Nonlocalized cluster dynamics and nuclear molecular structure

A container picture is proposed for understanding cluster dynamics where the clusters make nonlocalized motion occupying the lowest orbit of the cluster mean-field potential characterized by the size parameter $``B$” in the Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave function. The nonlocalized cluster aspects of the inversion-doublet bands in ${}^{20}\mathrm{Ne}$ which have been considered as a typical manifestation of localized clustering are discussed. An as-yet-unexplained puzzling feature of the THSR wave function, namely that after angular-momentum projection for two-cluster systems the prolate THSR wave function is almost 100$\%$ equivalent to an oblate THSR wave function, is clarified. It is shown that the true intrinsic two-cluster THSR configuration is nonetheless prolate. The proposal of the container picture is based on the fact that typical cluster systems, 2$\alpha$, 3$\alpha$, and ${}^{16}\text{O}+\alpha$, are all well described by a single THSR wave function. It is shown for the case of linear-chain states with 2- and 3$\alpha$ clusters, as well as for the ${}^{16}\text{O}+\alpha$ system, that localization is entirely of kinematical origin, that is, attributable to the intercluster Pauli repulsion. It is concluded that this feature is general for nuclear cluster states.

Figure 1

Contour map of the energy surface of the intrinsic wave function of ${}^{20}\mathrm{Ne}$ in the two-parameter space, $S_z$ and ${\beta }_x={\beta }_y={\beta }_z$.

Figure 2

Contour map of the energy surface of the $J^{\pi }=0^{+}$ state in the two-parameter space, $S_z$ and ${\beta }_x={\beta }_y={\beta }_z$.

Figure 3

Contour map of the energy surface of the $J^{\pi }=2^{+}$ state in the two-parameter space, $S_z$ and ${\beta }_x={\beta }_y={\beta }_z$.

Figure 4

Contour map of the energy surface of the $J^{\pi }=4^{+}$ state in the two-parameter space, $S_z$ and ${\beta }_x={\beta }_y={\beta }_z$.

Figure 5

Contour map of the energy surface of the $J^{\pi }=1^{-}$ state in the two-parameter space, $S_z$ and ${\beta }_x={\beta }_y={\beta }_z$.

Figure 6

Contour map of the energy surface of the $J^{\pi }=3^{-}$ state in the two-parameter space, $S_z$ and ${\beta }_x={\beta }_y={\beta }_z$.

Figure 7

Contour map of the energy surface of the $J^{\pi }=5^{-}$ state in the two-parameter space, $S_z$ and ${\beta }_x={\beta }_y={\beta }_z$.

Figure 8

Energy curves of $J^{\pi }=0^{+}$, $2^{+}$, $1^{-}$, and $3^{-}$ states with different widths of Gaussian relative wave functions in the hybrid model.

Figure 9

Contour map of the energy surface of the $J^{\pi }=0^{+}$ state in the two-parameter space, ${\beta }_x={\beta }_y$ and ${\beta }_z$.

Figure 10

Contour map of the energy surface of the $J^{\pi }=2^{+}$ state in the two-parameter space, ${\beta }_x={\beta }_y$ and ${\beta }_z$.

Figure 11

Contour map of the energy surface of the $J^{\pi }=4^{+}$ state in the two-parameter space, ${\beta }_x={\beta }_y$ and ${\beta }_z$.

Figure 12

Contour map of the energy surface of the $J^{\pi }=1^{-}$ state in the two-parameter space, ${\beta }_x={\beta }_y$ and ${\beta }_z$.

Figure 13

Contour map of the energy surface of the $J^{\pi }=3^{-}$ state in the two-parameter space, ${\beta }_x={\beta }_y$ and ${\beta }_z$.

Figure 14

Contour map of the energy surface of the $J^{\pi }=5^{-}$ state in the two-parameter space, ${\beta }_x={\beta }_y$ and ${\beta }_z$.

Figure 15

Contour map of the squared overlap between the $0^{+}$ wave function with ${\beta }_x={\beta }_y=0.9$ fm, ${\beta }_z=2.5$ fm and the $0^{+}$ wave function with variable ${\beta }_x={\beta }_y$ and ${\beta }_z$. Numbers attached to the contour lines are squared-overlap values.

Figure 16

Contour map of the squared overlap between the $2^{+}$ wave function with ${\beta }_x={\beta }_y=0.0$ fm, ${\beta }_z=2.2$ fm, and the $2^{+}$ wave function with variable ${\beta }_x={\beta }_y$ and ${\beta }_z$. Numbers attached to the contour lines are squared-overlap values.

