Abstract
The physics of neutron star crusts is vast, involving many different research fields, from nuclear and condensed matter physics to general relativity. This review summarizes the progress, which has been achieved over the last few years, in modeling neutron star crusts, both at the microscopic and macroscopic levels. The confrontation of these theoretical models with observations is also briefly discussed.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Constructing models of neutron stars requires knowledge of the physics of matter with a density significantly exceeding the density of atomic nuclei. The simplest picture of the atomic nucleus is a drop of highly incompressible nuclear matter. Analysis of nuclear masses tells us that nuclear matter at saturation (i.e. at the minimum of the energy per nucleon) has the density ρ_{0} = 2.8 × 10^{14} g cm^{−3}, often called normal nuclear density. It corresponds to n_{0} = 0.16 nucleons per fermi cubed. The density in the cores of massive neutron stars is expected to be as large as ∼ 5 − 10ρ_{0} and in spite of decades of observations of neutron stars and intense theoretical studies, the structure of the matter in neutron star cores and in particular its equation of state remain the wellkept secret of neutron stars (for a recent review, see the book by Haensel, Potekhin and Yakovlev [184]). The physics of matter with ρ ∼ 5 − 10ρ_{0} is a huge challenge to theorists, with observations of neutron stars being crucial for selecting a correct densematter model. Up to now, progress has been slow and based overwhelmingly on scant observation [184].
The outer layer of neutron stars with density ρ < ρ_{0} — the neutron star crust — which is the subject of the present review, represents very different theoretical challenges and observational opportunities. The elementary constituents of the matter are neutrons, protons, and electrons — like in the atomic matter around us. The density is “subnuclear”, so that the methods developed and successfully applied in the last decades to terrestrial nuclear physics can be applied to neutron star crusts. Of course, the physical conditions are extreme and far from terrestrial ones. The compression of matter by gravity crushes atoms and forces, through electron captures, the neutronization of the matter. This effect of huge pressure was already predicted in the 1930s (Sterne [390], Hund [202, 203]). At densities ρ ≳ 4 × 10^{11} g cm^{−3} a fraction of the neutrons is unbound and forms a gas around the nuclei. For a density approaching 10^{14} g cm^{−3}, some 90% of nucleons are neutrons while nuclei are represented by proton clusters with a small neutron fraction. How far we are taken from terrestrial nuclei with a moderate neutron excess! Finally, somewhat above 10^{14} g cm^{−3} nuclei can no longer exist — they coalesce into a uniform plasma of nearlypure neutron matter, with a few percent admixture of protons and electrons: we reach the bottom of the neutron star crust.
The crust contains only a small percentage of a neutron star’s mass, but it is crucial for many astrophysical phenomena involving neutron stars. It contains matter at subnuclear density, and therefore there is no excuse for the theoretical physicists, at least in principle: the interactions are known, and manybody theory techniques are available. Neutron star crusts are wonderful cosmic laboratories in which the full power of theoretical physics can be demonstrated and hopefully confronted with neutron star observations.
To construct neutron star crust models we have to employ atomic and plasma physics, as well as the theory of condensed matter, the physics of matter in strong magnetic fields, the theory of nuclear structure, nuclear reactions, the nuclear manybody problem, superfluidity, physical kinetics, hydrodynamics, the physics of liquid crystals, and the theory of elasticity. Theories have to be applied under extreme physical conditions, very far from the domains where they were originally developed and tested. Therefore, caution is a must!
Most of this review is devoted to theoretical descriptions of various aspects of neutron star crusts. The plural “crusts” in the title is well justified; depending on the scenario of their formations, the crusts may be very different in their composition and structure as sketched in Figure 1. In Section 2 we briefly describe the basic plasma parameters relevant to crust physics and delineate various plasma regimes in the temperaturedensity plane. We also address the important question of the magnetic field.
The ground state of the crust (one of the possible “crusts”) is reviewed in Section 3. We also discuss there the uncertainties concerning the densest bottom layers of the crust and we mention possible deviations from the ground state. As we describe in Section 4, a crust formed via accretion is expected to be very different from that formed during the aftermath of a supernova explosion. We show how it can be a site for nuclear reactions. We study its thermal structure during accretion, and briefly review the phenomenon of Xray bursts. We quantitatively analyze the phenomenon of deep crustal heating.
To construct a neutron star model one needs the equation of state (EoS) of the crust, reviewed in Section 5. We consider separately the groundstate crust, and the accreted crust. For the sake of comparison, we also describe another EoS of matter at subnuclear density — that relevant for the collapsing type II supernova core.
Section 6 is devoted to the stellarstructure aspects of neutron star crusts. We start with the simplest case of a sphericallysymmetric static neutron star and derive approximate formulae for crust mass and moment of inertia. Then we study the deformation of the crust in a rotating star. Finally, we consider the effects of magnetic fields on the crust structure. Apart from isotropic stress resulting from pressure, a solid crust can support an elastic strain. Elastic properties are reviewed in Section 7. A separate subsection is devoted to the elastic parameters of the socalled “pasta” layers, which behave like liquid crystals. The inner crust is permeated by a neutron superfluid. Various aspects of crustal superfluidity are reviewed in Section 8. After a brief introduction to superconductivity and its relevance for neutron star crusts, we start with the static properties of neutron superfluidity, considering first a uniform neutron gas, and then discussing the effects of the presence of the nuclear crystal lattice. In the following, we consider superfluid hydrodynamics. We stress those points, which have been raised only recently. We consider also the important problem of the critical velocity above which superfluid flow breaks down. The interplay of superfluid flow and vortices is reviewed. The section ends with a discussion of entrainment effects. Transport phenomena are reviewed in Section 9. We present calculational methods and results for electrical and thermal conductivity, and shear viscosity of neutron star crusts. Differences between accreted and groundstate crusts, and the potential role of impurities are illustrated by examples. Finally, we discuss the very important effects of magnetic fields on transport parameters. In Section 10, we review macroscopic models of the crust, and we describe in particular a twofluid model, which takes into account the stratification of crust layers, as well as the presence of a neutron superfluid. We show how entrainment effects between the superfluid and the charged components can be included using the variational approach developed by Brandon Carter [70]. Section 11 is devoted to a description of the wealth of neutrino emission processes associated with crusts. We limit ourselves to the basic mechanisms, which, according to existing calculations, are the most important ones at subsequent stages of neutronstar cooling.
The confrontation of theory with observations is presented in Section 12. Neutron stars are born very hot, and we briefly describe in Section 12.1 the present status of the theory of hot dense matter at subnuclear densities; this layer of the protoneutron star will eventually become the neutron star crust. The crust is crucial for neutron star cooling, as observed by a distant observer. Namely, the crust separates the neutron star core from its surface, where the observed Xray radiation is produced. The relation between crust physics and observations of cooling neutron stars is studied in Section 12.2. In Section 12.3 we briefly consider possible rprocesses associated with the ejection and subsequent decompression of the neutron star crusts. Pulsar glitches are thought to originate in neutron star crust and glitch models are confronted with observations of glitches in pulsar timing data in Section 12.4. The asteroseismology of neutron stars from their gravitational wave radiation is discussed in Section 12.5. Due to its elasticity, the solid crust can support mountains and shear (torsional) oscillations, both associated with gravitational wave emission. The crustcore interface can be crucial for the damping of rmodes, which, if unstable, could be a promising source of gravitational waves from rotating neutron stars. Observations of oscillations in the giant flares from Soft Gamma Repeaters are confronted with models of torsional oscillations of crusts in Section 12.6. As discussed in Section 12.7, the modeling of phenomena associated with lowmass Xray binaries (LMXB) requires a rather detailed knowledge of the physics of neutron star crusts. New phenomena discovered in the last decade (and some very recently) necessitate realistic physics of accreted neutron star crusts, including deep crustal heating and the correct degree of purity. All aspects of accreted crusts, relevant for soft Xray transients, Xray superbursts, and persistent Xray transients, are discussed in this section.
2 Plasma Parameters
2.1 No magnetic field
In this section we introduce several parameters that will be used throughout this review. We follow the notations of the book by Haensel, Potekhin and Yakovlev [184].
We consider a onecomponent plasma model of neutron star crusts, assuming a single species of nuclei at a given density ρ. We restrict ourselves to matter composed of atomic nuclei immersed in a nearly ideal and uniform, strongly degenerate electron gas of number density n_{e}. This model is valid at ρ > 10^{5} g cm^{−3}. A neutron gas is also present at densities greater than neutron drip density ρ_{ND} ≈ 4 × 10^{11} g cm^{−3}.
For an ideal, fully degenerate, relativistic electron gas, the Fermi energy (using the letter F to denote quantities evaluated at the Fermi level and the letter e for electrons) is given by
where x_{r} is a dimensionless relativity parameter defined in terms of the Fermi momentum
by
and m*_{e} is the electron effective mass at the Fermi surface. The Fermi velocity is
Electrons are strongly degenerate for T ≪ T_{Fe}, where the electron Fermi temperature is defined by
k_{B} is the Boltzmann constant and
Let the mass number of nuclei be A and their proton number be Z. The electric charge neutrality of matter implies that the number density of ions (nuclei) is
for ρ < ρ_{ND}, the quantities n_{N} and A are related to the mass density of the crust by
where m_{u} is the atomic mass unit, m_{u} = 1.6605 × 10^{−24} g. For ρ > ρ_{ND}, one has to replace A in Equation (8) by A′ = A + A″, where A is the number of nucleons bound in the nucleus and A″ is the number of free (unbound) neutrons per ion. A′ is, thus, the number of nucleons per ion. The electron relativity parameter can be expressed as
where ρ_{6}≡ ρ/10^{6} g cm−^{3}.
The ion plasma temperature T_{pi} is defined, in terms of the the ion plasma frequency
by
Quantum effects for ions become very important for T ≪ T_{pi}. The electron plasma frequency is
so that the electron plasma temperature
which can be rewritten as
The crystallization of a Coulomb plasma of ions occurs at the temperature
where the ion sphere (also called the unit cell or WignerSeitz cell) radius is
For classical ions (T ≳ T_{pi}) one has Γ_{m} ≈ 175. For T ≲ T_{pi} the zeropoint quantum vibrations of ions become important and lead to crystal melting at lower Γ_{m}.
2.2 Effects of magnetic fields
Typical pulsars have surface magnetic fields B ∼ 10^{12} G. Magnetars have much higher magnetic fields, B ∼ 10^{14}−10^{15} G. The properties of the outer envelope of neutron stars can be drastically modified by a sufficiently strong magnetic field B. It is convenient to introduce the “atomic” magnetic field B_{0}
It is the value of B for which the electron cyclotron energy is equal to e^{2}/a_{0} = 2 × 13.6 eV (a_{0} is the Bohr radius). Putting it differently, at B = B_{0} the characteristic magnetic length a_{m} = (ħc/eB)^{1/2} equals the Bohr radius. For typical pulsars and magnetars the surface magnetic field is significantly stronger than B_{0}. As a result, the atomic structure at low pressure is expected to be strongly modified. The motion of electrons perpendicular to B is quantized into Landau orbitals. Assuming that B = [0, 0, B], the electron energy levels are given by \({\epsilon_n}({p_z}) = c{(m_e^2{c^2} + 2\hbar {\omega _{{\rm{c}}e}}{m_e}n\; + p_z^2)^{1/2}}\), where n is the Landau quantum number and p_{z} the zcomponent of the electron momentum. The ground state Landau level n = 0 is nondegenerate with respect to the spin (the spin is antiparallel to B, with spin quantum number s = −1), while the higher levels n > 0 are doubly degenerate (s = ±1). The cyclotron frequency for electrons is ω_{ce} = eB/m_{e}c; it is 1836 times larger than for protons. The Coulomb binding of electrons by the atomic nucleus is significantly less effective along B, while in the plane perpendicular to B the electron motion is confined to the n = 0 Landau level. Therefore atoms get a cylindrical shape and can form linear chains along B. The attraction between these chains can lead to a phase transition into a “magnetically condensed” phase (for a recent review on this topic, see Medin & Lai [287]). The density of the condensed phase at zero pressure P = 0 (i.e. at the stellar surface) and zero temperature T = 0 is
where B_{12} = B/10^{12} G. For each element, there is a critical temperature at given B, below which a magnetically condensed phase exists at P = 0. The values of T_{crit} at several B for ^{56}Fe, ^{12}C, and ^{4}He are given in Table 1.
We now briefly consider the effects of the magnetic field on plasma properties at finite pressure P > 0. The magnetic field strongly quantizes the motion of electrons, if it confines most of them to the ground Landau state n = 0. Parameters relevant to a strong quantization regime are
and
The field B is strongly quantizing if ρ < ρ_{B} and T ≪ T_{ce}. On the contrary, a magnetic field is called weakly quantizing if many Landau levels are occupied, but still T ≪ T_{B}. Finally, B is nonquantizing if T ≫ T_{B}. The temperature T_{B} and density ρ_{B} are shown in Figure 2.
The magnetic field can strongly modify transport properties (Section 9.5) and neutrino emission (Section 11.7). Its effect on the equation of state is significant only if it is strongly quantizing (see Section 6.4).
3 The Ground State Structure of Neutron Star Crusts
According to the cold catalyzed matter hypothesis, the matter inside cold nonaccreting neutron stars is assumed to be in complete thermodynamic equilibrium with respect to all interactions at zero temperature and is therefore supposed to be in its ground state with the lowest possible energy. The validity of this assumption is discussed in Section 3.4.
The ground state structure of a neutron star crust is sketched in Figure 4. The outer crust (Section 3.1) consists of a bodycentered cubic lattice of iron ^{56}Fe. At ρ ∼ 10^{4} g cm^{−3} the atoms are fully ionized owing to the high density. At densities above 10^{7} g cm^{−3}, the composition of the nuclei becomes more neutron rich as a result of electron captures. The inner crust (Section 3.2), which extends from ρ_{ND} ≃ 4 × 10^{11} g cm^{−3} to ∼ ρ_{0}/3 ≃ 10^{14} g cm^{−3}, is characterized by the presence of free neutrons, which may condense into a superfluid phase in some layers (see Section 8). At the bottom of the crust, some calculations predict various “pasta” phases of nonspherical nuclei, such as slabs or cylinders as discussed in Section 3.3.
The ground state of a neutron star crust is obtained by minimizing the total energy density ε_{tot} for a given baryon density n_{b} under the assumption of βequilibrium and electric charge neutrality. For simplicity, the crust is assumed to be formed of a perfect crystal with a single nuclear species at lattice sites (see Jog & Smith [221] and references therein for the possibility of heteronuclear compounds).
3.1 Structure of the outer crust
Matter at densities below neutron drip ρ_{ND} is not only relevant for neutron star crusts but also for white dwarfs. Following the classical paper of Baym, Pethick and Sutherland [42], the total energy density in a given layer can be written as
where n_{N} is the number density of nuclei, E{A, Z} is the energy of a nucleus with Z protons and A − Z neutrons, ε_{e} is the electron kinetic energy density and εL is the lattice energy density, which accounts for the electronelectron, electronion and ionion Coulomb interactions.
In dense, cold, neutron star crust, electronchargescreening effects are negligible and the electron density is essentially uniform [418]. The reason is that the electron ThomasFermi screening length is larger than the lattice spacing [326]. Charge screening effects are much more important in neutron star matter with large proton fractions, such as encountered in supernovae and newly born hot neutron stars [282]. At densities ρ ≫ 10AZ g cm^{−3} (∼ 10^{4} g cm^{−3} for iron), the electrons can be treated as a quasiideal Fermi gas so that
with x_{r} defined by Equation (3).
The lattice energy density can be estimated from the WignerSeitz approximation illustrated on Figure 5. The crust is decomposed into a set of independent spheres centered around each nucleus, with a radius R_{cell} defined by Equation (16). Each sphere is electrically neutral and therefore contains Z protons and Z electrons. The lattice energy density is then given by the density of nuclei times the Coulomb energy of one such sphere (excluding the Coulomb energy of the nucleus, which is already taken into account in E{A, Z}). Assuming pointlike nuclei since the lattice spacing is much larger than the size of the nuclei^{Footnote 1}, the lattice energy can be expressed as (see, for instance, Shapiro & Teukolsky [373], p3031)
An exact calculation of the lattice energy for cubic lattices yields similar expressions except for the factor 9/10, which is replaced by 0.89593 and 0.89588 for bodycentered and facecentered cubiclattices, respectively (we exclude simple cubic lattices since they are generally unstable; note that polonium is the only known element on Earth with such a crystal structure under normal conditions [258]). This shows that the equilibrium structure of the crust is expected to be a bodycentered cubic lattice, since this gives the smallest lattice energy. Other lattice types, such as hexagonal closed packed for instance, might be realized in neutron star crusts. Nevertheless, the study of Kohanoff and Hansen [242] suggests that such noncubic lattices may occur only at small densities, meaning that r_{e} ∼ a_{0}, while in the crust r_{e} ≪ a_{0}, where r_{e} ≡ (3/4πn_{e})^{1/3} and a_{0} = ħ^{2}/m_{e}e^{2} is the Bohr radius. Equation (23) shows that the lattice energy is negative and therefore reduces the total Coulomb energy. The lattice contribution to the total energy density is small but large enough to affect the equilibrium structure of the crust by favoring large nuclei. Corrections due to electronexchange interactions, electron polarization and quantum zero point motion of the ions are discussed in the book by Haensel, Potekhin and Yakovlev [184].
The main physical input is the energy E{A, Z}, which has been experimentally measured for more than 2000 known nuclei [25]. Nevertheless, this quantity has not been measured yet for the very neutron rich nuclei that could be present in the dense layers of the crust and has therefore to be calculated. The most accurate theoretical microscopic nuclear mass tables, using selfconsistent mean field methods, have been calculated by the Brussels group and are available on line [211].
According to the first law of thermodynamics, the total pressure P is given by
Using Equation (21), the total pressure can be expressed as
The electron pressure P_{e} is defined by
where the electron chemical potential μ_{e} is simply given by the electron Fermi energy ϵ_{Fe}, Equation (1), and p_{Fe}, the electron Fermi momentum given by Equation (2). The electrons make the dominant contribution to the total pressure in the outer crust.
The structure of the ground state crust is determined by minimizing the total energy density ε_{tot} for a given baryon density n_{b} = An_{N} imposing charge neutrality, n_{p} = n_{e}. However n_{b} (or the average mass density ρ) can suffer jumps at some values of the pressure. The pressure, on the contrary, should be continuous and monotonically increasing with increasing depth below the stellar surface. Therefore we will look for a ground state at T = 0 and at a fixed P. This corresponds to minimization of the Gibbs free energy per nucleon, g(P) = (ε_{tot} + P)/n_{b}, under the condition of electric charge neutrality. For a completelyionized onecomponent plasma, one constructs a table g(P; A, Z) and then finds an absolute minimum in the (A, Z) plane. The procedure, based on the classical paper of Baym, Pethick and Sutherland [42], is described in detail in the book by Haensel, Potekhin and Yakovlev [184]. Every time that the ground state shifts to a new nucleus with a smaller proton fraction, (A, Z) → (A′, Z′), there is a few percent jump of density at the (A, Z) and (A′, Z′) shell interface,
which stems from a strict continuity of the pressure. It should be stressed that these density discontinuities are the direct consequence of the onecomponent plasma approximation. Jog & Smith [221] have shown that the transition between two adjacent layers with a single nuclear species is much smoother due to the existence of mixed lattices. In particular, they have found that between a layer with a pure bodycentered cubic (bcc) lattice of (Z, A) nuclei (lower density) and a layer with a pure bcc lattice of (Z′, A′) nuclei, a bcc lattice with (Z, A) nuclei at the corners of the conventional cube and (Z′, A′) at the center is energetically favored.
The structure of the crust is completely determined by the experimental nuclear data up to a density of the order ρ ∼ 6 × 10^{10} g cm^{−3}. At higher densities the nuclei are so neutron rich that they have not yet been experimentally studied, and the energy E{A, Z} must be extrapolated. Consequently the composition of the nuclei in these dense layers is model dependent. Nevertheless most models show the predominance of nuclei with the magic neutron numbers N = 50, 82, thus revealing the crucial role played by the quantum shell effects. The structure of the outer crust is shown in Table 2 for one particularly representative recent model. Uptodate theoretical mass tables are available online [211].
3.2 Structure of the inner crust
With increasing density, the groundstate value of Z/A decreases and neutrons become less and less bound. Let us define the “net neutron chemical potential” of a neutron in a nucleus
As long as \(\mu _n^{\prime} < 0\), all neutrons are bound within nuclei. The neutron drip point corresponds to μ′_{n} = 0; beyond this point neutrons “drip out of nuclei”, i.e. they begin to fill states in the continuous part of the energy spectrum. We can roughly localize the neutron drip point using the approximate mass formula for E′{A, Z} = E{A, Z} − Amc^{2}, where, for simplicity, we neglect the neutronproton mass difference, putting m_{n} ≈ m_{p} ≈ m = 939 MeV/c^{2}. Neglecting surface and Coulomb terms, we have
where δ ≡ (N − Z)/A, and E_{vol} and E_{sym} are nuclear volume and symmetry energies, respectively. Experimentally, E_{vol} ≃ −16 MeV and E_{sym} ≃ 32 MeV. Using Equation (29), we can easily show that the value of δ at which μ′_{n} = 0 is
Using experimental values of E_{vol} and E_{sym}, we find δ_{ND} = 0.225.
Neglecting neutronproton mass difference, the beta equilibrium condition
reads
At the density under consideration, electrons are ultrarelativistic, so that μ_{e} ≃ 5.16 (ρ9Z/A)^{1/3} MeV, where ρ_{9} = ρ/10^{9} g cm^{−3}. It is now easy to show that the neutron drip density is roughly ρ_{ND} ≈ 2 × 10^{11} g cm^{−3}, which is about half of the value obtained in complete calculations [42, 183, 357]. For ρ > ρ_{ND} ≈ 4 × 10^{11} g cm^{−3} a fraction of neutrons thus forms a gas outside the nuclei.
The inner crust of a neutron star is a unique system, which is not accessible in the laboratory due to the presence of this neutron gas. In the following we shall thus refer to the “nuclei” in the inner crust as “clusters” in order to emphasize these peculiarities. The description of the crust beyond neutron drip therefore relies on theoretical models only. Manybody calculations starting from the realistic nucleonnucleon interaction are out of reach at present due to the presence of spatial inhomogeneities of nuclear matter. Even in the simpler case of homogeneous nuclear matter, these calculations are complicated by the fact that nucleons are strongly interacting via twobody, as well as threebody, forces, which contain about twenty different operators. As a result, the inner crust of a neutron star has been studied with phenomenological models. Most of the calculations carried out in the inner crust rely on purely classical (compressible liquid drop) and semiclassical models (ThomasFermi approximation and its extensions). The stateoftheart calculations performed so far are based on selfconsistent mean field methods, which have been very successful in predicting the properties of heavy laboratory nuclei.
