The cut-off procedure for the contact pairing force

When using zero-range pairing forces such as the density-dependent delta force, one has to introduce the energy cut-off [35]. Within the unprojected HFB calculations, a pairing cut-off is introduced by using the so-called equivalent single-particle spectrum [31]. After each iteration, one calculates an equivalent spectrum $\bar{e}_{n}$ and corresponding pairing gaps $\bar{\Delta}_{n}$:

$$\bar{e}_{n}=(1-2P_{n})E_{n}+\lambda,~~~\bar{\Delta}_{n}=2E_{n}\sqrt{P_{n}(1-P_{n})},$$ (84)

where $E_{n}$ is the quasiparticle energy, $\lambda$ is the chemical potential, and $P_{n}$ denotes the norm of the lower component of the HFB wave function. The energy cut-off is practically realized by requesting that the phase space for the pair scattering is limited to those quasiparticle states for which $\bar{e}_{n}$ is less than the cut-off energy $\epsilon_{\text{cut}}$ (usually $\epsilon_{\text{cut}}=$ 60MeV) [36].

Obviously, the above procedure cannot be directly applied to the HFB+VAPNP method, where the intrinsic quantities, in particular the `quasiparticle' energies $E^N_{n}$, do not have obvious physical meaning. A reasonable practical prescription for $\epsilon_{\text{cut}}$ can be proposed in terms of intrinsic ($\phi=0$) HFB fields $h$ and $\tilde{h}$. After each iteration of Eq. (65), the average quasiparticle energies,

$$E_n = \left( \begin{array}{c} U^N \\ V^N \end{array} \right)_n^\d... ...a \end{array} \right) \left( \begin{array}{c} U^N \\ V^N \end{array} \right)_n,$$ (85)

together with $P_{n}$$\equiv$$P^N_{n}$, give the equivalent energies (84). Based on the spectrum of $\bar{e}_{n}$, the set of quasiparticle states appearing below the cut-off energy can now be easily defined. At the same time, the Fermi energy $\lambda$ (as an auxiliary quantity) can be recalculated in each iteration.

Figure 1: (color online) The neutron equivalent single-particle energies (84) for $N$=70 and $Z$=50 obtained in the HFB+LN method (first spectrum), HFB+VAPNP method using the average quasiparticle energies $E_n$ (second spectrum) (85), and by using the `quasiparticle' energies $E^N_k$ calculated in the HFB+VAPNP method for different values of the intrinsic neutron $\bar{N}$ and proton $\bar{Z}$ numbers (the remaining five spectra). The dashed line indicates the position of the LN Fermi energy $\lambda$.

The results of such a procedure are illustrated in Fig. 1. The left-most spectrum shows the neutron equivalent energies obtained within the LN method applied to $N$=70 and $Z$=50, and the dashed line shows the position of the corresponding LN neutron Fermi energy $\lambda$. For $\bar{e}_{n}<0$, this spectrum is very similar to the HF bound single-particle energies of this nucleus. Our method, based on the average quasiparticle energies (85), gives almost identical negative equivalent energies and quite similar positive ones. In particular, for highly positive equivalent energies, in the region of the cut-off energy $\epsilon_{\text{cut}}=$ 60MeV, similar continuum quasiparticle states appear in both methods; this guarantees the correct application of the cut-off procedure. The five equivalent spectra shown on the right hand side of Fig. 1 were calculated directly from the unphysical `quasiparticle' energies $E^N_{n}$ obtained for several selected values of the intrinsic particle numbers $\bar{N}$ and $\bar{Z}$. It is obvious that these spectra (even at $\bar{N}$=70 and $\bar{Z}$=50) bear no resemblance to the real single-particle spectra and cannot be used to define the cut-off procedure.