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),

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

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

where

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

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

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

 

$$B_{\eta\gg1}^{(2)}(N=2n) -\frac{1}{G\Omega}\langle L_{\mbox{\rm\scriptsize {max}}}L_0\vert... ...}^2_{\mbox{\rm\scriptsize {sp}}}\vert L_{\mbox{\rm\scriptsize {max}}}L_0\rangle$$ =

$$+\frac{1}{G\Omega}\langle L_{\mbox{\rm\scriptsize {max}}}L_0\vert... ..._{\mbox{\rm\scriptsize {sp}}}\vert L_{\mbox{\rm\scriptsize {max}}}L_0\rangle^2.$$ (30)

By introducing the variance of single-particle levels,

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

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

$$B_{\eta\gg1}^{(2)}(N=2n)=-\frac{1}{G\Omega^2}4n(\Omega-n)\sigma^2_e.$$ (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 L_max=$(\Omega$-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:

$$\bar{e}'\equiv \frac{1}{\Omega-1}\sum_{\raisebox{-1ex}{$\stac... ...$ }}^{\Omega} e_k =\bar{e}+\frac{1}{\Omega-1}(\bar{e}-e_{n+1})$$ (33)

and

$$\sigma^{\prime 2}_e \equiv \frac{1}{\Omega-2} \sum_{\raisebox{-1ex}{$\stackrel {\scriptstyle{k=1}}{\scriptstyle{k\ne n+1}}$ }}^{\Omega}(e_k-\bar{e}')^2$$

$$\sigma^2_e+\frac{1}{\Omega-2}\left[\sigma^2_e -\frac{\Omega}{\Omega-1} (\bar{e}-e_{n+1})^2 \right].$$ = (34)

The resulting corrections to the binding energies can be written as

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

and

$$B_{\eta\gg1}^{(2)}(N=2n+1)=-\frac{1}{G(\Omega-1)^2}4n(\Omega-n-1) \sigma^{\prime 2}_e.$$ (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 $\Delta^{(3)}$:

 

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

$${\Delta^{(3)}_{\eta\gg1}(2n) = {1\over 2} G\Omega + {1\over 2} G}~~~~~$$

$$+ {\Omega-2n \over \Omega-1} \left({e_{n+1}+e_n \over 2}-\bar{e}\right) - {e_{n+1}-e_n \over 2(\Omega-1)}.$$ (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).