# 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 Boltzmann 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 Boltzmann 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 Boltzmann 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 Boltzmann 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 Boltzmann approach itself is questionable, but see this answer). However, it seems to me that the Boltzmann 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 Boltzmann transport is clearly described for different phases of matter?

Possible solution: a comparison between the solid and liquid cases is discussed Baiko 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. The collision is between electrons and phonons, but phonons can still be introduced in the transport equation via the structure factor. In discussing this point, they also reference Flowers, Itoh, ApJ 206 (1976).

1 The possibility of finding the system in a given phase depends on the physical properties of $$X^+$$, density and temperature of the system, but this is just a theoretical comparison experiment, so we are not concerned with this point.

Interesting reference: "Collisions are the mechanisms that restore equilibrium, or allow the system to reach a steady state under stationary non-equilibrium conditions. For Bloch electrons, quasi-momentum is a conserved quantity in equilibrium, so the interaction with a perfectly periodic crystal lattice cannot produce relaxation. Collisions arise when electrons are scattered by imperfections of the lattice (crystal defects, impurities, vacancies...), by lattice vibrations (electron-phonon interactions), or by other electrons (if the electron-electron interaction is taken into account). Scattering by imperfections is nearly temperature independent and dominates the transport properties of metals at low temperatures (e.g., the residual resistivity of a metal). The other mechanisms usually depend on the temperature as a power law $$T^n$$ and are thus ineffective at low temperatures: for electron-phonon interactions usually $$n = 5$$ at low temperature and $$n = 1$$ above the Debye temperature; for electron-electron interactions usually $$n = 2$$." From: Notes on the semiclassical theory of transport phenomena in metals by S. Caprara. Regarding the validity of the semiclassical approximation (semiclassical Boltzmann equation): The Boltzmann equation is valid under assumptions of semi-classical transport (i.e., effective mass approximation, which incorporates the quantum effects due to periodicity of the crystal); Born approximation for the collisions (i.e. "plane waves"), in the limit of small perturbation for the electron-phonon interaction and instantaneous collisions; no memory effects (i.e. no dependence on initial condition). The phonons are usually treated as in equilibrium, although the condition of non-equilibrium phonons may be included through an additional equation (the whole system is an electron-phonon mixture: multicomponent Boltzmann equation).

• iopscience.iop.org/book/978-0-7503-1200-4/chapter/… Commented Feb 8, 2022 at 13:12
• Thank you, the article you linked made me discover this: "The Boltzmann Equation in the Theory of Electrical Conduction in Metals" by Greenwood, 1958 Proc. Phys. Soc. 71. He writes: "Given the basic model of electrons moving independently in Bloch states, the usual approach to the theory of electrical conduction in metals is to set up the Boltzmann equation... where the collision terms model the effect of the scattering of electrons by phonons and lattice imperfections, calculated by second-order perturbation theory. " Commented Feb 8, 2022 at 13:36
• However, why do we do this (i.e. changing framework: no more scattering with $X^+$ but with "phonons", no impurities in my example) when we go from G and L to S (solid)? Why does the "structure factor approach" does not work anymore? Commented Feb 8, 2022 at 13:38
• Related: physics.stackexchange.com/questions/242319/… (structure factor BCC lattice), physics.stackexchange.com/q/471088/226902 (structure factor homogeneous). Regarding the difference between "liquid" and "solid": physics.stackexchange.com/a/372649/226902 Commented Feb 9, 2022 at 14:05
• In general quantum mechanical models are picked to fit observations. There can be two models for the same data and the best in fitting and predicting is what is used by the people working with those data. It needs a specialist to tell what makes the difference, but in general the model that fits and predicts better is the one chosen. Commented Feb 9, 2022 at 14:30