Neglecting neutronproton mass difference, the beta equilibrium condition
reads At the density under consideration, electrons are ultrarelativistic, so that , where . It is now easy to show that the neutron drip density is roughly , which is about half of the value obtained in complete calculations [42, 183, 357]. For 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 (Thomas–Fermi 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.
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 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 corresponding to the density inside heavy nuclei. This implies that the volume and surface terms in Equation (34) are proportional to and , respectively. Each contribution to the energy, Equation (34), can then be parameterized in terms of the numbers and . The parameters are adjusted to the known experimental masses of nuclei, with . 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 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 and 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 is the energy density of homogeneous nuclear matter and and are respectively the neutron and proton densities inside the clusters. is the volume of the cluster. For consistency the energy density of the surrounding neutron gas is expressed in terms of the same function as where 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 and the chemical potential of neutrons adsorbed on the surface of the drop (forming a neutron skin) by where is the surface area of the cluster and 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 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 Wigner–Seitz approximation, assuming uniformlycharged spherical clusters of radius , the lattice energy is given by [39] The physical input of the liquid drop model is the energy density of homogeneous nuclear matter , the surface potential and the chemical potential . 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 is the surface tension and the neutron skin thickness. Unlike the surface area , 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 for a given baryon density imposing electric charge neutrality . The conditions of equilibrium are obtained by taking the partial derivative of the energy density 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 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 , ... 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 means the total pressure inside the drop, . 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 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 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 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 varies very little. Let us also notice that at the bottom of the crust the number 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.
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 , and 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 Thomas–Fermi 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 Thomas–Fermi 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 is the proton electric charge and is the electrostatic potential, which obeys Poisson’s equation, The quantum contribution 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 Slater–Kohn–Sham functional [243] The nuclear functional is less certain. Its local part is just a function of and , 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 Thomas–Fermi 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 and global electroneutrality
leads to Euler–Lagrange equations for the nucleon densities. In practice the unit cell is usually approximated by a sphere of the same volume . The boundary conditions are that the gradients of the densities and of the Coulomb potential vanish at the origin and on the surface of the sphere . Instead of solving the Euler–Lagrange 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 Hartree–Fock 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 .

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 Wigner–Seitz 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, , where for neutron and proton, respectively, and is the set of quantum numbers characterizing each single particle state. Inside the Wigner–Seitz sphere, these equations take the form
where is the single particle energy, is the dimensionless orbital–angularmomentum operator and is a vector composed of Pauli spin matrices. The effective masses , the mean fields and the spinorbit potentials depend on wave functions of all nucleons inside the sphere through the particle number densities the kinetic energy densities (in units of , where is the nucleon mass) and the spinorbit densitiesEquations (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 Hartree–Fock approximation, the Coulomb potential is the sum of a direct part , where 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 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 . 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 of neutrons and of protons and some initial guess of the effective masses and potentials, the equations are solved for the wave functions of neutrons and protons, which correspond to the lowest energies . 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 Wigner–Seitz sphere, and thus treating the electrons as a relativistic Fermi gas. Since the sphere is electrically neutral, the number of electrons is equal to and the electron energy is easily evaluated from Equation (22) with , where is the volume of the sphere. As for the choice of boundary conditions, Negele & Vautherin imposed that wave functions with even parity (even ) and the radial derivatives of wave functions with odd parity (odd ) vanish on the sphere . 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 and . The same magic numbers have been recently found by Onsi et al. [311] using a highspeed approximation to the Hartree–Fock 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 are strongly favored in neutron star crusts, while 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 energydensity–functional theory in the Wigner–Seitz 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 Wigner–Seitz 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 Wigner–Seitz 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).
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^{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 in the crust are characterized by a wave vector and obey the Floquet–Bloch theorem
where 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 having the full periodicity of the lattice, .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 Wigner–Seitz 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 Wigner–Seitz cell of a crystal lattice is a complicated polyhedron in general. For instance, the Wigner–Seitz 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 Floquet–Bloch theorem (61). This theorem also determines the boundary conditions to be imposed at the cell boundary.
For each wave vector , there exists only a discrete set of single particle energies , labeled by the principal quantum number , 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 (bands are labelled by increasing values of energy, so that if ). The band index is associated with the rotational symmetry of the nuclear clusters around each lattice site, while the wave vector 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 . Nevertheless, it can be shown by symmetry that the single particle states (and, therefore, the single particle energies) are periodic in space
where the reciprocal lattice vectors are defined by being any positive or negative integer. The discrete set of all possible reciprocal vectors defines a reciprocal lattice in space. Equation (64) entails that only the wave vectors lying inside the first Brillouin zone (i.e. Wigner–Seitz 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 , 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 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 Wigner–Seitz approximation [422]. The complicated Wigner–Seitz 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 Wigner–Seitz approximation overestimates the importance of neutron shell effects, as can be clearly seen in Figure 15. The energy spectrum is discrete in the Wigner–Seitz approximation (due to the neglect of the dependence 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 , at . Since these gaps scale approximately like (where is the neutron mass), they increase with density 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 Wigner–Seitz 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 . Besides the Wigner–Seitz 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 W–S cells to account for deformations of the nuclear clusters.
Whereas the Wigner–Seitz 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 dependence 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.
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