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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
:<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.
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