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Residual interaction in finite nuclei

Non-self-consistent calculations often use the residual Landau-Migdal interaction in the spin-isospin channel:
$\displaystyle {
v_{\rm res} (\vec{r}, \vec{r}')
  = $\displaystyle N_0
\Big[ g_0' \, \delta (\vec{r}- \vec{r}')
+ g_1' \, \vec{k}' \...
...sigma$}}}') \,
(\mbox{{\boldmath {$\tau$}}} \cdot \mbox{{\boldmath {$\tau$}}}')$ (13)

where N0 is a normalization factor [see Eq. (69)] and $\vec{k}$ and $\vec{k}'$ are defined in Appendix 8. In most applications, only the s-wave interaction with strength g0' is used, and the matrix elements of the force are not antisymmetrized. The underlying single-particle spectra are usually taken from a parameterized potential, e.g., the Woods-Saxon potential. Typical values for g0', obtained from fits to GT-resonance systematics, are $1.4 \leq g_0' \leq 1.6 $ [45,46,47]. (See Ref. [48] for an early compilation of data.) Sometimes this approach is formulated in terms of the residual interaction between antisymmetrized states. The results are similar, e.g., g0' = 1.54 in the double-$\beta$-decay calculations by Engel et al. [49]. More complicated residual interactions, like boson-exchange potentials, have been used as well; see, e.g., Refs. [50,51,52]. Borzov et al. use a renormalized one-pion exchange potential in connection with a $\ell = 0$ Landau-Migdal interaction of type (13) [53].

A much simpler residual interaction in the GT channel is a separable (or ``schematic") interaction, $v_{\rm res} =
\kappa_{\rm GT} \; (\mbox{{\boldmath {$\sigma$}}} \cdot \mbox{{\b...
...igma$}}}' )\;
(\mbox{{\boldmath {$\tau$}}} \cdot \mbox{{\boldmath {$\tau$}}}')$, where the strength $\kappa_{\rm GT}$ has to be a function of A. This interaction is widely used in global calculations of nuclear $\beta$-decay [54,55]. Sarriguren et al. [56] use it for a description of the GT resonances in deformed nuclei with quasiparticle energies obtained from self-consistent HF+BCS calculations. They estimate $\kappa_{\rm GT}$ from the Landau parameters of their Skyrme interaction. (The same prescription is used in their calculations of M1 resonances [57].) But however useful this approach may be from a technical point of view, it is not self-consistent. Nor is it equivalent to using the original residual Skyrme interaction; see, e.g., the discussion in [46].

A truly self-consistent calculation, by contrast, should interpret the QRPA as the small-amplitude limit of time-dependent HFB theory. The Skyrme energy functional used in the HFB should then determine the residual interaction between unsymmetrized states in the QRPA:

v_{\rm res}
= \frac{\delta^2 {\cal E}}
{\delta \rho (\vec{r...
...r}_1', \sigma_1', \tau_1'; \vec{r}_2', \sigma_2',
\tau_2')} ~.
\end{displaymath} (14)

The actual form of the residual interaction that contributes to the QRPA matrix elements of 1+ states is outlined in Appendix 12.
Table 1: Landau parameters for various Skyrme interactions from relations (34) and the Gogny forces D1 and D1s. Missing entries are zero by construction.
Force g0 g1 g2 g0' g1' g2'

0.33     0.94    
SGII 0.62     0.93    
SkP -0.23 -0.18   0.06 0.97  
SkI3 1.89     0.85    
SkI4 1.77     0.88    
SLy4 1.39     0.90    
SLy5 1.14 0.24   -0.15 1.05  
SLy6 1.41     0.90    
SLy7 0.94 0.47   0.02 0.88  
SkO 0.48     0.98    
SkO' -1.61 2.16   0.79 0.19  
SkX -0.63 0.18   0.51 0.53  

0.47 0.06 0.12 0.60 0.34 0.08
D1s 0.48 -0.19 0.25 0.62 0.62 -0.04

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Next: GT strength distributions from Up: Giant Gamow-Teller resonances Previous: Giant Gamow-Teller resonances
Jacek Dobaczewski