Unexpected Goings-on in the Structure of a Neutron Star Crust

We present a brief account of two phenomena taking place in a neutron star crust: the Fermionic Casimir effect and the major density depletion of the cores of the superfluid neutron vortices. FERMIONIC CASIMIR EFFECT AND NEUTRON STAR CRUST At a depth of about 500 m or so below the surface of a neutron crust the nuclear matter (which consists mostly of neutrons plus a small percentage of protons and electrons in β -equilibrium) organize themselves in some exotic inhomogeneous solid phase [1]. As a matter of fact, neutron star crusts seem to be just about the only other places in the entire Universe, apart from planets, where one can find condensed matter, in particular a solid phase [2]. Moving from the neutron star surface inward, one finds at first a Coulomb crystal lattice of nuclei immersed in a very low density neutron gas and even lower density electron gas. With increasing depth, the density and pressure increase, the nuclei get closer to each other and start evolving into some unusual elongated nuclei, which eventually become rods. These nuclear rods evolve gradually into plates, their place being taken later by tubes and bubbles (dubbed “inside out” nuclei) just before the average density becomes almost equal to the nuclear saturation density and the entire mixture of neutrons, protons and electrons become an homogeneous phase. The properties of this part of the neutron star have been the subject of a lot of studies, see Refs. [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] and other references therein. Most of these approaches however have missed a rather subtle and apparently important physical phenomenon, the fermionic counterpart of the Casimir interaction in such a medium [10, 11, 12, 13, 14]. In order to quickly explain the main physics ideas behind this new phenomenon, let us consider an over-simplified model of the neutron star crust. One can ask the rather innocuous question: “What is the ground state energy of an infinite homogeneous Fermi sea of noninteracting neutral particles with two hard spheres of radii a, separated by a distance r?” The naive and somewhat startling answer that perhaps one can place the two hard spheres almost anywhere with respect to each other and that the energy of the system will not be affected if one were to move the hard spheres around. The “theoretical argument” which can lead to such a conclusion is based on the same type of argumentation, which was used in Refs.[1, 5, 6, 7] and allowed these authors to establish that by going deeper and deeper into the interior of the neutron star one finds a well defined sequence of “exotic” nuclear shapes. This traditional argumentation is based essentially on liquid drop model, which includes the volume, surface, Coulomb contributions to the ground state energy only. This is basically “classical thinking.” For a person using “quantum reasoning” instead, the fact that the ground state of such a system in infinitely degenerate (corresponding to an arbitrary relative arrangement of the two hard spheres) will find such an answer most likely wrong. An indeed, a careful analysis of the problem reveals the fact that indeed a system of two hard spheres, immersed in an infinite Fermi see of noninteracting particles at zero temperature has a well defined ground state. The correct answer, namely that the “interaction energy” of the two hard spheres of radius R, at distance r from each other, is somewaht even more surprising.