We give the expression for particle density
, kinetic density
, spin density
, spin-kinetic density
,
current density
, and spin-current density
.
Tensor-kinetic density
is not used and its expression is
omitted.
- (a)
- Scalar particle density
![$\displaystyle \rho_{m}(\bbox{r}) = \rho_{m}(\bbox{r},\bbox{r}')\biggl\vert _{\bbox{r} = \bbox{r}'}$](img226.png) |
(21) |
where
takes values 0, 1, 2 and 3. Suffix 0 represents
isoscalar component of the density
and 1 to 3 are the isovector components.
Expressing the density in terms of HFB wavefunctions (20), we obtain
![$\displaystyle \rho_{m}(\bbox{r})= \sum_{tt'} \rho^{tt'}(rz) \tau_{t't}^{m}$](img228.png) |
(22) |
where
![$\displaystyle \rho^{tt'}(rz) = \sum_{k}\biggl[ V^{+\ast}_{kt} V^{+}_{kt'}+V^{-\ast}_{kt} V^{-}_{kt'}\biggr],$](img229.png) |
(23) |
and we use the abbreviation of HFB wave functions
![$\displaystyle V^{\pm}_{kt} \equiv V^{\pm}_k(rzt).$](img230.png) |
(24) |
Isospin components of the particle density are given by
The isospin structure of the particle density given by Eqs. (25)-(28) is identical
for all the following particle-hole
densities, and these expressions shall not be repeated in the following.
- (b)
- Kinetic density
![$\displaystyle \tau_m (\bbox{r}) = (\bbox{\nabla} \cdot \bbox{\nabla} ')\rho_m(\...
...\biggr\vert _{ \bbox{r} = \bbox{r}'} = \sum_{tt'}\tau^{tt'}(rz) \tau_{t't}^{m},$](img244.png) |
(29) |
where,
- (c)
- Pseudovector spin density
![$\displaystyle \bbox{s}_{m}(\bbox{r}) = \bbox{s}_{m}(\bbox{r},\bbox{r}') \biggl\vert _{\bbox{r} = \bbox{r}'} = \sum_{tt'} \bbox{s}^{tt'}(rz) \tau_{t't}^{m},$](img249.png) |
(31) |
where
- (d)
- Pseudovector spin-kinetic density
where
- (e)
- Vector current density
where
- (f)
- Tensor spin current density
![$\displaystyle {\sf J}_{m}^{ij}=\frac{1}{2i}(\nabla_i-\nabla'_i)s_m^j(\bbox{r}, ...
...gr\vert _{\bbox{r}=\bbox{r}'}
=\sum_{tt'}{\sf J}^{tt'}_{ij}(rz) \tau^m_{t't} .$](img277.png) |
(37) |
Explicit expressions of the components are
![$\displaystyle {\sf J}_{r\phi}^{tt'}(rz) =$](img278.png) |
![$\displaystyle \frac{1}{2}\sum_{k}\biggr\{[\partial_{r} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{r} V^{-}_{kt'}]$](img279.png) |
|
|
![$\displaystyle -[\partial_{r} V^{-\ast}_{kt}]V^{+}_{kt'}+V^{-\ast}_{kt}[\partial_{r} V^{+}_{kt'}]\biggr\},$](img280.png) |
(38) |
![$\displaystyle {\sf J}_{rz}^{tt'}(rz) =$](img281.png) |
![$\displaystyle \frac{1}{2i}\sum_{k}\biggr\{[\partial_{r} V^{+\ast}_{kt}]V^{+}_{kt'}-V^{+\ast}_{kt}[\partial_{r} V^{+}_{kt'}]$](img282.png) |
|
|
