Visualization of nuclear many-body correlations with the most probable configuration of nucleons

A method to visualize many-body correlations using the information of the full wave function is presented. The set of nucleon coordinates which maximizes the square of the wave function, that is, the most probable spatial configuration of nucleons, is visualized. The method is applied to Hartree-Fock (HF) and $\mathrm{HF}+\mathrm{BCS}$ wave functions of $p$- and $sd$-shell $N=Z$ even-even nuclei to analyze the many-body correlations in those systems. It is found that there are $\alpha$-cluster-like four-body correlations already at the HF level in some of the nuclei. The effects of pairing on the most probable configuration are also investigated. The method is useful to analyze the nuclear many-body correlations, and it suggests a new viewpoint to microscopic nuclear wave functions.

Figure 1

The most probable configuration of spin-up (-down) neutrons in the HF ground state of ${}^8\mathrm{Be}$ are shown with blue (sky blue) arrows together with (a) the neutron one-body density and (b) the neutron localization function ${\overline{\mathcal{C}}}_{n\uparrow }$. The density isosurface in (a) is drawn at half the maximum value, while the localization function in (b) is drawn at 0.8 times the maximum.

Figure 2

The most probable configurations of spin-up (-down) neutrons obtained by the HF and the $\mathrm{HF}+\mathrm{BCS}$ wave functions in the ground state of ${}^{12}\mathrm{C}$ nucleus are shown with blue (sky blue) arrows. We show (a) the HF result together with the neutron one-body density, (b) the HF result together with the neutron localization function, and (c) the $\mathrm{HF}+\mathrm{BCS}$ result together with the neutron one-body density. The density isosurface is drawn at half the maximum value, while the localization function is drawn at 0.8 times the maximum. See the Supplemental Material [36] for the animated views of (a) and (c).

Figure 3

The same as Fig. 2 but for the ground state of ${}^{16}\mathrm{O}$ nucleus. There are two surfaces for the localization function because it is peaked around 1.5 fm away from the center of the nucleus.

Figure 4

The same as Fig. 2 but for the ground state of the ${}^{20}\mathrm{Ne}$ nucleus.

Figure 5

The same as Fig. 2 but for the ground state of the ${}^{24}\mathrm{Mg}$ nucleus.

Figure 6

The same as Fig. 2 but for the ground state of the ${}^{28}\mathrm{Si}$ nucleus.

Figure 7

${\left|\mathrm{\Psi }\right|}^2$ of ${}^{20}\mathrm{Ne}$ as functions of the relative angle $\theta$ between spin-up and -down neutrons. The angle $\theta$ is defined as the rotation angle of the positions of spin-up neutrons about the $z$ axis while those of spin-down neutrons are fixed. $\theta =0$ corresponds to the most probable configuration shown in Fig. 4. The probabilities are normalized with their own maximum values. The solid curve represents the HF result, and the dashed and dotted curves show the $\mathrm{HF}+\mathrm{BCS}$ results with pairing gaps $\mathrm{\Delta }=6/A^{1/2}$ MeV and $\mathrm{\Delta }=12/A^{1/2}$ MeV, respectively.

Figure 8

The same as Fig. 7 but for the ground state of the ${}^{24}\mathrm{Mg}$ nucleus.

Figure 9

The $\mathrm{HF}+\mathrm{BCS}$ potential-energy curve of the ${}^{16}\mathrm{O}$ nucleus as a function of the quadrupole moment $Q_2$. The HF $+$ BCS result is obtained with the constant-gap approximation with the gap parameter $\mathrm{\Delta }=12/A^{1/2}$ MeV. The density isosurfaces at $Q_2=-90$, $-50$, 50, and $180{\mathrm{fm}}^2$ are also shown.

Figure 10

The most probable neutron configurations of the ${}^{16}\mathrm{O}$ nucleus at (a) $Q_2=50{\mathrm{fm}}^2$, (b) 2 ${\mathrm{fm}}^2$, (c) 50 ${\mathrm{fm}}^2$, and (d) 180 ${\mathrm{fm}}^2$. The lower panel shows the neutron single-particle energies and the Fermi energy as functions of $Q_2$. The background colors are inserted according to the four regimes of neutron configurations: square for $Q_2\lesssim -25{\mathrm{fm}}^2$, tetrahedron for $-25{\mathrm{fm}}^2\lesssim Q_2\lesssim 25{\mathrm{fm}}^2$, diamond for $25{\mathrm{fm}}^2\lesssim Q_2\lesssim 100{\mathrm{fm}}^2$, and linear chain for $100{\mathrm{fm}}^2\lesssim Q_2$. The vertical lines labeled by (a)–(d) correspond to the $Q_2$ values at which the upper figures are obtained. See the Supplemental Material [36] for an animation showing the evolution of the neutron configuration as $Q_2$ increases.

Figure 11

Values of ${\rho }_{d_s}^{\left(N\right)}$ for $d_s=0,\cdots ,4$ as functions of CG iterations for the neutron part of the wave function of the ${}^{20}\mathrm{Ne}$ nucleus. The initial spatial neutron coordinates are generated randomly.