# Electron gas

In solid-state physics , the term electron gas describes a model for the freely moving electrons in the conduction band of metals or semiconductors . Under this model, the freely moving electrons as a reason for conductivity understood metal, and the electrical resistance is determined by the scattering of electrons on phonons and crystal - imperfections described.

The electron gas is not a gas in the chemical sense.

## Delocalized electrons

Electrons in the conduction band are delocalized ; That is, they cannot be assigned to a specific lattice atom , as is the case in chemical compounds . In other words, such an electron has a non-vanishing probability of being at every lattice atom, i.e. it is distributed over the entire crystal . The kinetic energy and the (quantum mechanical) wave vector of a free, non-interacting electron are related to the dispersion relation${\ displaystyle E}$ ${\ displaystyle {\ vec {k}}}$

${\ displaystyle E = {\ frac {1} {2}} \ cdot {\ frac {\ hbar ^ {2} \ cdot {\ vec {k}} ^ {2}} {m}}}$

Relations of this kind determine the band structure in wave vector space. The so-called free electron gas described (with the parabolic band) is only a simple model to describe the electrons in the conduction band. In more complicated models (e.g. approximation of quasi-free electrons or tight-binding model), which describe reality better, the periodic potential of the crystal is taken into account, which leads to more complex band structures. These can, however, in the first approximation, to be described by the above parabolic dispersion when for the effective mass of the corresponding band is set. ${\ displaystyle {\ vec {k}} = 0}$${\ displaystyle m}$

Since electrons are fermions , no two electrons can match in all quantum numbers. As a result, the energy levels at temperature are filled from ( zero point energy ) to Fermi energy . The distribution of the energy is described by the Fermi-Dirac statistics , which are softened at the "Fermi edge" in a range of width . ${\ displaystyle T = 0 \, \ mathrm {K}}$${\ displaystyle E_ {0} = {\ tfrac {1} {2}} \ hbar \ omega}$${\ displaystyle T> 0 \, \ mathrm {K}}$${\ displaystyle \ sim 2 \, k _ {\ mathrm {B}} T}$

## Degenerate electron gas

An electron gas is referred to as degenerate if the (largely temperature-independent) Fermi energy of the electrons in a potential box is much greater than the absolute temperature , multiplied by the Boltzmann constant : ${\ displaystyle E _ {\ mathrm {F}}}$ ${\ displaystyle T}$ ${\ displaystyle k _ {\ mathrm {B}}}$

${\ displaystyle E _ {\ mathrm {F}} \ gg k _ {\ mathrm {B}} \ cdot T \ Leftrightarrow {\ frac {E _ {\ mathrm {F}}} {k _ {\ mathrm {B}} \ cdot T}} \ gg 1}$

In particular, every electron gas is degenerate at . The term degenerate is to be understood in such a way that almost all states have the same probability of being occupied. The distribution function is constant over a large area (compared to the Fermi edge ). ${\ displaystyle T \ to 0 \, \ mathrm {K}}$

Numerical examples:

• for the conduction electrons in copper applies (at room temperature ):${\ displaystyle \, E _ {\ mathrm {F}} \, / \, k _ {\ mathrm {B}} T \, \ approx \, 280}$
• for the electrons in the center of white dwarfs applies (despite the high temperature):${\ displaystyle \, E _ {\ mathrm {F}} \, / \, k _ {\ mathrm {B}} T \, \ approx \, 10 ^ {2} ... \, 10 ^ {3}}$
• for the electrons in the center of the sun, however , the ratio is: (i.e. non-degenerate).${\ displaystyle \, E _ {\ mathrm {F}} \, / \, k _ {\ mathrm {B}} T \, \ approx \, 1/2}$