Tensor part of the Skyrme energy density functional. III. Time-odd terms at high spin

This article extends previous studies on the effect of tensor terms in the Skyrme energy density functional by the breaking of time-reversal invariance. We have systematically probed the impact of tensor terms on properties of superdeformed rotational bands calculated within the cranked Hartree-Fock-Bogoliubov approach for different parametrizations covering a wide range of values for the isoscalar and isovector tensor coupling constants. We analyze in detail the contribution of the tensor terms to the energies and dynamical moments of inertia and study their impact on quasiparticle spectra. Special attention is devoted to the time-odd tensor terms, the effect of variations of their coupling constants, and finite-size instabilities.

Figure 1

Coupling constants $C_0^J$ and $C_1^J$, where $C_t^J=-C_t^T+\frac{1}{2}{C}_t^F$, for the parameterizations discussed in the article.

Figure 2

Coupling constants $C_t^T$ and $C_t^F$ for the parameterizations discussed in the article.

Figure 3

Coupling constants $C_t^{\Delta s}$ and $C_t^{\nabla s}$ as obtained in a force-generated EDF for the parametrizations discussed in the article.

Figure 4

(a) Dependence of the $C_0^{\Delta s}{\mathbf{s}}_0·\Delta {\mathbf{s}}_0$ term of a variant of the T22 parametrization on the value of $C_0^{\Delta s}$ for the $J_z=54\hslash$ state in the ground-state SD band of ${}^{194}$Hg (see text). (b) Dependence of all other time-odd terms containing the spin density ${\mathbf{s}}_t$ relative to their value at $C_0^{\Delta s}=0$ in the same calculations.

Figure 5

Change of the (a) contribution from the tensor terms relative to the values at the spherical shape and (b) deformation energy relative to the spherical shape for ${}^{194}\mathrm{Hg}$ obtained with the parametrizations T22, T26, T44, and T62. The energy scale is the same for the two panels.

Figure 6

Proton and neutron Nilsson diagrams in ${}^{194}\mathrm{Hg}$ for the T22, T26, T44, and T62 parametrizations.

Figure 7

(a) Proton ($\pi$), neutron ($\nu$), and total ($t$) dynamical moments of inertia as a function of the rotational frequency for the SD band in ${}^{194}$Hg with the SLy4, T22, and T44 parameterization. (b) The charge quadrupole moment as a function of the rotational frequency for the SLy4, T22, and T44 parametrizations.

Figure 8

Evolution of the time-even terms in the EDF as a function of the rotational frequency $\hslash \omega$ for SLy4, T22, and T44 for the ground-state SD band of ${}^{194}\mathrm{Hg}$.

Figure 9

Same as Fig. 8, but for the time-odd terms in the Skyrme EDF.

Figure 10

Dynamical moments of inertia of different terms in the EDF as a function of the rotational frequency for SLy4, T22, and T44 in the calculation of the ground-state SD band of ${}^{194}$Hg.

Figure 11

Same as Fig. 10, but for the time-odd terms in the Skyrme EDF.

Figure 12

Proton quasiparticle Routhians for the ground-state SD band of ${}^{194}$Hg for the SLy4, T22, and T44 parametrizations. The (parity, signature) combinations are indicated in the figure. All Routhians are additionally characterized by the $j$ component of the dominant single-particle state in the quasiparticle wave function along the axis of largest elongation at $\hslash \omega =0$ and by their particle (p) or hole (h) character.

Figure 13

Same as Fig. 12 but for neutrons.

Figure 14

(a) Proton ($\pi$), neutron ($\nu$), and total ($t$) dynamical moments of inertia in ${}^{194}$Hg as a function of the rotational frequency for the SD band in ${}^{194}$Hg with the T26, the T44, and the T62 parametrization. (b) The charge quadrupole moment in ${}^{194}$Hg as a function of the rotational frequency for the T26, the T44, and the T62 parametrizations.

Figure 15

Same as Fig. 12, but for the T26, T44, and T62 parametrizations.

Figure 16

Same as Fig. 15, but for neutrons.

Figure 17

Same as Fig. 10 but for the T26, T44, and T62 parametrizations.

