Pairing functional

The form of the most general pairing EDF that is quadratic in local isoscalar and isovector densities has been discussed in Refs.[25,26]. Because of the lack of nuclear observables that can constrain coupling constants of this general pairing functional, current realizations are much simpler. A commonly used effective pairing interaction is the zero-range pairing force with the density-dependent form factor [27,28,29,30,31]:

$$f_{\mbox{\rm\scriptsize {pair}}}(\bm{r}) = V_0\left\{1+x_0\h... ...(\bm{r})}{\rho_c}\right]^\alpha (1+x_3\hat{P}^\sigma)\right\},$$ (15)

where $\hat{P}^\sigma$ is the usual spin-exchange operator and $\rho_0$=0.16fm$^{-3}$. When only the isovector pairing is studied, the exchange parameters $x_0$ and $x_3$ are usually set to 0. However, in the general case of coexisting isoscalar and isovector pairing correlations, nonzero values of $x_0$ and $x_3$ have to be used. Pairing interactions corresponding to $\eta$=0, 0.5, and 1, are usually referred to as volume, mixed, and surface pairing, respectively[32,33,34]. The volume pairing interaction acts primarily inside the nuclear volume while the surface pairing generates pairing fields peaked around or outside the nuclear surface.

Another form of density dependence has been suggested in Ref.[35] and successfully applied [36] to explain odd-even effects in charge radii. As discussed in Refs.[31,37], different assumptions about the density dependence may result in differences of pairing fields in very neutron rich nuclei. However, the results of the global survey[38] suggest that - albeit there is a slight favoring of the surface interaction - one cannot reliably extract the density dependence of the effective pairing interaction (15) from the currently available experimental odd-even mass differences, limited to nuclei with a modest neutron excesses (see also Refs.[39,34,40]).

A timely question, related to the density dependence, is whether there is an effective isospin dependence of the pairing interaction. The global survey[38] of odd-even staggering of binding energy indicates that the effective pairing strength $V_0$ for protons is larger than for neutrons, and the recent large-scale optimizations of the nuclear EDF are consistent with this finding [41,42]. This can be attributed to the isospin-dependent contribution to pairing from the Coulomb interaction[43,44,45] or to induced pairing due to the coupling to collective excitations[46,47]. To account for those effects, an extended density dependence has been proposed[48,49,50] that involves the local isovector density $\rho_1(\bm{r})$.

Little is known about the isoscalar pairing functional. The local isovector pairing potential[25,26] $\vec{\breve{U}}(\bm{r})$ is proportional to the isovector pair density $\vec{\breve{\rho}}$ whereas the isoscalar pairing potential $\breve{\bm{\Sigma}}_0(\bm{r})$ is a vector proportional to the isoscalar-vector pairing spin density $\breve{\bm{s}}_0$. Then, the isoscalar pairing field,

$$\breve{h}_0(\bm{r}) = \breve{\bm{\Sigma}}_0\cdot\hat{\bm{\sigma}} \propto \breve{\bm{s}}_0\cdot\hat{\bm{\sigma}}$$ (16)

is the projection of the quasiparticle's spin on the proton-neutron pairing field. Physically, $\vec{\breve{\rho}}$ represents the density of $S$=0, neutron-neutron, proton-proton, and proton-neutron pairs, whereas the vector field $\breve{\bm{s}}_0$ describes the spin distribution of $S$=1 pn pairs (that is, it contains all magnetic components of $S$=1 pn pairing field).

Symmetries of the isoscalar pairing mean-fields have been studied in detail in Ref.[26]. As an example, lines of the solenoidal field $\breve{\bm{s}}_0$ - present in the generalized pairing theory that mixes proton and neutron orbits - are schematically shown in Fig. 2. It is interesting to note that for the geometry of Fig. 2, the third component $\breve{\bm{s}}_{0z}$, associated with the $M$=0 isoscalar pairing field vanishes. That is, the solenoidal pairing field is created by the two components with $M$=$\pm 1$. One can thus conclude that the assumption of axial symmetry, or signature, does not preclude the existence of isoscalar pairing.

Figure 2: (a) Schematic illustration of the isoscalar vector pairing field $\breve{\bm{s}}_0$ in the case of conserved axial and mirror symmetries[26]. The field is solenoidal, with vanishing third component. (b) Under rotation around the third (symmetry) axis, the field at the point $\bm{r}_A$ is transformed to the position $\bm{r}_B$. Likewise, under ${\cal{}R}_x$, the field is transformed to $\bm{r}_C$. While neither of these operations leave the individual vector $\breve{\bm{s}}_0(\bm{r}_A)$ invariant, the field as whole does not change.