From cluster structures to nuclear molecules: The role of nodal structure of the single-particle wave functions

The nodal structure of the density distributions of the single-particle states occupied in rod-shaped, hyper- and megadeformed structures of nonrotating and rotating $N\sim Z$ nuclei has been investigated in detail. The single-particle states with the Nilsson quantum numbers of the $\left[NN0\right]1/2$ (with $N$ from 0 to 5) and $[N,N-1,1]\mathrm{\Omega }$ (with $N$ from 1 to 3 and $\mathrm{\Omega }=1/2$, 3/2) types are considered. These states are building blocks of extremely deformed shapes in the nuclei with mass numbers $A\le 50$. Because of (near) axial symmetry and large elongation of such structures, the wave functions of the single-particle states occupied are dominated by a single basis state in cylindrical basis. This basis state defines the nodal structure of the single-particle density distribution. The nodal structure of the single-particle density distributions allows us to understand in a relatively simple way the necessary conditions for $\alpha$ clusterization and the suppression of the $\alpha$ clusterization with the increase of mass number. It also explains in a natural way the coexistence of ellipsoidal mean-field-type structures and nuclear molecules at similar excitation energies and the features of particle-hole excitations connecting these two types of the structures. Our analysis of the nodal structure of the single-particle density distributions does not support the existence of quantum liquid phase for the deformations and nuclei under study.

Figure 1

Total neutron densities [in ${\mathrm{fm}}^{-3}]$ of specified configurations in ${}^{12}\mathrm{C}, {}^{28}\mathrm{Si}, {}^{36}\mathrm{Ar}, {}^{40}\mathrm{Ca}$, and ${}^{42}\mathrm{Sc}$ at indicated spin values. The plotting of the densities starts with yellow color at $0.001{\mathrm{fm}}^{-3}$. The densities presented in panels (b)–(f) are based on the results of the calculations of Ref. [9].

Figure 2

Neutron single-particle energies (routhians) in the self-consistent rotating potential as a function of the rotational frequency ${\mathrm{\Omega }}_x$. They are given along the deformation path of the rod-shaped structure in ${}^{12}\mathrm{C}$ and yrast MD configuration [42,42] in ${}^{40}\mathrm{Ca}$. Long-dashed, solid, dot-dashed, and dotted lines indicate $(\pi =+,r=+i), (\pi =+,r=-i), (\pi =-,r=+i)$, and $(\pi =-,r=-i)$ orbitals, respectively. At ${\mathrm{\Omega }}_x=0.0$ MeV, the single-particle orbitals are labeled by the asymptotic quantum numbers $\left[N{n}_z\mathrm{\Lambda }\right]\mathrm{\Omega }$ (Nilsson quantum numbers) of the dominant component of the wave function. Solid circles indicate the occupied orbitals. Large shell gaps are indicated in the right panel.

Figure 3

The single-neutron density distributions due to the occupation of the indicated Nilsson states with signature $r=-i$ in the rod-shaped configuration of ${}^{12}\mathrm{C}$. The color map shows the densities as multiplies of $0.001{\mathrm{fm}}^{-3}$. The plotting of the densities starts with yellow color at $0.001{\mathrm{fm}}^{-3}$. The results of the calculations are shown at rotational frequency ${\mathrm{\Omega }}_x=3.2$ MeV which corresponds to spin $I=8.77\hslash$. For each state, the cross sections in the $yz$ and $xz$ planes are plotted at $x=0.234$ fm and $y=0.234$ fm, respectively. The shape and size of the nucleus are indicated by black solid line, which corresponds to total neutron density line of $\rho =0.001{\mathrm{fm}}^{-3}$. In addition, the current distributions ${\mathbf{j}}^n\left(\mathbf{r}\right)$ produced by these states are shown by arrows. The currents in panels (a), (d), and (g) are plotted at arbitrary units for better visualization. The currents in other panels are normalized to the currents in above mentioned panels by using factor $F$.

Figure 4

The same as Fig. 3 but for the $\left[330\right]1/2(r=-i)$ orbital in the HD [2,2] configuration of ${}^{28}\mathrm{Si}$. The densities in the $yz$ and $xz$ planes are taken at $x=0.326$ fm and $y=0.326$ fm, respectively. Top and bottom panels show the results at ${\mathrm{\Omega }}_x=0.0$ MeV and ${\mathrm{\Omega }}_x=1.8$ MeV, respectively.

Figure 5

The same as Fig. 4 but for the $\left[101\right]1/2(r=\pm i)$ orbitals. The top panels show the results for the $(r=-i)$ orbital at ${\mathrm{\Omega }}_x=0.0$ MeV. Because the density distributions in both signatures of the [101]3/2 orbital are identical at ${\mathrm{\Omega }}_x=0.0$ MeV, the results for the $\left[101\right]3/2(r=+i)$ orbital are not shown here. Middle and bottom panels show the densities and currents at ${\mathrm{\Omega }}_x=1.8$ MeV for the $\left[101\right]1/2(r=-i)$ and $\left[101\right]1/2(r=+i)$ orbitals, respectively.

Figure 6

The same as Fig. 4 but for the $\left[211\right]3/2(r=-i)$ orbital.

Figure 7

Total neutron densities produced by six neutrons occupying the lowest orbitals of the nucleonic potential, namely, [000]1/2, [110]1/2, and [220]1/2 in considered configurations of ${}^{12}\mathrm{C}, {}^{28}\mathrm{Si}$ and ${}^{40}\mathrm{Ca}$. The plotting of the densities starts with yellow color at $0.001{\mathrm{fm}}^{-3}$.

Figure 8

The same as Fig. 3 but for the $\left[312\right]3/2(r=-i)$ orbital in the MD [42,42] configuration of ${}^{40}\mathrm{Ca}$ at rotational frequency ${\mathrm{\Omega }}_x=1.8$ MeV corresponding to spin $I=25\hslash$. The densities in the $yz$ and $xz$ planes are taken at $x=0.329$ fm and $z=0.329$ fm, respectively.

Figure 9

The same as Fig. 8 but for the $\left[440\right]1/2(r=\pm i)$ orbitals at ${\mathrm{\Omega }}_x=1.8$ MeV. Top and bottom panels show the results for the ($r=-i)$ and $(r=+i)$ branches, respectively. Note that the density distribution at ${\mathrm{\Omega }}_x=0.0$ MeV is very similar to what is seen in bottom panels.

Figure 10

The calculated moments of inertia for the indicated configurations.

Figure 11

Total neutron currents ${\mathbf{j}}^n\left(\mathbf{r}\right)$ in the $yz$ plane plotted at $x=0.234$ fm, $x=0.326$ fm, and $x=0.329$ fm for the considered configurations in ${}^{12}\mathrm{C}, {}^{28}\mathrm{Si}$, and ${}^{40}\mathrm{Ca}$, respectively. They are given at spin values indicated in Fig. 1. The currents in panel (a) are plotted at arbitrary units for better visualization. The currents in other panels are normalized to the currents in above mentioned panels by using factor $F$.