Superdeformation and clustering in ${}^{40}\mathrm{Ca}$ studied with antisymmetrized molecular dynamics

Deformed states in ${}^{40}\mathrm{Ca}$ are investigated with a method of antisymmetrized molecular dynamics. Above the spherical ground state, rotational bands arise from a normal deformation and a superdeformation as well as an oblate deformation. The calculated energy spectra and $E2$ transition strengths in the superdeformed band reasonably agree to the experimental data of the superdeformed band starting from the $0_3^{+}$ state at 5.213 MeV. By the analysis of single-particle orbits, it is found that the superdeformed state has particle-hole nature of an $8p\text{−}8h$ configuration. One of new findings is parity asymmetric structure with ${}^{12}\mathrm{C}+{}^{28}\mathrm{Si}$-like clustering in the superdeformed band. We predict that ${}^{12}\mathrm{C}+{}^{28}\mathrm{Si}$ molecular bands may be built above the superdeformed band due to the excitation of intercluster motion. They are considered to be higher nodal states of the superdeformed state. We also suggest negative-parity bands caused by the parity asymmetric deformation.

Figure 1

The binding energies of $Z=N$ nuclei from ${}^4\mathrm{He}$ to ${}^{40}\mathrm{Ca}$ obtained by the simple AMD calculations without constraints. The binding energies calculated by variation after parity projection but no total-angular momentum projection are shown in the upper panel (a), and the energies obtained by total-angular-momentum projection after the variation with parity projection are plotted in the lower panel (b). Squares are the experimental binding energies. Filled circles and crosses indicate the results by using interactions (i) and (ii), respectively. The results for interaction (ii) in Ref. 30 of AMD calculations with constraint and superposition are shown by triangles with dotted line.

Figure 2

The energy curves for the positive-parity states obtained by the constrained AMD calculations. The constraint of the total number of harmonic oscillator quanta, $\langle {\stackrel{\wedge }{N}}_{\mathrm{os}}\rangle =N_{\mathrm{os}}$, is imposed on the parity projected AMD wave function. The energies of the spin-parity projected states, $P_{M0}^{J=0(2)}{\Phi }_{\mathrm{AMD}}^{+}(N_{\mathrm{os}}^{(+)})$, are plotted (a), (c) as a function of $N_{\mathrm{os}}$ and (b), (d) as a function of deformation ${\beta }_2$ of the states: ${\Phi }_{\mathrm{AMD}}^{+}(N_{\mathrm{os}}^{(+)})$. The left panels [(a) and (b)] show the results with interaction (i), and the right panels [(c) and (d)] are for the results with interaction (ii).

Figure 3

The energy curves for the negative-parity states obtained by the constrained AMD calculations. The constraint of the total number of harmonic oscillator quanta, $\langle {\stackrel{\wedge }{N}}_{\mathrm{os}}\rangle =N_{\mathrm{os}}$, is imposed on the parity projected AMD wave function. The energies of the spin-parity projected states, $P_{M0}^{J=1,3}{\Phi }_{\mathrm{AMD}}^{-}(N_{\mathrm{os}}^{(-)})$, are plotted as a function of $N_{\mathrm{os}}$ in the upper panel (a) and as a function of the deformation ${\beta }_2$ of ${\Phi }_{\mathrm{AMD}}^{-}(N_{\mathrm{os}}^{(-)})$ in the lower panel (b). The results are obtained with interaction (i).

Figure 4

Density distributions of the intrinsic states, ${\Phi }_{\mathrm{AMD}}(N_{\mathrm{os}}^{(\pm )})$ for (a) the spherical state with $N_{\mathrm{os}}^{(\pm )}={62}^{(+)}$, and the prolate solutions with $N_{\mathrm{os}}^{(\pm )}=$ (b) ${65}^{(+)}$, (d) ${68}^{(+)}$, (e) ${70}^{(+)}$, (f) ${76}^{(+)}$, (g) ${62}^{(-)}$, (h) ${65}^{(-)}$, (i) ${68}^{(-)}$, and (c) the oblate solution with $N_{\mathrm{os}}^{(\pm )}={65}^{(+)}$, which are obtained by using interaction (i). The intrinsic system is projected onto the $Y\text{−}Z$ plane, and the density is integrated along the X axis, where $X,Y$, and $z$ axes are chosen as $\langle X^2\rangle \le \langle Y^2\rangle \le \langle Z^2\rangle$ and $\langle \mathit{XY}\rangle =\langle \mathit{YZ}\rangle =\langle \mathit{ZX}\rangle =0$. The deformation parameters ${\beta }_2$ are written at the bottom of the figures. The size of the frame box is 10 fm×10 fm.

Figure 5

Density distribution of the intrinsic states, ${\Phi }_{\mathrm{AMD}}(N_{\mathrm{os}}^{(\pm )})$ for (a) the the spherical state with $N_{\mathrm{os}}^{(\pm )}={62}^{(+)}$, the prolate solutions with $N_{\mathrm{os}}^{(\pm )}=$ (c) ${65}^{(+)}$ and (d) $70(+)$, (b) the oblate solution with $N_{\mathrm{os}}^{(\pm )}={65}^{(+)}$, and (e) the α-cluster state with $N_{\mathrm{os}}^{(\pm )}={68}^{(+)}$, which are obtained by using interaction (ii). The intrinsic system is projected onto the $Y\text{−}Z$ plane, and the density is integrated along the X axis, where $X,Y$, and $z$ axes are chosen as $\langle X^2\rangle \le \langle Y^2\rangle \le \langle Z^2\rangle$ and $\langle \mathit{XY}\rangle =\langle \mathit{YZ}\rangle =\langle \mathit{ZX}\rangle =0$. The deformation parameters ${\beta }_2$ are written at the bottom of the figures. The size of the frame box is 10 fm×10 fm.

