Theoretical models

In this section, we investigate several exactly solvable models to see the interplay between the particle-hole and particle-particle channels of interaction. In all cases, the Hamiltonian has the form

$$\hat H=\hat{H}_0 + \hat H_{\mbox{\rm\scriptsize {pair}}},$$ (13)

where $H=\hat{H}_0$ is either the intrinsic (i.e., deformed) single-particle Hamiltonian or the laboratory-system quadrupole-quadrupole Hamiltonian, and $\hat H_{\mbox{\rm\scriptsize {pair}}}$ is always the monopole-pairing (seniority) Hamiltonian:

$$\hat H_{\mbox{\rm\scriptsize {pair}}} = -G\hat P^\dagger \hat P.$$ (14)

In Eq. (14) G is the pairing strength parameter,

$$\hat P^\dagger =\sum_{k=1}^\Omega a_k^\dagger a_{\bar{k}}^\dagger$$ (15)

denotes the monopole-pair creation operator, and $\bar{k}$ denotes the time-reversed state.

Properties of the Hamiltonian (13) depend on the ratio

$$\eta=\frac{G}{\kappa},$$ (16)

where $\kappa$ represents the strength of $\hat H_0$. For both $\eta \ll 1$ (weak pairing) and $\eta \gg 1$ (strong pairing), one can treat the Hamiltonian (13) perturbatively. However, the situation encountered most often in the nuclear physics context is the intermediate case ($\eta$$\sim$0.4) in which pairing correlations are strongly influenced by the nuclear mean field.