Editing Free electron model (section)

== Properties of an electron gas ==
{{main|Fermi gas}}
Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. For a three-dimensional electron gas we can define the [[Fermi energy]] as<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=32-37}}</ref>
:<math>E_{\rm F} = \frac{\hbar^2}{2m_e}\left(3\pi^2n\right)^\frac{2}{3},</math>

where <math>\hbar</math> is the reduced [[Planck constant]]. The [[Fermi energy]] defines the energy of the highest energy electron at zero temperature. For metals the Fermi energy is in the order of units of [[electronvolt]]s above the free electron band minimum energy.<ref>{{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/fermi.html|title=Fermi Energies, Fermi Temperatures, and Fermi Velocities|last=Nave|first=Rod|publisher=[[HyperPhysics]]|access-date=2018-03-21}}</ref>

[[File:Free-electron DOS.svg|thumb|In three dimensions, the [[density of states]] of a gas of [[fermion]]s is proportional to the square root of the kinetic energy of the particles.]]

=== Density of states ===
The 3D [[density of states]] (number of energy states, per energy per volume) of a non-interacting electron gas is given by:<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=44}}</ref>
:<math>g(E) = \frac{m_e}{\pi^2\hbar^3}\sqrt{2m_eE} = \frac{3}{2}\frac{n}{E_{\rm F}}\sqrt{\frac{E}{E_{\rm F}}},</math>

where <math display="inline">E \geq 0</math> is the energy of a given electron. This formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the [[Valence and conduction bands|conduction band]]. For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy.

=== Fermi level ===
The [[chemical potential]] <math>\mu</math> of electrons in a solid is also known as the [[Fermi level]] and, like the related [[Fermi energy]], often denoted <math>E_{\rm F}</math>. The [[Sommerfeld expansion]] can be used to calculate the Fermi level (<math>T>0</math>) at higher temperatures as:<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=45-48}}</ref>
:<math>E_{\rm F}(T) = E_{\rm F}(T=0) \left[1 - \frac{\pi ^2}{12} \left(\frac{T}{T_{\rm F}}\right) ^2 - \frac{\pi^4}{80} \left(\frac{T}{T_{\rm F}}\right)^4 + \cdots \right], </math>

where <math>T</math> is the temperature and we define <math display="inline">T_{\rm F} = E_{\rm F}/k_{\rm B}</math> as the [[Fermi temperature]] (<math>k_{\rm B}</math> is [[Boltzmann constant]]). The perturbative approach is justified as the Fermi temperature is usually of about 10<sup>5</sup> K for a metal, hence at room temperature or lower the Fermi energy <math>E_{\rm F}(T=0)</math> and the chemical potential <math>E_{\rm F}(T>0)</math> are practically equivalent.

=== Compressibility of metals and degeneracy pressure ===
The total energy per unit volume (at <math display="inline">T = 0</math>) can also be calculated by integrating over the [[phase space]] of the system, we obtain<ref name=":3" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=38-39|p=}}</ref>
:<math>u(0) = \frac{3}{5}nE_{\rm F},</math>

which does not depend on temperature. Compare with the energy per electron of an ideal gas: <math display="inline">\frac{3}{2}k_{\rm B}T</math>, which is null at zero temperature. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. Thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by<ref name=":3" group="Ashcroft & Mermin" />
: <math>P = -\left(\frac{\partial U}{\partial V}\right)_{T,\mu} = \frac{2}{3}u(0),</math>

where <math display="inline">V</math> is the volume and <math display="inline">U(T) = u(T) V</math> is the total energy, the derivative performed at temperature and chemical potential constant. This pressure is called the [[electron degeneracy pressure]] and does not come from repulsion or motion of the electrons but from the restriction that no more than two electrons (due to the two values of spin) can occupy the same energy level. This pressure defines the compressibility or [[bulk modulus]] of the metal<ref name=":3" group="Ashcroft & Mermin" />
:<math>B = -V\left(\frac{\partial P}{\partial V}\right)_{T,\mu} = \frac{5}{3}P = \frac{2}{3}nE_{\rm F}.</math>

This expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal. For other metals the crystalline structure has to be taken into account.

=== Magnetic response ===
According to the [[Bohr–Van Leeuwen theorem]], a classical system at thermodynamic equilibrium cannot have a magnetic response. The magnetic properties of matter in terms of a microscopic theory are purely quantum mechanical. For an electron gas, the total magnetic response is [[paramagnetism|paramagnetic]] and its [[magnetic susceptibility]] given by{{Cn|date=April 2024}} 
:<math>\chi=\frac{2}{3}\mu_0\mu_\mathrm{B}^2g(E_\mathrm{F}),</math>
where <math display="inline">\mu_0</math> is the [[vacuum permittivity]] and the <math display="inline">\mu_{\rm B}</math> is the [[Bohr magneton]]. This value results from the competition of two contributions: a [[Diamagnetism|diamagnetic]] contribution (known as [[Diamagnetism#Theory|Landau's diamagnetism]]) coming from the orbital motion of the electrons in the presence of a magnetic field, and a paramagnetic contribution (Pauli's paramagnetism). The latter contribution is three times larger in absolute value than the diamagnetic contribution and comes from the electron [[Spin (physics)|spin]], an intrinsic quantum degree of freedom that can take two discrete values and it is associated to the [[electron magnetic moment]].