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

where is the electron velocity, and is the external force acting on the electrons (e.g., electrostatic force , where electron electric charge is ). The total collision integral , which is a functional of , is a sum of collision integrals corresponding to collisions of electrons with nuclei, electrons, and impurities in the crystal lattice, .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, , the gas of phonons is dilute, and e-N scattering, represented by , is actually the electron scattering by single phonons. These phonons form a Bose gas, and their number density and mean energy depend on .

In the absence of external forces, the solution of Equation (186) is the Fermi–Dirac distribution function, , corresponding to full thermodynamic equilibrium. The collision integrals then vanish, , with . We shall now show how to calculate the conductivities and . Let us consider small stationary perturbations characterized by gradients of temperature , of the electron chemical potential , and let us apply a weak constant electric field . 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 and , as well as , will induce a deviation of from and will produce heat and charge currents.

The next step consists in writing , where is a small correction to , linear in , , and . We introduce the enthalpy per electron , where is the electron entropy density. The linearized left-hand side of Equation (186) is then

The general form of is [435] where is a slowly varying function of electron energy . This specific form of results from the BE, and deserves a comment. In the limit of strong electron degeneracy where is given by Equation (5), the derivative is strongly peaked at . Actually, for we get . On the contrary, is a slowly varying function of .In our case, the general form of linear in , and can be written as [435]

where and the coefficients and are functions of electron energy . They fully determine and . However, to determine them, we have to linearize the collision integrals with respect to , 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 . The collision integral then takes the simple form (see, e.g., Ziman [435])

This is the relaxation time approximation, and is an effective relaxation time for the electron distribution function at an energy . For strongly degenerate electrons we should put in the argument of .This simple relaxation time approximation breaks down at , when the quantum effects in the phonon gas become pronounced so that the typical energies transferred become (and the number of phonons becomes exponentially small). The dominance of electron-phonon scattering breaks down at very low . Simultaneously, has a characteristic low- behavior . All this implies the dominance of the e-impurity scattering in the low- limit, .

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 , denoted , is related to the corresponding transport scattering cross section by

where is electron velocity. The transport scattering cross section is calculated by the integration of the differential scattering cross section, where is the electron scattering angle.The electron-nucleus scattering is quasi-elastic at , with electron energy change . The function can be calculated, including screening and relativistic effects. After scattering, the electron momentum changes by within . Therefore, the formula for can be rewritten as

where 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, , by where is related to the screening and for strongly degenerate electrons. In a concise numerical form where is given by Equation (6). 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 (), but also by themselves (), 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,

Notice that, as scattering does not change the electric current, it will not contribute to electrical conductivity and to , so that the Matthiessen rule givesElectron scattering on randomly distributed impurities in some lattice sites is similar to the scattering by ions with charge . The scattering frequency of electrons by impurities is

where is the number density of impurities of a given type “imp” and the sum is over all types “imp”. Detailed calculations of and 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 [339]. There one can find analytic fitting formulae, which are useful for applications.Recently, the calculation of has been revised, taking into account the Landau damping of transverse plasmons [378]. This effect strongly reduces for ultrarelativistic electrons at .

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

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