We consider spin-unpolarized plasma. Then does not depend on
and the BE for electrons reads
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
In our case, the general form of linear in
,
and
can be written as [435
]
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 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
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
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 Electron 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
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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