$\alpha$ clustering from the quartet model

$\alpha$ clustering in nuclei is considered with the quartet model (QM) where four valence nucleons (the quartet) move on the top of the core (daughter) nucleus. In the QM approach, it is assumed that the intrinsic wave function of the quartet is changed from the pure cluster configuration to the shell-model configuration when it crosses some critical radius and enters into the core nucleus. The QM approach could give not only the level scheme, the electromagnetic transition, and the nuclear radius but also the $\alpha$-cluster formation probability. Numerical results are calculated for ${}^{20}\mathrm{Ne}$, ${}^{44}\mathrm{Ti}$, and ${}^{212}\mathrm{Po}$, where a quartet moves on top of a double magic nucleus. Good agreement with experimental data and previous theoretical studies is obtained. The QM approach is a useful complement to the present phenomenological and microscopic models and could help deepen our understanding of $\alpha$ clustering across the nuclide chart.

Figure 1

The comparison of the WSG potential between the quartet and ${}^{16}\mathrm{O}$ with the real parts of various optical potentials, including the Michel potential [28] and the Kumar potential [29].

Figure 2

Plots of the density profiles for the doubly magic nuclei ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$, and ${}^{208}\mathrm{Pb}$ taken from Ref. [30] and summarized in the Appendix. $R_{\text{Mott}}$ denotes the benchmark value of the critical radius determined by matching the tails of the density profiles with the Mott density for the homogeneous nuclear matter ${\rho }_{\text{Mott}}=0.02917{\text{fm}}^{-3}$, which is about one fifth of the nuclear saturation density.

Figure 3

The $\alpha$-cluster formation probability $P_{\alpha }$ vs the critical radius $R_{\text{crit}}$ for the ground-state bands of ${}^{20}\mathrm{Ne}$, ${}^{44}\mathrm{Ti}$, and ${}^{212}\mathrm{Po}$. In Fig. 3, the black solid line corresponds to the results for the $L=0$ state, while the dashed lines with increasing segment lengths correspond to the results for $L=2-8$, respectively. The data points in Fig. 3 denote the AMD results on $P_{\alpha }$ taken from Ref. [32], with the $L=0$ data point labeled by the empty up triangle, the $L=2$ data point labeled by the empty down triangle, the $L=4$ data point labeled by the filled up triangle, the $L=6$ data point labeled by the filled down triangle, and the $L=8$ data point labeled by the empty square. In Fig. 3, the black solid line corresponds to the results for the $L=0$ state, while the dashed lines with increasing segment lengths correspond to the results for $L=2-12$, respectively. The data points in panel (b) denote the AMD results on $P_{\alpha }$ taken from Ref. [33], with the $L=0$ data point labeled by the empty up triangle, the $L=2$ data point labeled by the empty down triangle, the $L=4$ data point labeled by the filled up triangle, the $L=6$ data point labeled by the filled down triangle, the $L=8$ data point labeled by the empty square, and the $L=10$ data point labeled by the empty diamond. In panel (c), the black solid line corresponds to the results for the $L=0$ state, while the dashed lines with increasing segment lengths correspond to the results for $L=2-18$, respectively.

Figure 4

The radial components of the quartet wave functions for the ground-state band of ${}^{20}\mathrm{Ne}$. The critical radius is taken to be $R_{\text{crit}}=1.2R_{\text{Mott}}$.

Figure 5

The same as Fig. 4, except that the target nucleus is ${}^{44}\mathrm{Ti}$.

Figure 6

The same as Fig. 4, except that target nucleus is ${}^{212}\mathrm{Po}$ and the critical radius is $R_{\text{crit}}=1.2R_{\text{Mott}}$.

Figure 7

The $\alpha$-cluster formation probability vs the $\alpha$-core overlap measured by the parameter $D$ for ${}^{20}\mathrm{Ne}$, ${}^{44}\mathrm{Ti}$, and ${}^{212}\mathrm{Po}$.