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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
,
the various densities
are defined as
![$\displaystyle \tilde\rho_{00} (\vec{k})$](img339.png) |
= |
![$\displaystyle \sum_{\sigma} \sum_{\tau}
\tilde\rho (\vec{k}\sigma \tau \sigma \tau ) ,$](img340.png) |
(55) |
![$\displaystyle \tilde\rho_{1 t_3} (\vec{k})$](img341.png) |
= |
![$\displaystyle \sum_{\sigma} \sum_{\tau,\tau'}
\tilde\rho (\vec{k}\sigma \tau \sigma \tau' ) \;
\tau^{t_3}_{\tau \tau'} ,$](img342.png) |
(56) |
![$\displaystyle \tilde{\vec{s}}_{00} (\vec{k})$](img343.png) |
= |
![$\displaystyle \sum_{\sigma, \sigma'} \sum_{\tau}
\tilde\rho (\vec{k}\sigma \tau \sigma' \tau ) \;
\mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'} ,$](img344.png) |
(57) |
![$\displaystyle \tilde{\vec{s}}_{1 t_3} (\vec{k})$](img345.png) |
= |
![$\displaystyle \sum_{\sigma, \sigma'} \sum_{\tau, \tau'}
\tilde\rho (\vec{k}\sig...
... \:
\mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'} \; \tau^{t_3}_{\tau \tau'} ,$](img346.png) |
(58) |
The kinetic densities are given by
,
.
The Landau-Migdal
interaction is defined as
The isoscalar-scalar, isovector-scalar, isoscalar-vector, and
isovector-vector channels of the residual interaction are given by
![$\displaystyle \tilde{f} (\vec{k}_1, \vec{k}_2)$](img353.png) |
= |
![$\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde\rho_{00} (\vec{k}_1) \delta \tilde\rho_{00}
(\vec{k}_2)}$](img354.png) |
(60) |
![$\displaystyle \tilde{f}{}' (\vec{k}_1, \vec{k}_2)$](img355.png) |
= |
![$\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde\rho_{1 t_3} (\vec{k}_1)
\delta \tilde\rho_{1 t_3} (\vec{k}_2)}$](img356.png) |
(61) |
![$\displaystyle \tilde{g} (\vec{k}_1, \vec{k}_2)$](img357.png) |
= |
![$\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde{\vec{s}}_{00} (\vec{k}_1)
\delta \tilde{\vec{s}}_{00} (\vec{k}_2)}$](img358.png) |
(62) |
![$\displaystyle \tilde{g}{}' (\vec{k}_1, \vec{k}_2)$](img359.png) |
= |
![$\displaystyle \frac{\delta^2 {\cal E}}
{\delta \tilde{\vec{s}}_{1 t_3} (\vec{k}_1)
\delta \tilde{\vec{s}}_{1 t_3} (\vec{k}_2)}$](img360.png) |
(63) |
Assuming that only states at the Fermi surface contribute, i.e.,
,
,
,
,
and
depend on
the angle
between
and
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}](img367.png) |
(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}](img368.png) |
(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 |
= |
![$\displaystyle N_0 \left( 2 C_0^{\rho}
+ 4 \frac{\partial C_0^{\rho}}{\partial \...
...tial \rho_{00}^2} \rho_0^2
+ 2 C_0^{\tau} \, \beta \, \rho_{00}^{2/3}
\right) ,$](img369.png) |
|
f0' |
= |
![$\displaystyle N_0 \big( 2 C_1^{\rho}
+ 2 C_1^{\tau} \, \beta \, \rho_{00}^{2/3}
\big) ,$](img370.png) |
|
g0 |
= |
![$\displaystyle N_0 \big( 2 C_0^{s}
+ 2 C_0^{T} \, \beta \, \rho_{00}^{2/3}
\big) ,$](img371.png) |
|
g0 |
= |
![$\displaystyle N_0 \big( 2 C_1^{s}
+ 2 C_1^{T} \, \beta \, \rho_{00}^{2/3}
\big) ,$](img372.png) |
|
f1 |
= |
![$\displaystyle - 2 N_0 \; C_0^{\tau} \, \beta \, \rho_{00}^{2/3} ,$](img373.png) |
|
f1' |
= |
![$\displaystyle - 2 N_0 \; C_1^{\tau} \, \beta \, \rho_0^{2/3} ,$](img374.png) |
|
g1 |
= |
![$\displaystyle - 2 N_0 \; C_0^{T} \, \beta \, \rho_{00}^{2/3},$](img375.png) |
|
g1' |
= |
![$\displaystyle - 2 N_0 \; C_1^{T} \, \beta \, \rho_{00}^{2/3}.$](img376.png) |
(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
,
,
,
or
is less
than
.
Next: Landau Parameters from the
Up: Gamow-Teller strength and the
Previous: Pressure, Incompressibility and Asymmetry
Jacek Dobaczewski
2002-03-15