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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 [184Jump To The Next Citation Point].

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

∗ 2 ∗ ( 2)1∕2 εFe = m ec , m e = me 1 + xr , (1 )
where xr is a dimensionless relativity parameter defined in terms of the Fermi momentum
pFe = ℏ(3π2ne)1∕3, (2 )
p xr = --Fe (3 ) mec
and m ∗ e is the electron effective mass at the Fermi surface. The Fermi velocity is
pFe xr vFe = --∗-= c--------1∕2-. (4 ) m e (1 + x2r)
Electrons are strongly degenerate for T ≪ TFe, where the electron Fermi temperature is defined by
εFe − mec2 TFe = -----------= 5.93 × 109 (γr − 1) K , (5 ) kB
kB is the Boltzmann constant and
∗ γ = me-= (1 + x2)1∕2 . (6 ) r me r
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
nN = ne∕Z . (7 )
For ρ < ρND, the quantities nN and A are related to the mass density of the crust by
ρ ≈ nNAmu , (8 )
where mu is the atomic mass unit, −24 mu = 1.6605 × 10 g. For ρ > ρND, one has to replace A in Equation (8View Equation) 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
( )1∕3 x = 1.00884 ρ6Z- , (9 ) r A′
where 6 −3 ρ6 ≡ ρ∕10 g cm.

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

( 2 2)1∕2 ωpi = 4πe-nNZ--- , (10 ) Amu
ℏ ω ( ρ Z2 )1∕2 Tpi = ---pi= 7.832 × 106 -6---- K . (11 ) kB A′ A
Quantum effects for ions become very important for T ≪ T pi. The electron plasma frequency is
( 4πe2n )1 ∕2 ωpe = ---∗-e- , (12 ) me
so that the electron plasma temperature
T = ℏ-ωpe = 3.300 × 108x3∕2γ−1∕2 K , (13 ) pe kB r r
which can be rewritten as
( )1∕2 [ ( )2 ∕3] −1∕4 T = 3.34 × 108 K ρ Z- × 1 + 1.02 ρ Z- . (14 ) pe 6A 6A
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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 [184Jump To The Next Citation Point].
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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 [183Jump To The Next Citation Point] for the outer crust, and Negele & Vautherin [303Jump To The Next Citation Point] for the inner crust. Dot lines: accreted crust, as calculated by Haensel & Zdunik [185Jump To The Next Citation Point]. Jumps result from discontinuous changes of Z and A. Dot-dash line: results obtained for the compressible liquid drop model of Douchin & Haensel [125Jump To The Next Citation Point] 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

Z2e2 ( ρ )1 ∕3 175 Tm = ----------≈ 1.3 × 105 Z2 -6′ ---- K , (15 ) RcellkB Γ m A Γ m
where the ion sphere (also called the unit cell or Wigner–Seitz cell) radius is
( ) 4 πnN − 1∕3 Rcell = --3--- . (16 )
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.
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