Effect of time-odd mean fields on inertial parameters of the quadrupole collective Hamiltonian

Most current implementations of the quadrupole collective Hamiltonian model do not include the contributions of time-odd mean fields to the moments of inertia and mass parameters (Thouless-Valatin dynamical rearrangement contributions). A hybrid model is introduced that allows a quantitative estimate of these contributions to the inertial functions of a five-dimensional collective Hamiltonian based on microscopic energy density functionals. Fully self-consistent constrained relativistic Hartree-Bogoliubov calculations of triaxial energy surfaces in the $\beta -\gamma$ plane are used to determine the parameters of an equivalent pairing-plus-quadrupole ($P+Q$) Hamiltonian. This Hamiltonian is employed in constrained Hartree-Fock-Bogoliubov (CHFB) plus local quasiparticle random-phase approximation (LQRPA) calculation of the deformation-dependent corrections to the Inglis-Belyaev moments of inertia and cranking mass parameters. This hybrid model is used to evaluate the influence of time-odd mean fields on vibrational and rotational collective masses and investigate their effect on the low-energy collective excitation spectra and transition rates of $\gamma$-soft Xe and Ba nuclei.

Figure 1

Self-consistent RHB triaxial quadrupole binding energy maps of even-even ${}^{128-132}$Xe and ${}^{130-134}$Ba in the $\beta -\gamma$ plane ($0^{\circ }⩽\gamma ⩽{60}^{\circ }$). All energies are normalized with respect to the corresponding minima. The energy gap between two neighboring contours is 1 MeV.

Figure 2

Quadrupole energy surfaces in the $\beta -\gamma$ plane ($0^{\circ }⩽\gamma ⩽{60}^{\circ }$) of the even-even ${}^{128-132}$Xe and ${}^{130-134}$Ba isotopes, calculated with the pairing + quadrupole Hamiltonian (see text for description). All energies are normalized with respect to the corresponding absolute minimum. The energy gap between two neighboring contours is 0.3 MeV.

Figure 3

Ratios of the moments of inertia $B_x$, $B_y$, and $B_z$, calculated with the LQRPA and the IB formula, as functions of the deformations $\beta$ and $\gamma$, for the nuclei ${}^{128-132}$Xe.

Figure 4

Ratios of the vibrational mass parameters $B_{\beta \beta }$ and $B_{\gamma \gamma }$ calculated with the LQRPA and in the cranking approximation, as functions of the deformations $\beta$ and $\gamma$, for the nuclei ${}^{128-132}$Xe.

Figure 5

Same as described in the caption to Fig. 3 but for ${}^{130-134}$Ba.

Figure 6

Same as described in the caption to Fig. 4 but for ${}^{130-134}$Ba.

Figure 7

The IB moments of inertia for ${}^{128}$Xe calculated from Eq. (29) using the fully self-consistent RHB quasiparticle states (left column) and multiplied by the CHFB + LQRPA ratios that represent the contribution of time-odd mean fields (right column).

Figure 8

The cranking mass parameters $B_{\beta \beta }^{\left(C\right)}$ and $B_{\gamma \gamma }^{\left(C\right)}$ [Eq. (30)] for ${}^{128}$Xe, based on the self-consistent RHB ground-state solution (left column) and multiplied by the CHFB + LQRPA ratios that represent the contribution of time-odd mean fields (right column).

Figure 9

Low-energy level schemes of ${}^{128-132}$Xe calculated with the five-dimensional collective Hamiltonian, in comparison to available data (right column). The spectra are calculated with the IB moments of inertia and cranking mass parameters (left column) and with the approximate inclusion of the contribution of time-odd mean fields using the CHFB + LQRPA method (middle column). $B\left(E2\right)$ values are given in Weisskopf units.

Figure 10

Same as described in the caption to Fig. 9 but for ${}^{130-134}$Ba.