts; performed model-independent combined analysis of muonic x-ray data and elastic electron scattering data; compared charge parameters with Hartree-Fock and other calculations.