Entrainment coefficient and effective mass for conduction neutrons in neutron star crust: simple microscopic models

In the inner crust of a neutron star, at densities above the ``drip'' threshold, unbound ``conduction'' neutrons can move freely past through the ionic lattice formed by the nuclei. The relative current density $n^i= n \bar v^i$ of such conduction neutrons will be related to the corresponding mean particle momentum $p_i$ by a proportionality relation of the form $n^i= {\cal K}p^i$ in terms of a physically well defined mobility coefficient $\cal K$ whose value in this context has not been calculated before. Using methods from ordinary solid state and nuclear physics, a simple quantum mechanical treatment based on the independent particle approximation, is used here to formulate $\cal K$ as the phase space integral of the relevant group velocity over the neutron Fermi surface. The result can be described as an ``entrainment'' that changes the ordinary neutron mass m to a macroscopic effective mass per neutron that will be given -- subject to adoption of a convention specifying the precise number density n of the neutrons that are considered to be ``free'' -- by $m_\star=n/{\cal K}$. The numerical evaluation of the mobility coefficient is carried out for nuclear configurations of the ``lasagna'' and ``spaghetti'' type that may be relevant at the base of the crust. Extrapolation to the middle layers of the inner crust leads to the unexpected prediction that $m_\star$ will become very large compared with m.