Friday, March 2, 2012
Relaxation Time
The idea in the Drude model that I found to be interesting while reading chapter 1 was the idea of introducing the relaxation time, t -the average time between collisions and the fact that we only need one new variable in this model. It turns out that the original interpretation that it is time between electrons simply bouncing off ions was wrong but the idea of relaxation is till used in the later models. Quoting Ashcroft and Mermin "for in many respects the precise quantitative treatment of the relaxation time remains the weakest link in modern treatments o f metallic conductivity" which is why relaxation time independent quantities are of much interest. Even so, I can't think of any other way to develop a solid state theory without using this concept. The later chapters of the book may prove me wrong.
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I agree that the simplicity of this model is very appealing. Especially in this field, where it seems that one must always balance simplicity and accuracy. I wonder if the concept of relaxation time becomes less important when we start talking about periodic potentials and other things of that nature. Thoughts?
ReplyDeleteOne thing I noticed: the relaxation time is not necessarily the same in all directions. So there are at least some applications where a matrix representation would be necessary. Leading too multiple varibles. Still this is a basic and simple generalisation.
ReplyDeleteDavid:
So when we talk about quantum mechanics the concept of two particles colliding is a little ill defined. So I guess on the lowest level collisions are an approximation of a quantum system. So anything that brings out enough quantumness of a system will make the concept of relaxation time useless.
Inhomogeneity seems to bring out the quantumness. For example the hall effect. So ima guess your right.
Anyway that was a lot of reasoning with little reason. Did it make sense?
I was initially not sure what you mean by relaxation time being different in different directions. But now I guess you mean if you direct the electric field in one direction the relaxation time of the material would be different than if you direct the electric field in a different direction. The relaxation time being not isotropic means that we have to consider the anisotropic crystal lattice of the material which means we need to go beyond the Drude model...again.
ReplyDeleteI find it interesting that there is a time that characterises the collision between particles but otherwise we completely ignore any interaction.
ReplyDeleteThinking about whether relaxation time would be different in particular directions under a potential:
Consider a classical system of particles with an applied potential to push it towards a direction with an increase in density -> analogy of (disordered) crowded train platform with many people trying to board/leave a train.
On the platform away from the train, there would be the odd collision, but close to the train, there would be an increased density of people => collisions more likely.
=> From this change in density, I would expect the mean time between collisions to decrease.
Also, if there was a general force (so potential, I suppose this one's uniform) pushing towards everyone towards the, say, train, then I would expect that everyone's pushed together in the same general direction => an applied uniform potential would not change the relaxation time.
However, if potential going in different direction, e.g. towards a sink in the centre, then I would expect the mean collision time to decrease...
Perhaps this is too classical of thinking...
Ann:
ReplyDeleteYour analogy is interesting but I think it is more of a description of the edges of the system, which are ignored by the Drude model. Here, the people in the train are in the middle of the material, but the boundary between the inside of the train and the platform (the door :) ) is the point that you're concerned with. So, inside the train, the relaxation time is constant but at the boundary it changes.
What if you considered quantum people?
Quantum people?
ReplyDeleteJosh: yeah, we do tend to ignore boundary effects -> we tend to pretend something is infinitely large or sufficiently large such that boundary effects are negligible.
DeleteJoseph: http://g.co/maps/4a3q3
I was thinking along the lines of Andy; does the relaxation time have a meaning for more advanced treatments of systems? I suppose, for a quantum system, relaxation time would represent (depending on the derivation) the characteristic timescale of the system but that would be more of a system property than a property of electrons bouncing into each other. Is there some way to make sense of two electrons colliding? Perhaps something to do with an interaction probability radius?
ReplyDeleteAlso, quantum people. People don't like to collide, so perhaps you can model them as (spin-less, I don't see many upside down people on trains... although spin could represent some kind of relationship i.e. friendship) quantum particles. However, you'd need a big crowd to avoid the boundary problems....