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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 C1s, which is directly related to the Landau parameters via Eqs. (16) and (18):
= \frac{1}{2 N_0} \, ( g_0' + g_1' ) ,
\end{displaymath} (19)

where g1' is fixed by C1T [also in Eq. (18)]. Figure 6 shows results for the GT resonance energy when g0' is systematically varied from its SkO' value by altering C1s. 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$\equiv$C1s[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.

next up previous
Next: Study of C1s [0]. Up: GT resonances from generalized Previous: GT resonances from generalized
Jacek Dobaczewski