Effect of BCS pairing on entrainment in neutron superfluid current in neutron star crust

Abstract

The relative current density $n^i$ of conduction neutrons in a neutron star crust beyond the neutron drip threshold can be expected to be related to the corresponding particle momentum covector $p_i$ by a linear relation of the form $n^i=K^{ij}p_j$ in terms of a physically well defined mobility tensor $K^{ij}$. This result is describable as an ``entrainment'' whose effect - wherever the crust lattice is isotropic - will simply be to change the ordinary neutron mass m to an effective mass $m_\star$ such that in terms of the relevant number density n of unconfined neutrons we shall have $K^{ij}=n/m_\starγ^{ij}$. In a preceding analysis based on an independent particle treatment using Bloch boundary conditions to obtain the distribution of energy $E_k$ and associated group velocity $v_k^i=\partial E_k/\partial\hbar k_i$ as a function of wavenumber $k_i$, it was shown that the mobility tensor would be given by $K^{ij}\propto\int d^3 k v_k^i v_k^jδ\{E_k-μ\}$, where $μ$ is the Fermi energy. Using the approach due to Bogoliubov, it is shown here that the effect of BCS pairing with an energy gap Δ_F and corresponding quasiparticle energy function \EE_k=\sqrt{(E_k-μ)^2+Δ_F^2}$ will just be to replace the Dirac distributional integrand by a smoother distribution : $K^{ij}\propto \int d^3 k v_k^i v_k^j Δ_F^2 /\EE^3_k$. It is also shown how the pairing condensation gives rise to superfluidity in the technical sense of providing (meta) stability against resistive perturbations for a current that is not too strong (its momentum $p_i$ must be small enough to give $2|p_i v_k^i|<\EE_k^2/|E_k-μ|$ for all modes). It is concluded that the prediction of a very large effective mass enhancement in the middle layers of the crust will not be significantly affected by the pairing mechanism.