Three-nucleon forces and spectroscopy of neutron-rich calcium isotopes

We study excited-state properties of neutron-rich calcium isotopes based on chiral two- and three-nucleon interactions. We first discuss the details of our many-body framework, investigate convergence properties, and for two-nucleon interactions benchmark against coupled-cluster calculations. We then focus on the spectroscopy of ${}^{47-56}\mathrm{Ca}$, finding that with both 3N forces and an extended $pf{g}_{9/2}$ valence space, we obtain a good level of agreement with experiment. We also study electromagnetic transitions and find that experimental data are well described by our calculations. In addition, we provide predictions for unexplored properties of neutron-rich calcium isotopes.

Figure 1

Convergence of the (a) ${}^{42}\mathrm{Ca}$ and (b) ${}^{48}\mathrm{Ca}$ ground-state energies as a function of increasing intermediate-state excitations $N\hslash \omega$ and perturbative order. Calculations are based on NN forces in 13 major harmonic-oscillator shells.

Figure 2

Convergence of (a) neutron and (b) proton SPEs as a function of increasing intermediate-state excitation $N\hslash \omega$. Calculations are based on NN forces in 13 major harmonic-oscillator shells.

Figure 3

Comparison of MBPT and CC ground-state energies of calcium isotopes relative to ${}^{40}\mathrm{Ca}$ based on the same NN interaction (for details see text). The MBPT results use the SPEs obtained in CC theory.

Figure 4

Calculated ground-state energies of calcium isotopes in (a) $pf$ shell and (b) $pf{g}_{9/2}$ shell compared with experimental data (solid points) and AME2012 extrapolated values (open circles) [59]. Calculations are performed in the extended $pf{g}_{9/2}$ valence space and based on NN forces only, $\text{NN}+\text{3N}$ forces with empirical SPEs, and $\text{NN}+3\text{N}$ forces with calculated (MBPT) SPEs.

Figure 5

Evolution of SPEs as a function of mass number. Calculations are based on $\text{NN}+3\text{N}$ forces in the extended $pf{g}_{9/2}$ space.

Figure 6

Excitation energies of bound excited states in ${}^{47}\mathrm{Ca}$ compared with experiment [62] and phenomenological GXPF1A [22] and KB3G [20] interactions. The NN-only results are calculated in the $pf$ and $pf{g}_{9/2}$ spaces with empirical SPEs. The $\text{NN}+3\text{N}$ results are obtained in the same spaces. In the $pf$ shell, empirical SPEs are used, while the $pf{g}_{9/2}$-space results use the consistently calculated MBPT SPEs.

Figure 7

Excitation energies of bound excited states in ${}^{48}\mathrm{Ca}$ compared with experiment [62] and phenomenological interactions (labels are as in Fig. 6).

Figure 8

Excitation energies of bound excited states in ${}^{49}\mathrm{Ca}$ compared with experiment [18, 62] and phenomenological interactions (labels are as in Fig. 6).

Figure 9

Excitation energies of bound excited states in ${}^{50}\mathrm{Ca}$ compared with experiment [18, 62] and phenomenological interactions (labels are as in Fig. 6).

Figure 10

Excitation energies of bound excited states in ${}^{51}\mathrm{Ca}$ compared with experiment [14, 17] and phenomenological interactions (labels are as in Fig. 6).

Figure 11

Excitation energies of bound excited states in ${}^{52}\mathrm{Ca}$ compared with experiment [14, 17] and phenomenological interactions (labels are as in Fig. 6).

Figure 12

Excitation energies of bound excited states in ${}^{53}\mathrm{Ca}$ compared with experiment [15] and phenomenological interactions (labels are as in Fig. 6).

Figure 13

Excitation energies of bound excited states in ${}^{54}\mathrm{Ca}$ compared with experiment [15] and phenomenological interactions (labels are as in Fig. 6).

Figure 14

Excitation energies in ${}^{55}\mathrm{Ca}$ compared with phenomenological interactions (labels are as in Fig. 6).

Figure 15

Excitation energies in ${}^{56}\mathrm{Ca}$ compared with phenomenological interactions (labels are as in Fig. 6).

Figure 16

Magnetic-dipole transition rates from the ground state to $1^{+}$ excited states in ${}^{48}\mathrm{Ca}$ compared with experiment [67]. The $B\left(M1\right)$ values are calculated in the $pf{g}_{9/2}$ space with $\text{NN}+3\text{N}$ interactions. Spin $g$ factors are quenched by 0.75, except for the empty blue bars.

Figure 17

First-order contribution from residual 3N forces to the energies of the ground state (bars) and first-excited state (solid line) of the calcium isotopes in the extended $pf{g}_{9/2}$ valence space.

Figure 18

Excitation energies of bound excited states in ${}^{42}\mathrm{Ca}$ compared with experiment and phenomenological interactions (labels are as in Fig. 6).

Figure 19

Excitation energies of bound excited states in ${}^{43}\mathrm{Ca}$ compared with experiment and phenomenological interactions (labels are as in Fig. 6).

Figure 20

Excitation energies of bound excited states in ${}^{44}\mathrm{Ca}$ compared with experiment and phenomenological interactions (labels are as in Fig. 6).

Figure 21

Excitation energies of bound excited states in ${}^{45}\mathrm{Ca}$ compared with experiment and phenomenological interactions (labels are as in Fig. 6).

Figure 22

Excitation energies of bound excited states in ${}^{46}\mathrm{Ca}$ compared with experiment and phenomenological interactions (labels are as in Fig. 6).