Hartree-Fock-Bogoliubov densities and mean fields

This section contains a very brief description of the HFB formalism in the coordinate representation. Since the method is standard, our discussion is limited to the most essential definitions and references. For more details, we would like to refer the reader to the previous work [18,19,2].

The HFB approach is a variational method which uses nonrelativistic independent-quasiparticle states as trial wave functions [9]. The total binding energy of a nucleus is obtained self-consistently from the energy functional [20]:

$${\cal E} = {\cal E}_{\rm kin} + {\cal E}_{Sk} + {\cal E}_{C} +{\cal E}_{\rm pair},$$ (1)

where ${\cal E}_{\rm kin}$ is the kinetic energy functional, ${\cal E}_{Sk}$ is the Skyrme functional, ${\cal E}_C$ is the Coulomb energy (including the exchange term in the Slater approximation), and ${\cal E}_{\rm pair}$ is the pairing energy.

By minimizing the functional (1), one arrives at the HFB equation [18,19,2] for the two-component single-quasiparticle HFB wave function $\{\phi_1,\phi_2\}$:

$$\int\mbox{\rm\scriptsize {d}}^3\mbox{{\boldmath {$r$ }}}' ... ...hi_2 (E,\mbox{{\boldmath {$r$ }}}\sigma) \end{array}\right),$$ (2)

where $\lambda$ is the Fermi energy and h and $\tilde{h}$ are the particle-hole (p-h) and particle-particle (p-p) mean-field Hamiltonians.

Properties of the HFB equation in the spatial coordinates, Eq. (2), and the asymptotic properties of HFB wave functions and density distributions have been analyzed in Refs. [18,19,2]. In particular, it has been shown that the spectrum of quasi-particle energies E is continuous for |E|>$-\lambda$ and discrete for |E|<$-\lambda$. Since for $\lambda$<0 (bound system) the lower components $\phi_2(E,\mbox{{\boldmath {$r$ }}}\sigma)$ are localized functions of $\mbox{{\boldmath {$r$ }}}$, the particle and pairing density matrices,

  

$$\rho(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r$ }}}'\sigma') \sum_{0<E_n<-\lambda} \phi_2 (E_n,\mbox{{\boldmath {$r$ }}} \sigm... ...x{{\boldmath {$r$ }}} \sigma ) \phi^*_2(E ,\mbox{{\boldmath {$r$ }}}'\sigma') ,$$ = (3)

$$\tilde\rho(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r$ }}}'\sigma') - \sum_{0<E_n<-\lambda} \phi_2 (E_n,\mbox{{\boldmath {$r$ }}} \si... ...x{{\boldmath {$r$ }}} \sigma ) \phi^*_1(E ,\mbox{{\boldmath {$r$ }}}'\sigma') ,$$ = (4)

are localized as well. For the case of a discretized continuum, the integral over the energy reduces to a discrete sum [19].

The p-h mean-field Hamiltonian can be expressed through the kinetic energy and the p-h mean-field potential:

$$h(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r$ }}}'\... ...{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r$ }}}'\sigma'),$$ (5)

where

$$\Gamma(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r... ...math {$r$ }}}_2'\sigma_2',\mbox{{\boldmath {$r$ }}}_2\sigma_2)$$ (6)

and V is the two-body density-dependent effective p-h interaction (in our case it is a contact Skyrme interaction). The pairing mean-field potential can be expressed through the p-p density

$$\tilde h(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {... ... {$r$ }}}_1'\sigma_1',\mbox{{\boldmath {$r$ }}}_2'\sigma_2'),$$ (7)

where $V_{\rm pair}$ is the two-body density-dependent effective p-p interaction. (If one adopts the philosophy of the energy density functional theory, V and $V_{\rm pair}$can be different since they result from different variations of the energy functional.)

The local HFB densities discussed in this work,

  

$$\rho(\mbox{{\boldmath {$r$ }}}) \sum_\sigma \rho(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r$ }}}\sigma),$$ = (8)

$$\tilde\rho(\mbox{{\boldmath {$r$ }}}) \sum_\sigma \tilde\rho(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r$ }}}\sigma),$$ = (9)

have a simple physical interpretation [2]. Namely, $\rho(\mbox{{\boldmath {$r$ }}})$ represents the probability density of finding a particle at a given point. On the other hand, $\vert\tilde\rho\vert^2(\mbox{{\boldmath {$r$ }}})$ is the probability of finding a pair of nucleons in excess of the probability of finding two uncorrelated nucleons.

In this work, in the p-h channel the SLy4 Skyrme parametrization [21] has been used. This force performs well for the total energies, radii, and moments, and it is also reliable when it comes to predictions of long isotopic sequences [21].