Figure 18

Same as Fig. 10, but for the time-odd terms in the Skyrme EDF with the T26, T44, and T62 parametrizations.

Figure 19

Dynamical moments of inertia as a function of the rotational frequency for different choices of the strength $V=V_p=V_n$ of the density-dependent $\delta$ pairing interaction [Eq. (18)]. The SLy4 parametrization was chosen in the particle-hole channel of the interaction and the pairing strength is given in units of MeV fm${}^{-3}$.

Figure 20

Dynamical moments of inertia as a function of the rotational frequency for different density-independent choices of the $C_t^s$ coupling constant. If the isoscalar $C_0^s$ is varied, then the density-independent $C_1^s$ is chosen to be equal to $C_1^s\left[\text{nm}\right]$ and vice versa. All other coupling constants in ${\mathcal{E}}_{\text{Sk}}$ are determined by the SLy4 parametrization and the $C_t^s$ coupling constant is given in units MeV fm${}^3$.

Figure 21

Dynamical moments of inertia as a function of the rotational frequency for variations of (a) $C_t^{\Delta s}$, (b) $C_t^{\nabla s}$, and (c) $C_t^s$ coupling constants. All coupling constants of the Skyrme EDF not explicitly mentioned in the legend are determined by the T22 parametrization. For the density-independent variations of the $C_t^s$ coupling constant (c), the same prescription as for SLy4 and explained in the caption of Fig. 20 is followed. The $C_t^{\nabla s}$ and $C_t^{\Delta s}$ coupling constants are expressed in units MeV fm${}^5$ and the $C_t^s$ coupling constant is given in units MeV fm${}^3$.

Figure 22

Same as Fig. 21, but for the T26 parametrization.

Figure 23

Same as Fig. 21, but for the T44 parametrization.

Figure 24

Same as Fig. 21, but for the T62 parametrization.

Figure 25

Contributions to the dynamical moment of inertia coming from the time-even terms of the EDF as a function of the rotational frequency for variations of the $C_t^s$ coupling constant. All coupling constants not explicitly mentioned are determined by the T44 parametrization. The $C_t^s$ coupling constants are expressed in units of MeV fm${}^3$.

Figure 26

Same as Fig. 25, but for the time-odd terms in the Skyrme EDF.

Figure 27

(a) Proton ($\pi$), neutron ($\nu$), and total ($t$) dynamical moments of inertia as a function of the rotational frequency for the SD ground band in ${}^{152}$Dy with the SLy4, T22, and T44 parametrizations. (b) The charge quadrupole moment in ${}^{152}$Dy as a function of the rotational frequency for the SLy4, T22, and T44 parametrizations.

Figure 28

(a) Proton ($\pi$), neutron ($\nu$), and total ($t$) dynamical moments of inertia as a function of the rotational frequency for the SD ground band in ${}^{152}$Dy with the T26, T44, and T62 parametrizations. (b) The charge quadrupole moment in ${}^{152}$Dy as a function of the rotational frequency for the T26, T44, and T62 parametrizations.

Figure 29

Evolution of different terms in the EDF as a function of the rotational frequency for the SLy4, T22, and T44 parametrizations in the calculation of the ground state SD band of ${}^{152}$Dy.

Figure 30

Same as Fig. 29, but for the time-odd terms in the Skyrme EDF.

Figure 31

Dynamical moments of inertia of different terms in the EDF as a function of the rotational frequency for the SLy4, T22, and T44 parametrizations in the calculation of the ground-state SD band of ${}^{152}$Dy.

Figure 32

Same as Fig. 31, but for the time-odd terms in the Skyrme EDF.

Figure 33

Proton quasiparticle Routhians for the ground-state SD band of ${}^{152}$Dy for the SLy4, T22, and T44 parametrizations. The (parity, signature) combinations are indicated in the figure. All Routhians are additionally characterized by the $j$ component of the dominant single-particle state in the quasiparticle wave function along the axis of largest elongation at $\hslash \omega =0$ and by their particle (p) or hole (h) character.

Figure 34

Same as Fig. 33, but for neutrons.