3.2.1 Liquid drop models
We will present in detail the liquid drop model because this approach provides very useful insight despite its simplicity. As in Section 3.1, we first start by writing the total energy density including the contribution ε_{n} of the neutron gas
The nuclear clusters are treated as liquid drops of nuclear matter whose energy can be decomposed into volume, surface and Coulomb terms
In the simplest version, the drop is supposed to be incompressible with a density on the order of ρ_{0} corresponding to the density inside heavy nuclei. This implies that the volume and surface terms in Equation (34) are proportional to A and A^{2/3}, respectively. Each contribution to the energy, Equation (34), can then be parameterized in terms of the numbers A and Z. The parameters are adjusted to the known experimental masses of nuclei, with Z/A ∼ 0.5. The first models of neutron star crust were based on such semiempirical mass formulae, see for instance the book by Haensel, Potekhin and Yakovlev [184]. However, such formulae can not be reliably extrapolated to the neutron rich nuclear clusters in neutron star crusts, where Z/A varies from ∼ 0.3 at the neutron drip threshold to ∼ 0.1 at the bottom of the crust. Besides, the presence of the neutron liquid has a profound effect on the clusters. First, it reduces the surface energy of the clusters as compared to that of isolated nuclei. Second, it exerts a pressure on the clusters. A major breakthrough was reached by Baym, Bethe and Pethick [39], who applied a compressible liquiddrop model, which included the results of microscopic manybody calculations, to describe consistently both the nucleons in the clusters and the “free” neutrons. The energy Equation (34) of the cluster then depends not only on A and Z but on a few additional parameters, such as, for instance, the size of the cluster and the density of the neutrons and protons inside it.
The volume contribution in Equation (34) is given by
where ε{n_{n}, n_{p}} is the energy density of homogeneous nuclear matter and n_{ni} and n_{pi} are respectively the neutron and proton densities inside the clusters. \({\mathcal{V}_{\rm{N}}}\) is the volume of the cluster. For consistency the energy density of the surrounding neutron gas is expressed in terms of the same function ε{n_{n}, n_{p}} as
where n_{no} is the number density of free neutrons outside the clusters and
is the volume fraction of the cluster. Let us define the surface thermodynamic potential per unit area σ and the chemical potential μ_{ns} of neutrons adsorbed on the surface of the drop (forming a neutron skin) by
where \(\mathcal{A}\) is the surface area of the cluster and N_{s} is the number of adsorbed neutrons. Since energy is an extensive thermodynamic variable, it follows from Euler’s theorem about homogeneous functions that
The Coulomb energy of a uniformlycharged spherical drop of radius r_{p} is given by
Corrections due to the diffuseness of the cluster surface and due to quantum exchange can be found, for instance, in reference [277]. The electron energy density is approximately given by Equation (22). In the WignerSeitz approximation, assuming uniformlycharged spherical clusters of radius r_{p}, the lattice energy is given by [39]
The physical input of the liquid drop model is the energy density of homogeneous nuclear matter ε{n_{n}, n_{p}}, the surface potential σ and the chemical potential μ_{ns}. For consistency, these ingredients should be calculated from the same microscopic nuclear model. The surface properties are usually determined by considering two semiinfinite phases in equilibrium (nucleons in clusters and free neutrons outside) separated by a plane interface (for curvature corrections, see, for instance, [273, 126]). In this approximation, the surface energy is proportional to the surface area
where σ_{s} is the surface tension and s_{n} the neutron skin thickness. Unlike the surface area \(\mathcal{A}\), the proportionality coefficient does not depend on the actual shape of the nuclear clusters.
The structure of the inner crust is determined by minimizing the total energy density ε_{tot} for a given baryon density n_{b} imposing electric charge neutrality n_{p} = n_{e}. The conditions of equilibrium are obtained by taking the partial derivative of the energy density ε_{tot} with respect to the free parameters of the model. In the following we shall neglect curvature corrections to the surface energy. In this approximation, the surface potential σ_{s} is independent of the shape and size of the drop. The variation of the surface tension with the neutron excess is illustrated in Figure 6.
The conditions of chemical equilibrium are
where
and X = ni, no, ns, …
The mechanical equilibrium of the cluster is expressed by
which can be easily recognized as a generalization of Laplace’s law for an isolated drop. We have introduced bulk nuclear pressures by
Note that \(P_{\rm{i}}^{{\rm{bulk}}}\) means the total pressure inside the drop, \(P_{\rm{i}}^{{\rm{bulk}}} \equiv P_{n{\rm{i}}}^{{\rm{bulk}}} + P_{p{\rm{i}}}^{{\rm{bulk}}}\). The last term in Equation (47) comes from the pressure due to Coulomb forces between protons inside the cluster and between clusters of the lattice.
The mechanical equilibrium of the crystal lattice can be expressed as
where E_{Coul} is the total Coulomb energy
Relation (49) is referred to as a “virial” theorem [39].
Combining Equations (40) and (41), the total Coulomb energy can be written as
where the dimensionless function f_{3}{w} is given by Equation (46). Let us emphasize that the above equilibrium conditions are only valid if nuclear surface curvature corrections are neglected.
Equation (49) shows that the equilibrium composition of the cluster is a result of the competition between Coulomb effects, which favor small clusters, and surface effects, which favor large clusters. This also shows that the lattice energy is very important for determining the equilibrium shape of the cluster, especially at the bottom of the crust, where the size of the cluster is of the same order as the lattice spacing. Even at the neutron drip, the lattice energy reduces the total Coulomb energy by about 15%.
The structure of the inner crust, as calculated from a compressible liquid drop model by Douchin & Haensel [125], is illustrated in Figures 7 and 8. One remarkable feature, which is confirmed by more realistic models, is that the number Z ∼ 40 of protons in the clusters is almost constant throughout the inner crust. It can also be seen that, as the density increases, the clusters get closer and closer, while their size r_{p} varies very little. Let us also notice that at the bottom of the crust the number N_{s} of neutrons, adsorbed on the surface of the clusters, decreases with increasing density, because the properties of the matter inside and outside the clusters become more and more alike. The results of different liquid drop models are compared in Figure 9.
The liquid drop model is very instructive for understanding the contribution of different physical effects to the structure of the crust. However this model is purely classical and consequently neglects quantum effects. Besides, the assumption of clusters with a sharp cut surface is questionable, especially in the high density layers where the nuclei are very neutron rich.
3.2.2 Semiclassical models
Semiclassical models have been widely applied to study the structure of neutron star crusts. These models assume that the number of particles is so large that the quantum numbers describing the system vary continuously and instead of wave functions one can use the number densities of the various constituent particles. In this approach, the total energy density is written as a functional of the number densities of the different particle species
where ε_{N}, ε_{e} and ε_{Coul} are the nuclear, electron and Coulomb contributions, respectively.
The idea for obtaining the energy functional is to assume that the matter is locally homogeneous: this is known as the ThomasFermi or local density approximation. This approximation is valid when the characteristic length scales of the density variations are much larger than the corresponding interparticle spacings. The ThomasFermi approximation can be improved by including density gradients in the energy functional.
As discussed in Section 3.1, the electron density is almost constant so that the local density approximation is very good with the electron energy functional given by Equation (22).
The Coulomb part in Equation (52) can be decomposed into a classical and a quantum contribution. The classical contribution is given by
where e is the proton electric charge and ϕ is the electrostatic potential, which obeys Poisson’s equation,
The quantum contribution \(\varepsilon _{{\rm{Coul}}}^{{\rm{corr}}}\) accounts for the quantum correlations. For instance, the local part of the Coulomb exchange correlations induced by the Pauli exclusion principle is given by the SlaterKohnSham functional [243]
The nuclear functional ε_{N}{n_{n}(r), n_{p}(r)} is less certain. Its local part is just a function of n_{n} and n_{p}, and can, in principle, be inferred using the results of manybody calculations of the ground state of uniform asymmetric nuclear matter. However, the manybody calculations for the nonlocal part of the nuclear functional are much more difficult and have never been done in a fully satisfactory way. A simpler procedure is to postulate a purely phenomenological expression of the nonlocal part of the nuclear functional. The free parameters are then determined to reproduce some nuclear properties, for instance, the experimental atomic masses. Alternatively, the nuclear functional can be obtained from effective theories. In this case, the bare nucleonnucleon interaction is replaced by an effective phenomenological interaction. It is then possible to deduce the nuclear functional in a systematic way using the extended ThomasFermi approximation (see for instance [58]). This approach has been developed for neutron star crust matter by Onsi and collaborators [312, 311].
The total energy density is equal to the energy density of one unit cell of the lattice times the number of cells. The minimization of the total energy density under the constraints of a fixed total baryon density n_{b} and global electroneutrality
leads to EulerLagrange equations for the nucleon densities. In practice the unit cell is usually approximated by a sphere of the same volume \({\mathcal{V}_{{\rm{cell}}}}\). The boundary conditions are that the gradients of the densities and of the Coulomb potential vanish at the origin r = 0 and on the surface of the sphere r = R_{cell}. Instead of solving the EulerLagrange equations, the nucleon densities are usually parameterized by some simple analytic functions with correct boundary behavior. Free parameters are then determined by minimizing the energy as in the compressible liquid drop models discussed in Section 3.2.1. The proton number of the nuclear clusters in the inner crust vs. mass density is shown in Figure 10 for different models. In those semiclassical models (as well as in liquid drop models discussed in Section 3.2.1), the number of bound nucleons inside the clusters in neutron star crusts varies continuously with depth. However, the nuclear clusters are expected to exhibit specific magic numbers of nucleons as similarly observed in isolated terrestrial nuclei, due to the clustering of quantum singleparticle energy levels. The scattering of the unbound neutrons on the nuclear inhomogeneities leads also to “shell” (Casimir or band) effects [62, 278, 90]. The energy corrections due to shell effects have been studied perturbatively in semiclassical models [316, 130] and in HartreeFock calculations [278]. They have been found to be small. However, since the energy differences between different nuclear configurations are small, especially at high densities, these shell effects are important for determining the equilibrium structure of the crust. Calculations of the ground state structure of the crust, including proton shell effects, have recently been carried out by Onsi et al. [311]. As can be seen in Figure 11, these shell effects significantly change the composition of the clusters predicting proton magic numbers Z = 20, 40, 50.
3.2.3 Quantum calculations
Quantum calculations of the structure of the inner crust were pioneered by Negele & Vautherin [303]. These types of calculations have been improved only recently by Baldo and collaborators [29, 30]. Following the WignerSeitz approximation [422], the inner crust is decomposed into independent spheres, each of them centered at a nuclear cluster, whose radius is defined by Equation (16), as illustrated in Figure 5. The determination of the equilibrium structure of the crust thus reduces to calculating the composition of one of the spheres. Each sphere can be seen as an exotic “nucleus”. The methods developed in nuclear physics for treating isolated nuclei can then be directly applied.
Starting from manybody calculations of uniform nuclear matter with realistic nucleonnucleon interaction, and expanding the nucleon density matrix in relative and center of mass coordinates, Negele & Vautherin [302] derived a set of nonlinear equations for the single particle wave functions of the nucleons, \(\varphi _\alpha ^{(q)}({{r}})\), where q = n, p for neutron and proton, respectively, and α is the set of quantum numbers characterizing each single particle state. Inside the WignerSeitz sphere, these equations take the form
where \(\epsilon_\alpha ^{(q)}\) is the single particle energy, ℓ ≡ −ir × ∇ is the dimensionless orbitalangularmomentum operator and σ is a vector composed of Pauli spin matrices. The effective masses \(m_q^ \oplus (r)\), the mean fields U_{q}(r) and the spinorbit potentials W_{q}(r) depend on wave functions of all nucleons inside the sphere through the particle number densities
the kinetic energy densities (in units of ħ^{2}/2m, where m is the nucleon mass)
and the spinorbit densities
Equations (57) reduce to ordinary differential equations by expanding a wave function on the basis of the total angular momentum. Apart from the nuclear central and spinorbit potentials, the protons also feel a Coulomb potential. In the HartreeFock approximation, the Coulomb potential is the sum of a direct part eϕ(r), where ϕ(r) is the electrostatic potential, which obeys Poisson’s Equation (54), and an exchange part, which is nonlocal in general. Negele & Vautherin adopted the Slater approximation for the Coulomb exchange, which leads to a local proton Coulomb potential. As a remark, the expression of the Coulomb exchange potential used nowadays was actually suggested by Kohn & Sham [243]. It is smaller by a factor 3/2 compared to that initially proposed by Slater [380] before the formulation of the density functional theory. It is obtained by taking the derivative of Equation (55) with respect to the proton density n_{p}(r). Since the clusters in the crust are expected to have a very diffuse surface and a thick neutron skin (see Section 3.2.2), the spinorbit coupling term for the neutrons (which is proportional to the gradient of the neutron density) was neglected.
Equations (57) have to be solved selfconsistently. For a given number N of neutrons and Z of protons and some initial guess of the effective masses and potentials, the equations are solved for the wave functions of N neutrons and Z protons, which correspond to the lowest energies \(\epsilon_\alpha ^{(q)}\) (These wave functions are then used to recalculate the effective masses and potentials. The process is iterated until the convergence is achieved.
Negele & Vautherin [303] determined the structure of the inner crust by minimizing the total energy per nucleon in a WignerSeitz sphere, and thus treating the electrons as a relativistic Fermi gas. Since the sphere is electrically neutral, the number of electrons is equal to Z and the electron energy is easily evaluated from Equation (22) with \({n_e} = Z/{\mathcal{V}_{{\rm{cell}}}}\), where \({\mathcal{V}_{{\rm{cell}}}}\) is the volume of the sphere. As for the choice of boundary conditions, Negele & Vautherin imposed that wave functions with even parity (even ℓ) and the radial derivatives of wave functions with odd parity (odd ℓ) vanish on the sphere r = R_{cell}. This prescription leads to a roughly constant neutron density outside the nuclear clusters. However, the densities had still to be averaged in the vicinity of the cell edge in order to remove unphysical fluctuations. The structure and the composition of the inner crust is shown in Table 3. These results are qualitatively similar to those obtained with liquid drop models (see Figure 9 in Section 3.2.1) and semiclassical models (see Figure 10 in Section 3.2.2). The remarkable distinctive feature is the existence of strong proton quantumshell effects with a predominance of nuclear clusters with Z = 40 and Z = 50. The same magic numbers have been recently found by Onsi et al. [311] using a highspeed approximation to the HartreeFock method with an effective Skyrme force that was adjusted on essentially all nuclear data. Note however that the predicted sequence of magic numbers differs from that obtained by Negele and Vautherin as can be seen in Figure 11. Neutron quantum effects are also important (while not obvious from the table) as can be inferred from the spatial density fluctuations inside the clusters in Figure 12. This figure also shows that these quantum effects disappear at high densities, where the matter becomes nearly homogeneous. The quantum shell structure of nuclear clusters in neutron star crusts is very different from that of ordinary nuclei owing to a large number of neutrons (for a recent review on the shell structure of very neutronrich nuclei, see, for instance, [119]). For instance, clusters with Z = 40 are strongly favored in neutron star crusts, while Z = 40 is not a magic number in ordinary nuclei (however, it corresponds to a filled proton subshell).
Negele & Vautherin assume that nucleons can be described as independent particles in a mean field induced by all other particles. However, neutrons and protons are expected to form bound pairs due to the longrange attractive part of the nucleonnucleon interaction, giving rise to the property of superfluidity (Section 8). Baldo and collaborators [28, 32, 29, 30, 31] have recently studied the effects of these pairing correlations on the structure of neutron star crusts, applying the generalized energydensityfunctional theory in the WignerSeitz approximation. They found that the composition of the clusters differs significantly from that obtained by Negele & Vautherin [303], as can be seen from Table 4 and Figure 11. However, Baldo et al. stressed that these results depend on the moreorless arbitrary choice of boundary conditions imposed on the WignerSeitz sphere, especially in the deepest layers of the inner crust. Therefore, above 2 × 10^{13} g cm^{−3} results of Baldo et al. (and those of Negele & Vautherin) should be taken with a grain of salt. Another limitation of the WignerSeitz approximation is that it does not allow the calculation of transport properties, since neutrons are artificially confined inside the sphere. A more realistic treatment has been recently proposed by applying the band theory of solids (see [92] and references therein).
3.2.4 Going further: nuclear band theory
The unbound neutrons in the inner crust of a neutron star are closely analogous to the “free” electrons in an ordinary (i.e. under terrestrial conditions) metal^{Footnote 2}. Assuming that the ground state of cold dense matter below saturation density possesses the symmetry of a perfect crystal, which is usually taken for granted, it is therefore natural to apply the band theory of solids to neutron star crusts (see Carter, Chamel & Haensel [78] for the application to the pasta phases and Chamel [90, 91] for the application to the general case of 3D crystal structures).
The band theory is explained in standard solidstate physics textbooks, for instance in the book by Kittel [241]. Single particle wave functions of nucleon species q = n, p in the crust are characterized by a wave vector k and obey the FloquetBloch theorem
where T is any lattice translation vector (which transforms the lattice into itself). This theorem implies that the wave functions are modulated plane waves, called simply Bloch waves
with \(u_{{k}}^{(q)}({{r}})\) having the full periodicity of the lattice, \(u_{{k}}^{(q)}({{r}} + {{T}}) = u_{{k}}^{(q)}({{r}})\).
In the approach of Negele & Vautherin [302] (see Section 3.2.3), or in the more popular mean field method with effective Skyrme nucleonnucleon interactions [47, 391], single particle states are solutions of the equations
neglecting pairing correlations (the application of band theory including pairing correlations has been discussed in [77]). Despite their apparent simplicity, these equations are highly nonlinear, since the various quantities depend on the wave functions (see Section 3.2.3).
As a result of the lattice symmetry, the crystal can be partitioned into identical primitive cells, each containing exactly one lattice site. The specification of the primitive cell is not unique. A particularly useful choice is the WignerSeitz or Voronoi cell, defined by the set of points that are closer to a given lattice site than to any other. This cell is very convenient since it reflects the local symmetry of the crystal. The WignerSeitz cell of a crystal lattice is a complicated polyhedron in general. For instance, the WignerSeitz cell of a bodycentered cubic lattice (which is the expected ground state structure of neutron star crusts), shown in Figure 13, is a truncated octahedron.
Equations (63) need to be solved inside only one such cell. Indeed once the wave function in one cell is known, the wave function in any other cell can be deduced from the FloquetBloch theorem (61). This theorem also determines the boundary conditions to be imposed at the cell boundary.
For each wave vector k, there exists only a discrete set of single particle energies \(\epsilon_\alpha ^{(q)}({{k}})\), labeled by the principal quantum number α, for which the boundary conditions (61) are fulfilled. The energy spectrum is thus formed of “bands”, each of them being a continuous (but in general not analytic) function of the wave vector k (bands are labelled by increasing values of energy, so that \(\epsilon_\alpha ^{(q)}({{k}}) \le \epsilon_\beta ^{(q)}({{k}})\). The band index α is associated with the rotational symmetry of the nuclear clusters around each lattice site, while the wave vector k accounts for the translational symmetry. Both local and global symmetries are therefore properly taken into account. Let us remark that the band theory includes uniform matter as a limiting case of an “empty” crystal.
In principle, Equations (63) have to be solved for all wave vectors k. Nevertheless, it can be shown by symmetry that the single particle states (and, therefore, the single particle energies) are periodic in kspace
where the reciprocal lattice vectors G are defined by
N being any positive or negative integer. The discrete set of all possible reciprocal vectors G defines a reciprocal lattice in kspace. Equation (64) entails that only the wave vectors k lying inside the first Brillouin zone (i.e. WignerSeitz cell of the reciprocal lattice) are relevant. The first Brillouin zone of a bodycentered cubic lattice is shown in Figure 14.
An example of neutron band structure is shown in the right panel of Figure 15 from Chamel et al. [96]. The figure also shows the energy spectrum obtained by removing the nuclear clusters (empty lattice), considering a uniform gas of unbound neutrons. For comparison, the single particle energies, given in this limiting case by an expression of the form \(\epsilon({{k}}) = {\hbar ^2}{k^2}/(2m_n^ \oplus) + {U_n}\), have been folded into the first Brillouin zone (reduced zone scheme). It can, thus, be seen that the presence of the nuclear clusters leads to distortions of the parabolic energy spectrum, especially at wave vectors k lying on Bragg planes (i.e., Brillouin zone faces, see Figure 14).
The (nonlinear) threedimensional partial differential Equations (63) are numerically very difficult to solve (see Chamel [90, 91] for a review of some numerical methods that are applicable to neutron star crusts). Since the work of Negele & Vautherin [303], the usual approach has been to apply the WignerSeitz approximation [422]. The complicated WignerSeitz cell (shown in Figure 13) is replaced by a sphere of equal volume. It is also assumed that the clusters are spherical so that Equations (63) reduce to ordinary differential Equations (57). The WignerSeitz approximation has been used to predict the structure of the crust, the pairing properties, the thermal effects, and the lowlying energyexcitation spectrum of the clusters [303, 55, 360, 237, 413, 31, 294].
However, the WignerSeitz approximation overestimates the importance of neutron shell effects, as can be clearly seen in Figure 15. The energy spectrum is discrete in the WignerSeitz approximation (due to the neglect of the kdependence of the states), while it is continuous in the full band theory. The spurious shell effects depend on a particular choice of boundary conditions, which are not unique. Indeed as pointed out by Bonche & Vautherin [54], two types of boundary conditions are physically plausible yielding a moreorless constant neutron density outside the cluster: either the wave function or its radial derivative vanishes at the cell edge, depending on its parity. Less physical boundary conditions have also been applied, like the vanishing of the wave functions. Whichever boundary conditions are adopted, they lead to unphysical spatial fluctuations of the neutron density, as discussed in detail by Chamel et al. [96]. Negele & Vautherin [303] average the neutron density in the vicinity of the cell edge in order to remove these fluctuations, but it is not clear whether this ad hoc procedure did remove all the spurious contributions to the total energy. As shown in Figure 15, shell energy gaps are on the order of ∆∊ ∼ 100 keV, at ρ ≃ 7 × 10^{11} g cm^{−3}. Since these gaps scale approximately like \(\Delta \epsilon\infty {\hbar ^2}/(2{m_n}R_{cell}^2)\) (where m_{n} is the neutron mass), they increase with density ρ and eventually become comparable to the total energy difference between neighboring configurations. As a consequence, the predicted equilibrium structure of the crust becomes very sensitive to the choice of boundary conditions in the bottom layers [30, 96]. One way of eliminating the boundary condition problem without carrying out full band structure calculations, is to perform semiclassical calculations including only proton shell effects with the Strutinsky method, as discussed by Onsi et al. [311].
