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Landau Parameters from the Skyrme energy functional

A simple and instructive description of the residual interaction in homogeneous INM is given by the Landau interaction developed in the context of Fermi-liquid theory [50]. Landau parameters corresponding to the Skyrme forces are discussed in Refs. [21,27,36,37,38,73]. Starting from the full density matrix in (relative) momentum space $\tilde\rho (\vec{k}\sigma \tau \sigma' \tau')$, the various densities are defined as
$\displaystyle \tilde\rho_{00} (\vec{k})$ = $\displaystyle \sum_{\sigma} \sum_{\tau}
\tilde\rho (\vec{k}\sigma \tau \sigma \tau ) ,$ (55)
$\displaystyle \tilde\rho_{1 t_3} (\vec{k})$ = $\displaystyle \sum_{\sigma} \sum_{\tau,\tau'}
\tilde\rho (\vec{k}\sigma \tau \sigma \tau' ) \;
\tau^{t_3}_{\tau \tau'} ,$ (56)
$\displaystyle \tilde{\vec{s}}_{00} (\vec{k})$ = $\displaystyle \sum_{\sigma, \sigma'} \sum_{\tau}
\tilde\rho (\vec{k}\sigma \tau \sigma' \tau ) \;
\mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'} ,$ (57)
$\displaystyle \tilde{\vec{s}}_{1 t_3} (\vec{k})$ = $\displaystyle \sum_{\sigma, \sigma'} \sum_{\tau, \tau'}
\tilde\rho (\vec{k}\sig...
... \:
\mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'} \; \tau^{t_3}_{\tau \tau'} ,$ (58)

The kinetic densities are given by $\tau_{t t_3} = \rho_{t t_3} \,
k^2$, $\vec{T}_{t t_3} = \vec{s}_{t t_3} \, k^2$. The Landau-Migdal interaction is defined as

$\displaystyle {
\tilde{F}
(\vec{k}_1 \sigma_1 \tau_1 \sigma'_1 \tau'_1;
\vec{k}_2 \sigma_2 \tau_2 \sigma'_2 \tau'_2 )
}$
  = $\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde\rho (\vec{k}_1 \sigma_1 \...
...ma'_1 \tau'_1)
\delta \tilde\rho (\vec{k}_2 \sigma_2 \tau_2 \sigma'_2 \tau'_2)}$  
  = $\displaystyle \tilde{f} (\vec{k}_1, \vec{k}_2)
+ \tilde{f}{}' (\vec{k}_1, \vec{...
...}}}_1 \cdot \mbox{{\boldmath {$\tau$}}}_2
\phantom{\frac{\delta^2}{\tilde\rho}}$  
    $\displaystyle +
\tilde{g} (\vec{k}_1, \vec{k}_2) \; \mbox{{\boldmath {$\sigma$}...
...ldmath {$\tau$}}}_1 \cdot \mbox{{\boldmath {$\tau$}}}_2) .
\phantom{\bigg\vert}$ (59)

The isoscalar-scalar, isovector-scalar, isoscalar-vector, and isovector-vector channels of the residual interaction are given by
$\displaystyle \tilde{f} (\vec{k}_1, \vec{k}_2)$ = $\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde\rho_{00} (\vec{k}_1) \delta \tilde\rho_{00}
(\vec{k}_2)}$ (60)
$\displaystyle \tilde{f}{}' (\vec{k}_1, \vec{k}_2)$ = $\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde\rho_{1 t_3} (\vec{k}_1)
\delta \tilde\rho_{1 t_3} (\vec{k}_2)}$ (61)
$\displaystyle \tilde{g} (\vec{k}_1, \vec{k}_2)$ = $\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde{\vec{s}}_{00} (\vec{k}_1)
\delta \tilde{\vec{s}}_{00} (\vec{k}_2)}$ (62)
$\displaystyle \tilde{g}{}' (\vec{k}_1, \vec{k}_2)$ = $\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde{\vec{s}}_{1 t_3} (\vec{k}_1)
\delta \tilde{\vec{s}}_{1 t_3} (\vec{k}_2)}$ (63)

Assuming that only states at the Fermi surface contribute, i.e., $\vert \vec{k}_1 \vert = \vert \vec{k}_2 \vert = k_{\rm F}$, $\tilde{f}$, $\tilde{f}{}'$, $\tilde{g}$, and $\tilde{g}{}'$ depend on the angle $\theta$ between $\vec{k}$ and $\vec{k}$ only, and can be expanded into Legendre polynomials, e.g.
\begin{displaymath}
\tilde{f} (\vec{k}_1, \vec{k}_2)
= \frac{1}{N_0} \sum_{\ell = 0}^{\infty} f_\ell \; P_\ell (\theta) .
\end{displaymath} (64)

The normalization factor N0 is the level density at the Fermi surface
\begin{displaymath}
\frac{1}{N_0}
= \frac{\pi^2 \hbar^2}{2 m^* k_{\rm F} }
\appr...
...\rm\scriptsize {MeV}} \; \mbox{\rm\scriptsize {fm}}^3
\quad .
\end{displaymath} (65)

A variety of definitions of the normalization factor N0 are used in the literature and great care has to be taken when comparing values from different groups; see, e.g., Ref.[50] for a detailed discussion. We use the convention defined in [38]. The Landau parameters corresponding to the general energy functional (6) are
f0 = $\displaystyle N_0 \left( 2 C_0^{\rho}
+ 4 \frac{\partial C_0^{\rho}}{\partial \...
...tial \rho_{00}^2} \rho_0^2
+ 2 C_0^{\tau} \, \beta \, \rho_{00}^{2/3}
\right) ,$  
f0' = $\displaystyle N_0 \big( 2 C_1^{\rho}
+ 2 C_1^{\tau} \, \beta \, \rho_{00}^{2/3}
\big) ,$  
g0 = $\displaystyle N_0 \big( 2 C_0^{s}
+ 2 C_0^{T} \, \beta \, \rho_{00}^{2/3}
\big) ,$  
g0 = $\displaystyle N_0 \big( 2 C_1^{s}
+ 2 C_1^{T} \, \beta \, \rho_{00}^{2/3}
\big) ,$  
f1 = $\displaystyle - 2 N_0 \; C_0^{\tau} \, \beta \, \rho_{00}^{2/3} ,$  
f1' = $\displaystyle - 2 N_0 \; C_1^{\tau} \, \beta \, \rho_0^{2/3} ,$  
g1 = $\displaystyle - 2 N_0 \; C_0^{T} \, \beta \, \rho_{00}^{2/3},$  
g1' = $\displaystyle - 2 N_0 \; C_1^{T} \, \beta \, \rho_{00}^{2/3}.$ (66)

Higher-order Landau parameters vanish for the second-order energy functional (12), but not for finite-range interactions as the Gogny force discussed in the next Appendix. The Landau parameters provide a stability criterion for symmetric unpolarized INM: It becomes unstable for a given interaction when either $f_\ell$, $f_\ell'$, $g_\ell$, or $g_\ell'$ is less than $-(2 \ell + 1)$.
next up previous
Next: Landau Parameters from the Up: Gamow-Teller strength and the Previous: Pressure, Incompressibility and Asymmetry
Jacek Dobaczewski
2002-03-15