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9.3 Thermal and electrical conductivities

A small temperature gradient ∇∇∇T and a weak constant electric field EEE induce a heat current jjjT and an electric current jjje,
jjjT = − QT Tjjje − κ∇∇∇T , (200 )
jjje = σEEE ∗ + σQT ∇∇∇T , (201 )
where Q T is the thermopower, σ is electrical conductivity, and κ is thermal conductivity. Here, EEE ∗ is an “effective electric field”, defined in Equation (190View Equation).

Both conductivities can be expressed in terms of the corresponding effective scattering frequency, calculated in the preceding section as

e2n σ = --∗-e-, (202 ) m eνσ
2 2 κ = π--kBT-ne, (203 ) 3m ∗eν κ
where the electron effective mass is given by Equation (1View Equation).

In the relaxation time approximation and for strongly degenerate electrons we get

νκ = νσ ≃ νeN(μe), (204 )
so that the Wiedemann–Franz law is satisfied:
π2k2BT κ = ----2-σ . (205 ) 3e
Let us remind ourselves, that the equality (204View Equation) is violated when the ee scattering is not negligible compared to electron scattering by nuclei and by impurities. This and other effects leading to violation of the Wiedemann–Franz law are discussed in [339].
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Figure 54: Electrical conductivity σ and thermal conductivity κ of the outer and inner crust, calculated for the ground-state model of Negele & Vautherin [303Jump To The Next Citation Point]. Labels 7 and 8 refer to log T [K ] = 7 10 and 8, respectively. The thin line with label 7 corresponds to an impure crust, which contains in the lattice sites 5% impurities – nuclei with |Zimp − Z | = 4. Based on a figure made by A.Y. Potekhin.
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Figure 55: The same as in Figure 54View Image but calculated for the accreted-crust model of Haensel & Zdunik [185]. Results obtained assuming 5% of nuclei – impurities with |Zimp − Z | = 2. Based on a figure made by A.Y. Potekhin.

The role of neutron gas in the inner crust deserves comment. Its normal component contributes to κ, so that

κ = κe + κn . (206 )
However, there are actually two contributions to κn. The first one results from the scattering and is therefore of a standard “diffusive” nature. This contribution to κ n is
2 2 κdiff = π--kBT-nn , (207 ) n 3m ∗nνn
where nn is the number density of the gas of “neutron excitations” (neutrons are strongly degenerate and moreover superfluid), ∗ m n is their effective mass, and νn is their scattering frequency. Neutron excitations scatter by nuclear clusters and by themselves via strong interactions,
ν = ν + ν . (208 ) n nN nn
However, due to a much smaller neutron-neutron cross section as compared to the neutron-cluster one, and due to the low density of neutron excitations, we get νnN ≫ νnn, so that νn ≈ νnN. The second contribution to κ n is characteristic of superfluids and has a convective character (“convective counterflow”, see, e.g., Tilley & Tilley [403]); we denote it by conv κ n.

It can be noted that for T < Tm “neutron excitations” scatter by the lattice phonons. Complete calculation of κn, taking due account of the effect of the crystal lattice on neutron scattering and neutron superfluidity remains to be done.

The presence of impurities considerably decreases electrical and thermal conductivities at low temperature and high density; see Figure 54View Image. At 7 T = 10 K, 5% of impurities with |Zimp − Z | = 4 reduces σ and κ at ρ = 1013 g cm − 3 by two orders of magnitude. Accreted crusts are characterized by nuclides with lower values of A and Z than those in the ground-state crust. Accordingly, accreted crusts have higher electrical and thermal conductivities than the ground-state crust of the same ρ and T. This is illustrated in Figure 55View Image. Notice the differences between the σ and κ plots at 108 K and 109 K. They are due to an additional factor T in κ, reflected in the Wiedermann–Franz law (205View Equation).

Recent calculations of κee, taking into account the Landau damping of transverse plasmons, give a much larger contribution from ee scattering than the previous ones, using the static screening, on which Figures 54View Image and 55View Image are based. As shown by Shternin and Yakovlev [378], the Landau damping of transverse plasmons strongly reduces κ νe in the inner crust at 7 T ≲ 10 K.

The contribution of ions to κ was recently calculated by Chugunov and Haensel [100Jump To The Next Citation Point], who also quote older papers on this subject. As a rule, κN can be neglected compared to κe. A notable exception, relevant for magnetized neutron stars, is discussed in Section 9.5.

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