Electron transport in fluids VS solids ("structure factor" VS "phonons")
Assume to have a neutral plasma consisting of some ions $X^+$ in a neutralizing gas of electrons $e^-$. Imagine that the $X^+$ can be in a gaseous (G), liquid (L) or solid (S) phase. We want to calculate the conductivity (electrical and/or thermal due to the electron gas) for a given phase1 in the <PERSON> equation framework: what are the differences?
More precisely, I'd like to understand which are the collision processes that we have to include in the <PERSON> equation for the electrons in the three phases G, L, S:
$$ \frac{\partial f_e}{\partial t} + \frac{\mathbf{p} }{m_e} \cdot \nabla f_e + \mathbf{F} \cdot \frac{\partial f_e}{\partial \mathbf{p} } = \left(\frac{d f_e}{d t} \right)_\text{coll}$$
where $ f_e(\mathbf{x},\mathbf{p},t)$ is the distribution function over the one-electron phase space. This is a semiclassical extension (see this paper) of the original <PERSON> equation for a gas of classical particles, even though also a more appropriate quantum version exists. Question: Do we have to consider different collision integrals for the different phases G, L, S ?
* G: in this case, the system is dilute and there is randomization after each collision ("molecular chaos"), so <PERSON> should be applicable. I would say that we have to include $e^--e^-$ binary collisions and $X^+-e^-$ collisions, described by the scattering kernel $W_{eX}$ (there should be also $X^+-X^+$ collisions but I ignore them, as we are interested in the electrons):
$$ \left(\frac{d f_e}{d t} \right){\mathrm{e-X}} = \int W{eX}(\mathbf{p},\mathbf{p}')[f'_e (1-f_e) - f_e (1-f_e')] d^3\mathbf{p'}, $$
* L: I would say that the picture is similar, apart from the fact that now the system is denser, so multiple scattering may be possible (scattering events are not uncorrelated, mining the "molecular chaos hypothesis" and <PERSON> approach itself is questionable, but see this answer). 346
However, it seems to me that the <PERSON> framework is still used somehow for liquid metals (see e.g. this article): we have to consider the scattering kernel $W_{eX}$ for the collision with a single ion multiplied by the structure factor of the liquid (in the case of the gas the structure factor is just 1).
* S: Does the same approach work also when we have a perfect lattice of ions $X^+$? Is it possible to consider the scattering off a single ion $W_{eX}$ multiplied by the structure factor of the crystal? It seems to me that this is not the way to go, because not the collision terms are only $e^- - e^-$ and $e^- - phonon$: the lattice structure is "transparent" and the electrons can just scatter with deviations from the perfect lattice (i.e. the phonons, see this). In other words: a solid is a gas of excitations. Therefore, differently from the G and L cases based on the "structure factor approach", now the term $W_{eX}$ never appears: the electrons do not interact with $X^+$, unless the lattice can vibrate (or if it has impurities: dislocations, surfaces, domains, species other than $X^+$...). In fact, only "imperfections" (vibrations, impurities...) of the lattice can provide the perturbation to the Hamiltonian responsible for the scattering of the $e^-$.
Is this picture correct? If so, how does the electronic band structure (that is important to conductivity) enter the game? References where <PERSON> transport is clearly described for different phases of matter?
Possible solution: a comparison between the solid and liquid cases is discussed <PERSON> et al. PRL 81 1998: They write that the Bragg scattering of electrons results in the energy band structure of the electron Bloch states but does not contribute to the $X-e$ collision integral in the kinetic equation. 346