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To begin with, Eq. (23) can be written explicitly in terms of
the HFB eigenvectors:
|
|
![$\displaystyle {\dot {\cal R}_0} =
{\dot q} \frac {\partial} {\partial q}
\Big...
...& \kappa_0 \\
-\kappa_0^{\ast } \quad & 1-\rho_0^{\ast }
\end{array}\Bigg)
=$](img95.png) |
(35) |
|
|
![$\displaystyle = \left(
\begin{array}{cc}
A F B^T - B^\ast F^\ast A^\dagger ~~ ...
...st F^\ast A^\dagger ~~ &
B F A^T - A^\ast F^\ast B^\dagger
\end{array}\right).$](img96.png) |
|
Evaluating the matrix elements of (35) in the canonical
basis, we obtain
![\begin{displaymath}
\breve{F}^{i}_{\mu \bar \nu} =
\frac {s_{\bar \nu}} {(u_\m...
... \frac {\partial \rho_0} {\partial q_i} \biggr)_{\mu \nu}\,.
\end{displaymath}](img97.png) |
(36) |
By differentiating the HFB equation
with respect to
,
the derivative of the density matrix in (36) can be expressed in
terms of the derivatives of the particle-hole and the pairing mean-fields. The
resulting 2
2 matrix equation is
![\begin{displaymath}
\bigg[ {\cal A}^\dagger
{\dot q_i} \frac {\partial {\cal W}...
... A}, {\cal G} \bigg]
+ \bigg[ {\cal E}_0, {\cal F} \bigg] = 0.
\end{displaymath}](img101.png) |
(37) |
By employing the properties of
and
with respect to time
reversal, we obtain
and by approximating the HFB energy matrix in the canonical basis by
its diagonal matrix elements,
![$\displaystyle \breve{E}_{\mu\nu} \approx \delta_{\mu\nu}\breve{E}_{\mu},$](img106.png) |
|
|
(40) |
one arrives at an approximate ``BCS-equivalent'' expression for the
matrix elements of
in the canonical basis:
![\begin{displaymath}
\breve{F}^{i}_{\mu\nu} \approx \frac{-\dot q_i}{\breve{E}_\m...
...\mu\bar\nu}+\xi^+_{\mu\nu}
(\breve{\Delta}^i)_{\mu\nu}\right],
\end{displaymath}](img107.png) |
(41) |
where
with
,
, or
. In the following, the results obtained by
using this approximation will be called ATDHFB-C
.
Using relations (40) and (41), the
collective mass tensor (34) can now be expressed in terms of
the derivatives of the mean-field potentials with respect to the
collective coordinates
and BCS-like quasiparticle energies (11),
![\begin{displaymath}
{\cal M}^{C^{\rm c}}_{ij} \approx \frac {1} {2 {\dot q_i} {...
...{j\ast}_{\mu \nu} \right) } {\breve{E}_\mu + \breve{E}_\nu}.
\end{displaymath}](img111.png) |
(42) |
In the one-dimensional case, the resulting expression agrees with
that of Ref. [9].
Next: Quasiparticle basis
Up: Cranking approximation
Previous: Cranking approximation
Jacek Dobaczewski
2010-07-28