Absence of a shell closure in $^{140}$Sn

Technical Summary: Absence of a Shell Closure in $^{140}$Sn

Problem Statement
The evolution of shell closures in exotic, neutron-rich nuclei remains a central question in nuclear structure physics. While canonical magic numbers govern stable nuclei, shell closures can disappear or emerge in systems far from stability. Specifically, the neutron-rich nucleus $^{140}$Sn ($N=90$) has been the subject of conflicting theoretical predictions regarding the existence of a shell closure. Previous empirical shell-model trends suggested a shell closure at $N=90$, implying a significant rise in the first $2^+$ excitation energy. However, more realistic effective interaction calculations and recent spectroscopic data on neighboring nuclei (e.g., $^{135}$Sn) have suggested little variation in excitation energies between $^{132}$Sn and $^{140}$Sn, casting doubt on the $N=90$ closure. A definitive experimental test via direct measurement of the $2^+$ state in $^{140}$Sn is currently beyond experimental reach. This work addresses this controversy using ab initio computations to determine the structure of $^{140}$Sn from first principles.

Methodology
The authors employ two distinct ab initio many-body methods combined with a nuclear interaction derived from chiral effective field theory (chiral EFT):

1. Interaction: The study utilizes the chiral force 1.8/2.0 (EM) from Ref. [40]. This interaction includes leading-order chiral three-nucleon forces and has been benchmarked to accurately reproduce ground-state energies and shell closures across the nuclear chart (e.g., in Oxygen, Calcium, Nickel, and $^{208}$Pb) and predict properties of $^{78}$Ni and $^{100}$Sn prior to measurement.
2. Coupled-Cluster (CC) Theory: Calculations are performed using the Coupled-Cluster Singles and Doubles (CCSD) approximation for ground states and the Equation-of-Motion (EOM-CCSD) method for excited states. To assess higher-order correlations, the authors also employ the EOM-CCSDT-1 framework, which includes leading-order triples, restricted by computational resources to configurations where the sum of single-particle energy differences is less than 100 MeV.
3. Valence-Space In-Medium Similarity Renormalization Group (VS-IMSRG): This approach decouples the many-body problem into a valence space (neutron orbitals $2p_{1/2}, 2p_{3/2}, 1f_{5/2}, 1f_{7/2}, 0h_{9/2}, 0i_{13/2}$ with a closed proton core) using a unitary transformation. The study utilizes both VS-IMSRG(2) (truncation at the normal-ordered two-body level) and VS-IMSRG(3f2) (an approximation retaining specific three-body contributions). Excited states are computed via the EOM-IMSRG approximation.
4. Validation: The theoretical framework is first validated by computing the spectrum of $^{133}$Sn, comparing results against available experimental data.

Key Contributions and Results

• Validation in $^{133}$Sn: The authors successfully reproduce the low-lying spectrum of $^{133}$Sn. Both CC and VS-IMSRG methods yield binding energies within 1% of the experimental value. The calculated excitation energies for the $3/2^-, 1/2^-, 5/2^-$, and $9/2^-$ states agree well with data, confirming the interaction's suitability for neutron-rich Sn isotopes. The $13/2^+$ state is overestimated by a factor of two, a discrepancy the authors attribute to the state's neutron-unbound nature and the lack of continuum treatment in their calculations.
• Ground State of $^{140}$Sn: The calculated ground-state binding energies for $^{140}$Sn are consistent with estimates from nuclear mass models (e.g., FRDM(2012)), with values around $-1125$ MeV.
• Excitation Energy of the First $2^+$ State: The primary result concerns the energy of the first excited $2^+$ state.
  • Both CC and VS-IMSRG methods show a slow convergence pattern with increasing model space size ($N_{max}$).
  • The inclusion of higher-order correlations (3p-3h in CC, 3-body terms in IMSRG) generally reduces the excitation energy.
  • The final predicted excitation energy is smaller than 1.5 MeV, with coupled-cluster calculations at $N_{max}=16$ yielding a value below 1 MeV.
  • This value is less than 60% of the first $2^+$ excitation energy of the doubly-magic nucleus $^{78}$Ni and significantly lower than the pronounced rise expected for a shell closure.
• Shell Gap Analysis: Mean-field level computations indicate a neutron shell gap of only $\approx 2$ MeV at $N=90$ (compared to $\approx 7$ MeV for the proton gap), suggesting the reference state is not separated by a robust neutron shell gap. The failure of perturbative EOM-CCSD(T) calculations (yielding negative excitation energies) further indicates that the assumption of a closed-shell reference state is incorrect.

Significance and Conclusion
The paper concludes that the ab initio results do not support the hypothesis of a shell closure (or subshell closure) at $N=90$ in $^{140}$Sn. The small calculated $2^+$ excitation energy contradicts the assumption of a closed $1f_{7/2}$ neutron subshell beyond $^{132}$Sn. These findings are consistent with previous shell-model calculations using realistic interactions and recent ab initio studies of tin isotopes that found no clear gap in two-neutron separation energies or kinks in charge radii at $N=90$. The work resolves the theoretical controversy by demonstrating that $^{140}$Sn lacks the structural characteristics of a doubly-magic or subshell-closed nucleus.