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9.5 Transport in the presence of strong magnetic fields

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Figure 59: Longitudinal (∥) and transverse (⊥) electrical and thermal conductivities in the outer envelope composed of 56Fe for B = 1012 G and log10T [K] = 6,7. Quantum calculations (solid lines) are compared with classical ones (dash lines). Vertical bars: liquid-solid transition at T = 107 K. Based on Figure 5 from [338Jump To The Next Citation Point].

We consider a surface magnetic field to be strong if 9 B ≫ 10 G. Such magnetic fields affect the accretion of plasma onto the neutron star and modify the properties of atoms in the atmosphere. On the contrary, a magnetic field B ≲ 109 G, such as associated with millisecond pulsars or with most of the X-ray bursters, is considered to be weak. Typical pulsars are magnetized neutron stars, with the value of BBB near the magnetic pole ∼ 1012 G. Much stronger magnetic fields are associated with magnetars, ∼ 1014 – 1015 G; such magnetic fields with 14 B ≳ 10 G are often called “super-strong”. These magnetic fields can strongly affect transport processes within neutron star envelopes. Electron transport processes in magnetized neutron star envelopes and crusts are reviewed in [338Jump To The Next Citation Point412]. In the present section we limit ourselves to a very brief overview.

Locally, a magnetic field can be considered uniform. We will choose the z axis of a coordinate system along BBB, so that BBB = [0,0,B ]. We will limit ourselves to the case of strongly degenerate electrons and we will assume the validity of the relaxation time approximation. Let relaxation time for B = 0 be τ0. An important timescale associated with magnetic fields is the electron gyromagnetic frequency14

eB-- ωB = m∗c . (214 ) e
The magnetic field bends electron trajectories in the (x,y) plane, and suppresses the electron transport across BBB. Therefore, the transport properties become anisotropic, and we must consider tensors κij and σij, which can be written as
π2k2BT ne κ κij = -----∗--ξij , (215 ) 3m e
e2ne σij = ---∗ξiσj . (216 ) m e
In what follows we will consider three basic regimes of transport in magnetic fields.

9.5.1 Nonquantizing magnetic fields

Many Landau orbitals are populated and quantum effects are smeared by thermal effects because kBT > ℏ ωce. Transport along the magnetic field is not affected by BBB, while transport across BBB is fully described by the Hall magnetization parameters ωB τκ,σ 0,

B τ κ,σ ωB τκ0,σ ≈ 1760 --12---0----, (217 ) γr 10−16 s
where τκ,σ 0 are the effective relaxation times at BBB = 0 for the thermal and electric conductivities, respectively. The nonzero components of the κ,σ ξij tensors are
τκ,σ ω (τ κ,σ)2 ξzz = τκ0,σ , ξκx,σx = ξκy,yσ= ------0--κ,σ-- , ξκx,yσ = ξκy,σx = ---B--0-κ,σ--. (218 ) 1 + (ωB τ0 )2 1 + (ωB τ0 )2
As an example, consider strongly degenerate electrons and dominating eN scattering, τ ≈ τ 0 eN. Then, Equation (196View Equation) yields κ,σ 2 ωB τ0 ≈ 1003 B12 ∕(γrZΛeN ).

9.5.2 Weakly-quantizing magnetic fields

Many Landau levels are populated by electrons, but quantization effects are well pronounced because k T < ℏ ω B ce. There are two relaxation times, τ κ,σ ∥ and τ κ,σ ⊥, which oscillate with density (see below). As shown by Potekhin [338Jump To The Next Citation Point], the formulae for the nonzero components of the κ,σ ξij tensors can be written in a form similar to Equation (218View Equation):

κ,σ κ,σ κ,σ κ,σ -----τκ⊥,σ---- κ,σ κ,σ --ωB-(τκ⊥,σ)2-- ξzz = τ∥ , ξxx = ξyy = 1 + (ω τκ,σ)2 , ξxy = ξyx = 1 + (ω τκ,σ)2 . (219 ) B ⊥ B ⊥
At a fixed temperature and in the presence of a weakly quantizing BBB, the density dependence of the components of the σ and κ tensors exhibits characteristic oscillations around the nonquantized (classical) values. Each oscillation corresponds to the filling of a new Landau level. The amplitude of these oscillations decreases with decreasing density. An example of the density dependence of σ ∥ ≡ σzz and σ ⊥ ≡ σxx = σyy, and of the same components of the κij tensor, is presented in Figure 59View Image. As we see in Figure 59View Image, the “density period”of oscillation decreases with increasing ρ, and the oscillation amplitude decreases with increasing T. At T = 107 K, a magnetic field of 1012 G is weakly quantizing at 4.2 − 3 ρ > 10 g cm.

9.5.3 Strongly-quantizing magnetic fields

Not only is kBT < ℏωce, but also most of the electrons are populating the ground Landau level. Both the values of σij and κij and their density dependence are dramatically different from those of the nonquantizing (classical) case. As shown by Potekhin [338Jump To The Next Citation Point], the formulae for σij and κij are still given by Equations (219View Equation). Analytical fitting formulae for τ κ,σ ∥ and τ⊥κ,σ are given in [338]. As seen in Figure 59View Image, at 6 T = 10 K a field of 1012 G is strongly quantizing for 4.2 − 3 ρ < 10 g cm

9.5.4 Possible dominance of ion conduction

Thermal conduction by ions is much smaller than that by electrons along BBB. However, the electron conduction across BBB is strongly suppressed. In outer neutron star crust, heat flow across BBB can be dominated by ion/phonon conduction[100Jump To The Next Citation Point]. This is important for the heat conduction across BBB in cooling magnetized neutron stars. Correct inclusion of the ion heat conductivity then leads to a significant reduction of the thermal anisotropy in the envelopes of magnetized neutron stars[100].

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