Localization of pairing correlations in nuclei within relativistic mean field models

We analyze the localization properties of two-body correlations induced by pairing in the framework of relativistic mean field (RMF) models. The spatial properties of two-body correlations are studied for the pairing tensor in coordinate space and for the Cooper pair wave function. The calculations are performed both with relativistic Hartree-Bogoliubov (RHB) and RMF + projected-BCS (PBCS) models and taking as examples the nuclei ${}^{66}\mathrm{Ni}$, ${}^{124}\mathrm{Sn}$, and ${}^{200}\mathrm{Pb}$. It is shown that the coherence length has the same pattern as in previous nonrelativistic HFB calculations, i.e., it is maximum in the interior of the nucleus and drops to a minimum in the surface region. In the framework of RMF+PBCS we have also analyzed, for the particular case of ${}^{120}\mathrm{Sn}$, the dependence of the coherence length on the intensity of the pairing force. This analysis shows that (a) pairing is reducing the coherent length by about 25–$30\%$ compared to the RMF limit and (b) after the onset of pairing correlations, the coherent length depends only weakly on the strength of the pairing force.

Figure 1

Two-body correlations function ${\left|\kappa (R,r)\right|}^2$ provided by RHB (a,b,c) and RMF+PBCS (${\mathrm{a}}^{\prime },{\mathrm{b}}^{\prime },{\mathrm{c}}^{\prime }$) calculations. The results correspond to ${}^{66}\mathrm{Ni}$ (a and ${\mathrm{a}}^{\prime }$), ${}^{124}\mathrm{Sn}$ (b and ${\mathrm{b}}^{\prime }$) and ${}^{200}\mathrm{Pb}$ (c and ${\mathrm{c}}^{\prime }$).

Figure 2

The dependence of ${\left|\kappa (R,r)\right|}^2$ on relative distance $r$ shown for ${\mathrm{R}}_{\text{cm}}=1.0\mathrm{fm}$ (left), ${\mathrm{R}}_{\text{cm}}=r_n$ (middle), and ${\mathrm{R}}_{\text{cm}}={\mathrm{R}}_{\text{min}\left(\kappa \right)}\left(right\right)$. The results correspond to ${}^{66}\mathrm{Ni}$ (a), ${}^{124}\mathrm{Sn}$ (b) and ${}^{200}\mathrm{Pb}$ (c), calculated with RHB (line with diamonds) and RMF+PBCS (line with dots).

Figure 3

The coherence length corresponding to the pairing tensor [Eq. (7)] provided by RHB (upper panel) and RMF+PBCS (lower panel) calculations for ${}^{66}\mathrm{Ni}$ (line with diamonds), ${}^{124}\mathrm{Sn}$ (line with dots), and ${}^{200}\mathrm{Pb}$ (line with squares). The numerators and denominators of Eq. (7) are plotted on the right-hand side.

Figure 4

The ratio (7) between the coherence length and the local distance between the neutrons for ${}^{66}\mathrm{Ni}$, calculated with RHB (line with diamonds) and RMF+PBCS (line with dots). The vertical line corresponds to the neutron rms radius $r_n$.

Figure 5

The coherence length for ${}^{120}\mathrm{Sn}$ calculated in the RMF+PBCS approach for various intensities of the pairing force, scaled by the factors $\alpha$. The pairing energy is shown for each scaling factor.

Figure 6

The quantity $k_i=u_i{v}_i$ for the single-particle states of ${}^{120}\mathrm{Sn}$ from the major shell. The results correspond to the RMF+PBCS calculations done with the same intensities of the pairing force used in Fig. 5.

Figure 7

The coherence length corresponding to the Cooper pair wave function [Eq. (13)] provided by RHB (upper panel) and RMF+PBCS (lower panel) calculations for ${}^{66}\mathrm{Ni}$ (line with diamonds), ${}^{124}\mathrm{Sn}$ (line with dots), and ${}^{200}\mathrm{Pb}$ (line with squares). The numerators and denominators of Eq. (13) are plotted on the right-hand side.