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

### Strong-pairing limit,

In this case, the expansion parameter is . In the zero order, the wave function of an even system with N=2n is the ground state of ; i.e., it corresponds to the state with the maximal quasispin = and the third component of quasispin L0=(2n-. The binding energy is given by the seniority-model expression (see Refs. [27,28] and Sec. 3.2),

 (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 =0,1 selection rule for its matrix elements. The expectation value of in the lowest L= state is:

 (27)

where

 (28)

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

 (29)

In Eq. (28) denotes all the remaining quantum numbers other than L and L0, and E*LL0 is the unperturbed excitation energy of . Since can only connect the seniority-zero ground state with the seniority-two, L= -1 states at energy , Eq. (29) can be reduced to a simple form;

 = (30)

By introducing the variance of single-particle levels,

 (31)

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

 (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=-1)/2 and L0=(2n+1-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 levels only:

 (33)

and
 = (34)

The resulting corrections to the binding energies can be written as

 (35)

and

 (36)

The zero-order expressions for and 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 :

 (38)

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

Next: Degenerate shell: Seniority model Up: Limiting cases Previous: Weak-pairing limit,
Jacek Dobaczewski
2000-03-09