Shell-model study of titanium isotopic chain with chiral two- and three-body forces

The even-even Ti isotopic chain, from $A=42$ to 70, has been studied within the nuclear shell-model framework by employing an effective Hamiltonian which is derived by way of many-body perturbation theory from a chiral potential with two- and three-body forces, and includes three-body contributions which account for Pauli principle violations in nuclei with more than two valence particles. We consider ${}^{40}\mathrm{Ca}$ as a closed core and a model space spanned by the neutron and proton $0f1p$ orbitals with the addition of the $0g_{9/2}$ orbital for neutrons. Calculated two-neutron separation energies and excitation energies of the yrast $2^{+}$ states are reported and compared with the experimental data, which are available up to ${}^{62}\mathrm{Ti}$. The present study intends to investigate the effects of the adopted effective interactions on the evolution of the shell structure.

Figure 1

Second-order three-body diagrams. The sum over the intermediate lines runs over particle and hole states outside the model space, shown by $A$ and $B$, respectively. We report only one of the nine existing topologies which correspond to the permutations of the external lines.

Figure 2

Density-dependent two-body contribution that is derived from a three-body one. $\alpha$ is obtained by summing over one incoming and outgoing particle of the three-body graphs $A,B$ reported in Fig. 1.

Figure 3

Theoretical two-neutron separation energies for even-even Ti isotopes from $N=22$ to 48, with and without the 3-b correlations, are compared with the experimental data from Ref. [27] for $N=40$ and from AME2020 [58] for the other isotopes.

Figure 4

Theoretical $2_1^{+}$ excitation energies for even-even Ti isotopes from $N=20$ to 48, with and without the 3-b correlations, are compared with the experimental data from Ref. [29] for $N=40$ and from Ref. [59] for the other isotopes.

Figure 5

ESPE for neutrons in even-even Ti isotopes from $N=20$ to 48 resulting from calculations (a) without and (b) with 3-b correlations.

Figure 6

Neutron-neutron monopole matrix elements, ${\overline{V}}_{ab}^{\nu {\nu }^{\prime }}$, of the effective interactions with and without 3-b correlations (see text for details).