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9.2 Boltzmann equation for electrons and its solutions

The electron distribution function is f(ppp,rrr,t,s), where ppp and rrr are electron momentum and position vectors, respectively, t is time, and s is the electron spin projection on the spin quantization axis. The distribution function f(ppp,rrr,t,s) satisfies the Boltzmann equation (BE) for electrons. At first glance, the validity of the BE (originally derived for a gas of particles) for a super dense plasma of electrons may seem paradoxical. However, electrons are strongly degenerate, so that only electron states in a thin shell around the chemical potential μe with energies |ε − μe| ≲ kBT are involved in the transport phenomena. In other words, the gas of “electron excitations” is dilute. Also, the infinite range of Coulomb interactions in vacuum is no longer a problem in dense electron-nuclear plasma because of screening. Moreover, as the kinetic Fermi energy of electrons is much larger than the Coulomb energy per electron, the Coulomb energy can be treated as a small perturbation and the electrons can be considered as a quasi-ideal Fermi gas.

We consider spin-unpolarized plasma. Then f does not depend on s and the BE for electrons reads

∂f- ∂f- ∂f- ∂t + vvv ⋅∂rrr + FFF ⋅∂ppp = Ie[f], (186 )
where vvv is the electron velocity, and FFF is the external force acting on the electrons (e.g., electrostatic force FFF = − eEEE, where electron electric charge is − e ). The total collision integral Ie[f ], which is a functional of f, is a sum of collision integrals corresponding to collisions of electrons with nuclei, electrons, and impurities in the crystal lattice, I [f] = I [f ] + I [f ] + I [f] e eN ee imp.

The additivity of partial collision integrals is valid when the scatterers are uncorrelated. This assumption may seem surprising for a crystal. The electron scatters off a lattice by exciting it, i.e., transferring energy and momentum to the lattice. This process of electron-lattice interactions corresponds to the creation and absorption of phonons, which are the elementary excitations of the crystal lattice. In this way the electron-lattice interaction is equivalent to the scattering of electrons by phonons. At temperatures well below the Debye temperature, T < 1Θ 4, the gas of phonons is dilute, and e-N scattering, represented by I [f] eN, is actually the electron scattering by single phonons. These phonons form a Bose gas, and their number density and mean energy depend on T.

In the absence of external forces, the solution of Equation (186View Equation) is the Fermi–Dirac distribution function, f(0), corresponding to full thermodynamic equilibrium. The collision integrals then vanish, (0) Iej[f ] = 0, with j = N, e,imp. We shall now show how to calculate the conductivities κ and σ. Let us consider small stationary perturbations characterized by gradients of temperature ∇∇∇T, of the electron chemical potential ∇∇∇ μe, and let us apply a weak constant electric field EEE. The plasma will become slightly nonuniform, with weak charge and heat currents flowing through it. We assume that the length scale of this nonuniformity is much larger than the electron mean free path. Therefore, any plasma element will be close to a local thermodynamic equilibrium. However, gradients of T and μe, as well as EEE, will induce a deviation of f from f (0) and will produce heat and charge currents.

The next step consists in writing f = f(0) + δf, where δf is a small correction to f(0), linear in ∇∇∇T, ∇∇∇ μe, and EEE. We introduce the enthalpy per electron h = μe + (Se∕ne)T, where Se is the electron entropy density. The linearized left-hand side of Equation (186View Equation) is then

[ ] ε −-h ∇-μe- Se-∇∇∇T-- ∂f-(0) − T ∇∇∇T + EEE + e + ne e ⋅vvv ∂ ε . (187 )
The general form of δf is [435Jump To The Next Citation Point]
∂f (0) Φ (0) (0) δf = Φ----- = − ----f (1 − f ), (188 ) ∂ε kBT
where Φ is a slowly varying function of electron energy εe. This specific form of δf results from the BE, and deserves a comment. In the limit of strong electron degeneracy T ≪ TFe where TFe is given by Equation (5View Equation), the derivative ∂f (0)∕∂ε is strongly peaked at ε ≈ μ e. Actually, for T ∕TFe −→ 0 we get (0) ∂f ∕∂ε −→ − δ(ε − μe). On the contrary, Φ is a slowly varying function of ε.

In our case, the general form of Φ linear in ∇∇∇T, ∇∇∇ μe and EEE can be written as [435Jump To The Next Citation Point]

ε − h ∗ Φ = ----- AT(ε) vvv ⋅∇∇∇T + eAe(ε) vvv ⋅EEE , (189 ) T
where
∗ ∇-μe- Se-∇∇∇T-- EEE = EEE + e + ne e , (190 )
and the coefficients Ae(ε) and AT (ε) are functions of electron energy ε. They fully determine κ and σ. However, to determine them, we have to linearize the collision integrals with respect to Φ, and then solve the linearized BE.

