In dense, cold, neutron star crust, electron-charge–screening effects are negligible and the electron density is essentially uniform . The reason is that the electron Thomas–Fermi screening length is larger than the lattice spacing . 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 . At densities ( for iron), the electrons can be treated as a quasi-ideal Fermi gas so that
The lattice energy density can be estimated from the Wigner–Seitz approximation illustrated on Figure 5. The crust is decomposed into a set of independent spheres centered around each nucleus, with a radius defined by Equation (16). Each sphere is electrically neutral and therefore contains protons and 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 ). Assuming point-like nuclei since the lattice spacing is much larger than the size of the nuclei1, the lattice energy can be expressed as (see, for instance, Shapiro & Teukolsky , p30-31)
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 body-centered and face-centered cubic-lattices, 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 ). This shows that the equilibrium structure of the crust is expected to be a body-centered 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  suggests that such noncubic lattices may occur only at small densities, meaning that , while in the crust , where and 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 electron-exchange interactions, electron polarization and quantum zero point motion of the ions are discussed in the book by Haensel, Potekhin and Yakovlev .
The main physical input is the energy , which has been experimentally measured for more than 2000 known nuclei . 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 self-consistent mean field methods, have been calculated by the Brussels group and are available on line .
According to the first law of thermodynamics, the total pressure is given by
The structure of the ground state crust is determined by minimizing the total energy density for a given baryon density imposing charge neutrality, . However (or the average mass density ) 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 and at a fixed . This corresponds to minimization of the Gibbs free energy per nucleon, , under the condition of electric charge neutrality. For a completely-ionized one-component plasma, one constructs a table and then finds an absolute minimum in the plane. The procedure, based on the classical paper of Baym, Pethick and Sutherland , is described in detail in the book by Haensel, Potekhin and Yakovlev . Every time that the ground state shifts to a new nucleus with a smaller proton fraction, , there is a few percent jump of density at the and shell interface, 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 body-centered cubic (bcc) lattice of nuclei (lower density) and a layer with a pure bcc lattice of nuclei, a bcc lattice with nuclei at the corners of the conventional cube and 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 . At higher densities the nuclei are so neutron rich that they have not yet been experimentally studied, and the energy 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 , 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. Up-to-date theoretical mass tables are available online .
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