Calculation of Local Densities

After diagonalizing the HFB equation (37), local densities are calculated as

$$\begin{array}{lll} \rho ({\bf r}\sigma,{\bf r}^{\prime }\sig... ...\Phi_{\beta }({\bf r}^{\prime }\sigma^{\prime })\;, \end{array}$$ (46)

where $\Phi_\alpha({\bf r}\sigma)$ denotes the HO or THO basis wave functions, and the matrix elements of mean-field and pairing density matrices read

$$\rho _{\alpha \beta }= \sum\limits_{k}V_{\alpha k}^{\ast }V_... ... \beta }= - \sum\limits_{k} V_{\alpha k}^{\ast }U_{\beta k}\;.$$ (47)

The HFB calculations for zero-range pairing interaction give divergent energies when increasing the number of quasiparticle states in the sums of Eq. (47) (see discussion in Ref. [3]). Therefore, they invariably require a truncation of quasiparticle basis by defining a cut-off quasiparticle energy and including all quasiparticle states only up to this value.

The choice of an appropriate cut-off procedure has been discussed in [2]. After each iteration, performed with a given Fermi energy $\lambda$, one calculates an equivalent spectrum $\bar{e}_{k}$ and pairing gaps $\bar{\Delta}_{k}$:

$$\begin{tabular}{l} $\bar{e}_{k}=(1-2N_{k})E_{k},$\ \\ \ \\ $\bar{\Delta}_{k}=2E_{k}\sqrt{N_{k}(1-N_{k})},$% \end{tabular}$$ (48)

where $N_{k}$ denotes the norm (23) of the lower HFB wave function. Using this spectrum and pairing gaps, the Fermi energy is readjusted to obtain the correct value of particle number, and this new value is used in the next HFB iteration.

Due to the similarity between the equivalent spectrum $\bar{e}_{k}$ and the single-particle energies, one can take into account only those quasiparticle states for which

$$\bar{e}_{k}\leq \bar{e}_{\max },$$ (49)

where $\bar{e}_{\max }$$>$0 is a parameter defining the amount of the positive-energy phase space taken into account. Since all hole-like quasiparticle states, $N_{k}$$<$1/2, have negative values of $\bar{% e}_{k}$, condition (49) guarantees that they are all taken into account. In this way, a global cut-off prescription is defined 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. In the code, a default value of $\bar{e}_{\max }$=60 MeV is used.