Free electron model: Difference between revisions

Revision as of 11:31, 20 December 2024

Latest revision as of 17:10, 29 March 2025

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The free electron model is closer to the measured value of <math>L=2.44\times10^{-8} </math> V²/K² 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 |lastlast1=Xu |firstfirst1=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>

==Inaccuracies and extensions==

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;General

*{{cite book | last = Kittel | first = Charles | author-link=Charles Kittel | title = [[Introduction to Solid State Physics]] | ___location = University of Michigan | year = 1972|publisher=Wiley & Sons|isbn=978-0-471-49024-1}}

*{{cite book | lastlast1 = Ashcroft | firstfirst1 = Neil | author-linklink1 = Neil Ashcroft | last2 = Mermin | first2 = N. David | author-link2 = N. David Mermin | title = Solid State Physics | publisher = Holt, Rinehart and Winston | ___location = New York | year = 1976 | isbn = 978-0-03-083993-1 | url-access = registration | url = https://archive.org/details/solidstatephysic00ashc }}

*{{cite book | lastlast1 = Sommerfeld | firstfirst1 = Arnold | title = Elektronentheorie der Metalle | publisher = Springer Verlag | ___location = Berlin Heidelberg | year = 1933 | isbn = 978-3642950025 |first2=Hans|last2=Bethe|author-linklink1=Arnold Sommerfeld|author-link2=Hans Bethe}}

* {{cite book|last1=Ziman|first1=J.M.|title=Principles of the theory of solids|edition=2nd|publisher=Cambridge university press|year=1972|isbn=0-521-29733-8}}