Structures of exotic ${}^{131,133}$Sn isotopes and effect on $r$-process nucleosynthesis

Background: Four strong single-particle bound levels with strikingly similar level spacings have recently been measured in ${}^{131}$Sn and ${}^{133}$Sn. This similarity has not yet been addressed by a theoretical nuclear structure model. Information on these single-particle bound levels, as well as on resonant levels above the neutron capture threshold, is also needed to determine neutron capture cross sections—and corresponding capture reaction rates—on ${}^{130,132}$Sn. The ${}^{130}$Sn$(n,\gamma$) rate was shown in a recent sensitivity study to significantly impact the synthesis of heavy elements in the $r$-process in supernovae.

Purpose: Understand the structure of bound and resonant levels in ${}^{131,133}$Sn, and determine if the densities of unbound resonant levels are sufficiently high to warrant statistical model treatments of neutron capture on ${}^{130,132}$Sn.

Method: Single-particle bound and resonant levels for ${}^{131,133}$Sn are self-consistently calculated by the analytical continuation of the coupling constant (ACCC) method based on a relativistic mean field (RMF) theory with BCS approximation.

Results: We obtain four strong single-particle bound levels in both ${}^{131,133}$Sn with an ordering that agrees with experiments and spacings that, while differing from experiment, are consistent between the Sn isotopes. We also find at most one single-particle level in the effective energy range for neutron captures in the $r$-process.

Conclusions: Our RMF+ACCC+BCS model successfully reproduces observed single-particle bound levels in ${}^{131,133}$Sn and self-consistently predicts single-particle resonant levels with densities too low for widely used traditional statistical model treatments of neutron capture cross sections on ${}^{130,132}$Sn employing Fermi gas level density formulations.

Figure 1

Comparison of strong single-particle levels seen in recent ${}^{132}$Sn$(d,p)$${}^{133}$Sn [1] and ${}^{130}$Sn$(d,p)$${}^{131}$Sn [2] measurements.

Figure 2

Bound levels and resonant levels in ${}^{133}$Sn calculated by the RMF+ACCC+BCS approach with the NL3 effective interaction compared with the latest data [1, 3] and calculations from the LSS [13], Nilsson [10], FRDM [11], and HFB [11] models. The available neutron capture thresholds $S\left(n\right)$ are also shown.

Figure 3

(a) Similar to Fig. 2, but for ${}^{131}$Sn, with excitation energies relative to the 2$f_{7/2}$ level. The results of the RMF+ACCC+BCS approach with the NL3 effective interaction are shown with the latest data [2]. (b) Comparison of present calculations for bound levels in ${}^{131,133}$Sn.