1.Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 2.School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China 3.CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Received Date:2021-03-09 Available Online:2021-07-15 Abstract:The coexistence of neutron-neutron (n-n), proton-proton (p-p), and neutron-proton (n-p) pairings is investigated by adopting an effective density-dependent contact pairing potential. These three types of pairings can coexist only if the n-p pairing is stronger than the n-n and p-p pairings for the isospin asymmetric nuclear matter. In addition, the existence of n-n and p-p pairs might enhance n-p pairings in asymmetric nuclear matter when the n-p pairing strength is significantly stronger than the n-n and p-p ones. Conversely, the n-p pairing is reduced by the n-n and p-p pairs when the n-p pairing interaction approaches n-n and p-p pairings.
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II.FORMALISMThe Nambu-Gorkov propagator at finite temperatures, including the n-n, n-p, and p-p pairings [13], is expressed as:
where $ \omega_{\upsilon} = (2\upsilon+1)\pi k_{B}T $ with $ \upsilon\in \mathbb{Z} $ represents the Matsubara frequencies. $ \varepsilon_{n/p} = { p}^{2}/(2m)-\mu_{n/p} $ is the single particle (s.p.) energy with chemical potential $ \mu_{n/p} $. In addtion, $ \Delta_{nn} $, $ \Delta_{pp} $, and $ \Delta_{np} $ are the isospin-triplet n-n, isospin-triplet p-p, and isospin-singlet n-p pairing gaps, respectively. 2A.Gap equations
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A.Gap equations
The neutron-proton anomalous propagator, which corresponds to $ G_{13} $, reads
where $ E_{\pm} = \sqrt{\varepsilon_{+}^{2}\pm\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}} $ is the quasi-particle energy in the condensate with the definition $ \varepsilon_{\Delta}^{4} = $$ \Delta_{np}^{2}[(\varepsilon_{n}-\varepsilon_{p})^{2}+(\Delta_{nn}-\Delta_{pp})^{2}] $ and $ 2\varepsilon_{\pm}^{2} = \varepsilon_{n}^{2}+\Delta_{nn}^{2}+\Delta_{np}^{2}\pm $$ (\varepsilon_{p}^{2}+ \Delta_{pp}^{2}+\Delta_{np}^{2}) $. $ \delta\mu = (\varepsilon_{p}-\varepsilon_{n})/2 = (\mu_{n}-\mu_{p})/2 $ represents the Fermi surface mismatch. The summation over the Matsubara frequencies provides the density matrix of particles in the condensate, i.e, the n-p pairing probabilities,
Here $ f(E) = \left[1+\exp\left(\dfrac{E}{k_{B}T}\right)\right]^{-1} $ represents the well-known Fermi-Dirac distribution function under a temperature $ T $. Accordingly, the n-p gap equation is expressed as
In the absence of the n-n and p-p pairings, the quasi-particle energy $ E_{\pm} $ becomes $ E_{\pm} = \sqrt{[(\varepsilon_{n}+\varepsilon_{p})/2]^{2}+\Delta_{np}^{2}} $$ \pm\delta\mu = E_{\Delta}\pm\delta\mu $, and the gap equation is reduced to a more familiar form for the n-p pairing in asymmetric nuclear matter:
Notably, the n-n, p-p, and n-p pairing gaps couple to each other. For asymmetric nuclear matter at the fixed neutron and proton densities, these gap equations (Eqs. (4), (6), and (7)) should be solved self-consistently with the densities (Eq. (10)) at given densities and temperatures. 2B.Pairing interaction -->
B.Pairing interaction
In principle, the nucleon-nucleonpairing interaction in nuclear matter originates from the attractive component of the bare two-body potential and the three-body force, and this pairing interaction is modified by the nuclear medium, such as the polarization effect [26-32]. In this research, to obtain qualitative conclusions from the coexistence of n-n, p-p, and n-p pairs, we adopt the density-dependent contact interaction developed by Gorrido et al. [33] to model the pairing potential. For uniform nuclear matter, the potential takes the form
