Editing Free electron model (section)

=== Heat capacity ===
{{Further|Electronic specific heat}}
One open problem in solid-state physics before the arrival of quantum mechanics was to understand the [[heat capacity]] of metals. While most solids had a constant [[volumetric heat capacity]] given by [[Dulong–Petit law]] of about <math>3nk_{\rm B}</math> at large temperatures, it did correctly predict its behavior at low temperatures. In the case of metals that are good conductors, it was expected that the electrons contributed also the heat capacity.

The classical calculation using Drude's model, based on an ideal gas, provides a volumetric heat capacity given by
:<math>c^\text{Drude}_V = \frac{3}{2}nk_{\rm B}</math>.

If this was the case, the heat capacity of a metals should be 1.5 of that obtained by the Dulong–Petit law.

Nevertheless, such a large additional contribution to the heat capacity of metals was never measured, raising suspicions about the argument above. By using Sommerfeld's expansion one can obtain corrections of the energy density at finite temperature and obtain the volumetric heat capacity of an electron gas, given by:<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=47}} (Eq. 2.81)</ref>
:<math>c_V=\left(\frac{\partial u}{\partial T}\right)_{n}=\frac{\pi^2}{2}\frac{T}{T_{\rm F}} nk_{\rm B}</math>,

where the prefactor to <math>nk_B</math> is considerably smaller than the 3/2 found in <math display="inline">c^{\text{Drude}}_V</math>, about 100 times smaller at room temperature and much smaller at lower <math display="inline">T</math>.

Evidently, the electronic contribution alone does not predict the [[Dulong–Petit law]], i.e. the observation that the heat capacity of a metal is still constant at high temperatures. The free electron model can be improved in this sense by adding the contribution of the vibrations of the crystal lattice. Two famous quantum corrections include the [[Einstein solid]] model and the more refined [[Debye model]]. With the addition of the latter, the volumetric heat capacity of a metal at low temperatures can be more precisely written in the form,<ref group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=49}}</ref> 
:<math>c_V\approx\gamma T + AT^3</math>, 
where <math>\gamma</math> and <math>A</math> are constants related to the material. The linear term comes from the electronic contribution while the cubic term comes from Debye model. At high temperature this expression is no longer correct, the electronic heat capacity can be neglected, and the total heat capacity of the metal tends to a constant given by the Dulong–petit law.