Volume and surface pairing interactions

Without having at our disposal microscopic first-principle effective pairing interactions (with surface effects included as discussed in Sec. 2), one uses in the particle-particle (p-p) channel a phenomenological density-dependent contact interaction. As discussed in a number of papers (see, e.g., Refs. [8,9,4]), 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,

$$V^{\delta}_{\mbox{\scriptsize {vol}}}(\mbox{{\boldmath {$r$... ...\delta(\mbox{{\boldmath {$r$ }}}-\mbox{{\boldmath {$r$ }}}'),$$ (12)

leads to volume pairing. A simple modification of that force is the density-dependent delta interaction (DDDI) [10,11,12]

$$V_{\mbox{\scriptsize {surf}}}^{\delta}(\mbox{{\boldmath {$r... ...\delta(\mbox{{\boldmath {$r$ }}}-\mbox{{\boldmath {$r$ }}}'),$$ (13)

where the pairing-strength factor is

$$f_{\mbox{\scriptsize {pair}}}(\mbox{{\boldmath {$r$ }}})= V... ..._{0}(\mbox{{\boldmath {$r$ }}}) /\rho_c\right]^\alpha\right\}$$ (14)

and V₀, $\rho_c$, and $\alpha$ are constants. If $\rho_c$ is chosen such that it is close to the saturation density, $\rho_c$$\approx$ $\rho_{0}(\mbox{{\boldmath {$r$ }}}$=0), both the resulting pair density and the pairing potential 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. [13,14].

Apart from rendering the pairing weak in the interior, the specific functional dependence on $\rho_0$ used in Eq. (14) 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. [15,16], or equal to the power $\gamma$ of the Skyrme-force density dependence in the p-h channel [9,4].

  

Figure 3: Radial strength factor $f_{\mbox{\scriptsize{pair}}}$ of the density-dependent delta interaction, Eq. (14), as a function of $\rho$=$\rho_0$for several values of $\alpha$. The value of $\rho_c$ was assumed to be 0.16fm^-3. At each value of $\alpha$, the strength V₀was adjusted to reproduce the neutron pairing gap in ¹²⁰Sn. The inset shows $f_{\mbox{\scriptsize{pair}}}/\vert V_0\vert$ as a function of dimensionless normalized density $\rho_{0}/\rho_c$ (from Ref. [17]).

The dependence of results on $\alpha$ was studied in Ref. [17] within the Hartree-Fock-Bogoliubov (HFB) approach. We considered 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. 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. [9], i.e., so as to obtain in each case the value of 1.245MeV for the average neutron gap in ¹²⁰Sn. The resulting pairing-strength factors (14) are shown in Fig. 3 as functions of density $\rho$=$\rho_0$for the four values of the exponent $\alpha$. It is seen that for $\rho$$\geq$0.04fm^-3 the pairing-strength factor $f_{\mbox{\scriptsize{pair}}}$ 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. 3 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.

  

Figure 4: Comparison between the experimental two-neutron separation energies S_2N and neutron pairing gaps $\Delta_N$ (upper left panels, based on masses from Ref. [18]), and the corresponding results of the spherical HFB method for the Skyrme SLy4 force [3] and five different versions of the zero-range pairing interaction (see text).

The main conclusion of Ref. [17] is that, due to the self-consistent feedback between particle and pairing densities, the size of the neutron halo is indeed strongly influenced by pairing correlations; hence, by the pairing parametrization assumed. Consequently, experimental studies of neutron distributions in nuclei are extremely important for determining the density dependence of pairing interaction in nuclei. At the same time, the strong low-density dependence of the pairing force, simulated by taking very small values of $\alpha$ in DDDI, is unphysical. The present experimental data are consistent with about 1/2$\leq$$\alpha$$\leq$1. In this context, it is interesting to note that excellent fits to the data were obtained in Refs. [13,14] by taking $\alpha$=2/3. However, at present there is no theoretical argument why the density dependence should be even taken in a form of the power law.

Moreover, the pairing interaction is most likely of an intermediate character between the volume (12) and surface forms (13). (See Refs. [7,19,20] for recent analyses.) In particular, the force which is a fifty-fifty mixture of both types,

$$V^{\delta }_{\rm {mix}}({\bf r},{\bf r}^{\prime })= \frac{1}{... ...})}{2\rho _{0}}\right]\; \delta ({\bf r}- {\bf r}^{\prime }) ,$$ (15)

performs quite well [20] in reproducing the general mass-dependence of the odd-even mass staggering parameter $\Delta^{(3)}$ centered at odd particle numbers [21,22].

Figure 4 illustrates the role of using different types of the pairing interaction to predict the two-neutron separation energies and neutron pairing gaps, respectively, in very neutron-rich isotones around N=82. The experimental values were calculated based on the interim 2001 evaluation of atomic masses [18].

Figure 4 nicely illustrates the effect of the so-called shell quenching in heavy nuclei [23], i.e., the vanishing of the effective distance between the neutron single-particle levels above and below a magic neutron number when approaching the neutron drip line. The difference between the two-neutron separation energies above and below N=82 very well visualizes this effect. In fact, the experimental data show an apparent opposite effect; however, this is caused by the fact that the data are available only for Z$\geq$50. When approaching the magic proton number, the neutron magic gap is slightly enhanced [24]. This effect is entirely absent in calculations that do not include any effects of correlations and deformations.

Nevertheless, for Z<50 the effect of the shell quenching is very well visible in the calculations. Moreover, the magnitude of the effect is very strongly influenced by the type of pairing force used. For the volume pairing force (12), the effect is rather weak and the magic gap N=82 is still visible even at the very drip line. However, for the surface pairing force (13) the shell gap goes to zero much earlier, and this tendency is accentuated for pairing forces that are stronger at small densities (for smaller powers of $\alpha$).

For the neutron pairing gap (Fig. 4) the experimental data that exist for Z$\geq$50 do not indicate any definite change in the neutron pairing intensity with varying proton numbers. However, the surface pairing interactions (bottom panels) give a slow dependence for Z$\geq$50 that is dramatically accelerated after crossing the shell gap at Z=50. On the other hand, the volume and intermediate-type pairing forces predict a slow dependence all the way through to very near the neutron drip line. It is clear that measurements of only several nuclear masses for Z<50 will allow us to strongly discriminate between the pairing interactions that have different space and density dependencies.

  

Figure 5: Results of the HFB+THO+SLy4 deformed calculations for particle-bound even-even nuclei with Z$\leq$108 and N$\leq$188.