At the heart of BCS theory is the existence of an attractive interaction needed for pair formation. In conventional superconductors, this pairing interaction is indirect and weak. In the nuclear case the occurrence of superfluidity is a much less subtle phenomenon since the bare strong interaction between nucleons is naturally attractive at not too small distances in many channels (total angular momentum, orbital angular momentum, spin of nucleon pair). Apart from a proton superconductor similar to conventional electron superconductors, two different kinds of neutron superfluids are expected to be found in the interior of a neutron star (for a review, see, for instance, [363, 271, 116, 29, 367]). In the crust and in the outer core, the neutrons are expected to form an isotropic superfluid like helium4, while in denser regions they are expected to form a more exotic kind of (anisotropic) superfluid with each member of a pair having parallel spins, as in superfluid helium3. Neutronproton pairs could also exist in principle; however, their formation is not strongly favored in the asymmetric nuclear matter of neutron stars.
A central quantity in BCS theory is the gap function, which is related to the binding energy of a pair. Neglecting for the time being nuclear clusters in the inner crust and considering pure neutron matter, the gap equations at a given number density and at zero temperature read, in the simplest approximation [346],
together with where is the matrix element of the pairing interaction between timereversed planewave states with wave vectors and , is the chemical potential and is the single particle energy. Whenever the ratio is small, the concept of a Fermi surface remains well defined and the two equations can be decoupled by setting .Since the kernel in the gap integral peaks around the chemical potential , let us suppose that the pairing matrix elements remain constant within and zero elsewhere; is a cutoff energy. With this schematic interaction, the gap function becomes independent of . In conventional superconductors, the electron pairing is conveyed by vibrations in the ion lattice. The ion plasma frequency thus provides a natural cutoff (see Section 8.1). In the nuclear case however, there is no a priori choice of . A cutoff can still be introduced in the BCS equations, provided the pairing interaction is suitably renormalized, as shown by Anderson & Morel [11]. The BCS gap equations (128) can be solved analytically in the weak coupling approximation, assuming that the pairing interaction is small, , where is the density of single particle states at the energy . Considering that remains constant in the energy range , the gap at the Fermi momentum is given by
where the superscript C is to remind us that the strength of the pairing interaction depends on the cutoff. This expression is usually not very good for nuclear matter because the pairing interaction is strong. Nevertheless it illustrates the highly nonperturbative nature of the pairing gap. It also shows that the lower the density of states, the lower the gap. Consequently the neutron pairing gap in neutron star crust is expected to be smaller inside the nuclear clusters (discrete energy levels) than outside (continuous energy spectrum) as confirmed by detailed calculations (see Section 8.2.3).The pairing gap obtained by solving Equations (128) and (129) for neutron matter using a bare nucleonnucleon potential and assuming a free Fermi gas single particle spectrum
where is the neutron mass, is almost independent of the nucleonnucleon potential (provided it fits scattering data) and is shown in Figure 45.

