Asymptotic form of neutron Cooper pairs in weakly bound nuclei

The asymptotic form of a neutron Cooper pair penetrating to the exterior of a nuclear surface is investigated with the Bogoliubov theory for the superfluid fermions. Based on a two-particle Schrödinger equation governing the Cooper-pair wave function and systematic studies for both weakly bound and stable nuclei, the Cooper pair is shown to be spatially correlated even in the asymptotic large distance limit, and the penetration length of the pair condensate is revealed to be universally governed by the two-neutron separation energy $S_{2n}$ and the di-neutron mass $2m$.

Figure 1

(a) Single-particle levels and (b) quasiparticle spectra of neutrons in ${}^{138}\mathrm{Zr}$. The quasiparticle spectra are presented in terms of the pair number density ${\tilde{n}}_{lj}\left(E\right)$ [40]. The Fermi energy is denoted by the dashed curve. (c) Neutron pair condensate $\tilde{\rho }\left(R\right)$ in ${}^{138}\mathrm{Zr}$. The solid curve is the total pair condensate and the dotted curves are partial contributions ${\tilde{\rho }}_{lj}\left(R\right)$ from quasiparticle states with $l=0\sim 5$ ($j=l\pm 1/2$), and $l=10,20,30,40$ ($j=l-1/2$).

Figure 2

The asymptotic exponential constant $\tilde{\kappa }$ of the neutron pair condensate $\tilde{\rho }\left(R\right)$ obtained from the HFB calculation for ${}^{44-76}\mathrm{Ca}$, ${}^{60-88}\mathrm{Ni}$, ${}^{92-138}\mathrm{Zr}$, and ${}^{120-150}\mathrm{Sn}$, plotted with filled symbols as a function of $\sqrt{\left|\lambda \right|}$, where $\lambda$ is the Fermi energy. The results for ${}^{48}\mathrm{Ca}$, ${}^{78}\mathrm{Ni}$, ${}^{132}\mathrm{Sn}$, ${}^{110}\mathrm{Zr},$ and ${}^{122}\mathrm{Zr}$ are not included due to the absence of pairing. The open symbols denote the estimate ${\tilde{\kappa }}_{\mathrm{qp}}$, which is evaluated for the lowest quasiparticle state (see text for details).

Figure 3

Same as Fig. 1 but for ${}^{92}\mathrm{Zr}$. In the inset of panel (c), the partial pair condensate of $d_{5/2}$ states (thin solid curve) is further separated into those from the discrete $2d_{5/2}$ (dashed curve) and continuum states (dotted curve).