No magnetic field

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 one-component 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 $5 −3 ρ > 10 g cm$ . A neutron gas is also present at densities greater than neutron drip density $11 − 3 ρND ≈ 4 × 10 g cm$ .

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 $xr$ is a dimensionless relativity parameter defined in terms of the Fermi momentum

and $m ∗ e$ is the electron effective mass at the Fermi surface. The Fermi velocity is

Electrons are strongly degenerate for $T ≪ TFe$ , where the electron Fermi temperature is defined by

$kB$ 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 $ρ < ρND$ , the quantities $nN$ and $A$ are related to the mass density of the crust by

where $mu$ is the atomic mass unit, $−24 mu = 1.6605 × 10 g$ . For $ρ > ρND$ , 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 $6 −3 ρ6 ≡ ρ∕10 g cm$ .

The ion plasma temperature $Tpi$ is defined, in terms of the the ion plasma frequency

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

Figure 2:

Different parameter domains in the $ρ − T$ plane for ⁵⁶Fe plasma with magnetic field B = 10¹² G. Dash-dot line: melting temperature $Tm$ . Solid lines: $TF$ – Fermi temperature for the electrons (noted $TFe$ in the main text); $Tpi$ – ion plasma temperature. Long-dash lines: $TB$ and $ρB$ relevant for the quantized regime of the electrons (Section 2.2); for comparison we also show, by dotted lines, $TF$ , $Tm$ and $Tpi$ for B = 0. For further explanation see the text. From [184

Figure 3:

Left panel: melting temperature versus density. Right panel: electron and ion plasma temperature versus density. Solid lines: the ground-state composition of the crust is assumed: Haensel & Pichon [183

] for the outer crust, and Negele & Vautherin [303

] for the inner crust. Dot lines: accreted crust, as calculated by Haensel & Zdunik [185

]. Jumps result from discontinuous changes of Z and A. Dot-dash line: results obtained for the compressible liquid drop model of Douchin & Haensel [125

] for the ground state of the inner crust; a smooth behavior (absence of jumps) results from the approximation inherent in the compressible liquid drop model. Thick vertical dashes: neutron drip point for a given crust model. Figure made by A.Y. Potekhin.

The crystallization of a Coulomb plasma of ions occurs at the temperature

where the ion sphere (also called the unit cell or Wigner–Seitz cell) radius is

For classical ions ( $T ≳ Tpi$ ) one has $Γ m ≈ 175$ . For $T ≲ Tpi$ the zero-point quantum vibrations of ions becomes important and lead to crystal melting at lower $Γ m$ .