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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 C0T, 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 C0s.

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}
\end{displaymath} (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}
\end{displaymath} (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 C1s[0], $C_1^{s} [\rho _{\rm nm}]$, and C1T 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.
next up previous
Next: Summary, conclusions, and outlook Up: Gamow-Teller strength and the Previous: Regression analysis of the
Jacek Dobaczewski