### 8.1 Superconductivity in neutron star crusts

In the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity [36] the coupling of electrons with
lattice vibrations leads to an effective attractive interaction between electrons despite the repulsive Coulomb
force. As a result, the electrons of opposite spins form pairs with zero total angular momentum. These
Cooper pairs behave like bosons. Unlike fermions, bosons do not obey the Pauli exclusion principle, which
forbids multiple occupancy of single particle quantum states. Consequently, below some critical
temperature, bosons condense into the lowest-energy single-particle state, giving rise to superfluidity, as in
liquid helium-4. Loosely speaking, superconductivity can thus be seen as Bose–Einstein condensation of
bound electron pairs. However the analogy with Bose–Einstein condensation should not be
taken too far. Indeed, Cooper pairs do not exist above the superconducting transition, while in
Bose–Einstein condensation bosons always exist above and below the critical temperature. Besides the
electrons in a pair do not form well separated “molecules” like atoms in liquid helium, but instead
strongly overlap. The spatial extent of a Cooper pair in a conventional superconductor is typically
several orders of magnitude larger than the mean inter-electron spacing. The Bose–Einstein
condensation (BEC) and the BCS regime are now understood as two different limits of the
same phenomenon as illustrated in Figure 44. The transition between these two limits has
recently been studied in ultra-cold atomic Fermi gases, for which the interaction can be adjusted
experimentally [103].
In the outermost envelope of neutron stars, where the density is similar to that of ordinary
solids, the critical temperature for the onset of electron superconductivity is, at most, on the
order of a few K, which is many orders of magnitude smaller than the expected and observed
surface temperature (see Section 12.2). Besides, it is well known that iron, the most probable
constituent of the outer layers of the crust (Section 3.1), is not superconducting under normal
pressure. It was discovered in 2001 [376] that iron becomes superconducting at “high”
pressures
between 1.5 10^{11} and 3 10^{11} dyn cm^{–2}, when the temperature is below about 2 K. At higher
densities, assuming that the electrons are degenerate, we can estimate the critical temperature
from the BCS weak coupling approximation [36] (see also the discussion by Ginzburg [160]).

where is the ion plasma frequency, is the effective attractive electron-electron interaction and
is the density of electron states at the Fermi level (per unit energy and per unit volume).
Neglecting band structure effects (which is a very good approximation in dense matter; see, for
instance, the discussion of Pethick & Ravenhall [326]), the density of electron states is given by
so that the critical temperature takes the form
where is a numerical positive coefficient of order unity [160]. At densities much below ,
the electrons are nonrelativistic and their Fermi velocity is given by . As a
result, the critical temperature decreases exponentially with the average mass density as
where and is a typical density of ordinary matter ( is the
Bohr radius and the atomic mass unit). Considering, for instance, a plasma of iron and electrons, and
adopting the value calculated for the “jellium” model by Kirzhnits [240], the critical
temperature is approximately given by
This rough estimate shows that the critical temperature decreases very rapidly with increasing density,
dropping from 30 K at to 10^{–1} K at and to 10^{–7} K
at ! At densities above , electrons become relativistic, and
. According to Equation (124), the critical temperature of relativistic electrons is given by
where is the fine structure constant. Due to the exponential factor, the critical
temperature is virtually zero.
We can, thus, firmly conclude that electrons in neutron star crusts (and, a fortiori, in neutron star cores)
are not superconducting. Nevertheless, superconductivity in the crust is not completely ruled out. Indeed, at
the crust-core interface some protons could be free in the “pasta” mantle (Section 3.3), and could be
superconducting due to pairing via strong nuclear interactions with a critical temperature far higher than
that of electron superconductivity. Microscopic calculations in uniform nuclear matter predict transition
temperatures on the order of , which are much larger than typical temperatures
in mature neutron stars. Some properties of superconductors are discussed in Sections 8.3.3
and 8.3.4.