Existing parameterizations

The coupling constants of the time-odd Skyrme energy functional are usually taken from the (antisymmetrized) expectation value of a Skyrme force [2]. When so obtained, the 16 coupling constants of the energy functional (12) are uniquely linked to the 10 parameters t_i, x_i, W₀, and $\alpha$ of the standard Skyrme force (see Appendix 8 and Eq. (34)). Only a few parameterizations rigidly enforce these relations, however. Among them are the forces of Ref. [21] (e.g. Z$_\sigma$), SkP [19], the Skyrme forces of Tondeur [28], the recent parameterizations SLy5 and SLy7 [24], and SkX [29]. Most other parameterizations neglect the $\stackrel{\leftrightarrow}{J}^2$ term obtained from the two-body Skyrme force, setting C_t^T = 0. Some authors do this for practical reasons; the $\stackrel{\leftrightarrow}{J}^2$ term is time-consuming to calculate, and its contribution to the total binding energy is rather small. Other authors (see, e.g., [30]) find that including it with a coupling dictated from the HF expectation value of the Skyrme force can lead to unphysical solutions and/or unreasonable spin-orbit splittings. For spherical shapes, the $\stackrel{\leftrightarrow}{J}^2$ term contributes to the time-even energy density in the same way as the neglected tensor force. One might therefore argue that by including the tensor force one could counterbalance the unwanted $\stackrel{\leftrightarrow}{J}^2$ term exactly [30]. This argument, however, applies neither to deformed shapes nor to time-odd fields. Moreover, neglecting this term often violates self-consistency on the QRPA level (see below).

Although one might disagree with the rationale for neglecting the $\stackrel{\leftrightarrow}{J}^2$ terms, it is not easy to adjust the coupling constants C_t^T to spectral data. Large values for C_t^T can be ruled out because they spoil the previously obtained agreement for single-particle spectra, but there are broad regions of values where they influence the usual time-even observables too weakly to be uniquely determined [31]. Only once in the published literature has there been an attempt to do so [32].

All first-generation Skyrme interactions, e.g., SI, SII [33], and SIII [30], used a three-body delta force instead of a density-dependent two-body delta-force to obtain reasonable nuclear-matter properties. The three-body interactions led to $\alpha = 1$ for $C_t^{\rho}$ in Eq. (9), but a different density dependence of the C_t^s. $\alpha = 1$ is too large to get the incompressibility $K_\infty$ right, and causes a spin instability in infinite nuclear matter [34] and finite nuclei [35] (again only within a microscopic potential framework). Both problems are cured with smaller values of $\alpha$ (between 1/6 and 1/3 [23]) but the second-generation interactions that did so still had problems in the time-odd channels, giving a poor description of spin and spin-isospin excitations and prompting several attempts to describe finite nuclei with extended Skyrme interactions. Krewald et al. [36], Waroquier et al. [37], and Liu et al. [27], for example, introduced additional three-body momentum-dependent forces. Waroquier et al. added an admixture of the density-dependent two-body delta force and a three-body delta force, while Liu et al. considered a tensor force. But none of these interactions has been used subsequently.

Van Giai and Sagawa [38] developed the more durable parameterization SGII, which gave a reasonable description of GT resonance data known at the time and is still used today. The fit to ground state properties was made without the $\stackrel{\leftrightarrow}{J}^2$ terms, however, even though they were used in the QRPA. Consequently, in such an approach, the QRPA does not correspond to the small-amplitude limit of time-dependent HFB.

All these attempts to improve the description of the time-odd channels impose severe restrictions on the coupling by linking them to the HF expectation value of a Skyrme force, leading to one difficulty or another. The authors of Refs. [18,39] proceed differently, treating the Skyrme energy functional as the result of a local-density approximation. The interpretation of the Skyrme interaction as an energy-density functional, besides relaxing the restrictions on the time-odd couplings, endows the spin-orbit interaction with a more flexible isospin structure [40,41,42] than can be obtained from the standard Skyrme force [43]. Some of the parameterizations used here will take advantage of that freedom. But the authors of Ref. [39] include only time-odd terms that are determined by gauge invariance; the other couplings are tentatively set to zero ( $C_t^s = C_t^{\Delta s} = 0$). Such a procedure is reasonable when describing natural parity excitations within the (Q)RPA, but the neglected spin-spin terms are crucial for the unnatural parity states that we discuss.

In this study, we use the energy-functional approach (12) with fully independent time-even and time-odd coupling constants. Our hope is that this more general formulation will improve the description of the GT properties while leaving the good description of ground-state properties in even nuclei untouched.