In recent calculations [278, 304, 168] the WignerSeitz cell has been replaced by a cube with periodic boundary conditions instead of Bloch boundary conditions (61). Although such calculations allow for possible deformations of the nuclear clusters, the lattice periodicity is still not properly taken into account, since such boundary conditions are associated with only one kind of solutions with k = 0. Besides the WignerSeitz cell is only cubic for a simple cubic lattice and it is very unlikely that the equilibrium structure of the crust is of this type (the structure of the crust is expected to be a body centered cubic lattice as discussed in Section 3.1). Let us also remember that a simple cubic lattice is unstable. It is, therefore, not clear whether these calculations, which require much more computational time than those carried out in the spherical approximation, are more realistic. This point should be clarified in future work by a detailed comparison with full band theory. Let us also mention that recently Bürvenich et al. [64] have considered axiallydeformed spheroidal WS cells to account for deformations of the nuclear clusters.
Whereas the WignerSeitz approximation is reasonable at not too high densities for determining the equilibrium crust structure, full band theory is indispensable for studying transport properties (which involve obviously translational symmetry and, hence, the kdependence of the states). Carter, Chamel & Haensel [78] using this novel approach have shown that the unbound neutrons move in the crust as if they had an effective mass much larger than the bare mass (see Sections 8.3.6 and 8.3.7). This dynamic effective neutron mass has been calculated by Carter, Chamel & Haensel [78] in the pasta phases of rod and slablike clusters (discussed in Section 3.3) and by Chamel [90, 91] in the general case of spherical clusters. By taking consistently into account both nuclear clusters, which form a solid lattice, and the neutron liquid, band theory provides a unified scheme for studying the structure and properties of neutron star crusts.
3.3 Pastas
The equilibrium structure of nuclear clusters in neutron star crusts results from the interplay between the total Coulomb energy and the surface energy of the nuclei according to the virial Equation (49). At low densities, the lattice energy Equation (41) is a small contribution to the total Coulomb energy and nuclear clusters are therefore spherical. However, at the bottom of the crust the nuclei are very close to one another. Consequently, the lattice energy represents a large reduction of the total Coulomb energy, which vanishes in the liquid core when the nuclear clusters fill all space, as can be seen from Equation (51). In the densest layers of the crust the Coulomb energy is comparable in magnitude to the net nuclear binding energy (this situation also occurs in the dense and hot collapsing core of a supernova and in heavy ion collisions leading to multifragmentation [56, 57]). The matter thus becomes frustrated and can arrange itself into various exotic configurations as observed in complex fluids. For instance, surfactants are organic compounds composed of a hydrophobic “tail” and a hydrophilic “head”. In solutions, surfactants aggregate into ordered structures, such as spherical or tubular micelles or lamellar sheets. The transition between the different phases is governed by the ratio between the volume of the hydrophobic and hydrophilic parts as shown in Figure 16. One may expect by analogy that similar structures could occur in the inner crust of a neutron star depending on the nuclear packing.
According to the BohrWheeler fission condition [52] an isolated spherical nucleus in vacuum is stable with respect to quadrupolar deformations if
where \(E_{{\rm{N,Coul}}}^{(0)}\) and \(E_{{\rm{N,surf}}}^{(0)}\) are the Coulomb and surface energies of the nucleus, respectively. The superscript (0) reminds us that we are considering a nucleus in vacuum. The BohrWheeler condition can be reformulated in order to be applied in the inner crust, where both Coulomb and surface energies are modified compared to the “in vacuum” values. Neglecting curvature corrections and expanding all quantities to the linear order in w^{1}/^{3}, where w is the fraction of volume occupied by the clusters, it is found [326] that spherical clusters become unstable to quadrupolar deformation if w > w_{crit} = 1/8.
Reasoning by analogy with percolating networks, Ogasawara & Sato [308] suggest that as the nuclei fill more and more space, they will eventually deform, touch and merge to form new structures. A long time ago, Baym, Bethe and Pethick [39] predicted that as the volume fraction exceeds 1/2, the crust will be formed of neutron bubbles in nuclear matter. In the general framework of the compressible liquid drop model considering the simplest geometries, Hashimoto and his collaborators [191, 315] show that as the nuclear volume fraction w increases, the stable nuclear shape changes from sphere to cylinder, slab, tube and bubble, as illustrated in Figure 17. This sequence of nuclear shapes referred to as “pastas” (the cylinder and slab shaped nuclei resembling “spaghetti” and “lasagna” respectively) was found independently by Ravenhall et al. [345] with a specific liquid drop model. The volume fractions at which the various phases occur are in good agreement with those predicted by Hashimoto and collaborators on purely geometrical grounds. This criterion, however, relies on a liquid drop model, for which curvature corrections to the surface energy are neglected. This explains why some authors [274, 124] find within the liquid drop model that spherical nuclei remain stable down to the transition to uniform nuclear matter, despite volume fractions exceeding the critical threshold (see in particular Figure 7), while other groups found the predicted sequence of pasta phases [419, 420, 206, 207]. The nuclear curvature energy is, thus, important for predicting the equilibrium shape of the nuclei at a given density [327].
These pasta phases have been studied by various nuclear models, from liquid drop models to semiclassical models, quantum molecular dynamic simulations and HartreeFock calculations (for the current status of this issue, see, for instance, [417]). These models differ in numerical values of the densities at which the various phases occur, but they all predict the same sequence of configurations shown in Figure 17 (see also the discussion in Section 5.4 of [326]). Some models [274, 97, 124, 282] predict that the spherical clusters remain energetically favored throughout the whole inner crust. Generalizing the BohrWheeler condition to nonspherical nuclei, Iida et al. [206, 207] showed that the rodlike and slablike clusters are stable against fission and proton clustering, suggesting that the crust layers containing pasta phases may be larger than that predicted by the equilibrium conditions. It has also been suggested that the pinning of neutron superfluid vortices in neutron star crusts might trigger the formation of rodlike clusters [293]. Nevertheless, the nuclear pastas may be destroyed by thermal fluctuations [419, 420]. Quite remarkably, Watanabe and collaborators [417] performed quantum molecular dynamic simulations and observed the formation of rodlike and slablike nuclei by cooling down hot uniform nuclear matter without any assumption of the nuclear shape. They also found the appearance of intermediate spongelike structures, which might be identified with the ordered, bicontinuous, doublediamond geometry observed in block copolymers [284]. Those various phase transitions leading to the pasta structures in neutron star crusts are also relevant at higher densities in neutron star cores, where kaonic or quark pastas could exist [281].
The pasta phases cover a small range of densities near the crustcore interface with ρ ∼ 10^{14} g cm^{−3}. Nevertheless, by filling the densest layers of the crust, they may represent a sizable fraction of the crustal mass [274] and thus may have important astrophysical consequences. For instance, the existence of nuclear pastas in hot dense matter below saturation density affects the neutrino opacity [201, 384], which is an important ingredient for understanding the gravitational core collapse of massive stars in supernova events and the formation of neutron stars (see Section 12.1). The dynamics of neutron superfluid vortices, which is thought to underlie pulsar glitches (see Section 12.4), is likely to be affected by the pasta phase. Besides, the presence of nonspherical clusters in the bottom layers of the crust influences the subsequent cooling of the star, hence the thermal Xray emission by allowing direct Urca processes [274, 179] (see Section 11) and enhancing the heat capacity [112, 113, 135]. The elastic properties of the nuclear pastas can be calculated using the theory of liquid crystals [325, 419, 420] (see Section 7.2). The pasta phase could thus affect the elastic deformations of neutron stars, oscillations, precession and crustquakes.
3.4 Impurities and defects
There are many reasons why the real crust of neutron stars can be imperfect. In particular, apart from a dominating (A, Z) nuclide at a given density ρ, it can contain an admixture of different nuclei (“impurities”). The initial temperature at birth exceeds 10^{10} K. At such a high T, thermodynamic equilibrium is characterized by a statistical distribution of A and Z. With decreasing T, the A, Z peak becomes narrower [55, 63]. After crystallization at T_{m}, the composition is basically frozen. Therefore, the composition at T < T_{m} reflects the situation at T ∼ T_{m}, which can differ from that in the absolute ground state at T = 0. For example, between the neighboring shells, with nuclides (A_{1}, Z_{1}) and (A_{2}, Z_{2}), respectively, one might expect a transition layer composed of a binary mixture of the two nuclides [108, 109]. Another way of forming impurities is via thermal fluctuations of Z and N_{cell}, which, according to Jones [224, 225], might be quite significant at ρ ≳ 10^{12} g cm^{−3} and T ≳ 10^{9} K.
The real composition of neutron star crusts can also differ from the ground state due to the fallback of material from the envelope ejected during the supernova explosion and due to the accretion of matter. In particular, an accreted crust is a site of Xray bursts. The ashes of unstable thermonuclear burning at accretion rates \({10^{ 8}}{M_ \odot}{\rm{y}^{ 1}} \gtrsim \dot M \gtrsim {10^{ 9}}{M_ \odot}{\rm{y}^{ 1}}\) could be a mixture of A ≃ 60 − 100 nuclei and could therefore be relatively “impure” (heterogeneous and possibly amorphous) [365, 364]. If the initial ashes are a mixture of many nuclides, further compression under the weight of accreted matter can keep the heterogeneity. If the crust is weakly impure but rather amorphous, its thermal and electrical conductivities in the solid phase would be orders of magnitude lower than in the perfect crystal as discussed in Section 9. This would have dramatic consequences as far as the rate of the thermal relaxation of the crust is concerned (Section 12.7.3).
4 Accreting Neutron Star Crusts
4.1 Accreting neutron stars in lowmass Xray binaries
In this section we will concentrate on accreting neutron stars in lowmass Xray binaries (LMXB), where the mass of the companion is significantly less than M_{⊙}, and the accretion stage can last as long as ∼ 10^{9} y. A binary is sufficiently tight for the companion to fill its Roche lobe. The mass transfer proceeds through the inner Lagrangian point, and the transferred plasma flows in a deep gravitational potential well, via an accretion disk, towards the neutron star surface, as illustrated in Figure 18.
A hydrogen atom falling on a neutron star surface from infinity releases ∼ 200 MeV of gravitational binding energy. Therefore, accretion onto a neutron star releases ∼ 200 MeV per accreted nucleon. Most of this energy is radiated in Xrays, so that the total Xray luminosity of an accreting neutron star can be estimated as LX ∼ (Ṁ/10^{−10} M_{⊙}/y) 10^{36} erg s^{−1}. Space Xray observations of accreting neutron stars were at the origin of Xray astronomy [159]. Accreted matter is usually hydrogen rich. It forms the outer envelope of a neutron star, which contains a hydrogen burning shell, with an energy release of about 5 MeV/nucleon in a stable burning. Helium ashes from hydrogen burning accumulate in the helium layer, which ignites under specific densitytemperature conditions. For some range of accretion rate, helium burning is unstable, so that its ignition triggers a thermonuclear flash, burning within seconds all the envelope into nuclear ashes composed of nuclides of the iron group and beyond it; the energy release in the flash is less than 5 MeV/nucleon. These flashes are observed as Xray bursts, with luminosity rising in a second to about 10^{38} ergs^{−1} (≈ Eddington limit for neutron stars, L_{Edd}), and then typically decaying in a few tens of seconds^{Footnote 3}.
Multiplying the burst luminosity by its duration we get an estimate of the total burst energy ∼ 10^{39}−10^{40} erg. The Xray bursts are quasiperiodic, with typical recurrence time ∼ hoursdays. Since their discovery in 1975 [177], about seventy Xray bursters have been found. Many bursters are of transient character, and form a group of soft Xray transients (SXTs), with typical active periods of days — weeks, separated by periods of quiescence of several months — years long. During quiescent periods, there is very little or no accretion, while during much shorter periods of activity there is an abundant accretion, due probably to disc flow instability. Some SXTs, with active periods of years separated by decades of quiescence, are called persistent SXTs. In 2000, a special rare type of Xray superbursts was discovered. Superbursts last for a few to twelve hours, with recurrence times of several years. The total energy radiated in a superburst is ∼ 10^{42} erg. Superbursts are explained by the unstable burning of carbon in deep layers of the outer crust.
In all cases, ignition of the thermonuclear flash takes place in the neutron star crust, and is sensitive to the crust structure and to the physical conditions within it. This aspect will be discussed in Section 4.4. An accreted crust has a different structure and composition than the ground state one, as discussed in Section 4.2. It has, therefore, a different equation of state than the groundstate crust (see Section 5.2). Moreover, it is a reservoir of nuclear energy, which is released in the process of deep crustal heating, accompanying accretion, reviewed in Section 4.3. Observations of SXTs in quiescence prove the presence of deep crustal heating (Section 12.7.2). Cooling of the neutron star surface in quiescence after long periods of accretion (years — decades) in persistent SXTs also allows one to test physical properties of the accreted crust (Section 12.7.3).
4.2 Nuclear processes and formation of accreted crusts
In this section we will describe the fate of Xray burst ashes, produced at < 10^{7} g cm^{−3}, and then sinking deeper and deeper under the weight of accreted plasma above them. We start at a few tens of meters below the surface, and we will end at a depth of ∼ 1 km, where the density ≳ 10^{13} g cm^{−3}. Under conditions prevailing in accreting neutron star crusts, at ρ > 10^{8} g cm^{−3} matter is strongly degenerate, and is “relatively cold” (T < 10^{8} K, see Figures 25 and 26), so that thermonuclear processes are strongly suppressed because interacting nuclei have to overcome a large Coulomb barrier. The structure of an accreted crust is shown in Figure 19.
In what follows we will use a simple model of the accreted crust formation, based on the onecomponent plasma approximation at T = 0 [185, 187]. The (initial) Xburst ashes are approximated by a onecomponent plasma with (A_{i}, Z_{i}) nuclei.
At densities lower than the threshold for pycnonuclear fusion (which is very uncertain, see Yakovlev et al. [427]), ρ_{pyc} ∼ 10^{12}−10^{13} g cm^{−3}, the number of nuclei in an element of matter does not change during the compression resulting from the increasing weight of accreted matter. Due to nucleon pairing, stable nuclei in dense matter have even N = A − Z and Z (eveneven nuclides). In the outer crust, in which free neutrons are absent, the electron captures proceed in two steps,
Electron captures lead to a systematic decrease in Z (therefore an increase in N = A − Z) with increasing density. The first capture, Equation (67), proceeds as soon as μ_{e} > E {A, Z − 1} − E{A, Z}, in a quasiequilibrium manner, with negligible energy release. It produces an oddodd nucleus, which is strongly unstable in a dense medium, and captures a second electron in a nonequilibrium manner, Equation (68), with energy release Q_{j}, where j is the label of the nonequilibrium process.
After the neutrondrip point (ρ > ρ_{ND} ≃ 6 × 10^{11} g cm^{−3}), electron captures trigger neutron emissions,
Due to the electron captures, the value of Z decreases with increasing density. In consequence, the Coulomb barrier prohibiting the nucleusnucleus reaction lowers. This effect, combined with the decrease of the mean distance between the neighboring nuclei, and a simultaneous increase of energy of the quantum zeropoint vibrations around the nuclear lattice sites, opens up a possibility for the pycnonuclear fusion associated with quantummechanical tunneling through the Coulomb barrier due to zeropoint vibrations (see, e.g., Section 3.7 of the book by Shapiro & Teukolsky [373]). The pycnonuclear fusion timescale τ_{pyc} is a very sensitive function of Z. The chain of the reactions (69) and (70) leads to an abrupt decrease of τ_{pyc} typically by 7 to 10 orders of magnitude. In the onecomponent plasma approximation, the accreted crust is composed of spherical shells containing a single nuclide (A, Z). Pycnonuclear fusion switches on as soon as τ_{pyc} is smaller than the time of the travel of a piece of matter (due to accretion) through the considered shell of mass M_{shell}(A, Z), τ_{acc} ≡ M_{shell}/Ṁ. The masses of the shells are on the order of 10^{−5} M_{⊙}. As a result, in the inner crust the chain of reactions (69 and 70) in several cases is followed by the pycnonuclear reaction, occurring on a timescale much shorter than τ_{acc}. Introducing Z′ = Z − 2, we then have
where dots in Equation (73) denote an actual nonequilibrium process, usually following reaction (72). The total heat deposition in matter, resulting from a chain of reactions involving a pycnonuclear fusion, Equations (71), (72) and (73), is Q_{j} = Q_{j,1} + Q_{j,2} + Q_{j,3}.
The composition of accreted neutron star crusts, obtained by Haensel & Zdunik [187], is shown in Figure 21. These results describe crusts built of accreted and processed matter up to the density 5 × 10^{13} g cm^{−3} (slightly before the crustcore interface). At a constant accretion rate Ṁ = Ṁ_{−9} × 10^{−9}M_{⊙}/yr this will take ∼ 10^{6} yr/Ṁ_{−9}. During that time, a shell of Xray burst ashes will be compressed from ∼ 10^{8} g cm^{−3} to ∼ 10^{13} g cm^{−3}.
Two different compositions of Xray burst ashes at ≲ 10^{8} g cm^{−3}, A_{i}, Z_{i}, were assumed. In the first case, A_{i} = 56, Z_{i} = 26, which is a “standard composition”. In the second scenario A_{i} = 106, to imitate nuclear ashes obtained by Schatz et al. [364]. The value of Z_{i} = 46 stems then from the condition of beta equilibrium at ρ = 10^{8} g cm^{−3}. As we see in Figure 21, after the pycnonuclear fusion region is reached, both curves converge (as explained in Haensel & Zdunik [187], this results from A_{i} and Z_{i} in two scenarios).
4.3 Deep crustal heating
A neutron star crust that is not in full thermodynamic equilibrium constitutes a reservoir of energy, which can then be released during the star’s evolution. The formation and structure of nonequilibrium neutron star crusts has been considered by many authors [410, 50, 362, 186, 187, 188, 178]. Such a crust can be produced by accretion onto a neutron star in compact LMXB, where the original crust built of a catalyzed matter (see Section 3) is replaced by a crust with a composition strongly deviating from that of nuclear equilibrium. However, building up the accreted crust takes time. The outer crust (Section 3.1), containing ∼ 10^{−5} M_{⊙}, is replaced by the accreted crust in (10^{4}/Ṁ_{−9}) y. To replace the whole crust of mass ∼ 10^{−2} M_{⊙} by accreted matter requires (10^{7}/Ṁ_{−9}) y. After that time has passed, the entire “old crust” is pushed down through the crustcore interface, and is molten into the liquid core. The time (10^{7}/Ṁ_{−9}) y may seem huge. However, LMXBs can live for ∼ 10^{9} y, so that a fully accreted crust on a neutron star is a realistic possibility.
Heating due to nonequilibrium nuclear processes in the outer and inner crust of an accreting neutron star (deep crustal heating) was calculated, using different scenarios and models [186, 187, 188]. The effect of crustal heating on the thermal structure of the interior of an accreting neutron star can be seen in Figures 25 and 26. In what follows, we will describe the most recent calculations of crustal heating by Haensel & Zdunik [188]. In spite of the model’s simplicity (onecomponent plasma, T = 0 approximation), the heating in the accreted outer crust obtained by Haensel & Zdunik [188] agrees nicely with extensive calculations carried out by Gupta et al. [178]. The latter authors considered a multicomponent plasma, a reaction network of many nuclides, and included the contribution from the nuclear excited states. They found that electron captures in the outer crust proceed mostly via the excited states of the daughter nuclei, which then deexcite, the excitation energy heating the matter; this strongly reduces neutrino losses, accompanying nonequilibrium electron captures. The total deep crustal heating obtained by Haensel & Zdunik [188] is equal to Q_{tot} = 1.5 and 1.9 MeV per accreted nucleon for A_{i} = 106 and A_{i} = 56, respectively.
In Figure 22, we show the heat deposited in the matter, per accreted nucleon, in the thin shells where nonequilibrium nuclear processes occur. Actually, reactions proceed at a constant pressure, and there is a density jump within a thin “reaction shell”. The vertical lines, whose height gives the heat deposited in matter, are drawn at the density of the bottom of the reaction shell. The number of heat sources and the heating power of a single source depend on the assumed A_{i}. In the case of A_{i} = 56 the number of sources is smaller, and their heatpernucleon values Q_{j} are larger, than for A_{i} = 106.
An important quantity is the integrated heat deposited in the crust in the outer layer with bottom density ρ. It is given by
where (α) labels the crustal heating model (specific A_{i}, Z_{i}, etc.). The quantity Q^{(α)}(ρ) for two models of compressional evolution is plotted in Figure 23. The second model illustrates the effect of switching off pycnonuclear reactions. This was done by artificially blocking pycnonuclear fusion until the nuclear charge went down to Z_{min} = 4, which occurred at ρ_{pyc} = 10^{13.25} g cm^{−3}. And yet, Q^{(2)} for ρ > 10^{13} g cm^{−3} is very similar to that obtained in the first scenario, which was the most advantageous, as far as crust heating was concerned. Heating by pycnonuclear fusion at ρ > 10^{13} g cm^{−3} is insignificant. Heat from pycnonuclear fusions at ρ ∼ 10^{12} g cm^{−3} is to a large extent replaced by an additional heat release associated with electron captures and neutron emissions within the density range 10^{12} − 10^{13} g cm^{−3}. The values of Q^{(1)} and Q^{(2)} saturate above 10^{13}.^{6} g cm^{−3}, where 80% of nucleons are in neutron gas phase. All in all, for two scenarios with A_{i} = 56, the total deep crustal heat release is 1.8 − 1.9 MeV/nucleon. For A_{i} = 106, these numbers are lowered by about 0.5 MeV/nucleon.
The quite remarkable weak dependence of the total heat release in the crust, Q_{tot}, on the nuclear history of an element of matter undergoing compression from ∼ 10^{8} g cm^{−3} to ∼ 10^{13.6} g cm^{−3} deserves an explanation [188]. One has to study the most relevant thermodynamic quantity, the Gibbs free energy per nucleon (baryon chemical potential). Its minimum determines the state of thermodynamic equilibrium. Moreover, its drop at reaction surface P = P_{j} yields the total energy release Q_{j} per one nucleon [341]. In the T = 0 approximation, we have \({\mu _{\rm{b}}}(P) = [\mathcal{E}(P) + P]/{n_{\rm{b}}}(P)\) = enthalpy per nucleon. Minimizing μ_{b}(P), at a fixed P, with respect to the independent thermodynamic variables (A, Z, mean free neutron density \({\bar n_n}\), mean baryon density n_{b}, size of the WignerSeitz cell, etc.), under the constraint of electroneutrality, \({\bar n_p} = {\bar n_e}\), we get the ground state of the crust at a given P. This “cold catalyzed matter” (Section 3) corresponds to \(\mu _{\rm{b}}^{(0)}(P)\). All other \(\mu _{\rm{b}}^{(\alpha)}(P)\) curves displaying discontinuous drops due to nonequilibrium reactions included in a given evolutionary model (α) lie above the \(\mu _{\rm{b}}^{(0)}(P)\); see Figure 24.
