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For the zero-range Skyrme forces, the HFB formalism can be written
directly in the coordinate representation [30,31,32] by introducing
particle and pairing densities
which explicitly depend on spin.
The use of the pairing density
,
![$\displaystyle \tilde{\rho}({\bm r}\sigma ,{\bm r^{\prime }}\sigma^{\prime })=-2\sigma^{\prime }\kappa ({\bm r,}\sigma ,{\bm r^{\prime },}-\sigma^{\prime }),$](img57.png) |
(15) |
instead of the pairing tensor
is convenient when
restricting to time-even quasiparticle states where both
and
are hermitian and time-even [31].
The densities
and
can be expressed in the
single-particle basis:
where
and
are
the corresponding density matrices. In this study,
we take
as a set of the
HO wave functions.
The building blocks of the Skyrme HFB method are the local densities, namely
the particle density
, kinetic energy
density
, and spin-current density
:
![\begin{displaymath}\begin{array}{ccl} \rho ({\bm r}) & = & \rho ({\bm r},{\bm r}...
...prime })\right\vert _{{\bm r}^{\prime }={\bm r}}\;, \end{array}\end{displaymath}](img67.png) |
(18) |
as well as the corresponding pairing densities
,
and
.
In the coordinate representation, the Skyrme HFB
energy (8) can be written as a
functional of the local particle and pairing densities:
![$\displaystyle E[\rho,\tilde{\rho}]=\frac{\langle \Phi \vert H\vert\Phi \rangle }{\langle \Phi \vert\Phi \rangle } =\int d{\bm r}~{\cal H}({\bm r}).$](img71.png) |
(19) |
The energy density
is a sum
of the particle
and pairing energy density
:
![$\displaystyle {\cal H}({\bm r})=H({\bm r})+\tilde{H}({\bm r}).$](img75.png) |
(20) |
The derivatives of
with respect to density
matrices
and
define the self-consistent particle
(
) and pairing
(
) fields, respectively.
The explicit expressions for
,
,
, and
have been given
in Ref. [31] and will not be repeated here.
The Skyrme HFB equations can be written in the matrix form as:
![$\displaystyle \left( \begin{array}{cc} h-\lambda & \tilde{h} \\ \tilde{h} & -h+...
...E_k \left( \begin{array}{c} \varphi_{1,k} \\ \varphi_{2,k} \end{array} \right),$](img81.png) |
(21) |
where
and
and
and
are the upper and lower
components, respectively, of the quasiparticle wave function corresponding to
the positive quasiparticle energy
. After solving the HFB equations
(21), one obtains the density matrices,
![$\displaystyle \rho_{nn^{\prime }} = \sum_{E_k>0} \varphi_{2,nk} \varphi^*_{2,n^{\prime }k} ,$](img90.png) |
|
|
(24) |
![$\displaystyle \tilde{\rho}_{nn^{\prime }} = -\sum_{E_k>0} \varphi_{2,nk} \varphi^*_{1,n^{\prime }k} ,$](img91.png) |
|
|
(25) |
which define the spatial densities (16) and (17) .
We note in passing that the derivation of the coordinate-space
HFB equations
[31] is strictly
valid only when the time-reversal symmetry is assumed.
When the time-reversal
symmetry is broken, one has to introduce additional real
vector particle densities
,
,
[33], while the pairing densities acquire
imaginary parts; see Ref. [32] for complete derivations.
Next: Variation after particle-number projection
Up: The HFB method
Previous: The HFB equations
Jacek Dobaczewski
2006-10-13