Exploring Maxwell-Boltzmann Statistics for Electrons in Metals

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Discussion Overview

The discussion revolves around the applicability of Maxwell-Boltzmann (MB) statistics to describe electrons in metals at room temperature. Participants explore the implications of the Fermi temperature and the characteristics of electron gases compared to other particles, such as muons.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions whether MB statistics can be applied to electrons in metals at room temperature, noting the high Fermi temperature of about 10^4 K.
  • Another participant states that the Fermi temperature is inversely proportional to particle mass, implying that heavier particles would have different statistical behavior.
  • Some participants argue that MB statistics cannot be used at room temperature since it is not considered a high temperature, and that Fermi-Dirac, Bose-Einstein, and MB statistics converge only at high temperatures.
  • One participant asserts that the electron density in metals is too high for MB statistics to apply, emphasizing that electrons remain below the Fermi temperature and are subject to the Pauli exclusion principle, which limits their statistical behavior.
  • Another participant reinforces the idea that MB statistics would not be valid even in extreme conditions like the surface of the sun, highlighting the significance of the Pauli exclusion principle at lower temperatures.

Areas of Agreement / Disagreement

Participants generally disagree on the applicability of MB statistics to electrons in metals at room temperature, with some asserting it cannot be used due to the high Fermi temperature and electron density, while others explore the conditions under which it might apply.

Contextual Notes

There are unresolved assumptions regarding the temperature thresholds for different statistical models and the implications of particle density on the applicability of MB statistics. The discussion does not reach a consensus on the conditions under which MB statistics could be valid for electrons or heavier particles like muons.

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Ok so my question is as follows:

Can Maxwell Boltzmann statistics be used to describe electrons in a metal at room temperature?

I know that the Fermi Temperature in metals is about 10^4 K or something rather high, so does that mean that the metal / electron gas would need to be at a temperature of over 10^4K to be described by MB Statistics? So at room temp of about 300k

What about if the electrons were all replaced with something much heavier, say muons (approx 200x mass). What would you use then? My understanding is quantum gases occur at low temperatures / high densities (when the concentration is higher than the quantum concentration?) so does that mean the fermi temperature would be higher?
 
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the fermi temp is inversely proportiional to particl mass
 


well, as far as i know, its only at high temperature that fermi-dirac, bose-einstein and maxwell-boltzmann statistics amount to the same thing-if u look at some graph. and room temperature isn't high temperature, right? so maxwell Boltzmann stats can't b used.
correct me if I am wrong
 


No it can't. The electron density is too high and the Fermi Temp is on the order of a few thousand Kelvin. Thus, electrons are not excited far above the Fermi Temperature and there is a fairly well defined fermi surface that can be examined experimentally. There simply is not enough thermal scattering to get you to the maxwell-boltzmann limit.
 


blueyellow said:
well, as far as i know, its only at high temperature that fermi-dirac, bose-einstein and maxwell-boltzmann statistics amount to the same thing-if u look at some graph. and room temperature isn't high temperature, right? so maxwell Boltzmann stats can't b used.
correct me if I am wrong

It is appropriate to use MB statistics on an electron system when the probability of two electrons ever vying for the same state is extremely small. In other words if the effect of pauli-exclusion is insignificant. This is definitely not the case below the Fermi temperature where the vast majority of electrons are stuck in the first available state dictated by the pauli-exclusion principle. In fact this wouldn't even be valid on the surface of the sun, much less room temperature.
 
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