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Sommerfeld's model for the electrons heat capacity works well on the extremes of high and low temperatures.

While the model is quantum and assumes the electrons are an ideal gas, the reality is VERY different, with the electrons bounded to protons and having interactions between them.

So why do this model even work?

EDIT: Original question was asked about Drude's model as well, but answering the question about Sommerfeld's model will answer why Drude's model works at high temperatures since quantum effects will be neglected.

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  • $\begingroup$ In Physics there are lucky (or, better, unlucky) coincidences that make us believe we guessed the Physics right. $\endgroup$ Jun 28, 2021 at 18:12
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    $\begingroup$ Because the conduction electrons in most metals can be treated as mostly-free in solid state physics. $\endgroup$
    – Jon Custer
    Jun 28, 2021 at 18:45
  • $\begingroup$ So does it mean that in these models we only concern ourselves with the conduction electrons? So THATS why it only works with metals, because only they have electrons in the conduction band? This actually makes a lot of sense now! $\endgroup$
    – dor00012
    Jun 28, 2021 at 18:51
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    $\begingroup$ I don't have my copy of Ashcroft and Mermin handy, but IIRC, they comment that the models make two big mistakes that happen to roughly cancel out. In other words, they (arguably) got lucky. I'll try to dig up the relevant bit when I get home. $\endgroup$
    – lnmaurer
    Jun 30, 2021 at 15:19
  • $\begingroup$ I thought the idea of electrons actually orbiting a nucleus was given up on because it didn’t make sense. $\endgroup$ Oct 23, 2021 at 16:47

2 Answers 2

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Landau Fermi liquid theory shows that excitations in metals can be treated as independent particles. These particles have mass, dispersion relation and other characteristics very different from those of "free electrons" that we could have predicted from simple band structure calculations, but this hardly metter for Drude-Sommerfeld theory, where many parameters are phenomenological (mass, scattering time/length, etc.)

While the theory works remarkably well for many materials, there are well-known cases where it is not obeyed (non-fermi liquid behavior):

  • In superconducting regime
  • In one dimension (Littinger liquid)
  • In magnetic alloys (Kondo effect)

Remark After re-reading the question, I think that my answer may imply the background that the questioner does not have. I recommend therefore looking up questions on the band structure and the difference between insulators and metals, e.g. here and here.

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Drude's (and consequently Sommerfeld's) models for the electrons heat capacity works well on the extremes of high and low temperatures.

Do they?

The two models give very different results:

  • Drude's model is that of a classical gas: $c_v = \frac{3}{2}nk_B$ (See Ashcroft and Mermin Chapter 1, Section "Thermal Conductivity of a Metal".)
  • Sommerfeld's model is that of a free electron gass: $c_v = \frac{\pi^2}{3} \left(\frac{k_B T}{\epsilon_F}\right) nk_B$ (Ashcroft and Mermin Eq. 2.81).

Drude's model isn't temperature dependent, and it does not work well at the extreme of low temperature.

Could you clarify which models you are thinking of?

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  • $\begingroup$ Yes, Drude's model does not work well on low temperatures because it is a kinetic theory (and in low temperatures, the quantum effects are not negligible). My question could simply be asked on sommerfeld's model since it takes into account the quantum effects with the fact that electrons are fermions. I asked about both of them, since they both assume that the electrons are an ideal gas - which in a solid they are obviously not. Also, at no point have I suggested that the model within themselves give the same results. $\endgroup$
    – dor00012
    Jul 3, 2021 at 14:10
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    $\begingroup$ "At no point have I suggested that the model within themselves give the same results" I disagree. You said that they both give good results at both high and low temperature. So you're claiming that they both give good results, but the results are not the same --- or even remotely close to each other? If they're not the same --- or even remotely close to each other, how are they both good? It sounds like you're really only asking about Sommerfeld's model, which is pretty good for heat capacity. (Drude's model is not.) If so, it would help to edit the question for clarity. $\endgroup$
    – lnmaurer
    Jul 4, 2021 at 1:06
  • $\begingroup$ You're right, the question was somewhat poorly stated. I fixed it, thank you. $\endgroup$
    – dor00012
    Jul 5, 2021 at 10:45

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