Microscopic model for the collective enhancement of nuclear level densities

A microscopic method for calculating nuclear level density (NLD) is developed, based on the framework of energy density functionals. Intrinsic level densities are computed from single-quasiparticle spectra obtained in a finite-temperature self-consistent mean-field (SCMF) calculation that takes into account nuclear deformation, and is specified by the choice of the energy density functional (EDF) and pairing interaction. The total level density is calculated by convoluting the intrinsic density with the corresponding collective level density, determined by the eigenstates of a five-dimensional quadrupole or quadrupole plus octupole collective Hamiltonian. The parameters of the Hamiltonian (inertia parameters, collective potential) are consistently determined by deformation-constrained SCMF calculations using the same EDF and pairing interaction. The model is applied in the calculation of NLDs of ${}^{94,96,98}\mathrm{Mo}$, ${}^{106,108}\mathrm{Pd}$, ${}^{106,112}\mathrm{Cd}$, ${}^{160,162,164}\mathrm{Dy}$, ${}^{166}\mathrm{Er}$, and ${}^{170,172}\mathrm{Yb}$, in comparison with available data. It is shown that the collective enhancement of the intrinsic level density, consistently computed from the eigenstates of the corresponding collective Hamiltonian, leads to total NLDs that are in very good agreement with data over the entire energy range of measured values.

Figure 1

The energy of the equilibrium (global) minimum (a), the pairing energy (b), entropy (c), and intrinsic level density ${\rho }_i$ (d), as functions of temperature for ${}^{94,96,98}\mathrm{Mo}$, ${}^{106,108}\mathrm{Pd}$, and ${}^{106,112}\mathrm{Cd}$. The results are obtained in finite-temperature triaxial RHB calculations with the DD-PC1 energy density functional and finite-range pairing interaction, as described in the previous section.

Figure 2

Self-consistent triaxial quadrupole deformation-constrained energy surfaces of ${}^{94,96,98}\mathrm{Mo}$, ${}^{106,108}\mathrm{Pd}$, and ${}^{106,112}\mathrm{Cd}$ in the $\beta -\gamma$ plane $(0\le \gamma \le {60}^{\circ })$. For each nucleus the energies are normalized with respect to the binding energy of the global minimum. The contours join points on the surface with the same energy, and the spacing between neighboring contours is 0.5 MeV.

Figure 3

The calculated positive-parity low-spin states of ${}^{94}\mathrm{Mo}$ (b) and their possible experimental counterparts (a). Not all known levels of these spins are shown in panel (a). For a detailed discussion see Ref. [57].

Figure 4

The calculated intrinsic level densities (dash-dotted blue) and total level densities (solid red), as functions of excitation energy for ${}^{94,96,98}\mathrm{Mo}$, ${}^{106,108}\mathrm{Pd}$, and ${}^{106,112}\mathrm{Cd}$. The data (black squares) are from Refs. [59, 60, 61].

Figure 5

The collective enhancement factors $K_{\mathrm{coll}}$ (a), and ratios ${\rho }_{\mathrm{tot}}/{\rho }_{\exp }$ of calculated and experimental level densities (b), as functions of excitation energy for ${}^{94,96,98}\mathrm{Mo}$, ${}^{106,108}\mathrm{Pd}$, and ${}^{106,112}\mathrm{Cd}$.

Figure 6

Self-consistent RHB axially symmetric deformation energy surfaces of ${}^{160,162,164}\mathrm{Dy}$, ${}^{166}\mathrm{Er}$, and ${}^{170,172}\mathrm{Yb}$ in the $({\beta }_2,{\beta }_3)$ plane. For each nucleus the energies are normalized with respect to the binding energy of the global minimum. The contours join points on the surface with the same energy, and the spacing between neighboring contours is 1.0 MeV.

Figure 7

Same as in the caption to Fig. 1 but for the axially symmetric and reflection asymmetric RHB calculations of ${}^{160,162,164}\mathrm{Dy}$, ${}^{166}\mathrm{Er}$, and ${}^{170,172}\mathrm{Yb}$.

Figure 8

The calculated intrinsic level densities (dash-dotted blue) and total level densities (solid red), as functions of excitation energy for ${}^{160,162,164}\mathrm{Dy}$, ${}^{166}\mathrm{Er}$, and ${}^{170,172}\mathrm{Yb}$. The data (black squares) are from Refs. [62, 63, 64, 65, 66].

Figure 9

Same as in the caption to Fig. 5 but for the axially symmetric and reflection asymmetric calculations of ${}^{160,162,164}\mathrm{Dy}$, ${}^{166}\mathrm{Er}$, and ${}^{170,172}\mathrm{Yb}$.