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As one is dealing with
protons and
neutrons, two gauge angles,
and
, must enter
the number projection operator:
![$\displaystyle P^{NZ}=\frac{1}{2\pi }\int d\phi_{n}\ e^{i\phi _{n}(\hat{N}-N)}\frac{1}{ 2\pi }\int d\phi_{p}\ e^{i\phi _{p}(\hat{Z}-Z)}.$](img218.png) |
(70) |
Consequently, the total projected energy (53) becomes
a double integral,
![$\displaystyle E^{N}=\int d\phi_{n}~d\phi_{p}~y_{n}(\phi_{n})~y_{p}(\phi_{p})~ E(\phi_{n},\phi_{p}),$](img219.png) |
(71) |
where the transition energy density
![$\displaystyle E(\phi_{n},\phi_{p})=\int d{\bm r}~{\cal H}({\bm r},\phi_{n},\phi _{p})$](img220.png) |
(72) |
depends on both gauge angles
,
.
To simplify notation, we use the isospin
label
=
(
=+1 for neutrons and -1 for protons)
and
=
. In the following, we shall employ the convention
![$ y(\phi_{q})$](img225.png)
,
![$ C(\phi_{q})$](img228.png)
,
and
![$ Y(\phi_{q})$](img230.png)
.
The isospin-dependent particle-hole and
particle-particle fields (66), (67) can be written as:
In numerical applications, the two-dimensional integrals
over the gauge angles are replaced by a sum over
points
using the Gauss-Chebyshev
quadrature method [34].
Next: Canonical representation
Up: Skyrme HFB+VAPNP procedure: practical
Previous: Skyrme HFB+VAPNP procedure: practical
Jacek Dobaczewski
2006-10-13