Investigation of ${}^{10}\mathrm{Be}$ and its cluster dynamics with the nonlocalized clustering approach

We extend the concept of nonlocalized clustering to the nucleus ${}^{10}\mathrm{Be}$ with proton number $Z=4$ and neutron number $N=6 \left(N=Z+2\right)$. The Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave function is formulated for the description of different structures of ${}^{10}\mathrm{Be}$. Physical properties such as energy spectrum and root-mean-square radii are calculated for the first two $0^{+}$ states and corresponding rotational bands. With only one single THSR wave function, the calculated results show good agreement with other models and experimental values. We apply, for the first time, the THSR wave function on the chain orbit ($\sigma$-orbit) structure in the $0_2{}^{+}$ state of ${}^{10}\mathrm{Be}$. The ring-orbit ($\pi$-orbit) and $\sigma$-orbit structures are further illustrated by calculating the density distribution of the valence neutrons. We also investigate the nonlocalized dynamics of $\alpha$ clusters and the correlations of two valence neutrons in ${}^{10}\mathrm{Be}$.

Figure 1

Generator coordinates and $\beta$ parameters used in the THSR wave function for the ground state $0_1{}^{+}$ of ${}^{10}\mathrm{Be}$. Left panel (a) shows the case with correlations between valence neutrons. Right panel (b) shows the case of independent valence neutron motion in ${\beta }_{\mathrm{pair}}=0$ limit. Vectors are corresponding generator coordinates. Dashed ellipses denote nonlocalized motion of $\alpha$ clusters. Solid ellipses denote the nonlocalized motion of the valence neutrons.

Figure 2

Single-nucleon wave function ${\varphi }_n\left(z\right)$ for the valence neutron in the $0_2{}^{+}$ state of ${}^{10}\mathrm{Be}$. Parameters are chosen as $D=2$ fm and ${\beta }_{n,z}=5$ fm.

Figure 3

The $0^{+}$ ground state of ${}^{10}\mathrm{Be}$ and its rotational band. “THSR” denotes calculated results with the THSR wave function. “THSR $\left({\beta }_{\mathrm{pair}}=0\right)$” denotes calculated results with the THSR wave function and the ${\beta }_{\mathrm{pair}}=0$ limit. “Ref. [21]” denotes the results of the AMD method [21]. “Exp.” denotes the experimental result. The dashed lines indicate the corresponding $\alpha +\alpha +n+n$ threshold −55.2 MeV.

Figure 4

Density distribution of the valence neutrons in the intrinsic ground state of ${}^{10}\mathrm{Be}$. The color scale of each point in the fig-ure is proportional to the nucleon density on the $x\text{−}z$ plane of the $y=0$ cross section. The unit of the density is ${\mathrm{fm}}^{-3}$.

Figure 5

Energy spectrum of the $0_2{}^{+}$ rotational band relative to the ground-state energy. The one labeled with Ref. [21] is the value calculated with the AMD+DC method [21]. The one labeled with Ref. [29] is the value calculated with the $\beta -\gamma$ constrained AMD+GCM method [29]. The dashed lines are the corresponding $\alpha +\alpha +n+n$ thresholds of −55.2 MeV.

Figure 6

Density distribution of valence neutrons in the intrinsic $0_2{}^{+}$ state of ${}^{10}\mathrm{Be}$. The color scale of each point in the figure is proportional to the nucleon density in the $x\text{−}z$ plane of the $y=0$ cross section. The unit of the density is ${\mathrm{fm}}^{-3}$.

Figure 7

Energy curves with different parameter ${\beta }_z$ for the $\alpha$-cluster motion in the hybrid wave function.

Figure 8

Contour map of the binding energy of the $0_1{}^{+}$ ground state of ${}^{10}\mathrm{Be}$ with different parameters ${\beta }_{\alpha ,z}$ and $S_z$. Other parameters for $\alpha$ clusters are ${\beta }_{\alpha ,xy}=0.1$ fm and $S_z=0$ fm. Parameters for valence neutrons are taken from the variational result in Sec. 3. The optimum value is marked on the map labeled with corresponding coordinates.

Figure 9

Contour map of the binding energy surface of the ground state with different $\beta$ parameters in the THSR wave function. The horizontal coordinates are the ${\beta }_{\text{pair}}$ parameters as ${\beta }_{\text{pair},xy}={\beta }_{\text{pair},z}$. The vertical coordinates are the ${\beta }_n$ parameters as ${\beta }_{n,xy}={\beta }_{n,z}$. Other parameters are ${\beta }_{\alpha ,xy}=0.1\mathrm{fm},{\beta }_{\alpha ,z}=2.0$ fm. The optimum value is marked on the map labeled with corresponding coordinates.