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Next: Degenerate shell: Seniority model Up: Limiting cases Previous: Weak-pairing limit,

Strong-pairing limit, $\eta \gg 1$

In this case, the expansion parameter is $\eta^{-1}$. In the zero order, the wave function of an even system with N=2n is the ground state of $\hat H_{\mbox{\rm\scriptsize {pair}}}$; i.e., it corresponds to the state with the maximal quasispin $L_{\mbox{\rm\scriptsize {max}}}$=$\Omega/2$ and the third component of quasispin L0=(2n-$\Omega)/2$. The binding energy is given by the seniority-model expression (see Refs. [27,28] and Sec. 3.2),

B_{\eta\gg1}^{(0)}(N=2n) =-G \left[ L_{\mbox{\rm\scriptsize {max}}}(L_{\mbox{\rm\scriptsize {max}}}+1) -
L_0(L_0-1) \right].
\end{displaymath} (26)

In the quasispin formalism, the single-particle Hamiltonian is a combination of a scalar and a vector operator (with respect to the quasispin group), and this implies the $\Delta L$=0,$\pm$1 selection rule for its matrix elements. The expectation value of $\hat H_{\mbox{\rm\scriptsize {sp}}}$ in the lowest L= $L_{\mbox{\rm\scriptsize {max}}}$ state is:

B_{\eta\gg1}^{(1)}(N=2n)=\langle L_{\mbox{\rm\scriptsize {ma...
L_{\mbox{\rm\scriptsize {max}}}L_0\rangle
= 2n\bar{e},
\end{displaymath} (27)


\bar{e} \equiv \frac{1}{\Omega}\sum_{k=1}^{\Omega} e_k
\end{displaymath} (28)

is the average single-particle energy. The second-order correction to the energy is given by

B_{\eta\gg1}^{(2)}(N=2n)=-\sum_{L<L_{\mbox{\rm\scriptsize {m...
...\scriptsize {sp}}}\vert LL_0\alpha\rangle\vert^2}{E^*_{LL_0}}.
\end{displaymath} (29)

In Eq. (28) $\alpha$ denotes all the remaining quantum numbers other than L and L0, and E*LL0 is the unperturbed excitation energy of $\vert LL_0\alpha\rangle$. Since $\hat H_{\mbox{\rm\scriptsize {sp}}}$ can only connect the seniority-zero ground state with the seniority-two, L= $L_{\mbox{\rm\scriptsize {max}}}$-1 states at energy $E^*=G\Omega$, Eq. (29) can be reduced to a simple form;
$\displaystyle B_{\eta\gg1}^{(2)}(N=2n)$ = $\displaystyle -\frac{1}{G\Omega}\langle
L_{\mbox{\rm\scriptsize {max}}}L_0\vert...
...}^2_{\mbox{\rm\scriptsize {sp}}}\vert L_{\mbox{\rm\scriptsize {max}}}L_0\rangle$  
    $\displaystyle +\frac{1}{G\Omega}\langle
L_{\mbox{\rm\scriptsize {max}}}L_0\vert...
..._{\mbox{\rm\scriptsize {sp}}}\vert L_{\mbox{\rm\scriptsize {max}}}L_0\rangle^2.$ (30)

By introducing the variance of single-particle levels,

\sigma^2_e \equiv \frac{1}{\Omega-1} \sum_{k=1}^{\Omega}(e_k-\bar{e})^2,
\end{displaymath} (31)

one can derive a simple expression for the second-order correction:

\end{displaymath} (32)

That is, the first- and second-order corrections to the binding energy are given by the first and second moments of the single-particle energy distribution. Note that for the degenerate j-shell, the second-order correction is zero, as expected.

For odd particle numbers, N=2n+1, one should take Lmax=$(\Omega$-1)/2 and L0=(2n+1-$\Omega)/2$in Eq. (26). Assuming that the odd particle occupies level n+1, this level is removed from the sum of Eqs. (28) and (31), i.e., one needs to consider the remaining $\Omega-1$ levels only:

\begin{displaymath}\bar{e}'\equiv \frac{1}{\Omega-1}\sum_{\raisebox{-1ex}{$\stac...
...$ }}^{\Omega} e_k
\end{displaymath} (33)

$\displaystyle \sigma^{\prime 2}_e$ $\textstyle \equiv$ $\displaystyle \frac{1}{\Omega-2}
{\scriptstyle{k=1}}{\scriptstyle{k\ne n+1}}$ }}^{\Omega}(e_k-\bar{e}')^2$  
  = $\displaystyle \sigma^2_e+\frac{1}{\Omega-2}\left[\sigma^2_e -\frac{\Omega}{\Omega-1}
(\bar{e}-e_{n+1})^2 \right].$ (34)

The resulting corrections to the binding energies can be written as

\begin{displaymath}B_{\eta\gg1}^{(1)}(N=2n+1) = 2n\bar{e}'+e_{n+1}
\end{displaymath} (35)


\sigma^{\prime 2}_e.
\end{displaymath} (36)

The zero-order expressions for $\Delta^{(3)}$ and $\delta{e}^\pm$in the strong pairing limit are given in Sec. 3.2. By adding the zero- and first-order contributions to the binding energy, one obtains the strong-pairing-limit expressions for $\Delta^{(3)}$:

$\displaystyle {\Delta^{(3)}_{\eta\gg1}(2n+1) = {1\over 2} G\Omega+ {\Omega-2n-1 \over
$\displaystyle {\Delta^{(3)}_{\eta\gg1}(2n) = {1\over 2} G\Omega + {1\over 2} G}~~~~~$
    $\displaystyle + {\Omega-2n \over \Omega-1}
\left({e_{n+1}+e_n \over 2}-\bar{e}\right) - {e_{n+1}-e_n \over 2(\Omega-1)}.$ (38)

It can easily be shown that the first-order correction to $\delta{e}$ vanishes. Consequently, the seniority-model expressions discussed below give a good approximation to the single-particle energy-splitting filters (7) and (8).

next up previous
Next: Degenerate shell: Seniority model Up: Limiting cases Previous: Weak-pairing limit,
Jacek Dobaczewski