Here, $ v_{I} $, $ \eta_{I} $, and $ \gamma_{I} $ are adjustable parameters and $ I = 0,1 $ denote the total isospin of the pairs. For the n-n (p-p) pairing, $ \rho_{I} = \rho_{n} $ ($ \rho_{I} = \rho_{p} $) and for the n-p pairing, $ \rho_{I} = \rho_{n}+\rho_{p} $. $ \rho_{0} = 0.17\; \text{fm}^{-3} $ reprsents the saturation density. Taking suitable values of the parameters, the pairing gap $ \Delta(k_{\rm F}) $ can be reproduced as a function of the Fermi momentum $ k_{F} = (3\pi^{2}\rho_{I})^{1/3} $ in the channel $ L = 0 $, $ I = 1 $, $ S = 0 $ (n-n and p-p) and $ k_{F} = (3\pi^{2}\rho_{I}/2)^{1/3} $ in channel $ L = 0 $, $ I = 0 $, $ S = 1 $ (n-p). We would like to emphasize that there is also a kind of n-p pairing in the channel $ L = 0 $, $ I = 1 $, $ S = 0 $ for the symmetric nuclear matter. In this channel, the n-p pairing force is approximately the same as the n-n or p-p pairing force. As will be discussed in Sec. III, even a minor asymmetry will destroy the n-p pairing in this channel. Therefore, the $ I = 1 $ pairings only represent neutron-neutron and proton-proton pairings hereafter. In addition to the polarization effect, the self-energy effect of the medium quenches the pairing gaps [14, 17]. Because the self-energy effect for nuclear pairing remains an open question in asymmetric nuclear matter, we adopt the calculated pairing gaps [14, 18] under the Hartree-Fock approaches to calibrate the parameters presented in Fig. 1. It should be noted that the self-energy [17] and polarization [32] effects should be included to obtain a more reliable pairing interaction. As is well known, to avoid the ultraviolet divergence, an energy cut is required for the contact interaction. Here, we fix the energy at approximately $ 80 $ MeV for both cases. The left and right panels correspond to the $ I = 1 $ and $ I = 0 $ pairings, respectively. Figure1. (color?online) The density-dependent contact pairing interaction with parameters calibrated to the calculated pairing gaps. The dots represent the pairing gaps in Refs. [14, 18], whereas the lines correspond to the calculation from the effective pairing interaction. The left and right panels are related to the isospin triplet and isospin singlet channels, respectively.
2C.Thermodynamics -->
C.Thermodynamics
Now, we are in a position to determine the key thermodynamic quantities. Because the occupation of the quasi-particle states is given by the Fermi-Dirac distribution function, the entropy of the system is obtained from
$ S = -2k_{B}\sum\limits_{ {p}}\sum\limits_{i}\big[f(E_{i}){\rm ln} f(E_{i})+\overline{f}(E_{i}){\rm ln} \overline{f}(E_{i})\big], $
(13)
where $ \overline{f}(E_{i}) = 1-f(E_{i}) $ and $ i = \pm $. The internal energy of the superfluid state is expressed as
$ \begin{aligned}[b] U =& 2\sum\limits_{ {p}}\big[\varepsilon_{n}n^{n}+\varepsilon_{p}n^{p}\big]\\&+\sum\limits_{ {p}}\big[g_{nn}\nu_{nn}^{2}+g_{pp}\nu_{pp}^{2}+2g_{np}\nu_{np}^{2}\big], \end{aligned}$
(14)
The factor $ 2 $ corresponds to the spin summation. The first term of Eq. (14) includes the kinetic energy of the quasi-particle as a function of the pairing gap and chemical potential. The BCS mean-field interaction among the particles in the condensate is embodied in the second term of Eq. (14). It should be noted that for asymmetric nuclear matter, the n-n and p-p pairing interactions can be different, i.e., $ g_{nn}\neq g_{pp} $, owing to $ \rho_{n}\neq\rho_{p} $. Accordingly, the thermodynamic potential can be given as
$ \begin{array}{l} \Omega = U-TS. \end{array} $
(15)
Once the contact pairing interaction is adopted, the pairing gap is momentum independent. Therefore, the thermodynamic potential can be obtained in a simple form:
Here, we consider the property $f(\omega){\rm ln} f(\omega) + \overline{f}(\omega){\rm ln}\overline{f}(\omega) = $$ -\dfrac{\omega}{k_{B}T} -{\rm ln}(1+{\rm e}^{-\omega/(k_{B}T)})$. The gap Eqs. (4), (6), and (7) and the densities of Eq. (10) can be equivalently expressed as
It should be noted that the solution of these equations corresponds to the global minimum of the free energy $ F = \Omega+\mu_{n}\rho_{n}+\mu_{p}\rho_{p} $, which is the essential quantity that describes the thermodynamics of asymmetric nuclear matter.