Figure 17

Contour map of the squared overlap between the $4^{+}$ wave function with ${\beta }_x={\beta }_y=0.0$ fm, ${\beta }_z=1.8$ fm, and the $4^{+}$ wave function with variable ${\beta }_x={\beta }_y$ and ${\beta }_z$. Numbers attached to the contour lines are squared-overlap values.

Figure 18

Contour map of the squared overlap between the $1^{-}$ wave function with ${\beta }_x={\beta }_y=3.7$ fm, ${\beta }_z=1.4$ fm, and the $1^{-}$ wave function with variable ${\beta }_x={\beta }_y$ and ${\beta }_z$. Numbers attached to the contour lines are squared-overlap values.

Figure 19

Contour map of the squared overlap between the $3^{-}$ wave function with ${\beta }_x={\beta }_y=3.7$ fm, ${\beta }_z=0.0$ fm, and the $3^{-}$ wave function with variable ${\beta }_x={\beta }_y$ and ${\beta }_z$. Numbers attached to the contour lines are squared-overlap values.

Figure 20

Contour map of the squared overlap between the $5^{-}$ wave function with ${\beta }_x={\beta }_y=3.3$ fm, ${\beta }_z=0.0$ fm, and the $5^{-}$ wave function with variable ${\beta }_x={\beta }_y$ and ${\beta }_z$. Numbers attached to the contour lines are squared-overlap values.

Figure 21

Rotation average of a prolate THSR wave function around an axis ($x$ axis) perpendicular to the symmetry axis ($z$ axis) of the prolate THSR wave function.

Figure 22

Squared overlap $|O(B_x=B_y,B_z){|}^2$ of the rotation-average wave function ${\Phi }^{\mathrm{AV}}(B_x=B_y,B_z)$ with the oblate ${}^{16}\text{O}+\alpha$ THSR wave function ${\Phi }^{\mathrm{obl}}({\tilde{B}}_x,{\tilde{B}}_y={\tilde{B}}_z)$ which gives the minimum energy of the $J^{\pi }=1^{-}$ state after the angular-momentum projection. The values of ${\tilde{B}}_k$ are $({\tilde{\beta }}_x,{\tilde{\beta }}_y,{\tilde{\beta }}_z)=(1.4,3.7,3.7\mathrm{fm})$, where ${\tilde{B}}_k^2=b^2+2{\tilde{\beta }}_k^2$.

Figure 23

Squared overlap $|\widehat{O}(B_x,B_y=B_z){|}^2$ of the oblate ${}^{16}\text{O}+\alpha$ THSR wave function ${\Phi }^{\mathrm{obl}}(B_x,B_y=B_z)$ with the rotation-average wave function ${\Phi }^{\mathrm{AV}}({\tilde{B}}_x={\tilde{B}}_y,{\tilde{B}}_z)$ obtained from the prolate wave function ${\Phi }^{\mathrm{prol}}({\tilde{B}}_x={\tilde{B}}_y,{\tilde{B}}_z)$, which gives the minimum energy of the $J^{\pi }=0^{+}$ ground state after the angular-momentum projection. The values of ${\tilde{B}}_k$ are $({\tilde{\beta }}_x,{\tilde{\beta }}_y,{\tilde{\beta }}_z)=(0.9,0.9,2.5\mathrm{fm})$, where ${\tilde{B}}_k^2=b^2+2{\tilde{\beta }}_k^2$.

Figure 24

Contour map of the squared overlap between the $\alpha$-$\alpha$ oblate THSR wave function with ${\beta }_x$ = 0.1 fm, ${\beta }_y$ = ${\beta }_z$ = 4.4 fm, and the rotation-average wave function constructed from the THSR wave function with various ${\beta }_x$, ${\beta }_y$ = ${\beta }_z$.

Figure 25

The contour map of the squared-overlap values of the rotation-average wave function obtained from the oblate $3\alpha$ THSR wave function with ${\beta }_x={\beta }_y$ = 5.7 and ${\beta }_z$ = 1.3 with $3\alpha$ THSR wave functions with various ${\beta }_x$ and ${\beta }_y={\beta }_z$.

Figure 26

Density distribution of a 2$\alpha$ prolate THSR wave function with $({\beta }_x,{\beta }_y,{\beta }_z)$ = (1.78, 1.78, 7.85 fm).

Figure 27

Density distribution of a 3$\alpha$ THSR wave function with strong prolate deformation with $({\beta }_x,{\beta }_y,{\beta }_z)$ = (0.01, 0.01, 5.1 fm). This figure is taken from Ref. [42].

Figure 28

Density distribution of the ${}^{16}\text{O}+\alpha$ hybrid-Brink-THSR wave function with $S_z$ = 0.6 fm and $({\beta }_x,{\beta }_y,{\beta }_z)$ = (0.9, 0.9, 2.5 fm).