After reading Chapter 1 of Ashcroft and Mermin, I started to wonder about classical models. The relaxation time, when calculated experimentally, turns out to be 10E-14 to 10E-15. This can be used to find the mean free path, as long as you know the average electronic speed. Drude estimated this speed using classical equipartition energy, i.e.
1/2mv^2=3/2kT
This leads to a mean free path length of 1 to 10 angstroms. This seems reasonable. However, the chapter then goes on to say that the mean free path length is orders of magnitude larger than this, debunking the theory that the only collisions are off ions.
I always find it interesting when I encounter situations where different, not entirely correct, assumptions cancel each other out. It really drives the point home that one must be rigorous with one's assumptions, even when the end result is reasonable.
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Nice post!
ReplyDeleteNot much I can directly add so Im going a little off topic.
I started trying to think how we could correct the model classically and what it would change.
I think the fact that collisions are actually electromagnetic forces the electron paths will be curved. So this means paths will actually be slightly shorter.
So hard collisions is probably a good assumption...
I can't think of any other classical effects that were missed...
Anyone?
I don't think there are other significant classical effects. Even if we just add electromagnetic forces, the model would grow to be quite complicated. I recall reading somewhere in the text that the neglecting electron-electron interactions is surprisingly good in most cases but neglecting the electron-ion interaction is very bad. I wonder what is the reason we can neglect one but not the other. Naively it might be something to do the mass of ions or the net charge of the ions being larger? Aargh makes me one to read ahead ... but I will savour it slowly.
ReplyDeleteIt would seem reasonable to suspect the different in mass to be related to the significance of the interaction.
ReplyDeleteConsider the change in momentum of a really light thing colliding with an equally light thing. Then consider the really light thing colliding with a really heavy thing. There's going to be a much larger change in momentum onto the light thing after hitting the heavy thing.
Going even more off-topic:
I like how we consider electron-electron interactions and electron-ion collisions separately.
It's probably natural to do so since we have a repulsive and attraction force.
This sort of mirrors central dogma of biology: gene becomes protein then does some function.
There exists protein-protein interactions and there exists gene-gene interactions. Yet, we can still consider gene going to protein to function with no interaction.
It certainly simplifies things to ignore the interactions -> ~10^23 electrons per mole -> that's a lot of interactions to consider!
However, I'd expect exciting things to come out when we acknowledge these interactions.
I'm sure this will be coming soon :)
We will learn that the key failure of Drude concerning the mean-free path is because it has the the wrong velocity. This is because the electrons are not classical particles following a Maxwellian distribution, but rather fermions. The average velocity is the Fermi velocity which is orders of magnitude larger than the classical velocity. This is because the fermi temperature is much larger than the temperature.
ReplyDeleteProf. Ross,
ReplyDelete(without reading ahead at the time of writing...) Is this Fermi velocity larger because of the interactions of fermions (forcing fermions to not occupy the same position)? Also, is this implying that the Fermi velocity is the same for all fermions of a given type at a given temperature? This seems like an interesting idea, but I suppose that when you're looking at electrons, they are the same as each other, just like in the kinetic/classical theory of gases....