Isospin Dependence of Pairing

The uniqueness of drip-line nuclei for studies of effective interactions is due to the very special role played by the pairing force. Correlations due to pairing, core polarization, and clustering are crucial in weakly bound nuclei. In a drip-line system, the pairing interaction and the presence of skin excitations (soft modes) could invalidate the picture of a nucleon moving in a single-particle orbit [10,11,12,13,14].

Surprisingly, rather little is known about the basic properties of the pairing force. In most nuclear structure calculations, the pairing Hamiltonian has been approximated by the state-independent seniority pairing force, or schematic multipole pairing interaction [15]. Such oversimplified forces, usually treated by means of the BCS approximation, perform remarkably well when applied to nuclei in the neighborhood of the stability valley, but they are inappropriate (and formally wrong) when extrapolating far from stability. The self-consistent mean-field models have meanwhile reached such a high level of precision that one needed to improve on the pairing part of the model. Presently, the most up-to-date models employ local pairing forces parametrized as contact interactions [16,17,10]. More flexible forms attach a density-dependence to the pairing strength [18,19,20]. There exist even more elaborate forms which include also gradient terms [21,22]. Although all these various density dependencies of pairing are widely used, very little is yet known about their relations to observable quantities.

Up to now, the microscopic theory of the pairing interaction has only seldom been applied in realistic calculations for finite nuclei. A ``first-principle" derivation of pairing interaction from the bare NN force still encounters many problems such as, e.g., treatment of core polarization [23,24]. Hence, phenomenological density-dependent pairing interactions are usually introduced. It is not obvious, how the density dependence should be parametrized [10], although nuclear matter calculations and experimental data on isotope shifts strongly suggest that pairing is a surface phenomenon, and that pairing interaction should be maximal in the surface region. This is why neutron-rich nuclei play such an important role in this discussion. Indeed, because of strong surface effects, the properties of these nuclei are sensitive to the density dependence of pairing.

Recent work [25], based on the spherical Skyrme-HFB model, contains the theoretical analysis of particle and pairing densities in neutron-rich nuclei and their dependence on the choice of pairing interaction. In the particle-particle (p-p) channel, the density-dependent delta interaction (DDDI) [18,19,20] has been employed:

$$V_{\rm pair}^{\delta\rho}({\bf r},{\bf r}') = f_{\rm pair}({\bf r})\delta({\bf r}-{\bf r}'),$$ (1)

where the pairing-strength factor is

$$f_{\rm pair}({\bf r})= V_0\left\{1-\left[\rho_{\rm IS}({\bf r}) /\rho_c\right]^\alpha\right\}$$ (2)

and V₀, $\rho_c$, and $\alpha$ are constants. The presence of the density dependence in the pairing channel has consequences for the spatial properties of pairing densities and fields [16,17,10]. In Eq. (2) $\rho_{\rm IS}({\bf r})$ stands for the isoscalar single-particle density $\rho_{\rm IS}({\bf r})$= $\rho_{n}({\bf r})$+ $\rho_{p}({\bf r})$. If $\rho_c$ is chosen such that it is close to the saturation density, $\rho_c$$\approx$ $\rho_{\rm IS}({\bf r}=0)$, both the resulting pair density and the pairing potential $\tilde{h}({\bf r})$ 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.

Apart from rendering the pairing weak in the interior, the specific functional dependence on $\rho_{\rm IS}$ used in Eq. (2) 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. [26,27], or equal to the power $\gamma$ of the Skyrme-force density dependence in the p-h channel [17,10]. As a typical example, the particle and pairing local HFB+SLy4 neutron densities $\rho_n(r)$ and $\tilde{\rho}_n(r)$calculated for several values of $\alpha$are displayed in Fig. 4 for ¹⁵⁰Sn.

  

Figure 4: Self-consistent spherical HFB+SLy4 local densities $\rho(r)$ (top) and $\tilde{\rho}(r)$ (bottom) for neutrons in ¹⁵⁰Sn for several values of $\alpha$. The insets show the same data in logarithmic scale.

With decreasing $\alpha$, the p-p density $\tilde{\rho}_n(r)$ develops a long tail extending towards large distances. This is a direct consequence of the attractiveness of DDDI at low densities when $\alpha$ is small. While in the nuclear interior, the p-h density $\rho_n(r)$ depends extremely weakly on the actual form of pairing interaction. Due to the self-consistent feedback between particle and pairing densities, the asymptotic values of $\rho_n(r)$ are significantly increased when $\alpha$ gets small (see inset). Moreover, one observes a clear development of a halo structure, i.e., a smooth exponential decrease, that for $\alpha$=1 starts at r$\simeq$6fm, is interrupted at r$\simeq$9fm for small $\alpha$, and replaced by a significantly slower decrease of the density. The general conclusion drawn from Fig. 4 is that experimental studies of neutron distributions in nuclei are extremely important for determining the density dependence of pairing interaction in nuclei.

While the analysis presented in Ref.[25] strongly suggested that the strong low-density dependence of pairing force, simulated by taking very small values of $\alpha$ in DDDI, is unphysical, it is only very recently that a realistic fit of DDDI to the odd-even staggering in nuclear masses has been carried out [28].

  

Figure 5: Comparison between the experimental staggering parameters (upper left panel, based on masses from Ref. [29]) and the average neutron pairing gaps calculated within the spherical HFB method for the Skyrme SLy4 force and three different versions of the zero-range pairing interaction.

Results of the spherical coordinate-space HFB calculations for semi-magic even-even nuclei for the average pairing gaps are shown in Figs. 5 and 6 for neutrons and protons, respectively. In the upper left panels we show the values of experimental three-point staggering parameters $\Delta^{(3)}$centered at odd particle numbers [30,31] and averaged over the two particle numbers adjacent to the even value. The experimental data from the interim 2001 atomic mass evaluation[29] were used.

  

Figure 6: Same as in Fig. 5 except for the average proton pairing gaps.

The lower left and right panels in Figs. 5-6 show the results obtained for the surface ($\alpha$=1) and volume pairing interactions, respectively. When compared with the experimental numbers, one sees that both types of pairing interaction are in clear disagreement with experiment. The surface interaction gives the pairing gaps that increase very rapidly in light nuclei, while the volume force gives the values that are almost independent of A. The surface pairing in light nuclei is so strong that pairing correlations do not vanish in doubly magic nuclei such as ¹⁶O or ⁴⁰Ca. The experimental data show the trend that is intermediate between surface and volume; hence, one may consider the intermediate-character pairing force that is half way in between, i.e., it is defined as:

$$V^{\delta }_{\rm {mix}}({\bf r},{\bf r}^{\prime })= \frac{1}{... ...r})}{2\rho _{0}}\right]\; \delta ({\bf r}- {\bf r}^{\prime }).$$ (3)

The upper right panels in Figs. 5 and 6 show the results obtained with the mixed pairing force. It can be seen that one obtains significantly improved agreement with the data, although a more precise determination of the balance between the surface and volume contributions still seems to be possible. One should note that similar intermediate-character pairing forces have recently been studied in Ref.[32] where it was concluded that pairing in heavy nuclei is of a mixed nature.

  

Figure 7: Similar as in Fig. 5, but for chains of isotones around N=82. The experimental neutron pairing gaps were estimated from the three-point[30,31] staggering parameters $\Delta^{(3)}$centered at N=81 and N=83 (from Ref. [33]).

Figure 7 illustrates the role of using different types of the pairing interaction to predict the neutron pairing gaps in very neutron-rich nuclei. 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 mixed 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.