Editing Free electron model (section)

=== Thermal conductivity and thermopower ===
While Drude's model predicts a similar value for the electric conductivity as the free electron model, the models predict slightly different thermal conductivities.

The thermal conductivity is given by <math>\kappa=c_V \tau\langle v^2\rangle/3 </math> for free particles, which is proportional to the heat capacity and the mean free path which depend on the model (<math>\langle v^2\rangle^{1/2} </math> is the mean (square) speed of the electrons or the Fermi speed in the case of the free electron model).<ref name=":6" group="Ashcroft & Mermin" />  This implies that the ratio between thermal and electric conductivity is given by the [[Wiedemann–Franz law]],

:<math>\frac \kappa \sigma = \frac{m_{\rm e}c_V \langle v^2 \rangle  }{3n e^2} = L T</math>

where <math>L </math> is the Lorenz number, given by<ref name=":10" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=23 and 52|ps=(Eq. 1.53 and 2.93)}}</ref>

:<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>

The free electron model is closer to the measured value of <math>L=2.44\times10^{-8} </math> V<sup>2</sup>/K<sup>2</sup> while the Drude prediction is off by about half the value, which is not a large difference. The close prediction to the Lorenz number in the Drude model was a result of the classical kinetic energy of electron being about 100 smaller than the quantum version, compensating the large value of the classical heat capacity.

However, Drude's mode predicts the wrong order of magnitude for the [[Seebeck coefficient]] (thermopower), which relates the generation of a potential difference by applying a temperature gradient across a sample <math>\nabla V =-S \nabla T</math>. This coefficient can be showed to be <math>S=-{c_{\rm V}}/{|ne|}</math>, which is just proportional to the heat capacity, so the Drude model predicts a constant that is hundred times larger than the value of the free electron model.<ref name=":7" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=|p=23|ps=}}</ref> While the latter get as coefficient that is linear in temperature and provides much more accurate absolute values in the order of a few tens of μV/K at room temperature.<ref name=":6" group="Ashcroft & Mermin" /><ref name=":7" group="Ashcroft & Mermin" /> However this models fails to predict the sign change<ref name=":4" group="Ashcroft & Mermin" /> of the thermopower in [[lithium]] and noble metals like gold and silver.<ref>{{Cite journal |last1=Xu |first1=Bin |last2=Verstraete |first2=Matthieu J. |date=2014-05-14 |title=First Principles Explanation of the Positive Seebeck Coefficient of Lithium |url=https://link.aps.org/doi/10.1103/PhysRevLett.112.196603 |journal=Physical Review Letters |volume=112 |issue=19 |pages=196603 |doi=10.1103/PhysRevLett.112.196603|pmid=24877957 |arxiv=1311.6805 |bibcode=2014PhRvL.112s6603X }}</ref>