Neutron occupancies and single-particle energies across the stable tin isotopes

The occupancies and vacancies of the valence neutron orbitals across the stable tin isotopic chain from $112\le A\le 124$ have been determined. These were inferred from the cross sections of neutron-adding and -removing reactions. In each case, the reactions were chosen to have good angular-momentum matching for transfer to the low- and high-$\ell$ orbitals present in this valence space. These new data are compared to older systematic studies. The effective single-neutron energies are determined by combining information from energy centroids determined from the adding and removing reactions. Two of the five orbitals are nearly degenerate, below $N=64$, and approximately 2 MeV more bound than the other three, which are also degenerate.

Figure 1

The filling of neutrons in the $0g_{7/2}, 1d_{5/2}, 2s_{1/2}, 1d_{3/2}$, and $0h_{11/2}$ orbitals across the stable, even-$A$ tin isotopes as derived from this work. The horizontal bars are the nominal number of neutrons above $N=50$ for each isotope. The missing $0g_{7/2}$ strength (hatched) and the uncertainties are discussed in the text.

Figure 2

The ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}$ and ${}^{116}\mathrm{Sn}(\alpha ,{}^3\mathrm{He}){}^{117}\mathrm{Sn}$ reactions at 15 MeV (${\theta }_{\mathrm{lab}}={18}^{\circ }$) and 41 MeV (${\theta }_{\mathrm{lab}}=10.9^{\circ }$), respectively, are shown in (a). Selected states are labeled by the transferred angular momentum $\ell$ to highlight the different matching conditions. (b) is the same ($d,p$)-reaction data but with a logarithmic $y$ axis to emphasize the details of the spectrum. Similarly, the ${}^{116}\mathrm{Sn}(p,d){}^{115}\mathrm{Sn}$ and ${}^{116}\mathrm{Sn}({}^3\mathrm{He},\alpha ){}^{115}\mathrm{Sn}$ reactions at 21 MeV (${\theta }_{\mathrm{lab}}={18}^{\circ }$) and 36 MeV (${\theta }_{\mathrm{lab}}=5.9^{\circ }$), respectively, in (c) and (d). The broad peak around 2.3 MeV in (a) and (b) is from reactions on the carbon target backing.

Figure 3

Example angular distributions for (a) $\ell =0$ and 2 transfer via the ($d,p$) and ($p,d$) reactions on ${}^{112}\mathrm{Sn}$ and ${}^{116}\mathrm{Sn}$ and (b) $\ell =4$ and 5 transfer via the (${}^3\mathrm{He},\alpha$) reaction on ${}^{122}\mathrm{Sn}$. The solid curves are the assigned $\ell$ value and the dashed curves are for alternative, albeit similar, $\ell$ values as discussed in the text

Figure 4

The summed strength for each orbital, highlighting the consistency. For the $0g_{7/2}$ orbital for $A>116$, there is reason to believe that strength at high excitation energies was outside the range studied here (see text for details), and this is emphasized by showing the missing portion as hatching.

Figure 5

The fractional occupancy of the neutron $0g_{7/2}, 1d_{5/2}, 2s_{1/2}, 1d_{3/2}$, and $0h_{11/2}$ orbitals as deduced in this work. The dashed lines are to guide the eye and the $0g_{7/2}$ data with open symbols have been adjusted using an approximation discussed in the text.

Figure 6

The fractional occupancy of the neutron $0g_{7/2}, 1d_{5/2}, 2s_{1/2}, 1d_{3/2}$, and $0h_{11/2}$ orbitals as deduced in the study of Ref. [5]. The dashed lines are are the same as in Fig. 5.

Figure 7

A comparison of the fractional occupancies determined in this work compared with shell-model calculations from Ref. [45], for which information is available for the $1d_{5/2}, 0g_{7/2}$, and $0h_{11/2}$ orbitals for $102\le A\le 132$, and for the $2s_{1/2}$ and $1d_{3/2}, 112\le A\le 124$ only.

Figure 8

The effective single-particle energies for neutron orbitals across the stable, even-$A$ tin isotopes.

Figure 9

The effective single-particle energies for neutron orbitals across the stable, even-$A$ tin isotopes contrasted with those of protons for the $0g_{7/2}$ and $0h_{11/2}$ orbitals, which are well determined from the ($\alpha ,t$) reaction [39].

Figure 10

The distribution of the effective single-neutron energies about the Fermi surface for ${}^{112-124}\mathrm{Sn}$ (a)–(g) as a function of fractional occupancy, $V^2$. The solid curves are the BCS occupation probabilities calculated with parameters of $\lambda$ varying smoothly from $-9.4$ MeV to $-7.3$ MeV across the range and $\mathrm{\Delta }=1.2$ MeV, as defined in the text, aside from the dashed lines (d-g) which are for $\mathrm{\Delta }=2$ MeV.