Introduction

Effective interactions for self-consistent nuclear structure calculations are usually adjusted to reproduce ground-state properties in even-even nuclei [1]. These properties depend only on terms in the corresponding energy functional that are bilinear in time-reversal-even (or ``time-even'') densities and currents [2]. But the functional also contains an equal number of terms bilinear in time-odd densities and currents (see [2,3] and refs. quoted therein), and these terms are seldom independently adjusted to experimental data. (For the sake simplicity we refer below to terms in the functional as time-even or time-odd, even though strictly speaking we mean the densities and currents on which they depend.) The time odd terms can be important as soon as time-reversal symmetry (and with it Kramers degeneracy) is broken in the intrinsic frame of the nucleus. Such breaking obviously occurs for rotating nuclei, in which the current and spin-orbit time-odd channels (linked to time-even channels by the gauge symmetry) play an important role. Time-odd terms also interfere with pairing correlations in the masses of odd-A and odd-odd nuclei [4,5,6] and contribute to single-particle energies [7,8,9] and magnetic moments [10]. Finally, the spin-isospin channel of the effective interaction determines distributions of the Gamow-Teller (GT) strength.

The latter are the focus of this paper. We explore the effects of time-odd couplings on GT resonance energies and strengths, with an eye toward fixing the spin-isospin part of the Skyrme interaction. As discussed in our previous study [11], there are many good reasons for looking at this channel first. For instance, a better description of the GT response should enable more reliable predictions for $\beta$-decay half-lives of very neutron-rich nuclei. Those predictions in turn may help us identify the astrophysical site of r-process nucleosynthesis, which produces about half of the heavy nuclei with A>70.

Our goal is an improved description of GT excitations in a fully self-consistent mean-field model. To this end, we treat excited states in the Quasiparticle Random Phase Approximation (QRPA), with the residual interaction taken from the second derivative of the energy functional with respect to the density matrix. This approach is equivalent to the small-amplitude limit of time-dependent Hartree-Fock-Bogoliubov (HFB) theory. We proceed by taking the time-odd coupling constants in the Skyrme energy functional to be free parameters that we can fit to GT distributions. We then check that the coupling constants so deduced do not spoil the description of superdeformed (SD) rotational bands.

Our formulation is nonrelativisitic. In relativistic mean-field theory (RMF) [12,13], the time-odd channels, referred to as ``nuclear magnetism,'' are not independent from the time-even ones because they arise from the small components of the Dirac wave functions. For rotational states, the time-odd effects have been extensively tested and shown to be important for reproducing experimental data (see, e.g., Ref. [14]). Only the current terms and spin-orbit terms play a role there, however, and the time-odd spin and spin-isospin channels of the RMF have never been tested against experimental data.

This paper is structured as follows: In Section 2 we review properties of the Skyrme energy functional. Section 3 reviews existing parameterizations of the functional, with particular emphasis on time-odd terms. Our main results are in Section 4, where we present calculations of GT strength and discuss the role played by the time-odd coupling constants. Section 5 describes calculations of moments of inertia for selected SD bands. Section 6 contains our conclusions. We supplement our results with six Appendices that provide more detailed information on local densities and currents (Appendix 7), early parameterizations of time-odd Skyrme functionals (Appendix 8), the limit of the infinite nuclear matter (Appendix 9), Landau parameters of Skyrme functionals (Appendix 10) and of the Gogny force (Appendix 11), and the residual interaction in finite nuclei from Skyrme functionals (Appendix 12).