As can be seen, the neutron pairs are most strongly bound at neutron densities around . At higher densities, the pairing gap decreases due to the short range repulsive part of the nucleonnucleon interaction. The pairing gap is almost independent of the nucleonnucleon potential. The reason is that nucleonnucleon potentials are constrained to reproduce the experimental phase shifts up to scattering energies of order , which corresponds to neutron densities of order . In fact, it can be shown that the pairing gap is completely determined by the experimental nucleonnucleon phase shifts [136]. At small relative momenta , the neutronneutron phase shifts are well approximated by the expansion
where fm and fm are the neutronneutron scattering length and effective range, respectively [381]. Large negative values of the scattering length are associated with attractive interactions, which nearly lead to the existence of a dineutron. In magneticallytrapped Bose atomic gases, the scattering length can be varied experimentally by tuning the magnetic field [127]. It can be shown that for Fermi wave vectors (), the pairing gap is completely determined by these two parameters only [136]. In the zero density limit (i.e., ), the gap equations are exactly solvable and the pairing gap can be expressed analytically by the exact formula [321] Let us emphasize that this formula is universal and valid for any fermion system with attractive interactions (). Equation (133) shows that the pairing gap strongly depends on the density and the nucleonnucleon interaction. It can, thus, be foreseen that modifications of the bare nucleonnucleon interaction due to medium polarization, which have been neglected, have a dramatic effect. Indeed, it can be rigorously shown [172] that in the low density limit, polarization effects reduce the gap value (133) by a factor of , independent of the strength of the interaction!The gap Equations (128) and (129) solved for the bare interaction with the free single particle energy spectrum, Equation (131), represent the simplest possible approximation to the pairing problem. A more consistent approach from the point of view of the manybody theory, is to calculate the single particle energies in the Hartree–Fock approximation (after regularizing the hard core of the bare nucleonnucleon interaction). The next step is to “dress” the pairing interaction by medium polarization effects. Calculations have been carried out with phenomenological nucleonnucleon interactions such as the Gogny force [117, 140], that are constructed so as to reproduce some properties of finite nuclei and nuclear matter. Another approach is to derive this effective interaction from a bare nucleonnucleon potential (twobody and/or threebody forces) using manybody techniques. Still the gap equations of form (128) neglect important manybody aspects.
In manybody theory, the general equations describing a superfluid Fermi system are the Nambu–Gorkov equations [3], in which the gap function depends not only on the wave vector but also on the frequency . This frequency dependence arises from dynamic effects. In this framework, it can be shown that BCS theory is a mean field approximation to the manybody pairing problem. The Gorkov equations cannot be solved exactly and some approximations have to be made. Over the past years, this problem has been tackled using different microscopic treatments and approximation schemes. Qualitatively these calculations lead to the conclusion that medium effects reduce the maximum neutron pairing gap compared to the BCS value (note that this includes the possibility that medium effects actually increase the pairing gap for some range of densities). However, these calculations predict very different density dependence of the pairing gap as illustrated in Figure 46.
Before concluding this section, we provide an analytic formula for a few representative neutron pairing gaps, using the following expression proposed by Kaminker et al. [229].
where and is the Heaviside step function if and otherwise. The parameters , , , and are given in Table 5. At low density, the pairing gap varies roughly as expected, Equation (133), remembering that .

The BCS gap Equations (128) at zero temperature can be generalized to finite temperature (adopting the standard notation where is the Boltzmann constant)
Superfluidity disappears whenever the temperature exceeds some critical threshold. Let us remark that isotropic neutron superfluidity can also be destroyed by a sufficiently strong magnetic field, since it would force each spin of a neutron pair to be aligned (as pointed out by Kirszshnits [239]). It can be shown on general grounds that the isotropic pairing gap at zero temperature (at Fermi momentum ) and the critical temperature are approximately related by [36]
where is the Euler constant. This wellknown result of conventional electron superconductivity applies rather well to nucleon superfluidity, especially for densities at which takes its maximum value [271].The temperature dependence of the pairing gap, for , can be approximately written as [428]
Zero temperature pairing gaps on the order of 1 MeV are therefore associated with critical temperatures of the order 10^{10} K, considerably larger than typical temperatures inside neutron stars except for the very early stage of their formation. The existence of a neutron superfluid in the inner crust of a neutron star is therefore well established theoretically. Nevertheless the density dependence of the critical temperature predicted by different microscopic calculations differ considerably due to different approximations of the manybody problem. An interesting issue concerns the cooling of neutron stars and the crystallization of the crust: do the neutrons condense into a superfluid phase before the formation of the crust or after?
Figure 47 shows the melting temperature of the inner crust of neutron stars compared to the critical temperature for the onset of neutron superfluidity. The structure of the crust is that calculated by Negele & Vautherin [303]. The melting temperature has been calculated from Equation (15) with . The temperature has been obtained from Equation (136), considering a uniform neutron superfluid, with the density of unbound neutrons given by Negele & Vautherin [303]. Several critical temperatures are shown for comparison. As discussed in Section 8.2.1, the BCS value represents the simplest approximation to the true critical temperature. The other two critical temperatures have been obtained from more realistic pairinggap calculations, which include medium effects using different manybody approximations. The calculation of Cao et al. [69] is based on diagrammatic calculations, while that of Schwenk et al. [366] relies on the renormalization group.
For the BCS and Brueckner calculations of the pairing gap, in the density range of 10^{12} – 10^{14} g cm^{–3}, the neutrons may become superfluid before the matter crystallizes into a solid crust. As discussed in Section 8.3.2, as a result of the rotation of the star, the neutron superfluid would be threaded by an array of quantized vortices. These vortices might affect the crystallization of the crust by favoring nuclear clusters along the vortex lines, as suggested by Mochizuki et al. [293]. On the contrary, the calculations of Schwenk et al. [366] indicate that, at any density, the solid crust would form before the neutrons become superfluid. Recently, it has also been shown, by taking into account the effects of the inhomogeneities on the neutron superfluid, that in the shallow layers of the inner crust, the neutrons might remain in the normal phase even long after the formation of the crust, when the temperature has dropped below 10^{9} K [294].