![$\displaystyle -[\partial_{r} V^{-\ast}_{kt}]V^{-}_{kt'}+V^{-\ast}_{kt}[\partial_{r} V^{-}_{kt'}]\biggr\},$](img283.png) |
(39) |
![$\displaystyle {\sf J}_{\phi z}^{tt'}(rz) =$](img284.png) |
![$\displaystyle -\sum_{k}\biggr\{ \frac{\Lambda^{-}}{r}V^{+\ast}_{kt}V^{+}_{kt'}-\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}V^{-}_{kt'}\biggr\},$](img285.png) |
(40) |
![$\displaystyle {\sf J}_{z\phi}^{tt'}(rz) =$](img286.png) |
![$\displaystyle \frac{1}{2}\sum_{k}\biggr\{[\partial_{z} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{z} V^{-}_{kt'}]$](img287.png) |
|
|
![$\displaystyle -[\partial_{z} V^{-\ast}_{kt}]V^{+}_{kt'}+V^{-\ast}_{kt}[\partial_{z} V^{+}_{kt'}]\biggr\},$](img288.png) |
(41) |
![$\displaystyle {\sf J}_{zr}^{tt'}(rz) =$](img289.png) |
![$\displaystyle \frac{1}{2i}\sum_{k}\biggr\{[\partial_{z} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{z} V^{-}_{kt'}]$](img290.png) |
|
|
![$\displaystyle +[\partial_{z} V^{-\ast}_{kt}]V^{+}_{kt'}-V^{-\ast}_{kt}[ \partial_{z}V^{+}_{kt'}]\biggr\},$](img291.png) |
(42) |
![$\displaystyle {\sf J}_{\phi r}^{tt'}(rz)=$](img292.png) |
![$\displaystyle -\frac{1}{2}\sum_{k}\biggr\{ \frac{\Lambda^{-}}{r}V^{+\ast}_{kt}V^{-}_{kt'}+\frac{\Lambda^{-}}{r}V^{-\ast}_{kt}V^{+}_{kt'}$](img293.png) |
|
|
![$\displaystyle +\frac{\Lambda^{+}}{r}V^{+\ast}_{kt}V^{-}_{kt'}+\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}V^{+}_{kt'}\biggr\},$](img294.png) |
(43) |
![$\displaystyle {\sf J}_{zz}^{tt'}(rz) =$](img295.png) |
![$\displaystyle \frac{1}{2i}\sum_{k}\biggr\{[\partial_{z} V^{+\ast}_{kt}]V^{+}_{kt'}-V^{+\ast}_{kt}[ \partial_{z}V^{+}_{kt'}]$](img296.png) |
|
|
![$\displaystyle -[\partial_{z} V^{-\ast}_{kt}]V^{-}_{kt'}+V^{-\ast}_{kt}[\partial_{z} V^{-}_{kt'}]\biggr\},$](img297.png) |
(44) |
![$\displaystyle {\sf J}_{\phi \phi}^{tt'}(rz)=$](img298.png) |
![$\displaystyle -\frac{1}{2i}\sum_{k}\biggr\{ \frac{\Lambda^{-}}{r}V^{-\ast}_{kt}V^{+}_{kt'}+\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}V^{+}_{kt'}$](img299.png) |
|
|
![$\displaystyle -\frac{\Lambda^{+}}{r}V^{+\ast}_{kt}V^{-}_{kt'}-\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}V^{-}_{kt'}\biggr\},$](img300.png) |
(45) |
![$\displaystyle {\sf J}_{rr}^{tt'}(rz) =$](img301.png) |
![$\displaystyle \frac{1}{2i}\sum_{k}\biggr\{[\partial_{r} V^{+\ast}_{kt}]V^{-}_{kt'}-V^{+\ast}_{kt}[\partial_{r} V^{-}_{kt'}]$](img302.png) |
|
|
![$\displaystyle +[\partial_{r} V^{-\ast}_{kt}]V^{+}_{kt'}-V^{-\ast}_{kt}[\partial_{r} V^{+}_{kt'}]\biggr\}.$](img303.png) |
(46) |
The trace, antisymmetric, and symmetric parts of the tensor spin-current density are given by
Jacek Dobaczewski
2014-12-07