BCS-BEC crossover of neutron pairs in symmetric and asymmetric nuclear matter

We propose new types of density-dependent contact pairing interactions which reproduce the pairing gaps in symmetric and neutron matters obtained by a microscopic treatment based on realistic nucleon nucleon interaction. These interactions are able to simulate the pairing gaps of either the bare interaction or the interaction screened by the medium polarization effects. It is shown that the medium polarization effects cannot be cast into the usual density power law function of the contact interaction and require the introduction of another isoscalar term. The BCS-BEC crossover of neutron pairs in symmetric and asymmetric nuclear matters is studied by using these contact interactions. This work shows that the bare and screened pairing interactions lead to different features of the BCS-BEC crossover in symmetric nuclear matter. For the screened pairing interaction, a two-neutron BEC state is formed in symmetric matter at $k_{\mathit{Fn}}~0.2$ fm${}^{-1}$ (neutron density ${\rho }_n/{\rho }_0~{10}^{-3}$). In contrast, the bare interaction does not form the BEC state at any neutron density.

Figure 1

Phase shifts for $s$-wave nucleon-nucleon scattering as a function of the center of mass energy. Left panel: $\mathit{nn}$ phase shifts obtained from Argonne $v_{18}$ potential [16] (stars) and the result of the best adjustment obtained with a contact interaction for a set of cutoff energies (from 10 to 120 MeV). Right panel: $s$-wave phase shifts for various channels: $\mathit{nn},\mathit{np}$, and $\mathit{pp}$. The $\mathit{np}$ and $\mathit{pp}$ phase shifts have been provided by the Nijmegen group (http://nn-online.org).

Figure 2

Pairing gap in symmetric, asymmetric ($x_p={\rho }_p/\rho$), and neutron matter adjusted to the “bare gap” (upper panel) or to the “screened gap” (lower panel) for various cutoff energies $E_c$. The pairing gap calculated from microscopic theory in Ref. [11] is also shown as star symbols.

Figure 3

Pairing gap calculated in symmetric, asymmetric, and neutron matters with the screened-II interaction ($g=g_1+g_2$) and for several values of ${\eta }_2$ as indicated in the legend. The corresponding parameters are given in Table . See the text for details.

Figure 4

Comparison of the neutron effective mass $m_n^{*}$, the effective chemical potential ${\nu }_n={\mu }_n-U_n$, and the difference ${\nu }_n-{\epsilon }_{\mathit{Fn}}$ calculated with the bare and the screened-II contact interactions. The parameters of the pairing interactions are taken from Table with the cutoff energy $E_c=40$ MeV. The effective mass $m_n^{*}$ and the neutron potential $U_n$ are taken from the SLy4 Skyrme interaction. The two arrows indicate the lower and upper limits for the condition ${\nu }_n<0$ with the screened-II interaction in symmetric nuclear matter.

Figure 5

Neutron Cooper pair wave function $r^2|{\Psi }_{\mathrm{pair}}(r)|{}^2$ as a function of the relative distance $r$ between the pair partner at the Fermi momenta $k_{\mathit{Fn}}=1.1$, 0.8, 0.5, and 0.2 fm${}^{-1}$.

Figure 6

Occupation probability $n_n(k)$ for symmetric matter defined by Eq. (8) as a function of the ratio of the single-particle kinetic energy to the Fermi energy ${\epsilon }_n(k)/{\epsilon }_{\mathit{Fn}}$ for a set of Fermi momenta $k_{\mathit{Fn}}=1.1$, 0.8, 0.5, and 0.2 fm${}^{-1}$. We compare the results of the bare and screened-II interactions. The values of the effective neutron chemical potential ${\nu }_n/{\epsilon }_{\mathit{Fn}}$ are indicated by the filled circles.

Figure 7

Probability $P$ for the partner neutrons within two typical lengths, 3 fm and $d_n$, as a function of the neutron Fermi momentum $k_{\mathit{Fn}}$ in symmetric, asymmetric, and neutron matters. The boundary of the BCS-BEC crossover is denoted by the dashed line.

Figure 8

Top panels: Comparison between the rms radius ${\xi }_{\mathrm{rms}}$ of the neutron pair and the average interneutron distance $d_n={\rho }_n^{-1/3}$ (thin line) as a function of the neutron Fermi momentum $k_{\mathit{Fn}}$ in symmetric, asymmetric, and neutron matter. Bottom panels: The order parameter ${\xi }_{\mathrm{rms}}/d_n$ as a function of $k_{\mathit{Fn}}$. The boundaries of the BCS-BEC crossover are represented by the two dashed lines, while the unitary limit is shown by the dotted line. The two pairing interactions are used for the calculations.

Figure 9

Ratios ${\Delta }_n/{\epsilon }_{\mathit{Fn}}$ and ${\nu }_n/{\epsilon }_{\mathit{Fn}}$ plotted as a function of the neutron Fermi momentum $k_{\mathit{Fn}}$ in symmetric, asymmetric, and neutron matter. The boundaries of the BCS-BEC crossover are shown by the two dashed lines; the unitary limit is given by the dotted line. See the text for details.

Figure 10

Pairing gap of symmetric nuclear matter as a function of the neutron Fermi momentum $k_{\mathit{Fn}}$ for the three prescriptions (see symbols key), for three fixed values for the cutoff energy, $E_c=10$, 20, or 60 MeV.

Figure 11

Cooper pair wave function $r^2{\Psi }_{\mathrm{pair}}^2$ as well as its individual contributions $r^2{\Psi }_1^2$ and $r^2{\Psi }_2^2$ for different values of the ratio $k_{\infty }/k_0$ and for two values of the neutron Fermi momentum. Notice that the convergence with respect to the ratio $k_{\infty }/k_0$ is very fast.