Monopole-optimized effective interaction for tin isotopes

We present a systematic configuration-interaction shell-model calculation on the structure of light tin isotopes with a global optimized effective interaction. The starting point of the calculation is the realistic CD-Bonn nucleon-nucleon potential. The unknown single-particle energies of the $1d_{3/2}$, $2s_{1/2}$, and $0h_{11/2}$ orbitals and the $T=1$ monopole interactions are determined by fitting to the binding energies of 157 low-lying yrast states in ${}^{102-132}$Sn. We apply the Hamiltonian to analyze the origin of the spin inversion between ${}^{101}$Sn and ${}^{103}$Sn that was observed recently and to explore the possible contribution from interaction terms beyond the normal pairing.

Figure 1

Dimensions of the $M^{\pi }=0^{+}$ states in even-even Sn (circle) and Te (triangle) isotopes as a function of valence neutron numbers, where $M$ and $N$ denote the total magnetic quantum number and the number of valence neutrons, respectively. The open symbols connected by dashed lines give the dimensions of the corresponding $I^{\pi }=0^{+}$ states.

Figure 2

The convergence of the mean-square deviations ${\chi }^2$ as a function of iteration number within the singular value decomposition (SVD) and Monte Carlo global optimization (MC) approaches starting from a random monopole Hamiltonian. See text for the difference between the red (solid) and green (open) triangles.

Figure 3

The convergence of the mean-square deviations ${\chi }^2$ as a function of iteration number for fittings that are started from the realistic CD-Bonn interaction. The inset shows the maximal deviation $r=\max |E_i^{\mathrm{Cal}.}-E_i^{\mathrm{Expt}.}|$ in each step.

Figure 4

Differences between experimental and calculated binding energies $E_i^{\mathrm{Expt}.}-E_i^{\mathrm{Cal}.}$ as a function of valence neutron number.

Figure 5

The calculated shell-model energies (solid circles) $E^{\text{SM}}=\langle H\rangle$, and contributions from monopole Hamiltonian $\langle H_m\rangle$ (open squares).

Figure 6

Experimental [17, 18, 67, 76] (open symbols) and calculated (solid symbols with dotted lines) excitation energies of the low-lying even spin states in nuclei ${}^{102-130}$Sn.

Figure 7

Shell-model occupancies $\langle {\widehat{N}}_j\rangle /(2j+1)$ of the three higher-$j$ shells in the ground states of even tin isotopes.

Figure 8

Experimental [79, 80] and calculated $B$($E2$) values on the transitions of the ${10}_1^{+}$ states in even Sn isotopes. The lower panel gives the calculated occupancies of the $0h_{11/2}$ orbital in the ${10}^{+}$ and $8^{+}$ states.

Figure 9

Neutron pairing gaps in Sn isotopes extracted from the experimental and calculated binding energies.

Figure 10

The calculated shell-model energies of the first $5/2^{+}$ and $7/2^{+}$ states in ${}^{103}$Sn with a Hamiltonian $H^{\prime }$ containing the single-particle terms and two-body matrix elements with $J\le J_{\max }$. The solid symbols denote the eigenvalues of $H^{\prime }$ derived by diagonalizing the Hamiltonian matrix. The open symbols denote the expectation values of $H^{\prime }$, $\langle {\Psi }_I\left|H^{\prime }\right|{\Psi }_I\rangle$, with respect to the corresponding wave functions $|{\Psi }_I\rangle$ of the full Hamiltonian $H$, where all matrix elements are taken into account.

Figure 11

Solid symbol: The overlaps between the wave functions $|{\Psi }_I\rangle$ of the full Hamiltonian $H$ and those of the pairing Hamiltonian with $J_{\max }=0$ for the first $5/2^{+}$ and $7/2^{+}$ states in light odd-$A$ Sn isotopes. Open symbol: Same as above, but only the nondiagonal pairing matrix elements are considered.