Influence of pairing correlations on the radius of neutron-rich nuclei

The influence of pairing correlations on the neutron root mean square (rms) radius of nuclei is investigated in the framework of self-consistent Skyrme Hartree-Fock-Bogoliubov calculations. The continuum is treated appropriately by the Green's function techniques. As an example the nucleus ${}^{124}\mathrm{Zr}$ is treated for a varying strength of pairing correlations. We find that, as the pairing strength increases, the neutron rms radius first shrinks, reaches a minimum, and beyond this point it expands again. The shrinkage is due to the the so-called pairing antihalo effect, i.e., due to the decrease of the asymptotic density distribution with increasing pairing. However, in some cases, increasing pairing correlations can also lead to an expansion of the nucleus due to a growing occupation of so-called halo orbits, i.e., weakly bound states and resonances in the continuum with low-$\ell$ values. In this case, the neutron radii are extended just by the influence of pairing correlations, since these halo orbits cannot be occupied without pairing. The term “antihalo effect” is not justified in such cases. For a full understanding of this complicated interplay, self-consistent calculations are necessary.

Figure 1

Total energy $E_{\text{tot}}$ and neutron root mean square (rms) radius $r_{\mathrm{rms}}$ obtained by the continuum (filled circle) and the discretized (open circle) Hartree-Fock-Bogoliubov (HFB) calculations for ${}^{124}\mathrm{Zr}$ with the SkI4 interaction as a function of the strength $\eta V_0$ of the DDDI pairing force, where $V_0=-458.4$ MeV ${\mathrm{fm}}^{-3}$.

Figure 2

Total neutron densities $4\pi r^2\rho \left(r\right)$ of ${}^{124}\mathrm{Zr}$ obtained by the continuum HFB calculation with the SkI4 interaction and the DDDI pairing force strengths $\eta V_0$ for $\eta =0.6$ (dashed line), 1.0 (solid line), 1.4 (dotted line). The inset shows the asymptotic behavior of the density in a logarithmic scale.

Figure 3

(a) Single-neutron energies of ${}^{124}\mathrm{Zr}$ around the Fermi energy for $3p$, $2f$, and $1h$ orbits, which are the eigenvalues of the single-particle Hamiltonian $h$ after the final convergence of the continuum Skyrme HFB calculation, and the dashed line denotes the Fermi energy $\lambda$; (b) neutron occupation probabilities $v_{nlj}^2$, and (c) the contributions to the rms radius $r_{nlj}^{\mathrm{rms}}$ from the quasiparticle states corresponding to the orbits shown in panel (a) within the energy $E=0–6$ MeV in the continuum HFB calculation obtained with different pairing strength $\eta V_0$.

Figure 4

Neutron density $4\pi r^2{\rho }_{nlj}\left(r\right)$ of $3p$, $2f$, and $1h$ orbits in ${}^{124}\mathrm{Zr}$ contributed from the quasiparticle states within the energy $E=0–6$ MeV calculated by the continuum HFB approach with different pairing strengths $\eta V_0$. The insets give the same density distributions with logarithmic scale. In panel (f) for the $1h_{11/2}$ orbit, the neutron density is scaled by a factor 0.1 compared to the original value.

Figure 5

(a) Total energy $E_{\text{tot}}$ and the neutron rms radius $r_{\mathrm{rms}}$ obtained by the continuum (filled circle) and the discretized (open circle) HFB calculations for ${}^{122}\mathrm{Zr}$, and (b) occupation probabilities of the $3p$, $2f$, and $1h$ orbits around the Fermi energy contributed from the quasiparticle states within $E=0–7$ MeV obtained by the continuum HFB calculation with the SkI4 interaction and the DDDI pairing force with different strengths $\eta V_0$.