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8.1 Superconductivity in neutron star crusts

In the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity [36Jump To The Next Citation Point] 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 44View Image. 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].
View Image

Figure 44: Schematic picture illustrating the difference between the BCS regime (left) of overlapping loosely bound fermion pairs and the BEC regime (right) of strongly bound pairs.

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” pressures5 between 1.5 × 1011 and 3 × 1011 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 [36Jump To The Next Citation Point] (see also the discussion by Ginzburg [160Jump To The Next Citation Point]).

Tce ∼ Tpiexp (− 2∕veff gFe), (122 )
where Tpi is the ion plasma frequency, veff is the effective attractive electron-electron interaction and g Fe 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
--k2Fe-- gFe = π2ℏvFe , (123 )
so that the critical temperature takes the form
2 Tce ∼ Tpiexp (− ζℏvFe ∕e ), (124 )
where ζ is a numerical positive coefficient of order unity [160]. At densities much below ∼ 106 g cm −3, the electrons are nonrelativistic and their Fermi velocity is given by v = ℏk ∕m Fe Fe e. As a result, the critical temperature decreases exponentially with the average mass density ρ as
Tce ∼ Tpiexp (− ζ′(ρ∕ρord)1∕3) , (125 )
where ζ′ ≡ ζ (9 πZ∕4A )1∕3 and ρord = mu ∕(4πa30∕3) is a typical density of ordinary matter (a0 is the Bohr radius and mu the atomic mass unit). Considering, for instance, a plasma of iron and electrons, and adopting the value ζ = 8 ∕π calculated for the “jellium” model by Kirzhnits [240], the critical temperature is approximately given by
( ρ )1∕2 ( ( ρ )1 ∕3) Tce ∼ 3.6 × 103 ------−-3 exp − 2.7 -------−3 K. (126 ) 1 g cm 1 g cm
This rough estimate shows that the critical temperature decreases very rapidly with increasing density, dropping from ∼ 30 K at ρ = 10 g cm −3 to 10–1 K at ρ = 102 g cm− 3 and to 10–7 K at 3 −3 ρ = 10 g cm! At densities above 6 −3 ∼ 10 g cm, electrons become relativistic, and vFe ∼ c. According to Equation (124View Equation), the critical temperature of relativistic electrons is given by
Tce ∼ Tpi exp(− ζ∕α) , (127 )
where 2 α = e ∕ℏc ≃ 1 ∕137 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 Tcp ∼ 109 –1010 K, 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.

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