Pairing effects on the collectivity of quadrupole states around ${}^{32}\mathrm{Mg}$

The first $2^{+}$ states in $N=20$ isotones including neutron-rich nuclei ${}^{32}\mathrm{Mg}$ and ${}^{30}\mathrm{Ne}$ are studied by the Hartree-Fock-Bogoliubov plus quasiparticle random phase approximation method based on the Green’s function approach. The residual interaction between the quasiparticles is consistently derived from the Hamiltonian density of Skyrme interactions with explicit velocity dependence. The $B(E2,0_1^{+}\to 2_1^{+})$ transition probabilities and the excitation energies of the first $2^{+}$ states are well described within a single framework. We conclude that pairing effects account largely for the anomalously large $B\left(E2\right)$ value and the very low excitation energy in ${}^{32}\mathrm{Mg}$.

Figure 1

HFB neutron pairing gaps in ${}^{26,28,30}\mathrm{Ne}$ calculated with SkM* and SkP. The pairing strengths $V_{\mathit{pair}}$ are fixed so as to reproduce the experimental neutron gap in ${}^{30}\mathrm{Ne}$. The experimental pairing gaps are extracted by using the three-point mass difference formula [38].

Figure 2

HF neutron single-particle levels in $N=20$ isotones calculated with SkM*. Solid lines correspond to bound and resonancelike states, dashed lines to positive energy discretized states.

Figure 3

The neutron and proton pairing gaps in $N=20$ isotones calculated in HFB with SkM* and SkP.

Figure 4

Average number of neutrons in the $fp$ shell in $N=20$ isotones calculated in HFB with SkM* and SkP.

Figure 5

The $B(E2,0_1^{+}\to 2_1^{+})$ transition probabilities and excitation energies of the first $2^{+}$ states in $N=20$ isotones calculated in QRPA with SkM*. For comparison the calculations of MCSM [14] and the available experimental data [2, 3, 6, 22] are shown.

Figure 6

The $B(E2,0_1^{+}\to 2_1^{+})$ transition probabilities and excitation energies of the first $2^{+}$ states calculated in QRPA with SkM* and SkP.

Figure 7

The unperturbed isoscalar quadrupole strength functions in $N=20$ isotones calculated with SkM*. The peaks indicated by solid (dotted) arrows correspond to proton (neutron) two-quasiparticle configurations.

Figure 8

The $B(E2,0_1^{+}\to 2_1^{+})$ values and the excitation energies of the first $2^{+}$ states in $N=20$ isotones calculated with∕without neutron pairing correlations. Proton pairing is included in both cases.

Figure 9

The $B\left(E2\right)$ value and excitation energy of the first $2^{+}$ state in ${}^{32}\mathrm{Mg}$ calculated in QRPA with SkM*, as a function of the pairing strength $V_{\mathit{pair}}$. Three types of approximations for the residual interaction are examined: the solid line is a calculation with the self-consistent residual interaction, Eq. (6); the dashed line is obtained with a residual interaction where terms in $\mathbf{\nabla }$ are dropped; the dotted line corresponds to the Landau-Migdal force.