As the nuclei are very heavy compared to the electrons, the typical electron energy transferred during a collision is much smaller than kBT. The collision integral then takes the simple form (see, e.g., Ziman [435])

δf IeN[f ] ≈ − τ-(ε) . (191 ) 0
This is the relaxation time approximation, and τ0(ε) is an effective relaxation time for the electron distribution function at an energy ε. For strongly degenerate electrons we should put ε = μe in the argument of τ (ε) 0.

This simple relaxation time approximation breaks down at T ≲ Tpi, when the quantum effects in the phonon gas become pronounced so that the typical energies transferred become ∼ kBT (and the number of phonons becomes exponentially small). The dominance of electron-phonon scattering breaks down at very low T. Simultaneously, Iee has a characteristic low-T ≪ TFe behavior 2 Iee ∝ (T∕TFe ). All this implies the dominance of the e-impurity scattering in the low-T limit, Iimp ≫ Iee,IeN.

The scattering of electrons on ions (nuclei) can be calculated from the Coulomb interaction, including medium effects (screening). An effective scattering frequency of an electron of energy ε , denoted ν (ε) eN, is related to the corresponding transport scattering cross section σ (ε) tr by

1 νeN(ε) = -----= nN v σtr(ε), (192 ) τ0(ε)
where v = ∂ε∕∂p is electron velocity. The transport scattering cross section is calculated by the integration of the differential scattering cross section,
∫ π σtr(ε) = 2π dϑ sin ϑ σ (ε,ϑ) (1 − cos ϑ), (193 ) 0
where ϑ is the electron scattering angle.

The electron-nucleus scattering is quasi-elastic at T > Tpi, with electron energy change δε ≪ kBT. The function σtr(ε) can be calculated, including screening and relativistic effects. After scattering, the electron momentum changes by ℏq within qmin ≤ q ≤ qmax. Therefore, the formula for σtr(ε) can be rewritten as

2 4 σ (ε) = 4πZ--e-Λ (ε) , (194 ) tr p2v2 eN
where ΛeN(ε) is the famous Coulomb logarithm of the plasma transport theory. The Coulomb logarithm is directly related to the Fourier transform of the complete electron-nucleus interaction, φq, by
∫ qmax Λ (ε) = dqq3 | φ |2, (195 ) eN qmin q
where qmin is related to the screening and qmax = 2pF∕ℏ for strongly degenerate electrons. In a concise numerical form
1 5.70 × 10−17 s τeN = --------= --------------, (196 ) νeN(μe) γrZ ΛeN
where γ r is given by Equation (6View Equation). In the relaxation time approximation the calculation of transport coefficients reduces to the calculation of the Coulomb logarithm.

A second important approximation (after the relaxation time one) is expressed as the Matthiessen rule. In reality, the electrons scatter not only off nuclei (eN), but also by themselves (ee), and off randomly distributed impurities (imp), if there are any. The Matthiessen rule (valid under strong degeneracy of electrons) states that the total effective scattering frequency is the sum of frequencies on each of the scatterers.

For heat conduction the Matthiessen rule gives, for the total effective scattering frequency of electrons,

κ κ κ νκ = ν eN + νee + νimp . (197 )
Notice that, as ee scattering does not change the electric current, it will not contribute to electrical conductivity and to νσ, so that the Matthiessen rule gives
σ σ νσ = νeN + νimp. (198 )

Electron scattering on randomly distributed impurities in some lattice sites is similar to the scattering by ions with charge Z − Zimp. The scattering frequency of electrons by impurities is

4 ∑ νimp = 4πe---- (Z − Zimp)2nimp Λeimp , (199 ) p2FevFe imp
where nimp is the number density of impurities of a given type “imp” and the sum is over all types “imp”. Detailed calculations of νκeN and νσeN in liquid and solid plasma of neutron star envelopes, taking into account additional effects, such as electron-band structures and multi-phonon processes, are presented in [339Jump To The Next Citation Point]. There one can find analytic fitting formulae, which are useful for applications.

Recently, the calculation of νκee has been revised, taking into account the Landau damping of transverse plasmons [378Jump To The Next Citation Point]. This effect strongly reduces νκee for ultrarelativistic electrons at T < T pe.

In the presence of a magnetic field BBB, transport properties become anisotropic, as briefly described in Section 9.5.


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