In this section, we will discuss the effects of the nuclear clusters on the pairing properties of the neutron superfluid in neutron star crusts. The relative importance of these effects is determined by the coherence length, defined as the root mean square radius of the pair wave function. Broadly speaking, the coherence length represents the size of a neutron pair. This is an important length scale, which determines many properties of the superfluid. For instance, the coherence length is of the order of the size of the superfluid vortex cores. According to Anderson’s theorem [114], the effects of the inhomogeneities (here  nuclear clusters) on the neutron superfluid are negligible whenever the coherence length is much larger than the characteristic size of the inhomogeneities. Assuming weak coupling, the coherence length can be roughly estimated from Pippard’s expression
where is the neutron pairing gap at the Fermi momentum (see Section 8.2.1). This expression gives only a lower bound for the coherence length, since medium effects tend to reduce the pairing correlations as discussed in Section 8.2.1. Nevertheless, this estimate is rather close to the value obtained in more detailed calculations [111, 283]. As can be seen in Figure 48, the coherence length is smaller than the lattice spacing^{6} except for the densest layers of the crust. Consequently the effects of the solid crust on neutron superfluidity cannot be neglected. This situation is in sharp contrast to that encountered in ordinary type I superconductors, where the electron Cooper pairs are spatially extended over mesoscopic distances of 10^{3} – 10^{4} Å and as a result the pairing gap is nearly insensitive to the details of the atomic crystal structure, since the typical lattice spacing is of order a few Å. In Figure 48 we also displayed the mean interneutron spacing defined byIn the denser layers of the crust, the coherence length is smaller than the mean interneutron spacing, suggesting that the neutron superfluid is a Bose–Einstein condensate of stronglybound neutron pairs, while in the shallower layers of the inner crust the neutron superfluid is in a BCS regime of overlapping looselybound pairs. Quite remarkably, for screened pairing gaps like those of Schwenk et al.[366], the coherence length is larger than the mean interneutron spacing in the entire inner crust, so that in this case, at any depth, neutron superfluid is in the BCS regime.

Since the formulation of the BCS theory, considerable theoretical efforts have been devoted to the microscopic calculation of pairing gaps in uniform nuclear matter using the many body theory. On the other hand, until recently the superfluidity in neutron star crusts has not attracted much attention despite its importance in many observational phenomena like pulsar glitches (see Section 12). The pairing correlations in an inhomogeneous superfluid system can be described in terms of a pairing field . In early studies [59, 107, 135] the pairing field has been calculated assuming that the matter is locally homogeneous (local density approximation). Such calculations predict, in particular, that the value of the pairing field inside the nuclear clusters is almost the same for different layers of the crust. The reason lies in the nuclear saturation: the density inside heavy nuclei is essentially constant, independent of the number of bound nucleons. In some cases, the pairing field was found to vanish inside the clusters [135]. The local density approximation is valid if the coherence length is smaller than the characteristic scale of density variations. However, this condition is never satisfied in the crust. As a result, the local density approximation overestimates the spatial variation of the pairing field. Due to “proximity effects”, the free superfluid neutrons induce pairing correlations of the bound neutrons inside clusters and vice versa leading to a smooth spatial variation of the neutron pairing field [38]. As a remarkable consequence, the value of the neutron pairing field outside (resp. inside) the nuclear clusters is generally smaller (resp. larger) than that obtained in uniform neutron matter for the same density [31]. In particular, the neutrons inside the clusters are also superfluid. The neutron superfluid in the crust should, therefore, be thought of as an inhomogeneous superfluid rather than a superfluid flowing past clusters like obstacles. The effects of nuclear clusters on neutron superfluidity have been investigated in the Wigner–Seitz approximation by several groups. These calculations have been carried out at the mean field level with realistic nucleonnucleon potentials [37], effective nucleonnucleon interactions [38, 295, 361, 360, 237] and with semimicroscopic energy functionals [29, 30, 31]. Examples are shown in Figure 49. The effects of medium polarization have been considered by the Milano group [169, 413], who found that these effects lead to a reduction of the pairing gap, as in uniform neutron matter. However, this quenching is less pronounced than in uniform matter due to the presence of nuclear clusters. Apart from uncertainties in the pairing interaction, it has recently been shown [30, 31] that the pairing field is very sensitive to the choice of boundary conditions, especially in the bottom layers of the crust (as also found by the other groups). Consequently, the results obtained in the Wigner–Seitz approximation should be interpreted with caution, especially when calculating thermodynamic quantities like the neutron specific heat, which depends exponentially on the gap.

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