This makes visual the fact that noncatalyzed matter is a reservoir of energy, released in nonequilibrium processes that move the matter closer to the absolute ground state. In spite of dramatic differences between different \(\mu _{\rm{b}}^{(\alpha)}(P)\) in the region where the bulk of the heating occurs, P = (10^{30}−10^{31.5}) erg cm^{−3}, the functions \(\mu _{\rm{b}}^{(\alpha)}(P)\) tend to \(\mu _{\rm{b}}^{(0)}(P)\) for P ≳ 10^{32} erg cm^{−3}. The general structure of different \(\mu _{\rm{b}}^{(\alpha)}(P)\) is similar. At the same P, their continuous segments have nearly the same slope. What differs between \(\mu _{\rm{b}}^{(\alpha)}\) (P)s are discontinuous drops, by \(Q_j^{(\alpha)}\), at reaction thresholds \(P_j^{(\alpha)}\) The functions \(\mu _{\rm{b}}^{(\alpha)}(P)\) can therefore be expressed as (see Haensel & Zdunik [188])
where \({\bar \mu _{\rm{b}}}(P)\) is a smooth function of P, independent of (α). For P > 10^{33.5} erg cm^{−3} the values of Q_{j} are negligibly small, and all \(\mu _{\rm{b}}^{(\alpha)}(P)\) come quite close to the ground state line. This implies that the sum \(\sum\nolimits_j {Q_j^{(\alpha)}}\) must be essentially independent of (α).
4.4 Thermal structure of accreted crusts and Xray bursts
A thermonuclear flash is triggered by an instability in the thermonuclear burning. The relevant quantities are the local heating rate due to thermonuclear fusion, \({\dot \varepsilon _{{\rm{nuc}}}}\), and cooling rate, \({\dot \varepsilon _{{\rm{cool}}}}\), resulting from heat diffusion, volume expansion, and neutrino emission. A steady state of an accreting neutron star corresponds to \({\dot \varepsilon _{{\rm{nuc}}}} = {\dot \varepsilon _{{\rm{cool}}}}\). It depends on the accretion rate, composition of accreted plasma, structure and physical properties (thermal conductivity, neutrino emissivity, etc. of the stellar interior. Examples of the steady thermal structure of accreting neutron stars are shown in Figures 25 and 26.
A stable steady state of an accreting neutron star satisfies
The inequality guarantees that any thermal perturbation will be damped by a selfregulated cooling. Under the increasing weight of accreted matter, an element of the crust in the He burning shell moves in the ρ − T plane, in the direction of increasing ρ. At some moment, after crossing the “ignition line” characterized by
burning becomes unstable, and a selfaccelerating thermonuclear flash is ignited because
In the standard picture, it is the instability in the helium burning via 3α →^{12} C, which triggers an Xray burst (see, e.g., [152]). The total energy release in a burst can then be easily estimated (neglecting the general relativistic correction) via
where Q_{nuc} is the mean energy per nucleon released in the thermonuclear flash (2 − 8 MeV, depending on composition of burnt material), M_{burn} = M_{env}(ρ_{ign}) is the mass of the envelope burnt in the flash (determined by the ignition density ρ_{ign}), and m_{u} is the atomic mass unit. The value of M_{env}(ρ_{ign}) can be read from Figure 39. A good model of Xray bursts should yield Eburst ∼ 10^{40} erg with a recurrence time of hours. This can be satisfied with ignition of helium flashes at ∼ 10^{7} g cm^{−3}.
5 Equation of State
In this section we discuss the Equation of State (EoS) of the neutron star crust. Three different cases will be considered: cold catalyzed matter in Section 5.1, accreted crust matter in Section 5.2 assuming the formation scenario described in Section 4, and hot dense matter in supernova cores in Section 5.4.
5.1 Ground state crust
The EoS of the outer crust in the ground state approximation (see Section 3.1) is rather well established, so that the pressure at any given density is determined within a few percent accuracy [42, 183, 357] as can be seen in Figure 27.
Uncertainties arise above the density ρ > 6 × 10^{10} g cm^{−3} as shown in Figure 28 because experimental data are lacking. However, one may hope that with the improvement of experimental techniques, experimental data on very exotic nuclei will become available in the future.
On the contrary, the inner crust nuclei cannot be studied in a laboratory because their properties are influenced by the gas of dripped neutrons, as reviewed in Section 3.2. This means that only theoretical models can be used there and consequently the EoS after neutron drip is much more uncertain than in the outer layers. The neutron gas contributes more and more to the total pressure with increasing density. Therefore, the problem of correct modeling of the EoS of a pure neutron gas at subnuclear densities becomes important. The true EoS of cold catalyzed matter stems from a true nucleon Hamiltonian, expected to describe nucleon interactions at ρ ≲ ρ_{0}, where ρ_{0} is the nuclear saturation density. To make the solution of the manybody problem feasible, the task is reduced to finding an effective nucleon Hamiltonian, which would enable one to calculate reliably both the properties of laboratory nuclei and the EoS of cold catalyzed matter for 10^{11} g cm^{−3} ≲ ρ ≲ ρ_{0}. The task also includes the calculation of the crustcore transition. We will illustrate the general results with two examples of the EoS of the inner crust, calculated in the compressibleliquiddrop model (see Section 3.2.1) using the effective nucleonnucleon interactions FPS (FriedmanPanharipandeSkyrme [320]) and SLy (SkyrmeLyon [86, 88, 87]).
As one can see in Figure 29, significant differences between the SLy and FPS EoS are restricted to the density interval 4 × 10^{11} − 4 × 10^{12} g cm^{−3}. They result mainly from the fact that the density at which neutron drip occurs for each is different: ρ_{ND}(SLy) ≃ 4 × 10^{11} g cm^{−3} (in good agreement with the “empirical EoS” of Haensel & Pichon [183]), while ρ_{ND}(FPS) ≃6 × 10^{11} g cm^{−3}. For 4 × 10^{12} g cm^{−3} ≲ ρ ≲ 10^{14} g cm^{−3} the SLy and FPS EoSs are very similar, with the FPS EoS being a little softer at the highest densities considered. The detailed behavior of the two EoSs near the crustcore transition can be seen in Figure 30. The FPS EoS is softer there than the SLy EoS (for pure neutron matter the FPS model is softer at subnuclear densities; see [184]).
In the case of the SLy EoS, the crustliquid core transition takes place as a very weak firstorder phase transition, with a relative density jump on the order of one percent. Notice that, for this model, spherical nuclei persist to the very bottom of the crust [126]. As seen from Figure 30, the crustcore transition is accompanied by a noticeable stiffening of the EoS. For the FPS EoS the situation is different. Namely, the crustcore transition takes place through a sequence of phase transitions with changes of nuclear shapes as discussed in Section 3.3. These phase transitions make the crustcore transition smoother than in the SLy case, with a gradual increase of stiffness (see Figure 35). While the presence of exotic nuclear shapes is expected to have dramatic consequences for the transport, neutrino emission, and elastic properties of neutron star matter, their effect on the EoS is rather small.
The overall SLy EoS of the crust, calculated including adjacent segments of the liquid core and the outer crust, is shown in Figure 33. In the outer crust segment, the SLy EoS cannot be visually distinguished from the EoSs of Haensel & Pichon [183] or Rüster et al. [357], which are based on experimental nuclear masses. An important dimensionless parameter, measuring the stiffness of an EoS at a given density, is the adiabatic index,
which at subnuclear density can be approximated by γ ≃ (ρ/P)dP/dρ. The total pressure P is defined by Equation (24). The adiabatic index γ is shown in Figures 31 and 32 as a function of mass density ρ. At ρ_{ND} > ρ ≳ 10^{9} g cm^{−3}, we have γ ≈ 4/3. This is because in these outer crust layers, pressure is very well approximated by the sum of the contribution of the ultrarelativistic electron gas (P_{e}) and of the lattice contribution (P_{l}), which both have the same density dependence ∝ρ^{4/3} (see Section 3.1). One notices in Figure 32 a dramatic softening in the density region following the neutron drip point, ρ ≳ ρ_{ND}. This means, in particular, that no stable stars can exist with central densities around ρ_{ND} because the compressibility of the matter is too low. Then the EoS stiffens gradually, with a significant increase of γ near the crustcore interface. A jump in γ on the core side is connected with the disappearance of nuclei, and a subsequent stiffening (due to the nucleonnucleon interaction) in the uniform npe liquid. Figures 31 and 32 show that the EoS of the inner crust is very different from the polytropic form P ∝ ρ^{γ0}, where γ_{0} is a constant. Tabulated and analytical EoSs of the ground state crust are available online [210, 118].
5.2 Accreted crust
A model of the EoS of accreted crusts was calculated by Haensel & Zdunik [185]. They used the compressible liquid drop model (see Section 3.2.1) with a “single nucleus” scenario.
In Figure 34 this EoS is compared with the SLy model of cold catalyzed matter described in Section 5.1. At ρ < ρ_{ND} both EoSs are very similar. The reason is that for ρ < ρ_{ND}, as discussed in Section 3.1, we have P ≃ P_{e}(n_{e}) with n_{e} = (Z/A)n_{b} and the ratio Z/A is quite similar for both accreted and ground state crusts. Large differences appear for ρ_{ND} ≲ ρ ≲ 10ρ_{ND}, where the EoS of accreted matter is stiffer than that of cold catalyzed matter. One also notices wellpronounced density jumps at constant pressure in the EoS of accreted matter. They are associated with discontinuous changes in nuclear composition, an artifact of the onecomponent plasma approximation. The jumps are particularly large for ρ_{ND} ≲ ρ ≲ 10ρ_{ND}.
The difference between the cold catalyzed and accreted matter EoSs decreases for large density. Both curves are very close to each other for ρ > 10^{13} g cm^{−3}. This is because for such a high density the pressure is mainly produced by the neutron gas and is not sensitive to the detailed composition of the nuclear clusters. In view of this, one can use the EoS of the catalyzed matter for calculating the hydrostatic equilibrium of the highdensity (ρ > 10^{13} g cm^{−3}) internal layer of the accreted crust.
5.3 Effect of magnetic fields on the EoS
Typical values of the surface magnetic field of radio pulsars are B ∼ 10^{12} g. For magnetars, surface magnetic fields can be as high as ∼ 10^{15} G. Effects of such fields on the EoS and structure of the crust are briefly reviewed in Section 6.4. A detailed study of the effect of B on neutron star envelopes can be found in Chapter 4 of [184].
5.4 Supernova core at subnuclear density
The outer layers of the supernova core, which after a successful explosion will become the envelope of a protoneutron star, display a similar range of densities ρ ≲ 10^{14} g cm^{−3} and are governed by the same nuclear Hamiltonian as the neutron star crust. This is why we include it in the present review.
The subnuclear density layer of supernova cores shows many similarities with that of neutron star crusts. In both cases, the matter is formed of nuclear clusters embedded in a sea of leptons and hadrons. Nevertheless, the conditions are very different since supernova cores are lepton rich (contain many electrons, positrons as well as trapped neutrinos and antineutrinos) with lepton fraction
and very hot with temperatures typically T ≲ 40 MeV/kB while the matter in neutron star crusts is cold, neutrino free, and in βequilibrium. Neutrinos remain trapped with the supernova core for several seconds, which is their diffusion timescale in dense hot matter. The presence of nuclei in dense hot matter is of utmost importance for the neutrino opacity of the supernova core. Namely, the scattering cross section of neutrinos off nuclei of mass number A is A times larger than the sum of the individual cross sections for A nucleons in a nucleon gas (see, e.g., [373]). As long as neutrinos are trapped, Y_{L} = const., and collapse is adiabatic.
Of particular importance for supernova simulations is the adiabatic index defined by
where s is the entropy per nucleon in units of k_{B} and Y_{e} is the electron fraction. For low entropies per nucleon on the order of the Boltzmann constant k_{B}, and for densities ρ < 5× 10^{13} g cm^{−3} the value of γ is mainly determined by the relativistic electrons and is thus close to 4/3 (adiabatic index of an ultrarelativistic Fermi gas); see Figure 35. The adiabatic index jumps to larger values when the nuclear clusters dissolve into a uniform mixture of nucleons and leptons, as discussed in Section 5.1. The adiabatic index is shown in Figure 35 for three different EoS that have been used in supernova simulations. The impact of the presence of nuclear pasta phases (Section 3.3) on the adiabatic index is illustrated in Figure 36.
The striking differences between the adiabatic index of supernova matter, γ_{SN}(ρ), and that for cold catalyzed matter in neutron stars, γ_{SN}(ρ), deserves additional explanation. In the core collapse, compression of the matter becomes adiabatic as soon as ρ≳ 10^{11} g cm^{−3}, so that the entropy per nucleon s = const. Simultaneously, due to neutrino trapping, the electronlepton fraction is frozen, Y_{l} = const. The condition s = const. ≈ 1k_{B} blocks evaporation of nucleons from the nuclei; the motion of nucleons have to remain ordered. Therefore, the fraction of free nucleons stays small and they do not contribute significantly to the pressure, which is supplied by the electrons, until the density reaches 10^{14} g cm^{−3}.
At ρ ≳ 10^{14} g cm^{−3}, nuclei coalesce forming uniform nuclear matter. Thus, there are two density regimes for γ_{SN}(ρ). For ρ ≳ 10^{14} g cm^{−3}, pressure is supplied by the electrons, while nucleons are confined to the nuclei, so that γ_{SN} ≃ 4/3 ≈ 1.3. Then, for ρ ≳ 10^{14} g cm^{−3} nuclei coalesce into uniform nuclear matter, and the supernova matter stiffens violently, with the adiabatic index jumping by a factor of about two, to γ_{SN} ≈ 2 −3. This stiffening is actually responsible for the bounce of infalling matter. An additional factor stabilizing nuclei at ρ ≲ 10^{14} g cm^{−3} in spite of a high T > 10^{10} K, is a large lepton fraction, Y_{l} ≈ 0.4, enforcing a relatively large proton fraction, \(Y_p^{{\rm{SN}}} \approx 0.3\), to be compared with \(Y_p^{{\rm{NS}}} \approx 0.05\) for neutron stars.
Finally, for supernova matter we notice the absence of a neutrondrip softening, so well pronounced in γ_{SN}, Figure 31. This is because neutron gas is present in supernova matter also at ρ < 10^{11} g cm^{−3}, and the increase of the free neutron fraction at higher density is prevented by strong neutron binding in the nuclei (large \(Y_p^{{\rm{SN}}}\)), and low s ≈ 1k_{B}.
6 Crust in Global Neutron Star Structure
The equilibrium structure of the crust of a neutron star results from the balance of pressure, electromagnetic and elastic stresses, and gravitational pull exerted by the whole star. The global structure of the crust can be calculated by solving Einstein’s equations
where G is the gravitational constant and c the speed of light. The Ricci scalar R_{μν} and the scalar curvature R are determined from the spacetime metric g_{μν}, which represents gravity. Pressure, electromagnetic and elastic stresses are taken into account in the energymomentum tensor T_{μν}.
As a first approximation, treating the star as an ideal fluid, the energymomentum tensor is given by (see, e.g., Landau & Lifshitz [248])
where ρ is the massenergy density and u^{μ} is the 4velocity of the fluid. Equation (84) can be written in an equivalent but more general form [70]
where n^{μ} = nu^{μ} is the total 4current, n is the total particle number density in the fluid rest frame and π_{ν} is the momentum per particle of the fluid given by
The quantity μ is a dynamic effective mass defined by the relation
where c is the speed of light. As shown by Carter & Langlois [80], Equation (85) can be easily transposed to fluid mixtures (in order to account for superfluidity inside the star) as follows
where X labels matter constituents and Ψ is a generalized pressure, which is not simply given by the sum of the partial pressures of the various constituents (see Section 10).
The electromagnetic field can be taken into account by including the following contribution to the stressenergy tensor
where F_{μν} is the electromagnetic 2form. Likewise elastic strains in the solid crust contribute through an additional term \(T_\nu ^{\mu ({\rm{elast}})}\). While the components of \(T_\nu ^{\mu ({\rm{elast}})}\) are small compared to P (remember that the shear modulus is ∼ 10^{−2} of P, Section 7), it can produce nonaxial deformations in rotating neutron stars, and nonsphericity in nonrotating stars, as discussed in Section 12.5. For the time being, we will use the idealfluid approximation, \(T_\nu ^\mu \approx T_\nu ^{\mu ({\rm{liq}})} + T_\nu ^{\mu ({\rm{em}})}\), and consider the effect of \(T_\nu ^{\mu ({\rm{elast}})}\) on neutron star structure in Section 12.5. We will, therefore, consider purelyhydrostatic equilibrium of the crust, instead of a more general hydroelastic equilibrium.
6.1 Spherical nonrotating neutron stars
A static (nonrotating) neutron star, built of a perfect fluid, has spherical symmetry. The spacetime is also spherically symmetric, with the metric (see, for instance, Landau & Lifshitz [248]; here we use the notation of Haensel, Potekhin and Yakovlev [184])
where t is a time coordinate, r is a radial coordinate called “circumferential radius” (see below), and θ and ϕ are polar and azimuthal angular coordinates. The dimensionless metric function Φ = Φ(r) and λ = λ(r) have to be determined from Einstein’s equations. For a flat (Minkowski) spacetime we would have Φ = λ = 0. One can show that λ(r) is determined by the massenergy contained within radius r, divided by c^{2}, to be denoted by m(r) [248],
Let us limit ourselves to the static case t = const. Fixing r, θ = π/2, and then integrating ds over ϕ from zero to 2π, we find that the proper length of the equator of the star, i.e., its circumference, as measured by a local observer, is equal to 2πr. This is why r is called the circumferential radius. Notice that Equation (90) implies that the infinitesimal proper radial distance (corresponding to the infinitesimal difference of radial coordinates dr) is given by dℓ = e^{λ}dr.
From Einstein’s equations, we get the (relativistic) equations of hydrostatic equilibrium for a static sphericallysymmetric star
Equation (92) is the famous TolmanOppenheimerVolkoff equation of hydrostatic equilibrium [407, 313]. Equation (93) enables one to calculate m(r) within a radius r. Finally, Equation (94) determines the metric function Φ(r). Equations (92)–(94) have to be supplemented with an equation of state (EoS) P = P(ρ).
Let us consider the differential Equations (92) and (93), which determine the global structure of a neutron star. They are integrated from the star center, r = 0, with the boundary conditions ρ(0) = ρ_{c} [P(0) = P(ρ_{c})] and m(0) = 0. It is clear from Equation (92), that pressure is strictly decreasing with increasing r. The integration is continued until P = 0, which corresponds to the surface of the star, with radial coordinate r = R, usually called the star radius.
The gravitational mass of the star is defined by M = m(R). The mass M is the source of the gravitational field outside the star (r > R), and creates an outer spacetime described by the Schwarzschild metric,
The crust corresponds to the layer r_{cc} < r < R, where r_{cc} determines the crustcore interface. The depth below the stellar surface, z, is defined as the proper radial distance between the star surface and a given surface of radius r. It is given by
The structure of the crust depends on its EoS, stellar mass, and the EoS of its liquid core. In Figures 37 and 38 we present the structure of the crust of a 1.4 M_{⊙} star, for two EoSs of the neutron star interior.
An accreted crust has a different composition and thus a different EoS (stiffer) than the groundstate crust (see Sections 4 and 5). For the comparison to be meaningful, however, these two EoSs have to be calculated from the same nuclear Hamiltonian. We satisfied this by using in both cases the same compressible liquid drop model of Mackie & Baym [277]. The plots of ΔM(z) for the ground state and accreted crust of a 1.4 M_{⊙} neutron star are shown in Figure 39.
6.2 Approximate formulae
For astrophysically relevant neutron star masses M > M_{⊙}, the gravitational mass of the crust M_{cr} = M − m(r_{cc}), where r_{cc} is the radial coordinate of the crustcore interface, constitutes less than 3% of M. Moreover, for realistic EoSs and M > M_{⊙}, the difference ΔR = R − r_{cc} does not exceed 15% of R. Clearly, M_{cr}/M ≪ 1 and an approximation in which the terms \(\mathcal{O}({M_{{\rm{cr}}}}/M)\) are neglected is usually sufficiently precise. Neglecting the terms \({\mathcal O}(\Delta R/R)\) gives a less accurate but still useful approximation. In what follows we will use the above “light and thin crust approximation” to obtain useful approximate expressions for the crustal parameters.
Let us first derive an approximate equation for hydrostatic equilibrium within the crust. Let z be the proper depth below the neutron star surface, Equation (96). Within the crust, one can approximate z by
where r_{g} = 2GM/c^{2} is the Schwarzschild radius of mass M.
In Equation (92) one can use the approximation m ≃ M, and neglect P/ρc^{2} and 4πPr^{3} /Mc^{2} as compared to one. Then the equation for hydrostatic equilibrium can be rewritten in a Newtonian form,
where g_{s} is the surface gravity,
Using P(z = 0) = 0, we obtain a formula relating the pressure at depth z to the mass ΔM(z) of the crust layer above z,
In particular, putting z = z_{cc}, and denoting the pressure at the crustcore interface by P_{cc}, we get
Consider a slow rigid rotation, meaning that the angular frequency of rotation, Ω, as measured by an observer at infinity, is small compared to the mass shedding frequency, Ω_{ms}. It can be shown that for slow rigid rotation the moment of inertia, I, then involves only the structure of the nonrotating neutron star [190] (see also [166, 184]). Using our notation,
where \(\bar \omega\) is the local rotation frequency, as measured in the local inertial frame. It can be calculated from an ordinary differential equation derived by Hartle [190], with boundary conditions,
where J is the stellar angular momentum.
The contribution of the crust comes from the spherical outer shell, r_{cc} < r < R,
where P/ρc^{2} was neglected compared to one. Approximating the integrand by its value at r = R, and using \({{\rm{e}}^{\Phi (R)}} = {{\rm{e}}^{ \lambda (R)}} = {(1  {r_{\rm{g}}}/R)^{1/2}}\) and Equations (103) and (101), we get
6.3 Crust in rotating neutron stars
We consider stationary neutronstar rotation and assume, for the time being, perfect axial symmetry. Consequently, the spacetime metric is axially symmetric too. Using coordinates t, r, θ, and ϕ, we can write the spacetime metric in the form (we use the notation of Haensel, Potekhin and Yakovlev [184])
where the metric functions Φ, λ, ω, and α depend on r and θ. The metric function ω(r, θ) has the asymptotic behavior ω(r → ∞) = 0. It is usually referred to as the “angular velocity of the local inertial frames”. Einstein’s equations for a stationary rigid rotation of perfect fluid stars can be reduced to a set of 2D coupled partial differential equations [53, 389]. Their numerical solution can now be obtained using publicly available domain codes, for example, the code rotstar from the LORENE library [270] and the code RNS [388].
