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Gaussian overlap approximation
To compare cranking expressions with those obtained within the GOA, it is
convenient to introduce the
matrices [23]:
![\begin{displaymath}
{\cal S}_{ij}^{(K)}=\sum_{\mu,\nu}
\frac{\langle\mu \vert ...
...gle}
{(\breve{E}_\mu+\breve{E}_\nu)^{K}} (\eta^+_{\mu\nu})^2.
\end{displaymath}](img169.png) |
(61) |
It is immediately seen that for the mass tensor of Eq. (60)
one has
. In the case of GOA, also
assuming weak state dependence of pairing and neglecting the
derivatives of
and
, one obtains
[24,23]
![\begin{displaymath}
{\cal M}^{\rm GOA}=
{\cal S}^{(2)}\left[{\cal S}^{(1)}\right]^{-1}{\cal S}^{(2)}.
\end{displaymath}](img172.png) |
(62) |
Evaluating the matrix elements of
entering Eq. (61)
perturbatively, one can express
explicitly through the
matrix elements of the constraining field operators
:
![\begin{displaymath}
{\cal S}^{(K)}=\frac{1}{4}\left[{ M}^{(1)}\right]^{-1}{ M}^{(K)}
\left[{ M}^{(1)}\right]^{-1},
\end{displaymath}](img175.png) |
(63) |
where the energy-weighted moments
are given by
![\begin{displaymath}
M^{(K)}_{ij} = \sum_{\mu\nu}
\frac{\langle\mu\vert\hat Q...
...gle}
{(\breve{E}_\mu+\breve{E}_{\nu})^K} (\eta_{\mu\nu}^+)^2.
\end{displaymath}](img177.png) |
(64) |
Jacek Dobaczewski
2010-07-28