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Strongpairing 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
L_{0}=(2n.
The binding energy is given by the senioritymodel
expression (see Refs. [27,28] and Sec. 3.2),

(26) 
In the quasispin formalism, the singleparticle 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 singleparticle energy.
The secondorder correction to the energy is given by

(29) 
In Eq. (28)
denotes all the remaining quantum numbers other than
L and L_{0},
and
E^{*}_{LL0} is the unperturbed excitation energy of
.
Since
can only connect the
seniorityzero
ground state with the senioritytwo, L=
1 states at
energy
,
Eq. (29) can be reduced to a simple form;
By introducing the variance of singleparticle levels,

(31) 
one can derive a simple expression for the secondorder correction:

(32) 
That is, the first and secondorder corrections to the binding energy are
given by the first and second moments of the singleparticle energy
distribution.
Note that for the degenerate jshell, the secondorder
correction is zero, as expected.
For odd particle numbers, N=2n+1, one should take
L_{max}=1)/2 and L_{0}=(2n+1in 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
The resulting corrections to the binding energies can be written as

(35) 
and

(36) 
The zeroorder expressions for
and
in the strong pairing limit are given in
Sec. 3.2.
By adding the zero and firstorder contributions to
the binding energy, one obtains
the strongpairinglimit expressions for
:
It can easily be shown that the firstorder correction to
vanishes.
Consequently, the senioritymodel expressions discussed below give a good
approximation to
the singleparticle energysplitting filters (7)
and (8).
Next: Degenerate shell: Seniority model
Up: Limiting cases
Previous: Weakpairing limit,
Jacek Dobaczewski
20000309