Description of superdeformed bands in light $N=Z$ nuclei using the cranked Hartree-Fock-Bogoliubov method

Superdeformed states in light $N=Z$ nuclei are studied by means of the self-consistent cranking calculation (i.e., the P+QQ model based on the cranked Hartree-Fock-Bogoliubov method). Analyses are given for two typical cases of superdeformed bands in the $A\simeq 40$ mass region, that is, bands where backbending is absent (${}^{40}\mathrm{Ca}$) or present (${}^{36}\mathrm{Ar}$). Investigations are carried out particularly for the following points: cross-shell excitations in the $\mathit{sd}$ and $\mathit{pf}$ shells; the role of the $g_{9/2}$ and $d_{5/2}$ orbitals; the effect of nuclear pairing; and the interplay between triaxiality and band termination.

Figure 1

Proton (left) and neutron (right) occupation numbers as a function of the total angular momentum for ${}^{40}\mathrm{Ca}$. In this case, the $g_{9/2}$ orbital is excluded from the model space.

Figure 2

Calculated and observed rotational energy of the SD band in ${}^{40}\mathrm{Ca}$. The calculated ground-state energy (at $J=0$) is normalized with the experimental value, $E(J=0)=5.218$ MeV.

Figure 3

Excitation energies relative to a rigid-rotor energy, $E_R=\mathrm{aJ}(J+1)$, where $a=0.06909$ (MeV) for ${}^{40}\mathrm{Ca}$. The four calculated results (HFB-Np,Nn,Gp, and Gn) are compared with the experimental data. The HFB calculations with (without) the $g_{9/2}$ orbital are denoted as G (N). In addition, two different sets of the initial value for the pairing are chosen for each case. The set p corresponds to $({\Delta }_p^0,{\Delta }_n^0)=(1.360,1.513)$ MeV; set n to $({\Delta }_p^0,{\Delta }_n^0)=(0.15,0.15)$ MeV.

Figure 4

Proton (left) and neutron (right) occupation numbers as a function of the total angular momentum for ${}^{40}\mathrm{Ca}$. In this case, the $g_{9/2}$ orbital is included in the model space.

Figure 5

Calculated pairing-gap energies as a function of the total angular momentum for ${}^{40}\mathrm{Ca}$. The identification of the curves are the same as Fig. 3.

Figure 6

Backbending plot of the SD band of ${}^{40}\mathrm{Ca}$. HFB-Np (HFB-Gp) corresponds to the calculation without (with) the $g_{9/2}$ orbital in the model space.

Figure 7

Proton (left) and neutron (right) single-particle spin components along the cranking axis for ${}^{40}\mathrm{Ca}$. In this case, the $g_{9/2}$ orbital is excluded from the model space.

Figure 8

Same as Fig. 7, but for the case in which the $g_{9/2}$ orbital is included in the model space.

Figure 9

Calculated triaxiality γ (left) and elongation β (right), as functions of the total angular momentum for ${}^{40}\mathrm{Ca}$. Captions HFB-GP and HFB-NP denote the cases with and without the $g_{9/2}$ orbital, respectively.

Figure 10

Pairing-gap energies at the total angular momentum $0\le J\le 16\hslash$. The $g_{9/2}$ orbital is included in the model space.

Figure 11

Backbending plot of the SD band of ${}^{36}\mathrm{Ar}$. HFB-Np (HFB-Gp) corresponds to the calculation without (with) the $g_{9/2}$ orbital in the model space.

Figure 12

Proton (left) and neutron (right) occupation numbers as a function of the total angular momentum for ${}^{36}\mathrm{Ar}$. In this case, the $g_{9/2}$ orbital is included in the model space.

Figure 13

Calculated triaxiality γ (left) and elongation β (right), as functions of the total angular momentum for ${}^{36}\mathrm{Ar}$.

Figure 14

Single-particle spin components along the cranking axis. Only the $f_{7/2}$ and $d_{5/2}$ orbitals for protons are plotted to compare the two cases (with and without the $g_{9/2}$ in the model space).

Figure 15

Calculated and observed rotational energies of the SD band in ${}^{36}\mathrm{Ar}$. The $g_{9/2}$ orbital is taken into account in the model space. The calculated ground-state energy (at $J=0$) is normalized with the experimental value, $E(J=0)=4.3291$ MeV.

Figure 16

Excitation energies relative to a rigid-rotor energy, $E_R=\mathrm{aJ}(J+1)$, where $a=0.08234$ MeV, for ${}^{36}\mathrm{Ar}$.

Figure 17

Proton (left) and neutron (right) single-particle spin components along the cranking axis for ${}^{38}\mathrm{Ar}$. The $g_{9/2}$ orbital is included in the model space.

Figure 18

Calculated triaxiality γ (left) and elongation β (right), as functions of the total angular momentum for ${}^{36}\mathrm{Ar}$, when the pairing correlation is switched off.