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 helium-4, 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 helium-3. Neutron-proton 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],
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
The pairing gap obtained by solving Equations (128) and (129
) for neutron matter using a bare
nucleon-nucleon potential and assuming a free Fermi gas single particle spectrum
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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
nucleon-nucleon interaction. The pairing gap is almost independent of the nucleon-nucleon potential.
The reason is that nucleon-nucleon 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
nucleon-nucleon phase shifts [136
]. At small relative momenta
, the neutron-neutron
phase shifts
are well approximated by the expansion
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 many-body theory, is to calculate the
single particle energies in the Hartree–Fock approximation (after regularizing the hard core
of the bare nucleon-nucleon interaction). The next step is to “dress” the pairing interaction
by medium polarization effects. Calculations have been carried out with phenomenological
nucleon-nucleon 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 nucleon-nucleon potential (two-body and/or three-body forces) using
many-body techniques. Still the gap equations of form (128
) neglect important many-body
aspects.
In many-body 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 many-body 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
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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]
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 1010 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 many-body 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 pairing-gap calculations, which include medium
effects using different many-body 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
1012 – 1014 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
109 K [294
].
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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 In the denser layers of the crust, the coherence length is smaller than the mean inter-neutron
spacing, suggesting that the neutron superfluid is a Bose–Einstein condensate of strongly-bound
neutron pairs, while in the shallower layers of the inner crust the neutron superfluid is in a BCS
regime of overlapping loosely-bound pairs. Quite remarkably, for screened pairing gaps like
those of Schwenk et al.[366], the coherence length is larger than the mean inter-neutron spacing
in the entire inner crust, so that in this case, at any depth, neutron superfluid is in the BCS
regime.
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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
nucleon-nucleon potentials [37], effective nucleon-nucleon interactions [38, 295, 361
, 360, 237
] and with
semi-microscopic 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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