The cut-off procedure

 

HFB calculations in configurational representation invariably require a truncation of the single-particle basis and a truncation in the number of quasiparticle states. The latter is usually realized by defining a cut-off quasiparticle energy $E_{\max}$ and then including quasiparticle states only up to this value. When the finite-range Gogny force is used both in the p-p and p-h channels, the cut-off energy $E_{\max}$ has numerical significance only. In contrast, HFB calculations based on Skyrme forces in the p-h and p-p channels, as well as any other calculations based on a zero-range force in the p-p channel

$$V^{\delta }({\mbox{{\boldmath {$r$ }}}},{\mbox{{\boldmath {$r... ...{\boldmath {$r$ }}}}-{ \mbox{{\boldmath {$r$ }}}}^{\prime })\;$$ (55)

require a finite space of states. This is because, for any value of the coupling constant V₀, they give divergent energies with increasing $E_{\max}$ (see the discussion in Ref. [7]).

The choice of an appropriate cut-off procedure has been discussed in the case of coordinate-space HFB calculations for spherical nuclei [6]. It was demonstrated there that one must sum up contributions from all states close in quasiparticle energy to the bound particle states to obtain correct density matrices in the HFB method. Since the bound particle states are associated with quasiparticle energies smaller than the absolute value D of the depth of the effective potential well, one had to take the cut-off energy $E_{\max}$comparable to D.

In the case of deformed HFB calculations, and especially when performing configurational HFB calculations, it is difficult to look for the depth of the effective potential well in each $\Omega^{\pi }$ subspace. Thus, an alternative criterion with respect to the above cut-off procedure used in spherical calculations is needed. For this purpose, we have adopted the following procedure (see Appendix B of [6]). After each iteration, performed with a given chemical potential $\lambda$, we calculate an auxiliary spectrum $\bar{e}_{n}$and pairing gaps $\bar{\Delta}_{n}$ by using for each quasiparticle state the BCS-like formulae,

 

$$\sqrt{\left(\bar{e}_{n}-\lambda\right)^{2}+\bar{\Delta}_{n}^{2}},$$ E_n = (56)

$$\frac{1}{2}-\frac{\bar{e}_{n}-\lambda}{2E_n},$$ N_n = (57)

or equivalently

  

$$\bar{e}_{n}$$ = (1-2N_n)E_n, (58)

$$\bar{\Delta}_{n} 2E_n\sqrt{N_n(1-N_n)}.$$ = (59)

Then, in the next iteration, we readjust the proton and neutron chemical potentials to obtain the correct values of the proton and neutron particle numbers (51), where again N_n is calculated for the equivalent spectrum, Eq. (57). Due to the similarity between the equivalent spectrum $\bar{e}_{n}$ and the single-particle energies, we are taking into account only those quasiparticle states for which

$$\bar{e}_{n}\leq\bar{e}_{\max},$$ (60)

where $\bar{e}_{\max}$>0 is a parameter defining the amount of the positive-energy phase space taken into account. At the same time, since all hole-like quasiparticle states, N_n<1/2, have negative values of $\bar{e}_{n}$(58), condition (60) quarantees that they are all taken into account. In this way, we have a global cut-off prescription independent of $\Omega^{\pi }$, which fulfills the requirement of taking into account the positive-energy phase space as well as all quasiparticle states up to the highest hole-like quasiparticle energy.