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In the calculation of <math>L=\left\{\begin{matrix}\displaystyle \frac{3}{2}\left(\frac{k_{\rm B}}{e}\right)^2\;, & \text{Drude}\\ \displaystyle\frac{\pi^2}{3}\left(\frac{k_{\rm B}}{e}\right)^2\;,&\text{free electron model.} \end{matrix}\right.</math> there |
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=== 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
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:<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>
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