A consistency check: Superdeformed rotational bands

Another phenomenon in which the time-odd part of the Skyrme energy density functional plays a role is the high-spin rotation of very elongated nuclei. In this section we demonstrate that reasonable values for the spin-isospin coupling constants found when analyzing the GT strength are consistent with the description of superdeformed rotational bands.

When a nucleus rotates rapidly, there appear strong current and spin one-body densities along with the usual particle densities that characterize stationary (time-even) states. The time-odd densities are at the origin of strong time-odd mean fields. There are already many self-consistent studies of high-spin states available; see, e.g., reviews in Refs. [12,65,66,67]. The role and significance of the time-odd mean-field terms, however, has not been carefully studied. Basic features of high-spin states can often be well described by models that use phenomenological mean fields of the Woods-Saxon or Nilsson type, where no time-odd terms are explicitly present in the one-body potential. (The time-odd densities are, however, present there through the time-odd cranking term.) For the Gogny interaction [59], or within the standard RMF models [12], they cannot be independently modified; the Gogny interaction is defined as a two-body force (where the time-odd terms show up as exchange terms), while all time-odd terms appearing in standard RMF models are fixed by Lorentz invariance. Within the Skyrme framework, the time-odd terms in superdeformed rotational states were analyzed in an exploratory way in Refs. [3,20].

Unlike the GT response, rotational bands are influenced by both isoscalar and isovector time-odd channels of the effective interaction. In fact, the large effects of time-odd coupling constants found in [3] are mainly due to the isoscalar channel; the isovector channel induces corrections that are smaller, though non-negligible. The SkO' Skyrme parameterization, which we use for GT calculations, is unstable when the original parameters from Eq. (34) are used in the isoscalar spin channel because g₀ < -1 (a fact that is related to the unusually high value of g₁ in Table 1). This leads to unphysical ferromagnetic solutions where all spins align when the nucleus is cranked. Of course, the value of g₀ does not influence the GT calculations for even-even nuclei presented in our study which focuses on the isovector time-odd coupling constants. Consequently, in the following we employ a simple spin energy functional using the Skyrme force value for C₀^T, setting $C_0^{\Delta s} = 0$ and neglecting density dependence. We adopt the value g₀ = 0.4 given in [52] (note that a different definition of the normalization factor is used there) to fix C₀^s.

We perform the calculations in exactly the same way as in Ref. [3] by using the code HFODD (v1.75r) described in Ref.[68]. We examine ¹⁵²Dy, which is a doubly magic superdeformed system. Pairing has a minor influence and we neglect it. We focus on the dynamic moment of inertia ${\cal J}_2$:

$${\cal J}^{(2)}(I) = \left[ \frac{d^2E}{dI^2} \right]^{-1} \simeq \frac{4\hbar^2 }{\Delta E_\gamma}$$ (29)

(from experimental data) or

$${\cal J}^{(2)}(\omega_1) = \frac{dI}{d \omega} \simeq \frac{I(\omega_1) - I(\omega_2)}{\omega_1 - \omega_2}$$ (30)

(in calculations). Figure 13 shows results of calculations when one of the four time-odd isovector coupling constants is varied, while the other ones are kept at the values mentioned above. Variations of the coupling constants C₁^s[0], $C_1^{s} [\rho _{\rm nm}]$, and C₁^T have little effect on the dynamic moments of inertia in ¹⁵²Dy. When $C_1^{\Delta s}$ is varied, the moments change noticeably, but the general trend with frequency is still the same. Thus, altering the isovector time-odd couplings does not appear to change the quality with which we describe superdeformed rotational bands. Of course, a consistent description of both the high-spin data and the GT resonance properties over a wide range of nuclei will require a much more detailed analysis.