Study of $C_1^{s} [\rho _{\rm nm}]$.

We begin with the simplest case, assuming that (i) the functional is gauge invariant, (ii) all time-odd coupling constants are density-independent, and (iii) the spin-surface term can be neglected, i.e., $C_1^{\Delta s} = 0$. The only remaining free parameter in the spin-isospin channel is C₁^s, which is directly related to the Landau parameters via Eqs. (16) and (18):

$$C_1^{s} = \frac{1}{2 N_0} \, ( g_0' + g_1' ) ,$$ (19)

where g₁' is fixed by C₁^T [also in Eq. (18)]. Figure 6 shows results for the GT resonance energy when g₀' is systematically varied from its SkO' value by altering C₁^s. We have chosen only nuclei that can be expected to exhibit a collective response to GT excitations. Non-collective contributions may show up, however, when the coupling constants are changed. In ¹²⁴Sn, for example, a state below the resonance collects a lot of strength for small values of g₀'. Only by increasing g₀' does one push that strength into the resonance. Similarly, in ¹¹²Sn a state about 5MeV above the resonance increasingly collects strength as g₀' grows.

As the underlying single-particle spectra are the same for all the cases in Fig. 6, the differences are due entirely to the value of g₀'. With increasing g₀', the resonance energy increases and more strength is pushed into the resonance. The increase of $E_{\rm res}$ is nearly linear, but the lines for different nuclei have different slopes. It is gratifying that the curves for $E_{\rm calc} - E_{\rm expt}$ all have a zero around the same point, $g_0' \approx 1.2$. This value is much smaller than the empirical value $g_0' \approx 1.8$ derived earlier [63,51,52] for at least two reasons: (i) the influence of the single-particle spectrum, and (ii) the inclusion in the residual interaction of a p-wave force characterized by g₁'. The latter means that g₀'=0 does not correspond to a vanishing interaction in the spin-isospin channel.

Figure 7: Spatial dependence of $C_1^{s} [\rho]$ for various values of x$\equiv$C₁^s[0]/ $C_1^{s} [\rho _{\rm nm}]$, cf. Eq. (10), and g₀' fixed at 1.2. The value x=1 corresponds to no density dependence. For larger values of x, the residual interaction becomes more repulsive outside the nucleus than inside. When x=0, $C_1^{s} [\rho]$ vanishes at large distances, and for negative values of x, the residual interaction becomes attractive outside the nucleus. The density profile $\rho(r)$ used in this plot corresponds to ²⁰⁸Pb.