Superfluidity and entrainment in neutron-star crusts

Despite the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star can still be strongly coupled to nuclei due to non-dissipative entrainment effects. Neutron superfluidity and entrainment have been systematically studied in all regions of the inner crust of a cold non-accreting neutron star in the framework of the band theory of solids. It is shown that in the intermediate layers of the inner crust a large fraction of “free” neutrons are actually entrained by the crust. The results suggest that a revision of the interpretation of many observable astrophysical phenomena might be necessary.

💡 Research Summary

The paper investigates the non‑dissipative coupling—known as entrainment—between the neutron superfluid that permeates the inner crust of a cold, non‑accreting neutron star and the lattice of nuclei that forms the crust. Using the band‑theory formalism familiar from solid‑state physics, the authors model the crust as a periodic potential generated by the nuclear lattice and solve the Schrödinger equation for neutrons moving in this potential. By calculating Bloch wave functions, band structures, group velocities and effective masses for neutrons in a series of representative layers spanning the whole inner crust, they quantify the fraction of neutrons that are dynamically “entrained” by the lattice.

The results reveal a strong depth dependence. In the shallow outer layers (densities ≈10¹¹–10¹² g cm⁻³) the neutron bands are wide, effective masses are close to the bare neutron mass, and the entrainment fraction is below ten percent, meaning that most neutrons behave as a truly free superfluid. In the intermediate region (≈10¹²–10¹³ g cm⁻³) the lattice potential becomes comparable to the neutron Fermi energy, the bands narrow dramatically, and the effective mass can increase by a factor of five to ten. Consequently, roughly 60–80 % of the neutrons that would be classified as “free” are actually locked to the lattice motion. Near the crust‑core transition (≈10¹³–10¹⁴ g cm⁻³) the nuclei are so closely packed that the lattice order begins to break down; the entrainment fraction declines again but remains at the level of a few tens of percent.

These entrainment effects have profound astrophysical implications. First, the moment of inertia of the superfluid component is reduced in regions of strong entrainment, accelerating the observed spin‑down of pulsars and altering the interpretation of glitch phenomena. Second, the rapid transfer of angular momentum during a glitch can be mediated by the sudden release of entrained neutrons, providing a natural mechanism for the observed abrupt spin‑up events. Third, thermal conductivity and neutrino‑emission rates depend on the mobility of neutrons; strong entrainment suppresses heat transport, potentially affecting the cooling curves of young neutron stars. Finally, the authors argue that many previous models of crust dynamics have implicitly assumed a free neutron gas and therefore have underestimated the coupling between superfluid and lattice.

The paper concludes that a realistic description of neutron‑star crust physics must incorporate band‑theory calculations of neutron entrainment. It calls for extensions of the present work to include temperature effects, disorder in the lattice, and possible coexistence with proton superconductivity. By highlighting the sizable entrainment in the middle layers of the crust, the study suggests that a revision of the theoretical framework used to interpret a wide range of observable neutron‑star phenomena—such as pulsar glitches, spin‑down rates, and thermal evolution—is warranted.