As can be seen on Figure 40, the outer crust is the most strongly deformed by the centrifugal force. The spin frequency of 1200 Hz, significantly larger than the highest detected up to now, 716 Hz, was chosen to make the effect more spectacular. Let us nevertheless point out that some evidence suggesting the existence of a rapidlyrotating neutron star has recently been found in XTE J1739−285 [227] with a spin frequency of about 1122 Hz. Let us consider, for example, the difference between the equatorial (maximal) and polar (minimal) thickness of the crust, ΔR_{eq−pol} = ΔR_{eq} − ΔR_{pol}. For a Ω, which is not too close to the massshedding limit, ΔR_{eq−pol} ∝ Ω^{2}. Therefore, ΔR_{eq−pol}_{716} _{Hz} ≈ 0.4ΔR_{eq−pol}_{1200} Hz.
The crustal baryon mass (not to be confused with the gravitational mass) M_{b,cr}(Ω) of a neutron star rotating at angular frequency Ω, is larger than the crustal baryon mass M_{b,cr}(0) of the static star (with the same total baryon mass). The baryon mass (also called the rest mass) of a star is equal to the number A of baryons it contains times an assumed baryon mass m_{b}. One may take m_{b} = m_{n} or m_{b} = m_{u}. We take M_{b,cr} = Am_{u}. For Ω not too close to Ω_{ms}, we have M_{b,cr}(Ω) − M_{b,cr}(0) ∝ Ω^{2}. Due to the radiation of electromagnetic waves and particles, a pulsar spins down, so that \(\dot \Omega < 0\). Consequently, the baryon mass of the pulsar crust decreases in time, \({\dot M_{{\rm{b,cr}}}}\infty \dot \Omega \Omega < 0\). Nucleons pass from the crust to the liquid core, releasing some heat. As shown in Figure 40, the crust is decompressed near the equator and compressed near the pole. These deformations trigger various nuclear reactions involving electrons, neutrons, and nuclei. These reactions tend to drive the deformed crust towards its equilibrium shape and release heat, which influences the cooling of a spinning down pulsar [205]. Additional heating results from crust cracking when local shear strain exceeds the maximal one.
6.4 Effects of magnetic fields on the crust structure
The structure of the outer layers of the crust (neutron star “envelope”) can be affected by the presence of a magnetic field. Effects of magnetic fields on the EoS were briefly mentioned in Section 2.2. Here we consider examples showing effects of B on the crust structure.
Typical values of the surface magnetic fields of radio pulsars are B ∼ 10^{12} G. In Figure 41 we compare plots of ρ(z) and ΔM(z) for the pure iron envelope with ρ ≲ 10^{5} g cm^{−3} of a neutron star with M = 1.4 M_{⊙} and R = 10 km with and without a magnetic field. The T = 0 approximation for the crust is used. Typical values of the surface magnetic field of radio pulsars are B ∼ 10^{12} G. At such magnetic fields, the effect of B is seen only in the outer envelope, which is ∼ 30 cm thick. We can see there quantum oscillations of the density as a function of depth. They are associated with the filling of the lowest Landau levels n = 0,1,... by the electrons (Section 2.2). Increasing B to 10^{13} G, associated with the most magnetized radio pulsars, leads to much deeper magnetization of the crust, down to the depth of 30 m, where the prevailing density reaches 10^{6} g cm^{−3}. The effect of the magnetic field in the outer 30 cm of the crust is dramatic; in spite of a gravitational acceleration g = GM/R^{2} = 2.43 × 10^{14} cm s^{−2}, the density is only slowly decreasing, and is still ∼ 10^{5} g cm^{−3} at 10 cm depth, ten times higher than in a nonmagnetized envelope at the same depth. So, magnetized iron plasma is “condensed”, and much less compressible than nonmagnetized (Section 2.2). At 10^{13} G, the n = 1 Landau level begins to be populated only at 4 × 10^{5} g cm^{−3} (depth 3 m), to be compared with 2 × 10^{4} g cm^{−3} (depth 30 cm) at 10^{12} G. It should be stressed that, as the surface temperature at pulsar age 10^{3} −10^{4} y is T_{s} ∼ 10^{6} K, inclusion of T will weaken the magnetization effects on the structure of the crust [184].
Typical surface magnetic fields of magnetars are ∼ 10^{15} G. In Figure 42 we see that for ρ < 10^{9} g cm^{−3}, the effect of such a huge B on the crust EoS is strong and becomes dramatic at lower ρ. For example, at 10^{7} g cm^{−3}, the matter pressure decreases by two orders of magnitude, compared to the B = 0 case. The n = 1 Landau level begins to be populated only at 10^{8.7} g cm^{−3}, at a depth of about 50 m.
7 Elastic Properties
A solid crust can sustain an elastic strain up to a critical level, the breaking strain. Neutron stars are relativistic objects, and therefore a relativistic theory of elastic media in a curved spacetime should be used to describe elastic effects in neutron star structures and dynamics. Such a theory of elasticity has been developed by Carter & Quintana [82], who applied it to rotating neutron stars in [83, 84] (see also Beig [44] and references therein). Recently, Carter and collaborators have extended this theory to include the effects of the magnetic field [73], as well as the presence of the neutron superfluid, which permeates the inner crust [72, 85]. For the time being, for the sake of simplicity, we ignore magnetic fields and free neutrons. However, in Section 7.2 the effect of free neutrons on the elastic moduli of the pasta phases is included, within the compressible liquid drop model. Since relativistic effects are not very large in the crust, we shall restrict ourselves to the Newtonian approximation (see, e.g., [249]).
The thermodynamic equilibrium of an element of neutronstar crust corresponds to equilibrium positions of nuclei, which will be denoted by a set of vectors {r}, which are associated with the lattice sites. Neutron star evolution, driven by spindown, accretion of matter or some external forces, like tidal forces produced by a close massive body, or internal electromagnetic strains associated with strong magnetic fields, may lead to deformation of this crust element as compared to the equilibrium state.
For simplicity, we will neglect thermal contributions to thermodynamic quantities and restrict ourselves to the T = 0 approximation. Deformation of a crust element with respect to the equilibrium configuration implies a displacement of nuclei into their new positions r′ = r + u, where u = u(r) is the displacement vector. In the continuum limit, valid for macroscopic phenomena, both r and u are treated as continuous fields. Nonzero u is associated with elastic strain (i.e., forces which tend to return the matter element to the equilibrium state of minimum energy density ε_{0}), and with the deformation energy density ε_{def} = ε(u) − ε_{0}^{Footnote 4}.
A uniform translation does not contribute to ε_{def}, and the true deformation is described by the (symmetric) strain tensor
where i,j = 1, 2, 3, and x_{1} = x, x_{2} = y, x_{3} = z. This formula for u_{ik} is valid if the displacement vector u is small, so that the terms quadratic in the components of u can be neglected compared to the linear ones [249].
Any deformation can be decomposed into compression and shear parts,
Under deformation, matter element volume changes according to dV′ = [1 + (∇ · u)]dV. A pure compression (no shear) of the matter element is described by u_{ik} = αδ_{ik}. For volume preserving deformations (∇ · u) = 0.
To lowest order, the deformation energy is quadratic in the deformation tensor,
Since εdef is a scalar, λ_{iklm} is a fourth rank tensor. The total number of λ_{iklm} components is eighty one; general symmetry relations reduce the maximum number of linearlyindependent components (elastic moduli) to twenty one. The number of independent elastic moduli decreases with increasing symmetry of the elastic medium. It is three in the case of a cubic crystal, and two for an isotropic solid (see, e.g., [249]).
The elastic contribution to the stress tensor \(\Pi _{ik}^{{\rm{elast}}} \equiv {\sigma _{ik}}\).
7.1 Isotropic solid (polycrystal)
Microscopically, the ground state corresponds to a bodycentered cubic (bcc) crystal lattice. However, one usually assumes that macroscopically, the neutron star crust is an isotropic bcc polycrystal. Elastic properties of an isotropic solid are described by two elastic moduli. The deformation energy can be expressed as
Here, μ is the shear modulus and K is the compression modulus. The elastic stress tensor is, therefore,
Considering a small pure uniform compression, one finds
where P is the total pressure given by Equation (24) and γ is the adiabatic index defined by Equation (80).
Monte Carlo calculations of the effective shear modulus of a polycrystalline bcc Coulomb solid were performed by Ogata & Ichimaru [309]. The deformation energy, resulting from the application of a specific strain u_{ik}, was evaluated through Monte Carlo sampling.
As we have already mentioned, for an ideal cubic crystal lattice there are only three independent elastic moduli, denoted traditionally as c_{11}, c_{12} and c_{44} (see Chapter 3, pp. 80–87 of the book by Kittel [241]). For pure shear deformation, only two independent elastic moduli are relevant,
because \({\sigma _{ik}} = \partial {\varepsilon _{{\rm{def}}}}/\partial {{\rm{u}}_{ik}}\). At T = 0, Ogata & Ichimaru [309] find
These values agree with the classical result of Fuchs [151].
The definition of an “effective” shear modulus of a bcc polycrystal deserves a comment. In numerous papers, a standard preferred choice was μ = c_{44} ([43, 319, 285], and references therein). However, replacing μ by a single maximal elastic modulus of a strongly anisotropic bcc lattice is not correct. An effective value of μ was calculated by Ogata & Ichimaru [309]. They performed directional averages over rotations of the Cartesian axes. At T = 0, they obtained
some 30% smaller than μ = c_{44} used in previous papers. Dependence of μ on temperature was studied, using the Monte Carlo method, by Strohmayer et al. [393]. As expected, the effective shear modulus decreases with increasing temperature.
The formula for μ, Equation (116), can be rewritten as
where P_{e} is the pressure of ultrarelativistic degenerate electrons. Therefore,
The Poisson coefficient σ ≃ 1/2, while the Young modulus E ≃ 3μ.
Let us remember that the formulae given above hold for the outer crust, where the size of the nuclei is very small compared to the lattice spacing and P ≃ P_{e}. For the inner crust these formulae are only approximate.
7.2 Nuclear pasta
Some theories of dense matter predict the existence of “nuclear pasta” — rods, plates, tubes, bubbles — in the bottom layer of the crust with ρ ≳ 10^{14} g cm^{−3} (see Section 3.3). In what follows we will concentrate on rods (spaghetti) and plates (lasagna). They are expected to fill most of the bottom crust layers. The matter phases containing rods and plates have properties intermediate between solids and liquids. The displacement of an element of matter parallel to the plane containing rods or plates is not opposed by restoring forces: this lack of a shear strain is typical for a liquid. On the contrary, an elastic strain opposes any bending of planes or rods: this is a property of a solid. Being intermediate between solids and liquids, these kinds of matter are usually called mesomorphic phases, or liquid crystals (see, e.g., [249]).
The elastic properties of rod and plate phases of neutron star matter were studied by Pethick & Potekhin [325]. As they stressed, the physical reasons for the forming of mesomorphic phases in neutron star crusts are very different from those relevant to liquid crystals in laboratory. For terrestrial liquid crystals it is the interaction between very nonspherical molecules, which drives them to form rods and plates. In a neutron star crust the mechanism consists in spontaneous symmetry breaking, resulting from competition between the Coulomb energy and nuclear surface energy. We will follow closely Pethick & Potekhin [325]. They calculated the energies of mesomorphic phases using the generalized liquid drop model. The plate phase has rotational symmetry about any axis perpendicular to the plates, and is therefore similar to the smectics A phase of liquid crystals [115]. Let the zaxis coincide with the symmetry axis of the equilibrium configuration. Only a displacement in the zdirection is opposed by a restoring force. Therefore, we consider only u = (0,0, u). The deformation energy of a unit volume is then [115]
where ∇_{⊥} ≡ (∂/∂x, ∂/∂y, 0). Using the generalized liquid drop model, Pethick & Potekhin [325] obtain
where ε_{Coul} is the Coulomb energy density (in equilibrium)
and where R_{cell} is the halfdistance between the plates, n_{pi} is the proton density in nuclear matter and w is the volume fraction occupied by nuclear matter (all values calculated for the relaxed system).
In the case of the rod phase, also called the columnar phase [115], the number of elastic moduli is larger. They describe the increase in energy density due to compression, dilatation, transverse shearing, and bending of the rod lattice. Elastic moduli were calculated within the liquid drop model by Pethick & Potekhin [325] and by Watanabe, Iida & Sato [419, 420].
At the microscopic scale (fermis), the elastic properties of the nuclear pastas are very different from those of a bodycenteredcubic crystal of spherical nuclei. Nevertheless, the effects of pasta phases on the elastic properties of neutron star crusts may not be so dramatic at large scales (let’s say meters). Indeed these nuclear pastas are necessarily of finite extent since one and twodimensional longrange crystalline orders cannot exist in infinite systems (see, for instance, [157] and references therein). How the nuclear pastas arrange themselves remains to be studied, but it is likely that the resulting configurations look moreorless isotropic at macroscopic scales.
8 Superfluidity and Superconductivity
Except for a brief period after their birth, neutron stars are expected to contain various superfluid and superconducting phases [363, 116, 29, 367]. In this section, after a brief discussion of superconductivity and its possible occurrence in neutron star crusts, we will review our current theoretical understanding of the static and dynamic properties of neutron superfluid in the inner crust of neutron stars. For a general introduction and a recent overview on superfluidity and superconductivity, see, for instance, the book by Annett [22].
8.1 Superconductivity in neutron star crusts
In the BardeenCooperSchrieffer (BCS) theory of superconductivity [36] the coupling of electrons with lattice vibrations leads to an effective attractive interaction between electrons despite the repulsive Coulomb force. As a result, the electrons of opposite spins form pairs with zero total angular momentum. These Cooper pairs behave like bosons. Unlike fermions, bosons do not obey the Pauli exclusion principle, which forbids multiple occupancy of single particle quantum states. Consequently, below some critical temperature, bosons condense into the lowestenergy singleparticle state, giving rise to superfluidity, as in liquid helium4. Loosely speaking, superconductivity can thus be seen as BoseEinstein condensation of bound electron pairs. However the analogy with BoseEinstein condensation should not be taken too far. Indeed, Cooper pairs do not exist above the superconducting transition, while in BoseEinstein condensation bosons always exist above and below the critical temperature. Besides the electrons in a pair do not form well separated “molecules” like atoms in liquid helium, but instead strongly overlap. The spatial extent of a Cooper pair in a conventional superconductor is typically several orders of magnitude larger than the mean interelectron spacing. The BoseEinstein condensation (BEC) and the BCS regime are now understood as two different limits of the same phenomenon as illustrated in Figure 44. The transition between these two limits has recently been studied in ultracold atomic Fermi gases, for which the interaction can be adjusted experimentally [103].
In the outermost envelope of neutron stars, where the density is similar to that of ordinary solids, the critical temperature for the onset of electron superconductivity is, at most, on the order of a few K, which is many orders of magnitude smaller than the expected and observed surface temperature (see Section 12.2). Besides, it is well known that iron, the most probable constituent of the outer layers of the crust (Section 3.1), is not superconducting under normal pressure. It was discovered in 2001 [376] that iron becomes superconducting at “high” pressures^{Footnote 5} between 1.5 × 10^{11} and 3 × 10^{11} dyn cm^{−2}, when the temperature is below about 2 K. At higher densities, assuming that the electrons are degenerate, we can estimate the critical temperature from the BCS weak coupling approximation [36] (see also the discussion by Ginzburg [160]).
where T_{pi} is the ion plasma frequency, v_{eff} is the effective attractive electronelectron interaction and g_{Fe} is the density of electron states at the Fermi level (per unit energy and per unit volume). Neglecting band structure effects (which is a very good approximation in dense matter; see, for instance, the discussion of Pethick & Ravenhall [326]), the density of electron states is given by
so that the critical temperature takes the form
where ζ is a numerical positive coefficient of order unity [160]. At densities much below ∼ 10^{6} g cm^{−3}, the electrons are nonrelativistic and their Fermi velocity is given by v_{Fe} = ħk_{Fe}/m_{e}. As a result, the critical temperature decreases exponentially with the average mass density ρ as
where ζ′ ≡ ζ(9πZ/4A)^{1/3} and \({{\rm{b}}_{11}} = {1 \over 2}({c_{11}}  {c_{12}})\) is a typical density of ordinary matter (a_{0} is the Bohr radius and m_{u} the atomic mass unit). Considering, for instance, a plasma of iron and electrons, and adopting the value ζ = 8/π calculated for the “jellium” model by Kirzhnits [240], the critical temperature is approximately given by
This rough estimate shows that the critical temperature decreases very rapidly with increasing density, dropping from ∼ 30 K at ρ = 10 g cm^{−3} to 10^{−1} K at ρ = 10^{2} g cm^{−3} and to 10^{−7} K at ρ = 10^{3} g cm^{−3}! At densities above ∼ 10^{6} g cm^{−3}, electrons become relativistic, and v_{Fe} ∼ c. According to Equation (124), the critical temperature of relativistic electrons is given by
where α = e^{2}/ħc ≃ 1/137 is the fine structure constant. Due to the exponential factor, the critical temperature is virtually zero.
We can, thus, firmly conclude that electrons in neutron star crusts (and, a fortiori, in neutron star cores) are not superconducting. Nevertheless, superconductivity in the crust is not completely ruled out. Indeed, at the crustcore interface some protons could be free in the “pasta” mantle (Section 3.3), and could be superconducting due to pairing via strong nuclear interactions with a critical temperature far higher than that of electron superconductivity. Microscopic calculations in uniform nuclear matter predict transition temperatures on the order of T_{cp} ∼ 10^{9}−10^{10} K, which are much larger than typical temperatures in mature neutron stars. Some properties of superconductors are discussed in Sections 8.3.3 and 8.3.4.
8.2 Static properties of neutron superfluidity
Soon after its formulation, the BardeenCooperSchrieffer (BCS) theory of electron superconductivity [36] was successfully applied to nuclei by Bohr, Mottelson and Pines [51] and Belyaev [46]. In a paper devoted to the moment of inertia of nuclei, Migdal [291] speculated about the possibility that superfluidity could occur in the “neutron core” of stars (an idea which was raised by Gamow and Landau in 1937 as a possible source of stellar energy; see, for instance, [184]). The superfluidity inside neutron stars was first studied by Ginzburg and Kirzhnits in 1964 [161, 162]. Soon after, Wolf [424] showed that the free neutrons in the crust are very likely to be superfluid. It is quite remarkable that the possibility of superfluidity inside neutron stars was raised before the discovery of pulsars by Jocelyn Bell and Anthony Hewish in 1967. Later, this prediction seemed to be confirmed by the observation of the long relaxation time, on the order of months, following the first glitch in the Vela pulsar [41]. The neutron superfluid in the crust is believed to play a key role in the glitch mechanism itself. Pulsar glitches are still considered to be the strongest observational evidence of superfluidity in neutron stars (see Section 12.4).
At the heart of BCS theory is the existence of an attractive interaction needed for pair formation. In conventional superconductors, this pairing interaction is indirect and weak. In the nuclear case the occurrence of superfluidity is a much less subtle phenomenon since the bare strong interaction between nucleons is naturally attractive at not too small distances in many JLS channels (Jtotal angular momentum, Lorbital angular momentum, Sspin of nucleon pair). Apart from a proton superconductor similar to conventional electron superconductors, two different kinds of neutron superfluids are expected to be found in the interior of a neutron star (for a review, see, for instance, [363, 271, 116, 29, 367]). In the crust and in the outer core, the neutrons are expected to form an isotropic superfluid like helium4, while in denser regions they are expected to form a more exotic kind of (anisotropic) superfluid with each member of a pair having parallel spins, as in superfluid helium3. Neutronproton pairs could also exist in principle; however, their formation is not strongly favored in the asymmetric nuclear matter of neutron stars.
8.2.1 Neutron pairing gap in uniform neutron matter at zero temperature
A central quantity in BCS theory is the gap function, which is related to the binding energy of a pair. Neglecting for the time being nuclear clusters in the inner crust and considering pure neutron matter, the gap equations at a given number density n and at zero temperature read, in the simplest approximation [346],
together with
where \(\zeta^\prime \equiv \zeta {(9\pi Z/4A)^{1/3}}\) is the matrix element of the pairing interaction between timereversed planewave states with wave vectors k and k’, μ is the chemical potential and ϵ(k) is the single particle energy. Whenever the ratio Δ(k)/μ is small, the concept of a Fermi surface remains well defined and the two equations can be decoupled by setting μ = ϵ(k_{F}) ≡ ϵ_{F}.
Since the kernel in the gap integral peaks around the chemical potential μ ≃ ϵ_{F}, let us suppose that the pairing matrix elements \({\tilde V_{{{k}},{{k}}^\prime}}\) remain constant within ϵ(k) − μ < ϵ_{C} and zero elsewhere; ϵ_{C} is a cutoff energy. With this schematic interaction, the gap function becomes independent of k. In conventional superconductors, the electron pairing is conveyed by vibrations in the ion lattice. The ion plasma frequency thus provides a natural cutoff ϵ_{C} = ħω_{pi} (see Section 8.1). In the nuclear case however, there is no a priori choice of ϵ_{C}. A cutoff can still be introduced in the BCS equations, provided the pairing interaction is suitably renormalized, as shown by Anderson & Morel [11]. The BCS gap equations (128) can be solved analytically in the weak coupling approximation, assuming that the pairing interaction is small, \({\tilde V_{{{k}},{{{k}}^\prime}}} \simeq {\tilde V_{k{\rm{F}},k{\rm{F}}}}\), where g(ϵ) is the density of single particle states at the energy e. Considering that g(ϵ) remains constant in the energy range ϵ − μ < ϵ_{C}, the gap Δ(kF) ≡ Δ_{F} at the Fermi momentum k_{F} is given by
where the superscript C is to remind us that the strength of the pairing interaction depends on the cutoff. This expression is usually not very good for nuclear matter because the pairing interaction is strong. Nevertheless it illustrates the highly nonperturbative nature of the pairing gap. It also shows that the lower the density of states, the lower the gap. Consequently the neutron pairing gap in neutron star crust is expected to be smaller inside the nuclear clusters (discrete energy levels) than outside (continuous energy spectrum) as confirmed by detailed calculations (see Section 8.2.3).