In the particle-particle (p-p) channel, we employ the density-dependent contact interaction. As discussed in a number of papers, see e.g. Refs. [10,11,2], the presence of the density dependence in the pairing channel has consequences for the spatial properties of pairing densities and fields. The commonly used density-independent contact delta interaction, $V^{\delta}_{\rm pair}(\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}}') = V_0 \delta(\mbox{{\boldmath {$r$ }}}-\mbox{{\boldmath {$r$ }}}'),$leads to volume pairing. A simple modification of that force is the density-dependent delta interaction (DDDI) [12,13,14]

$$V_{\rm pair}^{\delta\rho}(\mbox{{\boldmath {$r$ }}},\mbox{{... ...\delta(\mbox{{\boldmath {$r$ }}}-\mbox{{\boldmath {$r$ }}}'),$$ (10)

where the pairing-strength factor is

$$f_{\rm pair}(\mbox{{\boldmath {$r$ }}})= V_0\left\{1-\left[... ... IS}(\mbox{{\boldmath {$r$ }}}) /\rho_c\right]^\alpha\right\}$$ (11)

and V₀, $\rho_c$, and $\alpha$ are constants. In Eq. (11) $\rho_{\rm IS}(\mbox{{\boldmath {$r$ }}})$ stands for the isoscalar single-particle density $\rho_{\rm IS}(\mbox{{\boldmath {$r$ }}})$= $\rho_{n}(\mbox{{\boldmath {$r$ }}})$+ $\rho_{p}(\mbox{{\boldmath {$r$ }}})$. If $\rho_c$ is chosen such that it is close to the saturation density, $\rho_c$$\approx$ $\rho_{\rm IS}(\mbox{{\boldmath {$r$ }}}=0)$, both the resulting pair density and the pairing potential $\tilde{h}(\mbox{{\boldmath {$r$ }}})$ are small in the nuclear interior, and the pairing field becomes surface-peaked. By varying the magnitude of the density-dependent term, the transition from volume pairing to surface pairing can be probed. A similar form of DDDI, also containing the density gradient term, has been used in Refs. [15,16].

Apart from rendering the pairing weak in the interior, the specific functional dependence on $\rho_{\rm IS}$ used in Eq. (11) is not motivated by any compelling theoretical arguments or calculations. In particular, values of power $\alpha$ were chosen ad hoc to be either equal to 1 (based on simplicity), see e.g. Refs. [22,23], or equal to the power $\gamma$ of the Skyrme-force density dependence in the p-h channel [11,2]. The dependence of results on $\alpha$ was, in fact, never studied. In the present paper, we perform such an analysis by choosing four values of $\alpha$=1, 1/2, 1/3, and 1/6 that cover the range of values of $\gamma$ used typically for the Skyrme forces.

Calculations which are based on the contact force, such as that of Eq. (10), require a finite space of states in the p-p channel. Usually, one takes a limited configuration space determined by a cut-off in the single-particle energy or in the single-quasiparticle energy. In this work, we made a cut-off with respect to ``equivalent-spectrum" single-particle energies obtained from HFB quasiparticle energies and occupation coefficients, as defined in Ref. [19]. All the quasiparticle states with $j\le j_{\mbox{\rm\scriptsize {max}}}$=21/2 and equivalent energies up to $\bar{\epsilon}_{\mbox{\rm\scriptsize {max}}}$=60MeV were considered, and the HFB equations were solved in the spherical box of $R_{\rm box}$=30fm.

This procedure differs from the prescription applied in Ref. [19] where a $(j\ell)$-dependent cut-off was used. There, the quasi-particle states with quasiparticle energies up to the depth of the effective potential $D_{j\ell}$= $-{\displaystyle{\min_r}}U_{\mbox{\rm\scriptsize {eff}}}(r;j\ell)$ were considered, and in addition, at least one quasiparticle state was considered in each $(j\ell)$ block. The combination of these two rules, together with the value of $R_{\rm box}$=20fm used there, ensured that low-lying high-j resonances were always included in the phase space. However, systematic calculations across the complete table of nuclides revealed several pathological cases where a higher-lying resonance was missing from the included phase space. This could be especially true for some near-the-barrier j=l-1/2 resonances, for which the effective potentials $U_{\mbox{\rm\scriptsize {eff}}}$ may have small depths. Since in the present study we aim at a better description of the continuum phase space, we increased the size of the box to $R_{\rm box}$=30fm and changed the cut-off prescription so as to include sufficiently many states and to never miss a resonance. (For more discussion pertaining to the energy cut-off problem, see Appendix B of Ref. [2].)

For $\rho_c$ we took the standard value of 0.16fm^-3, and the strength V₀ of DDDI was adjusted according to the prescription given in Ref. [11], i.e., so as to obtain in each case the value of 1.256MeV for the average neutron gap in ¹²⁰Sn. For $\alpha$=1, 1/2, 1/3, and 1/6 the adjusted values are V₀=-520.5, -787.7, -1041.3, and -1772.5Mevfm^-6, respectively. The resulting pairing-strength factors (11) are shown in Fig. 1 as functions of density $\rho$for the four values of the exponent $\alpha$. It is seen that for $\rho$>0.04fm^-3 the pairing-strength factor $f_{\rm pair}$ is almost independent of the power $\alpha$. At low densities, however, the pairing interaction becomes strongly dependent on $\alpha$ and very attractive at $\rho$ $\rightarrow$0. The pattern shown in Fig. 1 indicates that pairing forces characterized by small values of $\alpha$ should give rise to pair fields peaked at, or even beyond, the nuclear surface (halo region) where the nucleonic density is low.