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

Brandon Carter; Nicolas Chamel; Pawel Haensel

doi:10.1016/j.nuclphysa.2005.05.151

paper

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

arXiv:astro-ph/0406228 · doi:10.1016/j.nuclphysa.2005.05.151

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 $m_{⋆}$ such that in terms of the relevant number density n of unconfined neutrons we shall have $K^{ij} = n / m_{⋆} γ^{ij}$ . 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 $v_{k}^{i} = \partial E_{k} / \partial ℏ k_{i}$ as a function of wavenumber $k_{i}$ , it was shown that the mobility tensor would be given by $K^{ij} \propto \int d^{3} k v_{k}^{i} v_{k}^{j} δ {E_{k} - μ}$ , 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} $w i l l j u s t b e t or e pl a ce t h eD i r a c d i s t r ib u t i o na l in t e g r an d b y a s m oo t h er d i s t r ib u t i o n :$ K^{ij}\propto \int d^3 k v_k^i v_k^j Δ_F^2 /\EE^3_k $. I t i s a l sos h o w nh o w t h e p ai r in g co n d e n s a t i o n g i v esr i se t os u p er f l u i d i t y in t h e t ec hni c a l se n seo f p r o v i d in g (m e t a) s t abi l i t y a g ain s t r es i s t i v e p er t u r ba t i o n s f or a c u r r e n tt ha t i s n o tt oos t r o n g (i t s m o m e n t u m$ p_i $m u s t b es ma l l e n o ug h t o g i v e$ 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.

arXiv comment30 pages, revised published version

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

Abstract

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