The pairing gap obtained by solving Equations (128) and (129) for neutron matter using a bare nucleonnucleon potential and assuming a free Fermi gas single particle spectrum
where m_{n} is the neutron mass, is almost independent of the nucleonnucleon potential (provided it fits scattering data) and is shown in Figure 45.
As can be seen, the neutron pairs are most strongly bound at neutron densities around n_{n} ≃ 0.02 fm^{−3}. At higher densities, the pairing gap decreases due to the short range repulsive part of the nucleonnucleon interaction. The pairing gap is almost independent of the nucleonnucleon potential. The reason is that nucleonnucleon potentials are constrained to reproduce the experimental phase shifts up to scattering energies of order E_{lab} ∼ 350 MeV, which corresponds to neutron densities of order n_{n} ≃ 0.3 fm^{−3}. In fact, it can be shown that the pairing gap is completely determined by the experimental ^{1}S_{0} nucleonnucleon phase shifts [136]. At small relative momenta k, the neutronneutron ^{1}S_{0} phase shifts δ(k) are well approximated by the expansion
where a_{nn} = −18.5±0.3 fm and r_{nn} = 2.75±0.11 fm are the neutronneutron scattering length and effective range, respectively [381]. Large negative values of the scattering length are associated with attractive interactions, which nearly lead to the existence of a dineutron. In magneticallytrapped Bose atomic gases, the scattering length can be varied experimentally by tuning the magnetic field [127]. It can be shown that for Fermi wave vectors k_{F} ≪ 0.5 fm^{−1} (n_{b} ≪ 4× 10^{−3} fm^{−3}), the pairing gap Δ_{F} is completely determined by these two parameters only [136]. In the zero density limit k_{F}α_{nn} ≪ 1 (i.e., nn ≪ 5× 10^{−6} fm^{−3}), the gap equations are exactly solvable and the pairing gap can be expressed analytically by the exact formula [321]
Let us emphasize that this formula is universal and valid for any fermion system with attractive interactions (a_{nn} < 0). Equation (133) shows that the pairing gap strongly depends on the density and the nucleonnucleon interaction. It can, thus, be foreseen that modifications of the bare nucleonnucleon interaction due to medium polarization, which have been neglected, have a dramatic effect. Indeed, it can be rigorously shown [172] that in the low density limit, polarization effects reduce the gap value (133) by a factor of 4^{1/3} exp(1/3) ∼ 2, independent of the strength of the interaction!
The gap Equations (128) and (129) solved for the bare interaction with the free single particle energy spectrum, Equation (131), represent the simplest possible approximation to the pairing problem. A more consistent approach from the point of view of the manybody theory, is to calculate the single particle energies in the HartreeFock approximation (after regularizing the hard core of the bare nucleonnucleon interaction). The next step is to “dress” the pairing interaction by medium polarization effects. Calculations have been carried out with phenomenological nucleonnucleon interactions such as the Gogny force [117, 140], that are constructed so as to reproduce some properties of finite nuclei and nuclear matter. Another approach is to derive this effective interaction from a bare nucleonnucleon potential (twobody and/or threebody forces) using manybody techniques. Still the gap equations of form (128) neglect important manybody aspects.
In manybody theory, the general equations describing a superfluid Fermi system are the NambuGorkov equations [3], in which the gap function Δ(k, ω) depends not only on the wave vector k but also on the frequency ω. This frequency dependence arises from dynamic effects. In this framework, it can be shown that BCS theory is a mean field approximation to the manybody pairing problem. The Gorkov equations cannot be solved exactly and some approximations have to be made. Over the past years, this problem has been tackled using different microscopic treatments and approximation schemes. Qualitatively these calculations lead to the conclusion that medium effects reduce the maximum neutron pairing gap compared to the BCS value (note that this includes the possibility that medium effects actually increase the pairing gap for some range of densities). However, these calculations predict very different density dependence of the pairing gap as illustrated in Figure 46.
Before concluding this section, we provide an analytic formula for a few representative neutron pairing gaps, using the following expression proposed by Kaminker et al. [229].
where k_{F} = (3π^{3}n_{n})^{1/3} and θ is the Heaviside step function θ(x) = 1 if x > 0 and θ(x) = 0 otherwise. The parameters Δ_{0}, k_{1}, k_{2}, k_{3} and k_{max} are given in Table 5. At low density, the pairing gap \(g(\mu){\tilde V_{k{\rm{F,}}k{\rm{F}}}} \ll 1\) varies roughly as expected, Equation (133), remembering that \({\Delta _{\rm{F}}}\infty k_{\rm{F}}^2\).
8.2.2 Critical temperature for neutron superfluidity
The BCS gap Equations (128) at zero temperature can be generalized to finite temperature T (adopting the standard notation β ≡ 1/k_{B}T where k_{B} is the Boltzmann constant)
Superfluidity disappears whenever the temperature exceeds some critical threshold. Let us remark that isotropic neutron superfluidity can also be destroyed by a sufficiently strong magnetic field, since it would force each spin of a neutron pair to be aligned (as pointed out by Kirszshnits [239]). It can be shown on general grounds that the isotropic pairing gap Δ_{F}(T = 0) at zero temperature (at Fermi momentum k_{F}) and the critical temperature T_{c} are approximately related by [36]
where γ is the Euler constant. This wellknown result of conventional electron superconductivity applies rather well to nucleon superfluidity, especially for densities at which Δ_{F}(0) takes its maximum value [271].
The temperature dependence of the pairing gap, for T ≤ T_{c}, can be approximately written as [428]
Zero temperature pairing gaps on the order of 1 MeV are therefore associated with critical temperatures of the order 10^{10} K, considerably larger than typical temperatures inside neutron stars except for the very early stage of their formation. The existence of a neutron superfluid in the inner crust of a neutron star is therefore well established theoretically. Nevertheless the density dependence of the critical temperature predicted by different microscopic calculations differ considerably due to different approximations of the manybody problem. An interesting issue concerns the cooling of neutron stars and the crystallization of the crust: do the neutrons condense into a superfluid phase before the formation of the crust or after?
Figure 47 shows the melting temperature T_{m} of the inner crust of neutron stars compared to the critical temperature T_{c} for the onset of neutron superfluidity. The structure of the crust is that calculated by Negele & Vautherin [303]. The melting temperature has been calculated from Equation (15) with Γ_{m} = 175. The temperature T_{c} has been obtained from Equation (136), considering a uniform neutron superfluid, with the density \(\mu \sim {\epsilon _{\rm{F}}}\infty k_{\rm{F}}^2\) of unbound neutrons given by Negele & Vautherin [303]. Several critical temperatures are shown for comparison. As discussed in Section 8.2.1, the BCS value represents the simplest approximation to the true critical temperature. The other two critical temperatures have been obtained from more realistic pairinggap calculations, which include medium effects using different manybody approximations. The calculation of Cao et al. [69] is based on diagrammatic calculations, while that of Schwenk et al. [366] relies on the renormalization group.
For the BCS and Brueckner calculations of the pairing gap, in the density range of ∼ 10^{12} − 10^{14} g cm^{−3}, the neutrons may become superfluid before the matter crystallizes into a solid crust. As discussed in Section 8.3.2, as a result of the rotation of the star, the neutron superfluid would be threaded by an array of quantized vortices. These vortices might affect the crystallization of the crust by favoring nuclear clusters along the vortex lines, as suggested by Mochizuki et al. [293]. On the contrary, the calculations of Schwenk et al. [366] indicate that, at any density, the solid crust would form before the neutrons become superfluid. Recently, it has also been shown, by taking into account the effects of the inhomogeneities on the neutron superfluid, that in the shallow layers of the inner crust, the neutrons might remain in the normal phase even long after the formation of the crust, when the temperature has dropped below 10^{9} K [294].
8.2.3 Pairing gap in neutron star crusts
In this section, we will discuss the effects of the nuclear clusters on the pairing properties of the neutron superfluid in neutron star crusts. The relative importance of these effects is determined by the coherence length, defined as the root mean square radius of the pair wave function. Broadly speaking, the coherence length represents the size of a neutron pair. This is an important length scale, which determines many properties of the superfluid. For instance, the coherence length is of the order of the size of the superfluid vortex cores. According to Anderson’s theorem [114], the effects of the inhomogeneities (here — nuclear clusters) on the neutron superfluid are negligible whenever the coherence length is much larger than the characteristic size of the inhomogeneities. Assuming weak coupling, the coherence length can be roughly estimated from Pippard’s expression
where Δ_{F} is the neutron pairing gap at the Fermi momentum kF (see Section 8.2.1). This expression gives only a lower bound for the coherence length, since medium effects tend to reduce the pairing correlations as discussed in Section 8.2.1. Nevertheless, this estimate is rather close to the value obtained in more detailed calculations [111, 283]. As can be seen in Figure 48, the coherence length is smaller than the lattice spacing^{Footnote 6} except for the densest layers of the crust. Consequently the effects of the solid crust on neutron superfluidity cannot be neglected. This situation is in sharp contrast to that encountered in ordinary type I superconductors, where the electron Cooper pairs are spatially extended over mesoscopic distances of ∼ 10^{3} − 10^{4} Å and as a result the pairing gap is nearly insensitive to the details of the atomic crystal structure, since the typical lattice spacing is of order a few Å. In Figure 48 we also displayed the mean interneutron spacing defined by
In the denser layers of the crust, the coherence length is smaller than the mean interneutron spacing, suggesting that the neutron superfluid is a BoseEinstein condensate of stronglybound neutron pairs, while in the shallower layers of the inner crust the neutron superfluid is in a BCS regime of overlapping looselybound pairs. Quite remarkably, for screened pairing gaps like those of Schwenk et al. [366], the coherence length is larger than the mean interneutron spacing in the entire inner crust, so that in this case, at any depth, neutron superfluid is in the BCS regime.
Since the formulation of the BCS theory, considerable theoretical efforts have been devoted to the microscopic calculation of pairing gaps in uniform nuclear matter using the many body theory. On the other hand, until recently the superfluidity in neutron star crusts has not attracted much attention despite its importance in many observational phenomena like pulsar glitches (see Section 12). The pairing correlations in an inhomogeneous superfluid system can be described in terms of a pairing field Δ(r). In early studies [59, 107, 135] the pairing field has been calculated assuming that the matter is locally homogeneous (local density approximation). Such calculations predict, in particular, that the value of the pairing field inside the nuclear clusters is almost the same for different layers of the crust. The reason lies in the nuclear saturation: the density inside heavy nuclei is essentially constant, independent of the number of bound nucleons. In some cases, the pairing field was found to vanish inside the clusters [135]. The local density approximation is valid if the coherence length is smaller than the characteristic scale of density variations. However, this condition is never satisfied in the crust. As a result, the local density approximation overestimates the spatial variation of the pairing field. Due to “proximity effects”, the free superfluid neutrons induce pairing correlations of the bound neutrons inside clusters and vice versa leading to a smooth spatial variation of the neutron pairing field [38]. As a remarkable consequence, the value of the neutron pairing field outside (resp. inside) the nuclear clusters is generally smaller (resp. larger) than that obtained in uniform neutron matter for the same density [31]. In particular, the neutrons inside the clusters are also superfluid. The neutron superfluid in the crust should, therefore, be thought of as an inhomogeneous superfluid rather than a superfluid flowing past clusters like obstacles. The effects of nuclear clusters on neutron superfluidity have been investigated in the WignerSeitz approximation by several groups. These calculations have been carried out at the mean field level with realistic nucleonnucleon potentials [37], effective nucleonnucleon interactions [38, 295, 361, 360, 237] and with semimicroscopic energy functionals [29, 30, 31]. Examples are shown in Figure 49. The effects of medium polarization have been considered by the Milano group [169, 413], who found that these effects lead to a reduction of the pairing gap, as in uniform neutron matter. However, this quenching is less pronounced than in uniform matter due to the presence of nuclear clusters. Apart from uncertainties in the pairing interaction, it has recently been shown [30, 31] that the pairing field is very sensitive to the choice of boundary conditions, especially in the bottom layers of the crust (as also found by the other groups). Consequently, the results obtained in the WignerSeitz approximation should be interpreted with caution, especially when calculating thermodynamic quantities like the neutron specific heat, which depends exponentially on the gap.
8.3 Superfluid hydrodynamics
8.3.1 Superflow and critical velocity
The basic property of superfluid is that it can flow without dissipation. In a normal fluid, friction and viscosity arise because particles are randomly scattered. Such scattering events are forbidden in superfluid because energy and momentum cannot be simultaneously conserved. The key argument of Landau is that in a superfluid like helium4, the particles are strongly correlated so that the concept of single particles becomes meaningless. However, at low enough temperatures, the system is still assumed to be described in terms of noninteracting “quasiparticles”, which do not correspond to material particles but to manybody motions (excitations). The energy spectrum of these quasiparticles can be very different from that of single particles. Using this idea, Landau [247] was able to explain the origin of nondissipative superflow and the existence of a critical velocity beyond which superfluidity disappears. The argument is the following. Let us consider a macroscopic body of mass M flowing through the superfluid. At low temperatures, its velocity V can only be changed in scattering processes where one or more quasiparticles are created, assuming that the flow is not turbulent. For a quasiparticle of energy E(p) and momentum p to be created, energy conservation implies that
where V′ is the body’s velocity after the event. However, momentum must also be conserved,
These two conditions can only be satisfied if
Since the perturbing body contains a macroscopic collection of particles, the mass M is very large so that the second term can be neglected. The resulting inequality cannot be satisfied unless the velocity exceeds some critical value
where min{x} is the smallest value of x in a set {x}. This means that for velocities smaller than V_{c}, the creation of quasiparticles is forbidden and therefore the fluid flows without dissipation. In a normal liquid, the single particle energy is given by an expression of the form
where m is an effective mass suitably renormalized to include manybody effects. Consequently, the critical velocity, according to Landau’s criterion (143), is zero, V_{c} = 0. Since liquid helium4 is superfluid, Landau [247, 246] postulated a different energy spectrum. At low momenta, the quasiparticle excitations are sound waves (phonons) as illustrated in Figure 50. The dispersion relation is thus given by
where c_{s} is a sound speed.
At very high momenta, the dispersion relation coincides with that of a normal liquid, Equation (144). In between, the dispersion relation exhibits a local minimum and is approximately given by
The quasiparticles associated with this minimum were dubbed “rotons” by I.E. Tamm as reported by Landau [247]. Landau postulated that these rotons are connected with a rotational velocity flow, hence the name. These rotons arise due to the interactions between the particles. Feynman [145] argued that a roton can be associated with the motion of a single atom. As the atom moves through the fluid, it pushes neighboring atoms out of its way forming a ring of particles rotating backwards as illustrated in Figure 51. The net result is a vortex ring of an atomic size.
The roton local minimum has also been interpreted as a characteristic feature of density fluctuations marking the onset of crystallization [200, 334, 307]. According to Nozières [307], rotons are “ghosts of Bragg spots”. Landau’s theory has been very successful in explaining the observed properties of superfluid helium4 from the postulated energy spectrum of quasiparticles.
In weakly interacting dilute Bose gases, as in ultracold Bose atomic gases, the energy of the quasiparticles are given by (see for instance Section 21 of Fetter & Walecka [142])
where c_{s} is the speed of sound. At low momentum p, it reduces to Equation (145), while at high momentum it tends to Equation (144). Dilute Bose gases, thus, have only phonon excitations. Quite remarkably, the ideal Bose gas (which is characterized by the dispersion relation (147) with c_{s} = 0) exhibits a BoseEinstein condensation at low enough temperatures, but is not superfluid since its critical velocity is equal to zero.
Owing to the specific energy spectrum of quasiparticles in both atomic Bose gases and helium4, the critical velocity does not vanish, thus explaining their superfluid properties. Landau’s critical velocity, Equation (143), of superfluid helium4 due to the emission of rotons, is given by
This value has been confirmed by ion propagation experiments [137]. However, in most experiments, much smaller critical velocities are measured due to the existence of other kinds of excitations. The critical velocity of atomic Bose gases has also been measured, using laser beams instead of a macroscopic object [235]. Again, velocities smaller than Landau’s velocity (which is equal to the velocity of sound in this case) have been found.
The previous discussion of the critical velocity of Bose liquids can be easily extended to fermionic superfluids. In the BCS theory, fermions form bound pairs, which undergo Bose condensation when the temperature falls below a critical temperature (Section 8.2). The quasiparticle energies for a uniform Fermi system are given by
where p = ħk, ϵ(p) is the single particle energy, μ the chemical potential and Δ(p) the pairing gap at momentum p. According to Landau’s argument, the critical velocity (143) is equal to
This expression can be derived more rigorously from the microscopic BCS theory [35]. It shows that a system of fermions is superfluid (i.e. the critical velocity is not zero) whenever the interactions are attractive, so that the formation of pairs becomes possible. It is also interesting to note that the BCS spectrum can be interpreted in terms of rotons. Indeed, expanding Equation (149) around the minimum leads, to lowest order, to an expression similar to Equation (146). In this case, p_{0} is obtained by solving ϵ(p) = μ. The other parameters are given by Δ_{r} = Δ(p_{0}) and \({\tilde \rho _{\rm{G}}}\) where v_{0} = dϵ/dp is the group velocity evaluated at p_{0}.
The presence of an “external” potential affects superfluidity. This issue has recently attracted a lot of theoretical, as well as experimental, interest in the field of opticallytrapped ultracold atomic Bose gases [297]. It is also relevant in the context of neutron stars, where the solid crust is immersed in a neutron superfluid (and possibly a proton superconductor in the liquid crystal mantle, where nuclear “pastas” could be present; see Section 3.3). In the BCS approximation (128), considering a periodic potential (induced by the solid crust in neutron stars), the quasiparticle energies still take a form similar to Equation (149). However, the dependence on the momentum is no more isotropic. As shown by Carter, Chamel & Haensel [77], Equation (150) for the critical velocity should then be replaced by
where k is the Bloch wave vector (Section 3.2.4), v(k) = ħ ^{1}∇_{k} ϵ(k) is the group velocity of the fermions and m_{*} is an effective mass, which arises from the interactions between the particles and the lattice as discussed in Section 8.3.6. The subscript FS means that the minimum is to be searched on the Fermi surface^{Footnote 7}. In the neutron star crust, the effective mass of the unbound neutrons could be very large m_{★} ≫ m_{n}, as shown by Chamel [90, 91]. In the limit of homogeneous matter, the single particle energies are given by an expression of the form (144) with m = m_{n} and p = ħk. Consequently, m_{★} = m_{n} and υ = ħk/m_{n}, so that Equation (151) reduces to Equation (150) using the extended Brillouin zone scheme. Equation (151) shows that superfluidity disappears whenever the pairing gap vanishes at some point on the Fermi surface.
The real critical velocity is expected to be smaller than that given by Equation (151) due to finite temperature and manybody effects beyond the mean field. Likewise, the critical velocity of superfluid helium4 obtained in Landau’s quasiparticle model is only an upper bound because in this model, the quasiparticles are assumed to be noninteracting. The experimentallymeasured critical velocities are usually much smaller, in particular, due to the nucleation of vortices. Indeed, relative motions between superfluid and the vortices lead to mutual friction forces and, hence, to dissipative effects. In general, any curved vortex line does not remain at rest in the superfluid reference frame and, therefore, induces dissipation. Feynman [143] derived the critical velocity associated with the formation of a vortex ring in a channel of radius R,
where r_{υ} is the radius of the vortex core. This result shows that the critical velocity depends, in general, on experimental setup. More generally, the critical velocity scales like V_{c} ∝ ħ/mL, where L is a characteristic length scale in the experiment. The theoretical determination of the breakdown of superfluidity is still an open issue, which requires a detailed understanding of superfluid dynamics and, in particular, the dynamics of vortices.
8.3.2 Rotating superfluid and vortices
Superfluidity is closely related to the phenomenon of BoseEinstein condensation as first envisioned by Fritz London [272]. In the superfluid phase, a macroscopic collection of particles condense into the lowest quantum single particle state, which (for a uniform system) is a plane wave state with zero momentum (therefore a constant). Soon after the discovery of the superfluidity of liquid helium, Fritz London introduced the idea of a macroscopic wave function Ψ(r), whose squared modulus is proportional to the density n_{Ψ} of particles in the condensate. This density n_{Ψ}, which should not be confused with the superfluid density nS introduced in the twofluid model of superfluids (Section 8.3.6), can be rigorously defined from the oneparticle density matrix [322]. The wave function is defined up to a global phase factor. The key distinguishing feature of a superfluid is the symmetry breaking of this gauge invariance by imposing that the phase ϕ(r) be local. The macroscopic wave function thus takes the form
Applying the momentum operator − iħ∇ to this wave function shows that superfluid carries a net momentum (per superfluid particle)
This implies that the superflow is characterized by the condition
In the absence of any entrainment effects (as discussed in Section 8.3.6), the momentum is given by p = mυ, where m is the mass of the superfluid “particles”^{Footnote 8} and υ is the velocity of superfluid. Equation (155) thus implies that the flow is irrotational. This means, in particular, that a superfluid in a rotating bucket remains at rest with respect to the laboratory reference frame. However, this Landau state is destroyed whenever the rotation rate exceeds the critical threshold for the formation of vortices given approximately by Equation (152). Experiments show that the whole superfluid then rotates like an ordinary fluid. The condition (155) can therefore be locally violated as first suggested by Onsager [310] and discussed by Feynman [143]. Indeed, since the phase of the macroscopic wave function is defined modulo 2π, the momentum circulation over any closed path is quantized
where N is any integer and h is Planck’s constant. Equation (156) is simply the BohrSommerfeld quantization rule. This rule can be equivalently deduced from the following argument. Considering superfluid as a macroscopic quantum state, its momentum p is given by p = h/λ, where λ is the de Broglie wavelength. The quantization rule (156) thus follows from the requirement that the length of any closed path must be an integral multiple of the de Broglie wavelength.
In a rotating superfluid the flow quantization implies the formation of vortex lines, each carrying a quantum ħ of angular momentum, the quantum number N being the number of vortices (the formation of a single vortex carrying all the angular momentum is not energetically favored). The size of the core of a vortex line is roughly on the order of the superfluid coherence length (see Section 8.2.3 for estimates of the coherence length). In some cases, however, it may be much smaller [110, 134], so that the coherence length is only an upper bound on the vortex core size. In the presence of vortices, Equation (155) must therefore be replaced by
k being the circulation of the N vortex lines. The existence of quantized vortices was demonstrated by Vinen [415] in 1961, however, they were not observed until much later, in 1974 at Berkeley [423, 432]. At length scales much larger than the superfluid coherence length, the finite size of the vortex core can be neglected and the circulation is, thus, given by
where δ^{(2)} is the twodimensional Dirac distribution, and κ_{α} is the circulation of a given vortex α
\({m_{\rm{r}}} = \Delta ({p_0})/\upsilon _0^2\) being a unit vector directed along the vortex line. Equation (157) is formally similar to Ampere’s law in magnetostatics. The momentum p induced by the vortex lines is, thus, given by the BiotSavart equation, substituting 1/c with the mass m divided by 4π in Gaussian cgs units (or replacing the magnetic permeability µ_{0} by the mass m in SI units) and the electric current I by κ,
where the integral is taken along the vortex lines as shown in Figure 52. The analogy between hydrodynamics and magnetostatics shows, in particular, that a vortex ring should move along its symmetry axis with a velocity inversely proportional to its radius.
