Neutron $3s_{1/2}$ occupation change across the stable tin isotopes investigated using isotopic analysis of proton scattering at 295 MeV

The isotopic systematics of Sn isotopes in the range $A=116$–124 were investigated by combining the nuclear structure and reaction calculations for the analysis of ${}^A\mathrm{Sn}(p,p)$ reactions at 295 MeV. The ${}^A\mathrm{Sn}(p,p)$ reactions were calculated employing the relativistic impulse approximation (RIA) with theoretical densities obtained for the Sn isotopes from relativistic Hartree-Bogoliubov (RHB) calculations with the DD-ME2 interaction and nonrelativistic Skyrme Hartree-Fock-Bogoliubov (SHFB) calculations with the SKM* interaction. Calculation using the DD-ME2 density reproduced the experimental data for ${}^{122}\mathrm{Sn}(p,p)$ but overestimated the ${}^{116}\mathrm{Sn}(p,p)$ and ${}^{118}\mathrm{Sn}(p,p)$ cross sections at backward angles. In the isotopic analysis of the cross section ratio $R\left(\sigma \right)$ of ${}^{122}\mathrm{Sn}$ to ${}^{116}\mathrm{Sn}$, a calculation using the SKM* density reproduced the peak amplitudes of $R\left(\sigma \right)$ obtained from the experimental cross sections, whereas a calculation using the DD-ME2 density did not. The ratio $R\left(\sigma \right)$ was found to be sensitive to the isotopic change of the neutron $3s_{1/2}$ occupation through the isotopic difference of the surface neutron density around $r=4$–5 fm. Isotopic analysis indicated that a rapid increase of the $3s_{1/2}$ occupation from ${}^{116}\mathrm{Sn}$ to ${}^{122}\mathrm{Sn}$ obtained by the DD-ME2 calculation is unlikely. This result derived from the 295 MeV ${}^A\mathrm{Sn}(p,p)$ cross section data [S. Terashima et al., Phys. Rev. C 77, 024317 (2008).] is consistent with the direct measurement of the neutron occupations in Sn isotopes by neutron transfer and pickup reactions [S. V. Szwec et al., Phys. Rev. C 104, 054308 (2021).

Figure 1

rms neutron ($r_n$) and proton $r_p$ radii of Sn isotopes. The theoretical values are the RHB (me2 and pc1) and the SHFB (SKM*) calculations. The experimental $r_n$ values are those determined by the $(p,p)$ reactions at both 295 MeV[4] and 800 MeV [2], while the experimental $r_p$ values are obtained from the rms charge radii observed by isotope-shift measurements [18].

Figure 2

Neutron (${\rho }_n^A$) and proton (${\rho }_p^A$) density distributions for ${}^{116}\mathrm{Sn}$ and ${}^{122}\mathrm{Sn}$. (a) and (b) Neutron and proton density distributions for ${}^{116}\mathrm{Sn}$ and ${}^{122}\mathrm{Sn}$, respectively, as obtained from the me2, pc1, and SKM* calculations. (c) and (d) The corresponding values of $4\pi r^2\rho$. The experimental neutron and proton densities from Ref. [4] are also presented (“Exp.fit”). The neutron density (error envelopes surrounded by thin lines) was determined by proton elastic scattering at 295 MeV, while the proton density (thin lines) was obtained from the charge distribution determined by electron elastic scattering.

Figure 3

Rutherford ratios of the $\mathrm{Sn}(p,p)$ cross sections at 295 MeV obtained from the $\mathrm{RIA}+\mathrm{ddMH}$ calculations using the me2, pc1, and SKM* densities, together with the experimental data [4].

Figure 4

Isotopic cross section ratio $R\left(\sigma \right)$ of ${}^{122}\mathrm{Sn}(p,p)$ to ${}^{116}\mathrm{Sn}(p,p)$ at 295 MeV. Theoretical results obtained by the $\mathrm{RIA}+\mathrm{ddMH}$ calculations using the me2 and SKM* densities are compared with the experimental data [4].

Figure 5

Neutron single-particle energies in ${}^{116}\mathrm{Sn}$ and ${}^{122}\mathrm{Sn}$ obtained from the me2 and SKM* calculations compared with the experimental data from Ref. [19].

Figure 6

Occupation probabilities of neutron single-particle orbits in the Sn isotopes obtained from the (a) me2 and (b) SKM* calculations.

Figure 7

(a) Neutron densities of ${}^{116}\mathrm{Sn}$ for the me2 and $\mathrm{me}2+3\mathrm{s}$ densities. For reference, the neutron density of ${}^{122}\mathrm{Sn}$ is also presented. (b) Isotopic difference in the neutron density of ${}^{122}\mathrm{Sn}$ from ${}^{116}\mathrm{Sn}$ obtained for the me2, $\mathrm{me}2+3\mathrm{s}$, and SKM* densities. (c) Single-particle densities ${\rho }^{\text{s.p.}}\left(r\right)$ of the $3s_{1/2}, 2d_{3/2}$, and $1h_{11/2}$ orbits in ${}^{120}\mathrm{Sn}$ obtained from the me2 calculation.

Figure 8

(a) The ${}^{116}\mathrm{Sn}(p,p)$ cross sections at 295 MeV and (b) isotopic cross section ratios $R\left(\sigma \right)$ of ${}^{122}\mathrm{Sn}(p,p)$ to ${}^{116}\mathrm{Sn}(p,p)$ obtained by the $\mathrm{RIA}+\mathrm{ddMH}$ calculations using the me2 and $\mathrm{me}2+3\mathrm{s}$ densities, compared with the experimental data [4].