Editing Free electron model (section)

==Inaccuracies and extensions==
The free electron model presents several inadequacies that are contradicted by experimental observation. We list some inaccuracies below:<ref name=":4" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=58-59}}</ref>
; Temperature dependence: The free electron model presents several physical quantities that have the wrong temperature dependence, or no dependence at all like the electrical conductivity. The thermal conductivity and specific heat are well predicted for alkali metals at low temperatures, but fails to predict high temperature behaviour coming from ion motion and [[phonon]] scattering.
; Hall effect and magnetoresistance: The Hall coefficient has a constant value  <math>R_{\mathrm{H}} = -1/|ne|</math> in Drude's model and in the free electron model. This value is independent of temperature and the strength of the magnetic field. The Hall coefficient is actually dependent on the [[band structure]] and the difference with the model can be quite dramatic when studying elements like [[magnesium]] and [[aluminium]] that have a strong magnetic field dependence. The free electron model also predicts that the traverse magnetoresistance, the resistance in the direction of the current, does not depend on the strength of the field. In almost all the cases it does.
; Directional: The conductivity of some metals can depend of the orientation of the sample with respect to the electric field. Sometimes even the electrical current is not parallel to the field.  This possibility is not described because the model does not integrate the crystallinity of metals, i.e. the existence of a periodic lattice of ions. 
; Diversity in the conductivity: Not all materials are [[electrical conductor]]s,  some do not conduct electricity very well ([[Insulator (electricity)|insulators]]), some can conduct when impurities are added like [[semiconductor]]s. [[Semimetal]]s, with narrow conduction bands also exist. This diversity is not predicted by the model and can only by explained by analysing the [[valence and conduction bands]]. Additionally, electrons are not the only charge carriers in a metal, electron vacancies or [[Electron hole|holes]] can be seen as [[quasiparticle]]s carrying positive electric charge. Conduction of holes leads to an opposite sign for the Hall and Seebeck coefficients predicted by the model.

Other inadequacies are present in the Wiedemann–Franz law at intermediate temperatures and the frequency-dependence of metals in the optical spectrum.<ref name=":4" group="Ashcroft & Mermin" />

More exact values for the electrical conductivity and Wiedemann–Franz law can be obtained by softening the relaxation-time approximation by appealing to the [[Boltzmann equation|Boltzmann transport equations]].<ref name=":4" group="Ashcroft & Mermin" />

The [[exchange interaction]] is totally excluded from this model and its inclusion can lead to other magnetic responses like [[ferromagnetism]].{{Cn|date=April 2024}}

An immediate continuation to the free electron model can be obtained by assuming the [[empty lattice approximation]], which forms the basis of the band structure model known as the [[nearly free electron model]].<ref name=":4" group="Ashcroft & Mermin" />

Adding repulsive interactions between electrons does not change very much the picture presented here. [[Lev Landau]] showed that a Fermi gas under repulsive interactions, can be seen as a gas of equivalent quasiparticles that slightly modify the properties of the metal. Landau's model is now known as the [[Fermi liquid theory]]. More exotic phenomena like [[superconductivity]], where interactions can be attractive, require a more refined theory.{{Cn|date=April 2024}}