As shown by Tkachenko [405, 406], quantized vortices tend to arrange themselves on a regular triangular array. Such patterns of vortices have been observed in superfluid helium and more recently in atomic BoseEinstein condensates. The intervortex spacing d_{υ} is given by
where Ω is the angular frequency. At length scales much larger than the intervortex spacing d_{υ}, as a result of the superposition of the flow pattern of all the vortex lines, superfluid flow mimics rigid body rotation. Since at this scale a fluid element is threaded by many vortex lines, it is relevant to smoothly average the hydrodynamic equations governing the flow of the superfluid. In particular, Equation (157) now reads
where n_{υ} is the surface density of vortices given (in the absence of entrainment effects) by
and the vector κ, whose norm is equal to h/m, is aligned with the average angular velocity (generalization of Equation (163) to account for entrainment effects is discussed in Section 10.4).
8.3.3 Type II superconductors and magnetic flux tubes
Let us remark that the condition (155) for superfluids also applies to superconductors, like the proton superconductor in the liquid core and possibly in the “pasta” mantle of neutron stars (Section 3.3). The momentum of a superconductor is given by p ≡ mυ + qA (in this section, we use SI units), where m, q, and υ are the mass, electric charge and velocity of “superconducting” particles respectively^{Footnote 9}, and A the electromagnetic potential vector. Introducing the density n of superconducting particles and their electric current density (referred to simply as “supercurrent”) \({\hat \kappa _\alpha}\), Equation (155) leads to the London equation
where B = V × A is the magnetic field induction. According to the BohrSommerfeld quantization rule (156), the London Equation (164) corresponds to N = 0. Situations with N > 0 are encountered in type II superconductors. Considering a closed contour outside a sample of such a superconductor for which \(\mathcal{J} = nqv\) and integrating the momentum p along this contour, leads to the quantization of the total magnetic flux into N flux tubes
These flux tubes tend to arrange themselves into a triangular lattice, the Abrikosov lattice, with a spacing given by
Averaging at length scales much larger than d_{υ}, the surface density of flux tubes is given by
8.3.4 Superfluid vortices and magnetic flux tubes in neutron stars
For neutron superfluid in neutron stars, superfluid particles are neutron pairs, so that m = 2m_{n}. As early as in 1964, Ginzburg & Kirzhnits [161, 162] suggested the existence of quantized vortex lines inside neutron stars. The critical velocity for the nucleation of vortices can be roughly estimated from V_{c} ∼ ħ/m_{n}R where R is the radius of a neutron star. For R = 10 km, V_{c} ∼ 10^{3} fm/s, which is, by several orders of magnitude, smaller than characteristic velocities of matter flows within the star. The interior of neutron stars is thus threaded by a huge number of vortices. The intervortex spacing is
which is much larger than the coherence length, so that the assumption of infinitelythin vortex lines in Equation (158) is justified. Assuming that neutron superfluid is uniformly corotating with the star, the density of vortices per square kilometer is then given by n_{υ} ≃ 10^{14}/P, where P is the rotation period in seconds. With a rotation period of 33 milliseconds, a pulsar like the Crab is threaded by an array of about 10^{18} vortex lines (assuming a radius of 10 km)! Likewise, proton superconductors (assumed to be of type II [41]) in the core of neutron stars (and possibly in the pasta layers; see Section 3.3) is threaded by quantized magnetic flux tubes each carrying a magnetic field on the order of 10^{15} G. The surface density of flux tubes is about 10^{13} times that of neutron vortices. The interactions between the neutron vortices and the flux tubes are likely to affect the dynamic evolution of the star [354, 355].
8.3.5 Dynamics of superfluid vortices
In this section, we will discuss the nonrelativistic dynamics of superfluid vortices. The generalization to relativistic dynamics has been discussed in detail by Carter [71]. According to the Helmholtz theorem, the vortex lines are frozen in superfluid and move with the same velocity unless some force acts on them. The dynamics of a vortex line through the crust is governed by different types of forces, which depend on the velocities υ_{f}, υ_{υ} and υ_{c} of the bulk neutron superfluid, the vortex and the solid crust, respectively.

A viscous drag force (not to be confused with entrainment, which is a nondissipative effect; see Section 8.3.6) opposes relative motion between a vortex line and the crust, inducing dissipation. At sufficiently small relative velocities, the force per unit length of the vortex line can be written as
$${{\mathcal F}_{\rm{d}}} =  {\mathcal R}({v_\upsilon}  {v_{\rm{c}}})\,,$$(169)where \(\mathcal{J} = 0\) is a positive resistivity coefficient, which is determined by the interactions of the neutron vortex line with the nuclear lattice and the electron gas. The pinning of the vortex line to the crust is the limit of very strong drag entailing that υ_{υ} = υ_{c}.

Relative motion of a vortex line with respect to bulk superfluid (caused by drag or pinning) gives rise to a Magnus or lift force (analog to the Lorentz force), given by
$${{\mathcal F}_{\rm{m}}} = {\rho _{\rm{f}}}\kappa \times ({v_{\rm{f}}}  {v_\upsilon})\,,$$(170)where ρ_{f} is the mass density of the free superfluid neutrons and κ is a vector oriented along the superfluid angular velocity and whose norm is given by h/2m_{n} (see Carter & Chamel [75] for the generalization to multifluid systems).

A tension force resists the bending of the vortex line and is given by
$${{\mathcal F}_{\rm{t}}} =  {\rho _{\rm{f}}}\kappa {{\mathcal C}_{\rm{t}}}{{{\partial ^2}u} \over {\partial {z^2}}}\,,$$(171)where u is the twodimensional displacement vector of the vortex line directed along the zaxis. \(\mathcal{R}\) is a rigidity coefficient of order
$${{\mathcal C}_{\rm{t}}}\sim{\kappa \over {4\pi}}\ln {{{d_\upsilon}} \over {{r_\upsilon}}}\,,$$(172)where κ = h/2m_{n}, d_{υ} is the intervortex spacing and r_{υ} the size of the vortex core [383].
All forces considered above are given per unit length of the vortex line. Let us remark that even in the fastest millisecond pulsars, the intervortex spacing (assuming a regular array) of order d_{υ} ∼ 10^{−3} − 10^{−4} cm is much larger than the size of the vortex core r_{υ} ∼ 10 − 100 fermis. Consequently the vortexvortex interactions can be neglected.
The dynamic evolution of a vortex line is governed by
where m_{υ} is the inertial mass of the vortex line and L_{υ} its length. Since the free neutron density inside the vortex core is typically much smaller than outside (unless the line is pinned to nuclei) [134, 433, 27], the motion of the vortex line is accompanied by a rearrangement of the free neutrons. The inertial mass of the vortex line is approximately equal to the mass density of the neutron superfluid ρ_{f} times the volume of the line \({{\mathcal C}_t}\) so that \(\pi r_\upsilon ^2{L_\upsilon}\).
On a scale much larger than the intervortex spacing, the drag force \({m_\upsilon}/{L_\upsilon} \sim {\rho _{\rm{f}}}\pi r_\upsilon ^2\) acting on every vortex line leads to a mutual friction force between the neutron superfluid and the normal constituents. Assuming that the vortex lines are rigid and form a regular array, the mutual friction force, given by \({\mathcal{F}_{\bf{d}}}\), can be obtained from Equation (173) after multiplying by the surface density n_{υ}. Since \({{{f}}_{\bf{d}}} = {n_\upsilon}{{\mathcal F}_{\bf{d}}}\), the inertial term on the lefthand side of Equation (173) is proportional to (r_{υ}/d_{υ})^{2} ≪ 1 and can be neglected. Solving the force balance equation yields the mutual friction force (per unit volume) [18]
where n_{υ} is the surface density of vortices in a plane perpendicular to the axis of rotation and B is a dimensionless parameter defined by
Different dissipative mechanisms giving rise to a mutual friction force have been invoked: scattering of electrons/lattice vibrations (phonons)/impurities/lattice defects by thermally excited neutrons in vortex cores [141, 189, 222], electron scattering off the electric field around a vortex line [48], and coupling between phonons and vortex line oscillations (Kelvin modes) [139, 223]. In the weak coupling limit \({n_\upsilon} \sim 1/\pi d_\upsilon ^2\), the vortices corotate with the bulk superfluid (Helmholtz theorem), while in the opposite limit \(\mathcal{B} \to 0\), they are “pinned” to the crust. In between these two limits, in a frame corotating with the crust, the vortices move radially outward at angle atan(ℛ) with respect to the azimuthal direction. The radial component of the vortex velocity reaches a maximum at ℛ = 1.
Vortex pinning plays a central role in theories of pulsar glitches. The strength of the interaction between a small segment of the vortex line and a nucleus remains a controversial issue [5, 138, 333, 134, 120, 121, 122, 27]. The actual “pinning” of the vortex line (i.e., υ_{υ} = υ_{c}) depends not only on the vortexnucleus interaction, but also on the structure of the crust, on the rigidity of lines and on the vortex dynamics. For instance, assuming that the crust is a polycrystal, a rigid vortex line would not pin to the crust simply because the line cannot bend in order to pass through the nuclei, independent of the strength of the vortexnucleus interaction! Recent observations of longperiod precession in PSR 1828−11 [387], PSR B1642−03 [371] and RX J0720.4−3125 [180] suggest that, at least in those neutron stars, the neutron vortices cannot be pinned to the crust and must be very weakly dragged [372, 266].
Let us stress that the different forces acting on a vortex vary along the vortex line. As a consequence, the vortex lines may not be straight [198]. The extent to which the lines are distorted depends on the vortex dynamics. In particular, Greenstein [175] suggested a long time ago that vortex lines may twist and wrap about the rotation axis giving rise to a turbulent flow. This issue has been more recently addressed by several groups [323, 324, 288, 19]. In such a turbulent regime the mutual friction force takes the form [173]
where \({\mathcal B} \to + \infty\) is a dimensionless temperaturedependent coefficient, assuming a dense random tangle of vortex lines.
8.3.6 Superfluid hydrodynamics and entrainment
One of the striking consequences of superfluidity is the allowance for several distinct dynamic components. In 1938, Tisza [404] introduced a twofluid model in order to explain the properties of the newly discovered superfluid phase of liquid helium4, which behaves either like a fluid with no viscosity in some experiments or like a classical fluid in other experiments. Guided by the Fritz London’s idea that superfluidity is intimately related to BoseEinstein condensation (which is now widely accepted), Tisza proposed that liquid helium is a mixture of two components, a superfluid component, which has no viscosity, and a normal component, which is viscous and conducts heat, thus, carrying all the entropy of the liquid. These two fluids are allowed to flow with different velocities. This model was subsequently developed by Landau[247, 246] and justified on a microscopic basis by several authors, especially Feynman [144]. Quite surprisingly, Landau never mentioned BoseEinstein condensation in his work on superfluidity. According to Pitaevskii (as recently cited by Balibar [33]), Landau might have reasoned that superfluidity and superconductivity were similar phenomena (which is indeed true), incorrectly concluding that they could not depend on the Bose or Fermi statistics (see also the discussion by Feynman in Section 11.2 of his book [144]).
In Landau’s twofluid model, the normal part with particle density n_{N} and velocity υ_{N} is identified with the collective motions of the system or “quasiparticles” (see Section 8.3.1). The viscosity of the normal fluid is accounted for in terms of the interactions between those quasiparticles (see, for instance, the book by Khalatnikov [236] published in 1989 as a reprint of an original 1965 edition). Following the traditional notations, the superfluid component, with a particle density n_{S} and a “velocity” υ_{S}, is locally irrotational except at singular points (see the discussion in Section 8.3.2)
As pointed out many times by Brandon Carter, unlike υ_{N} the “superfluid velocity” is not a true velocity but is defined through
where p is the true momentum per particle of the superfluid and m is the mass of a helium atom. Although deeply anchored in the Lagrangian and Hamiltonian formulation of classical mechanics, the fundamental distinction between velocities and their canonical conjugates, namely momenta, has been traditionally obscured in the context of superfluidity. Note also that the superfluid density n_{S} coincides neither with the density n of helium atoms (except at T = 0) nor with the density n_{Ψ} of atoms in the condensate^{Footnote 10}. The density of helium atoms is given by
The confusion between velocity and momentum is very misleading and makes generalizations of the twofluid model to multifluid systems (like the interior of neutron stars) unnecessarily difficult. Following the approach of Carter (see Section 10), the twofluid model can be reformulated in terms of the real velocity υ of the helium atoms instead of the superfluid “velocity” υ_{S}. The normal fluid with velocity υ_{N} is then associated with the flow of entropy and the corresponding number density is given by the entropy density. At low temperatures, heat dissipation occurs via the emission of phonons and rotons. As discussed in Section 8.3.1, these quasiparticle excitations represent collective motions of atoms with no net mass transport (see, in particular, Figures 50 and 51). Therefore, the normal fluid does not carry any mass, i.e., its associated mass is equal to zero.
Following the general principles reviewed in Section 10, the momentum p of the superfluid helium atoms can be written as
Note that the momentum p of the helium atoms is not simply equal to mυ due to the scattering of atoms by quasiparticles. In the rest frame of the normal fluid, in which υ_{N} = 0, the momentum and the velocity of the superfluid are aligned. However, the proportionality coefficient is not the (bare) atomic mass m of helium but an effective mass m_{★}. This effective mass m_{★} is related to the hydrodynamics of superfluid and should not be confused with the definitions employed in microscopic manybody theories. Before going further, let us remark that in the momentum rest frame of the normal component, the relation p = mυ holds! This can easily be shown from Equation (230), by using the identity (234) and remembering that the normal fluid is massless.
Comparing Equations (180) and (178) shows that the “superfluid velocity” in the original twofluid model of Landau is not equal to the velocity of the helium atoms but is a linear combination of both velocities υ and υ_{N}
The current of helium atoms is given by the sum of the normal and superfluid currents
Substituting Equation (181) into Equation (182) yields the relations
which clearly satisfies n = n_{S} + n_{N}. The superfluid and normal densities can be directly measured in the experiment devised by Andronikashvili [21]. A stack of disks, immersed in superfluid, can undergo torsional oscillations about its axis. Due to viscosity, the normal component is dragged by motion of the disks, while the superfluid part remains at rest. The normal and superfluid densities can, thus, be obtained at any temperature by measuring the oscillation frequency of the disks. Since n_{S}(T)/n = m/m_{★}(T), the dynamic effective mass m_{★}(T) can be determined experimentally. In particular, it is equal to the bare mass at T = 0, m_{★}(T = 0) = m, and goes up as the temperature is raised, diverging at the critical point when superfluidity disappears. At any temperature T > 0 the dynamic effective mass of an helium atom is therefore larger than the bare mass.
Entrainment effects, whereby momentum and velocity are not aligned, exist in any fluid mixtures owing to the microscopic interactions between the particles. But they are usually not observed in ordinary fluids due to the viscosity, which tends to equalize velocities. Even in superfluids like liquid Helium II, entrainment effects may be hindered at finite temperature^{Footnote 11} by dissipative processes. For instance, when a superfluid is put into a rotating container, the presence of quantized vortices induces a mutual friction force between the normal and superfluid components (as discussed in Section 8.3.5). As a consequence, in the stationary limit the velocities of the two fluids become equal. Substituting υ = υ_{N} in Equation (180) implies that p = mυ, as in the absence of entrainment.
8.3.7 Entrainment effects in neutron stars
A few years after the seminal work of Andreev & Bashkin [20] on superfluid ^{3}He −^{4}He mixtures, it was realized that entrainment effects could play an important role in the dynamic evolution of neutron stars (see, for instance, [363] and references therein). For instance, these effects are very important for studying the oscillations of neutron star cores, composed of superfluid neutrons and superconducting protons [14]. Mutual entrainment not only affects the frequencies of the modes but, more surprisingly, (remembering that entrainment is a nondissipative effect) also affects their damping. Indeed, entrainment effects induce a flow of protons around each neutron superfluid vortex line. The outcome is that each vortex line carries a huge magnetic field ∼ 10^{14} G [8]. The electron scattering off these magnetic fields leads to a mutual friction force between the neutron superfluid and the charged particles (see [18] and references therein). This mechanism, which is believed to be the main source of dissipation in the core of a neutron star, could also be at work in the bottom layers of the crust, where some protons might be unbound and superconducting (as discussed in Section 3.3).
It has recently been pointed out that entrainment effects are also important in the inner crust of neutron stars, where free neutrons coexist with a lattice of nuclear clusters [79, 78]. It is well known in solid state physics that free electrons in ordinary metals move as if their mass were replaced by a dynamic effective mass \({\mathcal{C}_{\rm{d}}}\) (usually referred to as an optical mass in the literature) due to Bragg scattering by the crystal lattice (see, for instance, the book by Kittel [241]). The end result is that, in the rest frame of the solid, the electron momentum is given by
where υ_{e} is the electron velocity. This implies that in an arbitrary frame, where the solid (ion lattice) is moving with velocity υ_{I}, the electron momentum is not aligned with the electron velocity but is given by
which is similar to Equation (180) for the momentum of superfluid Helium II. The concept of dynamic effective mass was introduced in the context of neutron diffraction experiments ten years ago [434] and has only recently been extended to the inner crust of neutron stars by Carter, Chamel & Haensel [79, 78]. While the dynamic effective electron mass in ordinary metallic elements differs moderately from the bare mass \(m_\star ^e\) (see, for instance, [204]), Chamel [90] has shown that the dynamic effective mass of free neutrons in neutron star crust could be very large, \(m_ \star ^e \sim 1  2{m_e}\). The dynamics of the free neutrons is deeply affected by these entrainment effects, which have to be properly taken into account (see Section 10). Such effects are important for modeling various observed neutron star phenomena, like pulsar glitches (see Section 12.4) or neutron star oscillations (see Sections 12.5 and 12.6).
9 Conductivity and Viscosity
9.1 Introduction
In this section, we will consider the transport of heat, electric charge, and momentum in neutron star crusts. In the absence of magnetic fields and assuming that the solid crust is isotropic^{Footnote 12}, the transport of heat and electric charge is described by the thermal conductivity κ and the electrical conductivity σ (Section 9.3) respectively. Under the same conditions, the transport of momentum is characterized by the shear viscosity η and the bulk viscosity ζ (Section 9.4).
Except for the very outer envelope, the main carriers in the transport processes in the outer crust are electrons, and they scatter mainly off ions (exceptions will be mentioned at the end of the corresponding sections). Theoretical techniques for the calculation of the transport coefficients in neutron star crusts are to a large extent borrowed from solid state physics, the classical reference still remaining the book of Ziman [435]. However, one has to remember that the density/temperature conditions within neutron star crusts are tremendously different from those in terrestrial solids, so that special care concerning the approximations used should be taken.
9.2 Boltzmann equation for electrons and its solutions
The electron distribution function is f(p, r, t, s), where p and r are electron momentum and position vectors, respectively, t is time, and s is the electron spin projection on the spin quantization axis. The distribution function f(p, r, t, s) satisfies the Boltzmann equation (BE) for electrons. At first glance, the validity of the BE (originally derived for a gas of particles) for a super dense plasma of electrons may seem paradoxical. However, electrons are strongly degenerate, so that only electron states in a thin shell around the chemical potential μ_{e} with energies ϵ − μ_{e} ≲ k_{B}T are involved in the transport phenomena. In other words, the gas of “electron excitations” is dilute. Also, the infinite range of Coulomb interactions in vacuum is no longer a problem in dense electronnuclear plasma because of screening. Moreover, as the kinetic Fermi energy of electrons is much larger than the Coulomb energy per electron, the Coulomb energy can be treated as a small perturbation and the electrons can be considered as a quasiideal Fermi gas.
We consider spinunpolarized plasma. Then f does not depend on s and the BE for electrons reads
where υ is the electron velocity, and F is the external force acting on the electrons (e.g., electrostatic force F = −eE, where electron electric charge is −e). The total collision integral I_{e}[f], which is a functional of f, is a sum of collision integrals corresponding to collisions of electrons with nuclei, electrons, and impurities in the crystal lattice, I_{e}[f] = I_{eN}[f] + I_{ee}[f] + I_{imp}[f].
The additivity of partial collision integrals is valid when the scatterers are uncorrelated. This assumption may seem surprising for a crystal. The electron scatters off a lattice by exciting it, i.e., transferring energy and momentum to the lattice. This process of electronlattice interactions corresponds to the creation and absorption of phonons, which are the elementary excitations of the crystal lattice. In this way the electronlattice interaction is equivalent to the scattering of electrons by phonons. At temperatures well below the Debye temperature, \(m_ \star ^{\rm{f}} \sim 10  15{m_n}\), the gas of phonons is dilute, and eN scattering, represented by I_{eN}[f], is actually the electron scattering by single phonons. These phonons form a Bose gas, and their number density and mean energy depend on T.
In the absence of external forces, the solution of Equation (186) is the FermiDirac distribution function, f^{(0)}, corresponding to full thermodynamic equilibrium. The collision integrals then vanish, I_{ej}[f^{(0)}] = 0, with j = N, e, imp. We shall now show how to calculate the conductivities κ and σ. Let us consider small stationary perturbations characterized by gradients of temperature ∇T, of the electron chemical potential ∇μ_{e}, and let us apply a weak constant electric field E. The plasma will become slightly nonuniform, with weak charge and heat currents flowing through it. We assume that the length scale of this nonuniformity is much larger than the electron mean free path. Therefore, any plasma element will be close to a local thermodynamic equilibrium. However, gradients of T and μ_{e}, as well as E, will induce a deviation of f from f^{(0)} and will produce heat and charge currents.
The next step consists in writing f = f^{(0)} + δf, where δf is a small correction to f^{(0)} linear in ∇T, ∇μ_{e}, and E. We introduce the enthalpy per electron h = μ_{e} + (S_{e}/n_{e})T, where S_{e} is the electron entropy density. The linearized lefthand side of Equation (186) is then
The general form of δf is [435]
where Φ is a slowly varying function of electron energy ϵ_{e}. This specific form of δf results from the BE, and deserves a comment. In the limit of strong electron degeneracy T ≲ T_{Fe} where T_{Fe} is given by Equation (5), the derivative ∂f^{(0)}/∂ϵ is strongly peaked at ϵ ≈ μ_{e}. Actually, for T/T_{Fe} → 0 we get ∂f^{(0)}/∂ϵ → −δ(ϵ − μ_{e}). On the contrary, Φ is a slowly varying function of ϵ.
