Luneburg-lens-like universal structural Pauli attraction in nucleus-nucleus interactions: Origin of emergence of cluster structures and nuclear rainbows

The Pauli exclusion principle plays an important role in many-body fermion systems, preventing them from collapsing by repulsion. For example, the Pauli principle causes a repulsive potential at short distances between two $\alpha$ particles. On the other hand, the existence of nuclear rainbows demonstrates that the internuclear potential is sufficiently attractive in the internal region to cause refraction. The two concepts of repulsion and attraction are seemingly irreconcilable. Contrary to traditional understanding, it is shown that the Pauli principle causes a universal structural Pauli attraction between nuclei rather than a structural repulsive core. Through systematic studies of $\alpha +\alpha ,\alpha +{}^{16}\mathrm{O},\alpha +{}^{40}\mathrm{Ca}$, and ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ systems, it is shown that the emergence of cluster structures near the threshold energy at low energies and nuclear rainbows at high energies is a direct consequence of the Pauli principle.

Figure 1

The experimental angular distribution (points) [16] in $\alpha +{}^{40}\mathrm{Ca}$ rainbow scattering at $E_L=61$ MeV and the calculated one (red thick solid line), which is decomposed into the farside (red long-dashed line) and nearside (red medium-dashed line) contributions. The moduli of $S$ matrices, $|S_L|$, in the inset (red filled circles) are connected by lines to guide the eye. The angular distribution (brown thin solid line) and its farside component (brown long-dashed line) calculated by switching off the imaginary potential $\left(W=0\right)$ are also displayed. The cutoff calculations of $L=0\text{–}11$ partial waves with and without $W$ are shown by the blue short-dashed lines.

Figure 2

The global nuclear potential (black solid lines) and the corresponding Luneburg lens potential (black circles) for the $\alpha +{}^{40}\mathrm{Ca}$ system. The potential including the Coulomb potentials and corresponding Luneburg lens potential are indicated by long-dashed lines (pink) and squares (pink), respectively. The calculated eigenstates for $L=0$ with $N<N_0=12$ together with the $N=N_0$ band head $0^{+}$ of the $\alpha +{}^{40}\mathrm{Ca}$ cluster state in ${}^{44}\mathrm{Ti}$ are indicated by horizontal solid lines. The eigenenergies of the Luneburg potential including the Coulomb potential are indicated by horizontal dashed lines.

Figure 3

Same as Fig. 1 for panel (a) and Fig. 2 for panel (b) but for the $\alpha +{}^{16}\mathrm{O}$ system with $N_0=8$. In the inset of panel (a) $|S_L|$ of the internal waves are additionally indicated by unfilled circles. In panel (b) the triangles (red) are the double folding potentials derived from the HNY force with $-538$ MeV for the triplet even state in the intermediate range [25]. The $N=8$ (solid horizontal line) shows the $\alpha +{}^{16}\mathrm{O}$ cluster ground state in ${}^{20}\mathrm{Ne}$.

Figure 4

Same as Fig. 2 but for the $\alpha +\alpha$ system with $N_0=4$. The solid horizontal line shows the $N=4$ ground state with $\alpha +\alpha$ cluster structure in ${}^8\mathrm{Be}$.

Figure 5

Same as Fig. 2 but for the ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ system with $N_0=24$. The solid horizontal line shows the $N=240^{+}$ state with ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ cluster structure in ${}^{32}\mathrm{S}$.