Microscopic core-quasiparticle coupling model for spectroscopy of odd-mass nuclei

Background: Predictions of the spectroscopic properties of low-lying states are critical for nuclear structure studies but are problematic for nuclei with an odd nucleon due to the interplay of the unpaired single particle with nuclear collective degrees of freedom.

Purpose: To predict the spectroscopic properties of odd-mass medium-heavy and heavy nuclei with a model that treats single-particle and collective degrees of freedom within the same microscopic framework.

Method: A microscopic core-quasiparticle coupling (CQC) model based on the covariant density functional theory is developed that contains the collective excitations of even-mass cores and spherical single-particle states of the odd nucleon as calculated from a quadrupole collective Hamiltonian combined with a constrained triaxial relativistic Hartree-Bogoliubov model.

Results: Predictions of the new model for excitation energies, kinematic and dynamic moments of inertia, and transition rates are shown to be in good agreement with results of low-lying spectroscopy measurements of the axially deformed odd-proton nucleus ${}^{159}\mathrm{Tb}$ and the odd-neutron nucleus ${}^{157}\mathrm{Gd}$.

Conclusions: A microscopic CQC model based on covariant density functional theory is developed for odd-mass nuclei and shown to give predictions that agree with measurements of two medium-heavy nuclei. Future studies with additional nuclei are planned.

Figure 1

Triaxial energy surfaces of the core nuclei ${}^{160}\mathrm{Dy},{}^{158}\mathrm{Gd}$, and ${}^{156}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0\le \gamma \le {60}^0$) as calculated from a constrained triaxial RHB model. For each nucleus, energies are normalized with respect to the binding energy of the global minimum. The contours join points on the surface with the same energy (in MeV).

Figure 2

The excitation energies (left panels) and reduced electric quadrupole transitions $B\left(E2\right)$ (right panels) of the ground-state bands in the core nuclei ${}^{160}\mathrm{Dy},{}^{158}\mathrm{Gd}$, and ${}^{156}\mathrm{Gd}$. The theoretical results are obtained from the quadrupole collective Hamiltonian with collective parameters determined by the constrained triaxial RHB model using the PC-PK1 density functional. The experimental data are taken from Refs. [42, 43, 44].

Figure 3

The calculated low-energy positive-parity bands (panels a, b), kinematic moments of inertia (panel c), and dynamic moments of inertia (panel d) of the odd-proton nucleus ${}^{159}\mathrm{Tb}$, plotted in comparison with experimental data [45]. The theoretical results are calculated from CQC model with only the ground-state band of the core (panel a) and with both the ground-state and $\gamma$ bands of the core (panels b, c, d). The Fermi surface and coupling strength ($\lambda ,\chi$) in the CQC model are chosen as ($-8.50$ MeV, 9.40 MeV/$b^2$) in the calculations.

Figure 4

Same as Fig. 3 but for the negative-parity bands of ${}^{159}\mathrm{Tb}$. The excitation energies in the upper panels are shown relative to the lowest state. The Fermi surface and coupling strength ($\lambda ,\chi$) in the CQC model are chosen as ($-7.00$ MeV, 12.80 MeV/$b^2$).

Figure 5

Same as Fig. 3 but for the negative-parity bands of the odd-neutron nucleus ${}^{157}\mathrm{Gd}$. The Fermi surface and coupling strength ($\lambda ,\chi$) in the CQC model are chosen as ($-9.70$ MeV, 12.50 MeV/$b^2$).

Figure 6

Same as Fig. 3 but for the positive-parity bands of ${}^{157}\mathrm{Gd}$. The excitation energies in the upper panels are shown relative to the lowest state. The Fermi surface and coupling strength ($\lambda ,\chi$) in the CQC model are chosen as ($-7.70$ MeV, 15.10 MeV/$b^2$).

Figure 7

The core and single-particle contributions to the intraband $B(E2;J\to J-1),B(E2;J\to J-2)$, and $B(M1;J\to J-1)$ in the ground-state bands of ${}^{159}\mathrm{Tb}$ (left panels) and ${}^{157}\mathrm{Gd}$ (right panels). In ${}^{157}\mathrm{Gd}$, the single neutron does not contribute to the $E2$ transitions.

Figure 8

(left panel) Quasiparticle energies relevant for the band heads of the low-lying bands as functions of $\chi$ calculated by CQC model with the Fermi surface choosing as the average value of RHB calculation and (right panel) single quasiparticle energies as functions of axially deformed parameter $\beta$ calculated by RHB.