Pairing versus quarteting coherence length

We systematically analyze the coherence length in even-even nuclei. The pairing coherence length in the spin-singlet channel for the effective density-dependent delta (DDD) and Gaussian interaction is estimated. We consider in our calculations bound states as well as narrow resonances. It turns out that the pairing gaps given by the DDD interaction are similar to those of the Gaussian potential if one renormalizes the radial width to the nuclear radius. The correlations induced by the pairing interaction have, in all considered cases, a long-range character inside the nucleus and a decrease towards the surface. The mean coherence length is larger than the geometrical radius for light nuclei and approaches this value for heavy nuclei. The effect of the temperature and states in the continuum is investigated. Strong shell effects are put in evidence, especially for protons. We generalize this concept to quartets by considering similar relations, but between proton and neutron pairs. The quartet coherence length has a similar shape, but with larger values on the nuclear surface. We provide evidence of the important role of proton-neutron correlations by estimating the so-called alpha coherence length, which takes into account the overlap with the proton-neutron part of the $\alpha$-particle wave function. It turns out that it does not depend on the nuclear size and has a value comparable to the free $\alpha$-particle radius. We have shown that pairing correlations are mainly concentrated inside the nucleus, while quarteting correlations are connected to the nuclear surface.

Figure 1

Paring gap defined by the first two lines of Eq. (2.6) versus $\epsilon$ in ${}^{48}\mathrm{Cr}$ for DDD potential (squares) and Gaussian potentials with $r_0=2$ fm (circles) and $r_0=R_N$ (diamonds).

Figure 2

Proton coherence length divided by geometrical radius versus c.m. radius in ${}^{48}\mathrm{Cr}$ computed with (a) normal and (b) anomalous densities for DDD potential (solid line) and Gaussian potentials with $r_0=2$ fm (long dashes) and $r_0=R_N$ (short dashes).

Figure 3

The integrand of the proton coherence length versus the relative radius radius in ${}^{48}\mathrm{Cr},$ computed with (a) normal and (b) anomalous densities for different c.m. radii. Here, we used the Gaussian interaction with $r_0=2$ fm.

Figure 4

(a) Proton coherence length versus c.m. radius for different chemical potentials $\lambda =-2.96$ MeV (solid line), 0 MeV (long dashes), and 0.96 MeV (short dashes) in ${}^{48}\mathrm{Cr}$. (b) Neutron coherence length versus c.m. radius for different chemical potentials $\lambda =-13.75$ MeV (solid line), 0 MeV (long dashes), and 0.88 MeV (short dashes) in ${}^{48}\mathrm{Cr}$.

Figure 5

Strength parameter of the Gaussian interaction, corresponding to $r_0=2$ fm, versus neutron number for (a) proton and (b) neutron systems.

Figure 6

Ratio $\langle \xi \rangle /R_N$, corresponding to a Gaussian interaction with $r_0=2$ fm, versus neutron number for (a) proton and (b) neutron systems.

Figure 7

Ratio $\langle \xi \rangle /R_N$, corresponding to the DDD interaction (3.2) with $X=\gamma =1$, versus neutron number for (a) proton and (b) neutron systems.

Figure 8

Logarithm of the ratio $\langle \xi \rangle /R_N$ versus logarithm of the mass number for (a) protons and (b) neutrons.

Figure 9

Proton coherence length versus c.m. radius in ${}^{220}\mathrm{Ra}$ computed for $T=0$ (solid line) and $T=0.5675$ MeV $\lesssim T_c$ (long dashes), for a Gaussian potential with $r_0=2$ fm.

Figure 10

(a) Quarteting coherence length in ${}^{220}\mathrm{Ra}$ versus c.m. radius. (b) Same as in panel (a) but for alpha coherence length.

Figure 11

Averaged alpha coherence length versus mass number.

Figure 12

The two terms $I^{\left(p\right)}\left(R\right)$, $p=2$ (solid line) $p=1$ (dashed line) given by Eq. (2.17), defining the pairing coherence length for (a) protons, (b) neutrons, (c) quarteting coherence length, and (d) alpha coherence length versus the c.m. radius.