Figure 6

Excitation energies of the levels up to $J=8$ in ${}^{40}\mathrm{Ca}$. The experimental data of the levels observed in Ref. 12 are shown in (a). (b) and (c) show the theoretical results for the positive-parity and negative-parity states, respectively, calculated by the superposition of the AMD wave functions. The interaction (i) is used. The excitation energies of the higher spin ($J>8$) states will be shown in the next figures.

Figure 7

Excitation energies of the positive-parity states in ${}^{40}\mathrm{Ca}$ as a function of a spin $J(J+1)$. Theoretical values shown in the lower panel (b) are those obtained with the interaction (i). The energies are calculated by the superposition of the AMD wave functions. The upper panel (a) shows the experimental data for the normal-deformed band and the superdeformed band, which are labeled “band 2” and “band 1” in Ref. 12, respectively.

Figure 8

Excitation energies of the positive-parity states in ${}^{40}\mathrm{Ca}$ obtained with the interaction (ii). The energies are calculated by the superposition of the AMD wave functions.

Figure 9

The deformation parameters ${\beta }^t$ extracted from the $B(E2)$ values for the intraband transitions. The labels of the bands correspond to those in Fig. 6. The results are for the calculations with the interaction (i).

Figure 10

The overlap between the calculated excited states ${\Psi }_n^{J+}$ and the prolate AMD states $P_{M0}^J{\Phi }_{\mathrm{AMD}}^{+}(N_{\mathrm{os}}^{(+)})$. The values $|\langle {\Phi }_n^{J+}|P_{M0}^J{\Phi }_{\mathrm{AMD}}^{+}(N_{\mathrm{os}}^{(+)})\rangle |{}^2$ for the states in the bands (A1), (A2), (A3), (A4), or (A5) are shown by open circles, open triangles, open squares, filled circles, or filled triangles, respectively.

Figure 11

(Color online) Density distributions of the HF-like single-particle orbits ${\phi }_a^{\mathrm{HF}}$ for protons in ${\Phi }_{\mathrm{AMD}}(N_{\mathrm{os}}^{(\pm )})$. The results are for the calculations with interaction (i) except for panel (f). The first (${\phi }_{a=1}^{\mathrm{HF}}$) and the third (${\phi }_{a=3}^{\mathrm{HF}}$) highest proton orbits in the the dominant component [${\Phi }_{\mathrm{AMD}}(N_{\mathrm{os}}^{(\pm )}={70}^{(+)}$] of the superdeformation are shown in (a) and (b), respectively. (c) The highest orbit, ${\phi }_{a=1}^{\mathrm{HF}}$, in the oblate solution with $N_{\mathrm{os}}^{(\pm )}={65}^{(+)}$. (d) The first (${\phi }_{a=1}^{\mathrm{HF}}$) and (e) the third (${\phi }_{a=3}^{\mathrm{HF}}$) orbits in the prolate solutions with $N_{\mathrm{os}}^{(\pm )}={65}^{(+)}$. (f) The highest (${\phi }_{a=1}^{\mathrm{HF}}$) orbit in the α-cluster state with $N_{\mathrm{os}}^{(\pm )}={68}^{(+)}$ obtained in the results with interaction (ii). In the left side, the intrinsic state is projected onto the $Y\text{−}Z$ plane, and the density is integrated along the X axis, where $X,Y$, and Z axes are chosen as $\langle X^2\rangle \le \langle Y^2\rangle \le \langle Z^2\rangle$ and $\langle \mathit{XY}\rangle =\langle \mathit{YZ}\rangle =\langle \mathit{ZX}\rangle =0$. The right figures are for the surface cut of the density. The percentage $P(-)$ of the negative-parity component in each single-particle orbit is also listed. The box size is 10 fm.

Figure 12

The percentage $P(-)$ of the negative-parity component in each HF-like single-particle orbit for the protons in the prolate states ${\Phi }_{\mathrm{AMD}}(N_{\mathrm{os}}^{(+)})$ obtained with interaction (i). The values $P(-)$ for the highest ten levels ($a=1-10$) are shown in the upper panel, and those for the orbits ($a=11-18$) from the 11th to the 18th level are shown in the lower panel.

Figure 13

Density distributions summed up (a) for the protons in the higher single-particle levels (${\phi }_{a=1-10}^{\mathrm{HF}}$) and (b) for the protons in the lower single-particle levels (${\phi }_{a=11-20}^{\mathrm{HF}}$) in the intrinsic state of the superdeformation: ${\Phi }_{\mathrm{AMD}}(N_{\mathrm{os}}={70}^{(+)})$. The total density is also shown in (c). The densities in (a) and (b) are normalized by a factor 4.