In our case, the general form of Φ linear in ∇T, ∇μ_{e} and E can be written as [435]
where
and the coefficients A_{e}(ϵ) and A_{T}(ϵ) are functions of electron energy ϵ. They fully determine κ and σ. However, to determine them, we have to linearize the collision integrals with respect to Φ, and then solve the linearized BE.
As the nuclei are very heavy compared to the electrons, the typical electron energy transferred during a collision is much smaller than k_{B}T. The collision integral then takes the simple form (see, e.g., Ziman [435])
This is the relaxation time approximation, and τ_{0}(ϵ) is an effective relaxation time for the electron distribution function at an energy ϵ. For strongly degenerate electrons we should put ϵ = μ_{e} in the argument of τ_{0} (ϵ).
This simple relaxation time approximation breaks down at T ≲ T_{pi}, when the quantum effects in the phonon gas become pronounced so that the typical energies transferred become ∼ k_{B}T (and the number of phonons becomes exponentially small). The dominance of electronphonon scattering breaks down at very low T. Simultaneously, I_{ee} has a characteristic lowT ≪ T_{Fe} behavior I_{ee} ∝ (T/T_{Fe})^{2}. All this implies the dominance of the eimpurity scattering in the lowT limit, I_{imp} ≫I_{ee},I_{eN}.
The scattering of electrons on ions (nuclei) can be calculated from the Coulomb interaction, including medium effects (screening). An effective scattering frequency of an electron of energy ϵ, denoted v_{eN}(ϵ), is related to the corresponding transport scattering cross section σ_{tr}(ϵ) by
where υ = ∂ϵ/∂p is electron velocity. The transport scattering cross section is calculated by the integration of the differential scattering cross section,
where ϑ is the electron scattering angle.
The electronnucleus scattering is quasielastic at T > T_{pi}, with electron energy change δϵ ≪ k_{B}T. The function σ_{tr}(ϵ) can be calculated, including screening and relativistic effects. After scattering, the electron momentum changes by ħ_{q} within q_{min} ≤ q ≤ q_{max}. Therefore, the formula for σ_{tr}(ϵ) can be rewritten as
where Λ_{eN}(ϵ) is the famous Coulomb logarithm of the plasma transport theory. The Coulomb logarithm is directly related to the Fourier transform of the complete electronnucleus interaction, ϕ_{q}, by
where q_{min} is related to the screening and q_{max} = 2_{pF}/ħ for strongly degenerate electrons. In a concise numerical form
where γ_{r} is given by Equation (6). In the relaxation time approximation the calculation of transport coefficients reduces to the calculation of the Coulomb logarithm.
A second important approximation (after the relaxation time one) is expressed as the Matthiessen rule. In reality, the electrons scatter not only off nuclei (eN), but also by themselves (ee), and off randomly distributed impurities (imp), if there are any. The Matthiessen rule (valid under strong degeneracy of electrons) states that the total effective scattering frequency is the sum of frequencies on each of the scatterers.
For heat conduction the Matthiessen rule gives, for the total effective scattering frequency of electrons,
Notice that, as ee scattering does not change the electric current, it will not contribute to electrical conductivity and to ν_{σ}, so that the Matthiessen rule gives
Electron scattering on randomly distributed impurities in some lattice sites is similar to the scattering by ions with charge Z − Z_{imp}. The scattering frequency of electrons by impurities is
where n_{imp} is the number density of impurities of a given type “imp” and the sum is over all types “imp”. Detailed calculations of \(T < {1 \over 4}\Theta\) and \(\nu_{e{\rm{N}}}^\kappa\) in liquid and solid plasma of neutron star envelopes, taking into account additional effects, such as electronband structures and multiphonon processes, are presented in [339]. There one can find analytic fitting formulae, which are useful for applications.
Recently, the calculation of \(\nu _{e{\rm{N}}}^\sigma\) has been revised, taking into account the Landau damping of transverse plasmons [378]. This effect strongly reduces \(\nu _{ee}^\kappa\) for ultrarelativistic electrons at T < T_{pe}.
In the presence of a magnetic field B, transport properties become anisotropic, as briefly described in Section 9.5.
9.3 Thermal and electrical conductivities
A small temperature gradient ∇T and a weak constant electric field E induce a heat current j_{T} and an electric current j_{e},
where Q_{T} is the thermopower, σ is electrical conductivity, and κ is thermal conductivity. Here, E* is an “effective electric field”, defined in Equation (190).
Both conductivities can be expressed in terms of the corresponding effective scattering frequency, calculated in the preceding section as
where the electron effective mass is given by Equation (1).
In the relaxation time approximation and for strongly degenerate electrons we get
so that the WiedemannFranz law is satisfied:
Let us remind ourselves, that the equality (204) is violated when the ee scattering is not negligible compared to electron scattering by nuclei and by impurities. This and other effects leading to violation of the WiedemannFranz law are discussed in [339].
The role of neutron gas in the inner crust deserves comment. Its normal component contributes to κ, so that
However, there are actually two contributions to κ_{n}. The first one results from the scattering and is therefore of a standard “diffusive” nature. This contribution to κ_{n} is
where n_{n} is the number density of the gas of “neutron excitations” (neutrons are strongly degenerate and moreover superfluid), m*_{n} is their effective mass, and ν_{n} is their scattering frequency. Neutron excitations scatter by nuclear clusters and by themselves via strong interactions,
However, due to a much smaller neutronneutron cross section as compared to the neutroncluster one, and due to the low density of neutron excitations, we get ν_{nN} ≫ ν_{nn}, so that ν_{n} ≈ ν_{nN}. The second contribution to κ_{n} is characteristic of superfluids and has a convective character (“convective counterflow”, see, e.g., Tilley & Tilley [403]); we denote it by \(\nu _{ee}^\kappa\).
It can be noted that for T < T_{m} “neutron excitations” scatter by the lattice phonons. Complete calculation of κ_{n}, taking due account of the effect of the crystal lattice on neutron scattering and neutron superfluidity remains to be done.
The presence of impurities considerably decreases electrical and thermal conductivities at low temperature and high density; see Figure 54. At T = 10^{7} K, 5% of impurities with Z_{imp} − Z = 4 reduces σ and κ at ρ = 10^{13} g cm^{−3} by two orders of magnitude. Accreted crusts are characterized by nuclides with lower values of A and Z than those in the groundstate crust. Accordingly, accreted crusts have higher electrical and thermal conductivities than the groundstate crust of the same ρ and T. This is illustrated in Figure 55. Notice the differences between the σ and κ plots at 10^{8} K and 10^{9} K. They are due to an additional factor T in κ, reflected in the WiedermannFranz law (205).
Recent calculations of κ_{ee}, taking into account the Landau damping of transverse plasmons, give a much larger contribution from ee scattering than the previous ones, using the static screening, on which Figures 54 and 55 are based. As shown by Shternin and Yakovlev [378], the Landau damping of transverse plasmons strongly reduces \(\kappa _n^{{\rm{conv}}}\) in the inner crust at T ≲ 10^{7} K.
The contribution of ions to κ was recently calculated by Chugunov and Haensel [100], who also quote older papers on this subject. As a rule, κ_{N} can be neglected compared to κ_{e}. A notable exception, relevant for magnetized neutron stars, is discussed in Section 9.5.
9.4 Viscosity
In this section we consider the viscosity of the crust, which can be in a liquid or a solid phase. For strongly nonideal (Γ ≫ 1) and solid plasma the transport is mediated mainly by electrons. We will limit ourselves to this case only. For solid crust, we will assume a poly crystal structure, so that on a macroscopic scale the crust will behave as an isotropic medium.
Let us denote a stationary macroscopic hydrodynamic velocity field, imposed on the plasma, by U(r). In an isotropic plasma the viscous part of the stress tensor can be written as
where η is the shear viscosity and ζ is the bulk viscosity. The viscous component of the stress tensor enters the equations of neutron star hydrodynamics and is relevant for neutron star pulsations.
First, let us consider volume preserving flows, characterized by ∇ · U = 0. A schematic view of such a flow in the solid crust, characteristic of torsional oscillations of the crust, is shown in Figure 58. The dissipation resulting in the entropy production is determined by the shear viscosity η. In the outer crust, η is a sum of the electron and nuclei contributions, η = η_{e} + η_{N}, but for ρ > 10^{5} g cm^{−3} η_{e} ≫ η_{N} and η ≈ η_{e}.^{Footnote 13}. In the inner crust, an additional contribution from the normal component of the gas of dripped neutrons should be added.
The electrons are scattered on nuclei, on impurity nuclei, and on themselves, so that the effective frequency of their scattering is given by the sum \(\nu _e^\kappa\). However, as long as the temperature is not too low, the approximation ν_{e} ≈ ν_{eN} can be used.
To calculate η_{e} from the BE for electrons, we have to determine δf = f − f^{(0)} due to the presence of a weak plasma velocity field, U. The solution of the BE, linearized in U and in δf, has the form
where A_{η}(ϵ) is a function to be determined from the BE. In the relaxation time approximation and for strongly degenerate electrons \(\nu _e^\eta = \nu _{e{\rm{N}}}^\eta + \nu _{{\rm{imp}}}^\eta + \nu _{ee}^\eta\) is the effective relaxation time due to eN collisions, calculated at the electron Fermi surface.
The scattering frequency ν_{eN}, in turn, can be expressed in terms of the effective Coulomb logarithm Λ_{eN} by
Having calculated the Coulomb logarithm, we get the electron viscosity using a standard formula
At low temperature, impurities can become the main scatterers of electrons, with
In the inner crust, one must also consider the gas of dripped neutrons.
Calculations of the shear viscosity for the liquid phase were done by Flowers & Itoh [146, 147] and Nandkumar & Pethick [300]. Recently, calculations of the shear viscosity of the neutron star crust were done for both the liquid and the crystal phases, by Chugunov & Yakovlev [101]; their results are displayed in Figure 56. These authors also give analytic fitting formulae for the effective Coulomb logarithms, which can be used for different models of the crust. The electronimpurity scattering becomes dominant at low T, when electronlattice scattering (via phonons) is suppressed by quantum effects. This is visualized in Figure 57. Recently, the contribution to η_{e} resulting from the ee scattering, was recalculated by Shternin [377], who took into account the Landau damping of transverse plasmons. The Landau damping of transverse plasmons leads to a significant suppression of η_{ee} for ultrarelativistic electrons, and modifies the temperature dependence of η_{ee}.
We are not aware of any calculations of the bulk viscosity of the crust, ζ. We just mention that it is generally assumed that the bulk viscosity of the crust is much smaller than the shear one, ζ ≪ η.
9.5 Transport in the presence of strong magnetic fields
We consider a surface magnetic field to be strong if B ≫ 10^{9} G. Such magnetic fields affect the accretion of plasma onto the neutron star and modify the properties of atoms in the atmosphere. On the contrary, a magnetic field B ≲ 10^{9} G, such as associated with millisecond pulsars or with most of the Xray bursters, is considered to be weak. Typical pulsars are magnetized neutron stars, with the value of B near the magnetic pole ∼ 10^{12} G. Much stronger magnetic fields are associated with magnetars, ∼ 10^{14}⊥10^{15} G; such magnetic fields with B ≳ 10^{14} G are often called “superstrong”. These magnetic fields can strongly affect transport processes within neutron star envelopes. Electron transport processes in magnetized neutron star envelopes and crusts are reviewed in [338, 412]. In the present section we limit ourselves to a very brief overview.
Locally, a magnetic field can be considered uniform. We will choose the z axis of a coordinate system along B, so that B = [0, 0, B]. We will limit ourselves to the case of strongly degenerate electrons and we will assume the validity of the relaxation time approximation. Let relaxation time for B = 0 be τ_{0}. An important timescale associated with magnetic fields is the electron gyromagnetic frequency^{Footnote 14}
The magnetic field bends electron trajectories in the (x, y) plane, and suppresses the electron transport across B. Therefore, the transport properties become anisotropic, and we must consider tensors κ_{ij} and σ_{ij}, which can be written as
In what follows we will consider three basic regimes of transport in magnetic fields.
9.5.1 Nonquantizing magnetic fields
Many Landau orbitals are populated and quantum effects are smeared by thermal effects because k_{B}T > ħω_{ce}. Transport along the magnetic field is not affected by B, while transport across B is fully described by the Hall magnetization parameters \({A_\eta} = \tau _{e{\rm{N}}}^\eta = 1/{\nu _{e{\rm{N}}}}\),
where \({\omega _B}\tau _0^{\kappa, \sigma}\) are the effective relaxation times at B = 0 for the thermal and electric conductivities, respectively. The nonzero components of the \(\tau _0^{\kappa, \sigma}\) tensors are
As an example, consider strongly degenerate electrons and dominating eN scattering, τ_{0} ≈ τ_{eN}. Then, Equation (196) yields \(\xi _{ij}^{\kappa, \sigma}\).
9.5.2 Weaklyquantizing magnetic fields
Many Landau levels are populated by electrons, but quantization effects are well pronounced because k_{B}T < ħω_{ce}. There are two relaxation times, \({\omega _B}\tau _0^{\kappa, \sigma} \approx 1003{B_{12}}/(\gamma _{\rm{r}}^2{\Lambda _{e{\rm{N}}}})\) and \(\tau _\parallel ^{\kappa, \sigma}\), which oscillate with density (see below). As shown by Potekhin [338], the formulae for the nonzero components of the \(\tau _ \bot ^{\kappa, \sigma}\) tensors can be written in a form similar to Equation (218):
At a fixed temperature and in the presence of a weakly quantizing B, the density dependence of the components of the σ and κ tensors exhibits characteristic oscillations around the nonquantized (classical) values. Each oscillation corresponds to the filling of a new Landau level. The amplitude of these oscillations decreases with decreasing density. An example of the density dependence of σ_{∥} ≡ σ_{zz} and σ_{⊥} ≡ σ_{xx} = σ_{yy}, and of the same components of the κ_{ij} tensor, is presented in Figure 59. As we see in Figure 59, the “density period” of oscillation decreases with increasing ρ, and the oscillation amplitude decreases with increasing T. At T = 10^{7} K, a magnetic field of 10^{12} G is weakly quantizing at ρ > 10^{4.2} g cm^{−3}.
9.5.3 Stronglyquantizing magnetic fields
Not only is k_{B}T < ħω_{ce}, but also most of the electrons are populating the ground Landau level. Both the values of σ_{ij} and κ_{ij} and their density dependence are dramatically different from those of the nonquantizing (classical) case. As shown by Potekhin [338], the formulae for σ_{ij} and κ_{ij} are still given by Equations (219). Analytical fitting formulae for \(\xi _{ij}^{\kappa, \sigma}\) and \(\tau _\parallel ^{\kappa, \sigma}\) are given in [338]. As seen in Figure 59, at T = 10^{6} K a field of 10^{12} G is strongly quantizing for ρ < 10^{4.2} g cm^{−3}
9.5.4 Possible dominance of ion conduction
Thermal conduction by ions is much smaller than that by electrons along B. However, the electron conduction across B is strongly suppressed. In outer neutron star crust, heat flow across B can be dominated by ion/phonon conduction [100]. This is important for the heat conduction across B in cooling magnetized neutron stars. Correct inclusion of the ion heat conductivity then leads to a significant reduction of the thermal anisotropy in the envelopes of magnetized neutron stars [100].
10 Macroscopic Model of Neutron Star Crusts
The understanding of many observed phenomena occurring in neutron stars (and briefly reviewed in Section 12, for instance, pulsar glitches or torsional oscillations in Soft Gamma Repeaters) requires modeling the dynamic evolution of the crust. So far theoretical efforts have been mainly devoted to modeling the dynamic evolution of the liquid core by considering a mixture of superfluid neutrons and superconducting protons (see, for instance, the recent review by Andersson & Comer [15]).
Macroscopic models of neutron star crusts, taking into account the presence of the neutron superfluid at ρ > ρ_{ND} (see Section 8), have been developed by Carter and collaborators. They have shown how to extend the twofluid picture of neutron star cores [93] to the inner crust layers in the Newtonian framework [79, 94]. They have also discussed how to calculate the microscopic coefficients of this model [78, 77]. More elaborate models treating the crust as a neutron superfluid in an elastic medium and taking into account the effects of a frozenin magnetic field have been very recently developed both in general relativity [73, 85] and in the Newtonian limit [73, 72]. All these models are based on an action principle that will be briefly reviewed in Section 10.1. We will consider a simple nonrelativistic twofluid model of neutron star crusts in Section 10.2 using the fully4D covariant formulation of Carter & Chamel [74, 75, 76]. Entrainment effects and superfluidity will be discussed in Sections 10.3 and 10.4, respectively.
10.1 Variational formulation of multifluid hydrodynamics
In the convective variational approach of hydrodynamics developed by Carter [70, 80], and recently reviewed by Gourgoulhon [174] and Andersson & Comer [15], the dynamic equations are obtained from an action principle by considering variations of the fluid particle trajectories. First developed in the context of general relativity, this formalism has been adapted to the Newtonian framework in the usual 3+1 spacetime decomposition by Prix [342, 343]. As shown by Carter & Chamel in a series of papers [74, 75, 76], the Newtonian hydrodynamic equations can be written in a very concise and elegant form in a fully4D covariant framework. Apart from facilitating the comparison between relativistic and nonrelativistic fluids, this approach sheds a new light on Newtonian hydrodynamics following the steps of Elie Cartan, who demonstrated in the 1920’s that the effects of gravitation in Newtonian theory can be expressed in geometric terms as in general relativity.
The variational formalism of Carter provides a very general framework for deriving the dynamic equations of any fluid mixture and for obtaining conservation laws, using exterior calculus. In particular, this formalism is very well suited to describing superfluid systems, like laboratory superfluids or neutron star interiors, by making a clear distinction between particle velocities and the corresponding momenta (see the discussion in Section 8.3.6).
The dynamics of the system (either in relativity or in the Newtonian limit) is thus governed by a Lagrangian density Λ, which depends on the particle 4currents \(\tau _ \bot ^{\kappa, \sigma}\), where n_{X} and \(n_{\rm{X}}^\mu = {n_{\rm{X}}}u_{\rm{X}}^\mu\) are the particle number density and the 4velocity of the constituent X, respectively. We will use Greek letters for spacetime indices with the Einstein summation convention for repeated indices. The index X runs over the different constituents in the system. Note that repeated chemical indices X will not mean summation unless explicitly specified.
The dynamic equations for a mixture of several interacting fluids (coupled by entrainment effects) can be obtained by requiring that the action integral
(where \(u_{\rm{X}}^\mu\) is the 4volume element) be stationary under variations of the 4currents \(\mathcal{V}\). These variations are not arbitrary because they have to conserve the number of particles. In classical mechanics of pointlike particles, the equations of motion can be deduced from an action integral by considering displacements of the particle trajectories. Likewise, considering variations of the 4currents induced by displacements of the fluidparticle worldlines yield
where ∇_{μ} denotes the covariant derivative. \(n_{\rm{X}}^\mu\), defined by
is the 4momentum per particle associated with the 4current \(\pi _\mu ^{\rm{X}}\), \(n_{\rm{X}}^\mu\) is the vorticity 2form defined by the exterior derivative of the corresponding 4momentum
and \(\varpi _{\nu \mu}^{\rm{X}}\) is the (nongravitational) 4force density acting on the constituent X. Equation (222) is the generalization to fluids of the definition introduced in classical Lagrangian mechanics for the momentum of pointlike particles. This equation shows that momentum and velocity are intrinsically different mathematical objects since the former is a covector while the latter is a vector. The vorticity 2form is closely analogous to the electromagnetic 2form F_{μν}. Equation (221) is the covariant generalization of Euler’s equation to multifluid systems. The stressenergymomentum tensor of this multifluid system is given by
where the generalized pressure Ψ is defined by
Note that so far we have made no assumptions regarding the spacetime geometry so that Equations (221), (224) and (225) are valid for both relativistic and nonrelativistic fluids. The presence of a frozenin magnetic field and the elasticity of the solid crust can be taken into account within the same variational framework both in (special and general) relativity [73, 85] and in the Newtonian limit [73, 72].
10.2 Twofluid model of neutron star crust
In this section, we will review the simple model for neutron star crust developed by Carter, Chamel & Haensel [79] (see also Chamel & Carter [94]). The crust is treated as a twofluid mixture containing a superfluid of free neutrons (index f) and a fluid of nucleons confined inside nuclear clusters (index c), in a uniform background of degenerate relativistic electrons. This model includes the effects of stratification (variation of the crust structure and composition with depth; see Section 3) and allows for entrainment effects (Section 8.3.7) that have been shown to be very strong [90]. However, this model does not take into account either the elasticity of the crust or magnetic fields. For simplicity, we will restrict ourselves to the case of zero temperature and we will not consider dissipative processes (see, for instance, Carter & Chamel [76], who have discussed this issue in detail and have proposed a threefluid model for hot neutron star crust).
This model has been developed in the Newtonian framework, since relativistic effects are expected to be small in crust layers, but using a 4D fullycovariant formulation in order to facilitate the link with relativistic models of neutron star cores [93]. In Newtonian theory, the 4velocities are defined by
t being the “universal” time. The components of the 4velocity vectors have the form \(f_\mu ^{\rm{X}}\), \(u_{\rm{X}}^0 = 1\) in “Aristotelian” coordinates (representing the usual kind of 3+1 spacetime decomposition with respect to the rest frame of the star). This means that the time components of the 4currents are simply equal to the corresponding particle number densities \(u_{\rm{X}}^i = \upsilon _{\rm{X}}^i\), while the space components are those of the current 3vector \(n_{\rm{X}}^0 = {n_{\rm{X}}}\) (using Latin letters i, j,… for the space coordinate indices).
The basic variables of the twofluid model considered here are the particle 4current vectors \(n_{\rm{X}}^i = {n_{\rm{X}}}\upsilon _{\rm{X}}^i\), \(n_{\rm{c}}^\mu\) and the number density n_{N} of nuclear clusters, which accounts for stratification effects. For clusters with mass number A, we have the relation n_{c} = An_{N}. In the following we will neglect the small neutronproton mass difference and we will write simply m for the nucleon mass (which can be taken as the atomic mass unit, for example). The total mass density is thus given by ρ = m(n_{c} + n_{f}). The Lagrangian density Λ, which contains the microphysics of the system, has been derived by Carter, Chamel & Haensel [