Landau Parameters from the Gogny force

The residual interaction in INM from the Gogny force [59]

$${V_{\rm Gogny}(\vec{r}_1, \vec{r}_2)}$$

$$\sum_{i=1,2} \big( W_i + B_i \hat{P}_\sigma + H_i \hat{P}_\tau - M_i \hat{P}_\sigma \hat{P}_\tau \big) {\rm e}^{-(\vec{r}_{1}-\vec{r}_2)^2/\mu_i^2}$$ =

$$+ t_0 ( 1 + x_0 \hat{P}_\sigma) \; \rho_0^\alpha \left({\textstyl... ...ec{r}_2}}{{2}}}\right) \, \delta (\vec{r}_1 - \vec{r}_2) \phantom{\sum_{i=1,2}}$$

$$+ \mbox{\rm\scriptsize {i}}W_0 \, ( \hat{\mbox{{\boldmath {$\sigm... ...) \cdot \hat{\vec{k}}{}' \times \delta (\vec{r}_1 - \vec{r}_2) \, \hat{\vec{k}}$$ (67)

(see Appendix 8 for the definition of $\hat{\vec{k}}$, $\hat{\vec{k}}{}^\prime$, $\hat{P}_\sigma$, and $\hat{P}_\tau$) has been discussed in [74,75]. Evaluating the expressions given in [75] for $(k, k', q) = (k_{\rm F}, k_{\rm F}, 0)$, one obtains the usual Landau parameters

$$f_\ell \sum_{i=1,2} \big[ ( 4 W_i + 2 B_i - 2 H_i - M_i ) \Psi^{(i)}_\ell$$ =

$$\phantom{\sum} + ( - W_i - 2 B_i + 2 H_i + 4 M_i ) \Phi^{(i)}_\ell \big]$$

$$\phantom{\sum} + \delta_{\ell 0} \, {\textstyle\frac{{3}}{{8}}} t_0 \, (\alpha + 1) (\alpha + 2) \, \rho_0^\alpha$$

$$f_\ell' \sum_{i=1,2} \big[ - ( 2 H_i + M_i ) \Phi^{(i)}_\ell - ( W_i - 2 B_i ) \Psi^{(i)}_\ell \big]$$ =

$$+ \delta_{\ell 0} \, {\textstyle\frac{{1}}{{4}}} t_0 \, (1 + 2 x_0) \rho_0^\alpha$$

$$g_\ell \sum_{i=1,2} \big[ ( 2 B_i - M_i ) \Psi^{(i)}_\ell + ( - W_i + 2 H_i ) \Phi^{(i)}_\ell \big]$$ =

$$+ \delta_{\ell 0} \, {\textstyle\frac{{1}}{{4}}} t_0 \, (1 - 2 x_0) \rho_0^\alpha$$

$$g_\ell' - \sum_{i=1,2} \big( M_i \Psi^{(i)}_\ell + W_i \Phi^{(i)}_\ell \big) + \delta_{\ell 0} \, {\textstyle\frac{{1}}{{4}}} t_0 \, \rho_0^\alpha$$ = (68)

where

$$\Psi^{(i)}_\ell {\textstyle\frac{{1}}{{4}}} \pi^{3/2} \mu_i^3 \, N_0 \; \delta_{\ell 0}$$ =

$$\Phi^{(i)}_0 {\textstyle\frac{{1}}{{4}}} \pi^{3/2} \mu_i^3 \, N_0 \; {\rm e}^{-z} \; \frac{\sinh (z)}{z}$$ =

$$\Phi^{(i)}_1 {\textstyle\frac{{3}}{{4}}} \pi^{3/2} \mu_i^3 \, N_0 \; {\rm e}^{-z} \left(\frac{\cosh (z)}{z} - \frac{\sinh (z)}{z^2}\right)$$ =

$$\Phi^{(i)}_2 {\textstyle\frac{{5}}{{4}}} \pi^{3/2} \mu_i^3 \, N_0 \; {\rm e}^{... ...h(z) \left(\frac{1}{z} + \frac{3}{z^3} \right) - \frac{3 \cosh(z)}{z^2} \right]$$ =

with $z= \mu_i^2 k_{\rm F}^2 /2$. The normalization factor N₀ is again given by (69).