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Theoretical formalism: isospin restoration and Coulomb
rediagonalization scheme
The first step and the starting point of our approach is the determination
of the isospin-symmetry-broken
single-particle (s.p.) Slater determinant
calculated by using the Hartree-Fock (HF) theory including the isospin-invariant Skyrme
(
) and the isospin-symmetry-breaking Coulomb (
)
interactions:
![\begin{displaymath}
\hat H = \hat H^S + \hat V^C \quad \textrm{where} \quad
\hat H^S =\hat T + \hat V^S.
\end{displaymath}](img15.png) |
(1) |
The isospace-deformed state
admixes higher isospin components
:
![\begin{displaymath}
\vert\textrm{HF} \rangle = \sum_{T\geq \vert T_z\vert}b_{T,T_z}\vert\eta; T,T_z\rangle ,
\end{displaymath}](img17.png) |
(2) |
where
and
are the total isospin and its third component,
respectively,
labels all other quantum numbers
pertaining to the
state, and the coefficients
are such that
.
In the second step we create the good-isospin states
by projecting them out from the Slater determinant
:
![\begin{displaymath}
\vert\eta ; T,T_z\rangle = \frac{1}{b_{T,T_z}} \hat P^T_{T_z T_z}\vert\textrm{HF}
\rangle .
\end{displaymath}](img24.png) |
(3) |
In the following, we denote the mixing coefficients
and average energies
:
![\begin{displaymath}
\vert b_{T,T_z}\vert^2 = \langle \textrm{HF} \vert \hat P^T...
...} =
\langle\eta ; T,T_z\vert \hat{H}\vert \eta ; T,T_z\rangle,
\end{displaymath}](img26.png) |
(4) |
as being obtained before rediagonalization.
In the above formulae,
denotes the conventional[11] SO(3) projection
operator reduced to one dimension due to the
quantum number
conservation, that is:
![\begin{displaymath}
\hat P^T_{T_zT_z} = \frac{2T+1}{2} \int_0^\pi d\beta \sin\beta
d^{T}_{T_zT_z}(\beta) \hat R(\beta),
\end{displaymath}](img28.png) |
(5) |
where
denotes active-rotation operator by the Euler angle
in
the isospace and
is the Wigner
-function[12].
In the third step we mix the projected states,
![\begin{displaymath}
\vert\eta; n,T_z\rangle = \sum_{T\geq \vert T_z\vert}a^n_{T,T_z}\vert\eta; T,T_z\rangle ,
\end{displaymath}](img33.png) |
(6) |
and determine the mixing coefficients
by diagonalizing
Hamiltonian (1) in the space of projected states,
![\begin{displaymath}
\sum_{T'\geq \vert T_z\vert}\langle\eta; T,T_z\vert\hat{H}\v...
...T_z\rangle
a^n_{T',T_z}
= E^{\textrm{AR}}_{n,T_z}a^n_{T,T_z},
\end{displaymath}](img35.png) |
(7) |
where
enumerates the obtained eigenstates.
In the following, we denote the mixing coefficients
and eigenenergies
as being obtained after rediagonalization.
The lowest-energy solution, for
, corresponds to the isospin mixing in the
ground state.
The Skyrme Hamiltonian,
, is an isoscalar operator; hence, it contributes
only to the diagonal matrix elements of the Hamiltonian (1),
, which can be obtained from:
![\begin{displaymath}
\langle\textrm{HF}\vert \hat H^{S} \hat
P^T_{T_zT_z}\vert\...
...trm{HF}\vert \hat H^{S} \hat R(\beta)\vert\textrm{HF}\rangle .
\end{displaymath}](img41.png) |
(8) |
Similarly, calculation of the diagonal and non-diagonal
matrix elements of the Coulomb interaction,
,
can be efficiently performed after decomposing
into the isoscalar,
, isovector,
, and
isotensor,
, components,
and by making use of the SO(3) transformation rules for the
spherical tensors under rotations in the isospace[12].
In the particular case of one-dimensional projection we deal with in
this work, all matrix elements of axial spherical tensors
reduce to one-dimensional integrals over the Euler angle
:
where
and
denote standard
Clebsch-Gordan coefficients.
The Skyrme-Hamiltonian and Coulomb-interaction kernels,
and
,
respectively, can be evaluated by using expressions for the standard
diagonal kernels[13] (
) and replacing there the
isoscalar and isovector densities and currents with the so-called
transition densities and currents. Exact direct and exchange kernels
of the Coulomb interaction can be evaluated by using methods outlined
in Refs.[14,15,16].
Next: Numerical applications: the isospin
Up: Isospin mixing of isospin-projected
Previous: Introduction
Jacek Dobaczewski
2009-04-13