Density functional theory studies of cluster states in nuclei

The framework of nuclear energy density functionals is applied to a study of the formation and evolution of cluster states in nuclei. The relativistic functional DD-ME2 is used in triaxial and reflection-asymmetric relativistic Hartree-Bogoliubov calculations of relatively light $N=Z$ and neutron-rich nuclei. The role of deformation and degeneracy of single-nucleon states in the formation of clusters is analyzed, and interesting cluster structures are predicted in excited configurations of Be, C, O, Ne, Mg, Si, S, Ar, and Ca $N=Z$ nuclei. Cluster phenomena in neutron-rich nuclei are discussed, and it is shown that in neutron-rich Be and C nuclei cluster states occur that are characterized by molecular bonding of $\alpha$ particles by the excess neutrons.

Figure 1

Neutron single-particle levels of ${}^{36}\mathrm{Ar}$ that correspond to the SCMF solutions calculated with the Skyrme functional SLy4, the Gogny effective interaction D1S, and the relativistic density functional DD-ME2. The levels are labeled by the Nilsson quantum numbers, and dotted lines denote the position of the Fermi level.

Figure 2

Self-consistent binding energy curves of ${}^{36}\mathrm{Ar}$ as functions of the quadrupole deformation parameter ${\beta }_2$, calculated with the functionals (a) SLy4, (b) D1S and (c) DD-ME2. The insets display the corresponding intrinsic nucleon density distributions in the reference frame defined by the principal axes of the nucleus.

Figure 3

Mean value of the energy gap between consecutive occupied neutron levels as a function of the axial quadrupole deformation parameter ${\beta }_2$ for (a) ${}^{12}\mathrm{C}$ and (b) ${}^{20}\mathrm{Ne}$. The insets display the total nucleonic density at the corresponding deformation. To limit the vertical scale the maximum mean value of the energy gap in the plot does not exceed 5 MeV.

Figure 4

Positive-parity projected density plots obtained for a number of excited configurations in $N=Z$ nuclei. For each nucleus the density in the bottom row corresponds to the equilibrium configuration. Other selected densities are displayed in order of increasing excitation energy.

Figure 5

Self-consistent energy surfaces of ${}^8\mathrm{Be}$, calculated with DD-ME2 by (a) imposing constraints on both the axial quadrupole and octupole deformation parameters ${\beta }_2$ and ${\beta }_3$, and (b) the corresponding positive parity-projected energy surfaces.

Figure 6

Same as in Fig. 5 but for the isotopes ${}^{12}\mathrm{C}$.

Figure 7

Total, proton, and neutron SCMF equilibrium intrinsic densities for beryllium isotopes, calculated using the RHB model with the functional DD-ME2.

Figure 8

(a) Contour plot of the ${}^8\mathrm{Be}$ neutron density, and (b, c) surface plots of the partial densities of each of the two occupied Nilsson states in the $\left(Oxz\right)$ plane.

Figure 9

Intrinsic densities of ${}^{10}\mathrm{Be}$ (a) at equilibrium deformation and (b) for an excited configuration. Bottom to top: 3D density of the $\alpha +\alpha$ core, contour plots of the core density and the density of the valence neutrons in the $\left(Oxz\right)$ plane, and 3D density of the valence neutrons.

Figure 10

Same as in the caption to Fig. 9 but for (a) the equilibrium deformation and (b) an excited configuration of ${}^{14}\mathrm{Be}$.

Figure 11

Nucleonic densities for an excited configuration of ${}^{14}\mathrm{C}$. Bottom to top: 3D density of the $\alpha +\alpha +\alpha$ core, contour plots of the core density and the density of the valence neutrons in the $\left(Oxz\right)$ plane, and 3D density of the valence neutrons.

Figure 12

Same as in Fig. 11 but for an excited configuration of ${}^{16}\mathrm{C}$.

Figure 13

Nucleonic densities for an excited configuration of ${}^{10}\mathrm{C}$. Bottom to top: 3D density of the $\alpha +\alpha$ core, contour plots of the core density and the density of the valence protons in the $\left(Oxz\right)$ plane, and 3D density of the valence protons.

Figure 14

Simple approximation of the n-n potential used to derive the relation between the localization and quantality parameters, Eq. (A11).