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3.1 Structure of the outer crust

Matter at densities below neutron drip ρND is not only relevant for neutron star crusts but also for white dwarfs. Following the classical paper of Baym, Pethick and Sutherland [42Jump To The Next Citation Point], the total energy density in a given layer can be written as
ɛtot = nNE {A, Z} + ɛe + ɛL, (21 )
where nN is the number density of nuclei, E {A,Z } is the energy of a nucleus with Z protons and A − Z neutrons, ɛe is the electron kinetic energy density and ɛL is the lattice energy density, which accounts for the electron-electron, electron-ion and ion-ion Coulomb interactions.

In dense, cold, neutron star crust, electron-charge–screening effects are negligible and the electron density is essentially uniform [418]. The reason is that the electron Thomas–Fermi screening length is larger than the lattice spacing [326Jump To The Next Citation Point]. 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 [282Jump To The Next Citation Point]. At densities ρ ≫ 10AZ g cm −3 (∼ 104 g cm −3 for iron), the electrons can be treated as a quasi-ideal Fermi gas so that

m4 c5 ( ∘ ------- ∘ ------- ) ɛe = ---e2-3 xr(2x2r + 1) x2r + 1 − ln{xr + x2r + 1} , (22 ) 8 π ℏ
with xr defined by Equation (3View Equation).

The lattice energy density can be estimated from the Wigner–Seitz approximation illustrated on Figure 5View Image. The crust is decomposed into a set of independent spheres centered around each nucleus, with a radius Rcell defined by Equation (16View Equation). 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 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 [373Jump To The Next Citation Point], p30-31)

9 ( 4π )1∕3 ɛL = − --- --- Z2 ∕3e2n4∕e3 . (23 ) 10 3
View Image

Figure 5: In the Wigner–Seitz approximation the crystal (represented here as a two-dimensional hexagonal lattice) is decomposed into independent identical spheres, centered around each site of the lattice. The radius of the sphere is chosen so that the volume of the sphere is equal to 1∕nN, where nN is the density of lattice sites (ions).

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 [258]). 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 [242] suggests that such noncubic lattices may occur only at small densities, meaning that r ∼ a e 0, while in the crust r ≪ a e 0, where 1∕3 re ≡ (3∕4πne ) and 2 2 a0 = ℏ ∕mee is the Bohr radius. Equation (23View Equation) 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 [184Jump To The Next Citation Point].

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 self-consistent mean field methods, have been calculated by the Brussels group and are available on line [211Jump To The Next Citation Point].

According to the first law of thermodynamics, the total pressure P is given by

d (ɛtot) P = n2b---- ---- . (24 ) dnb nb
Using Equation (21View Equation), the total pressure can be expressed as
1 P = Pe + -ɛL . (25 ) 3
The electron pressure Pe is defined by
Pe = ɛe − μene, (26 )
where the electron chemical potential μe is simply given by the electron Fermi energy εFe, Equation (1View Equation), and pFe, the electron Fermi momentum given by Equation (2View Equation). 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 ɛtot for a given baryon density nb = AnN imposing charge neutrality, np = ne. However nb (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 ) = (ɛtot + P) ∕nb, under the condition of electric charge neutrality. For a completely-ionized one-component 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 [42Jump To The Next Citation Point], is described in detail in the book by Haensel, Potekhin and Yakovlev [184Jump To The Next Citation Point]. 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,

Δnb-- Δ-ρ- Z-A-′ n ≈ ρ ≈ A Z ′ − 1, (27 ) b
which stems from a strict continuity of the pressure. It should be stressed that these density discontinuities are the direct consequence of the one-component 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 body-centered 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 10 − 3 ρ ∼ 6 × 10 g cm. 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. Up-to-date theoretical mass tables are available online [211].

Table 2: Sequence of nuclei in the ground state of the outer crust of neutron star calculated by Rüster et al. [357Jump To The Next Citation Point] using experimental nuclear data (upper part), and the theoretical mass table of the Skyrme model BSk8 (lower part).
ρmax [g cm–3] Element Z N Rcell [fm]
8.02 × 106 56Fe 26 30 1404.05
2.71 × 108 62Ni 28 34 449.48
1.33 × 109 64Ni 28 36 266.97
1.50 × 109 66Ni 28 38 259.26
3.09 × 109 86Kr 36 50 222.66
1.06 × 1010 84Se 34 50 146.56
2.79 × 1010 82Ge 32 50 105.23
6.07 × 1010 80Zn 30 50 80.58
8.46 × 1010 82Zn 30 52 72.77
9.67 × 1010 128Pd 46 82 80.77
1.47 × 1011 126Ru 44 82 69.81
2.11 × 1011 124Mo 42 82 61.71
2.89 × 1011 122Zr 40 82 55.22
3.97 × 1011 120Sr 38 82 49.37
4.27 × 1011 118Kr 36 82 47.92

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