Drude Theory of Metals Poisson Distribution Problem

In summary, the conversation discusses the probability of an electron not having a collision in a given time period and the average time between collisions. The factor of \tau is necessary to balance the decreasing probability with increasing \tau. The first probability considers all possible time intervals between any two successive collisions, while the second probability only considers the time interval between two specific collisions. This results in a longer average time between specific collisions than between any two successive collisions. The final part of the problem may involve using the concept of a probability density function for the time between two collisions.
  • #1
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Homework Statement



This is the first problem from Ashcroft's Solid-State Physics which I recently picked up due to having far too much free time. The first two parts of the problem relate to the probability that an electron picked at random will have had no collision during the preceding t seconds is [itex]e^{-t/\tau}[/itex] and the following t seconds, as well as that che probability that the time interval between two successive collisions of an electron falls in the range between t and t+dt is [itex](dt/\tau)e^{-t/\tau}[/itex]. I was able to do these, but the following parts I find issue with. The first is to show that the first probability gives an average time back to or up to the next collision is [itex]\tau[/itex], and that the second gives an average time of [itex]\tau[/itex] as well (which I have successfully done).

Homework Equations



Listed above.

The Attempt at a Solution


So for the first one, I get the expression [tex]\langle t \rangle = \int_0^\infty e^{-t/\tau} dt = \tau^2[/tex]
And for the second one, I get the same expression but with the factor of [itex]\tau[/itex] inherent in the equation fixing this square term for it. Obviously throwing this factor into the first equation will fix it, but I'm having trouble motivating any reason for doing so.

The final part asks to explain why the fact that the first part gives an average time between collisions of [itex]2\tau[/itex] does not conflict with the second of [itex]\tau[/itex], and to derive an explicit derivation of the probability distribution of the times between two collisions, but I'd like to focus on the above for the moment.

It seems likely that this is a fairly simple question, so if it's best put in the other board, by all means move it.
 
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  • #2


Hello! Thank you for sharing your question with us. It seems like you have made a good attempt at solving the problem so far.

To address your first question about why the factor of \tau is present in the first equation, it is important to consider the physical meaning behind the equations. The first equation represents the probability that an electron has not had a collision in the preceding t seconds. This probability decreases as t increases, since the longer an electron goes without a collision, the less likely it is to continue without one. However, the probability also decreases as \tau increases, since a longer \tau means that collisions are less frequent and thus an electron is more likely to go longer without a collision. Therefore, the \tau factor is necessary to balance out the decreasing probability with increasing \tau.

For your second question, it is important to note that the first probability is for the time between two successive collisions, while the second probability is for the time interval between a specific collision and the next one. This means that the second probability only considers the time interval between two specific collisions, while the first probability considers all possible time intervals between any two successive collisions. Therefore, it is possible to have a longer average time between two specific collisions (given by the second probability) than the average time between any two successive collisions (given by the first probability).

I hope this helps to clarify the concepts behind the equations. As for the final part of the problem, it may be helpful to use the concept of the probability density function (PDF) for the time between two collisions. I would be happy to provide more guidance on this if needed. Best of luck with your studies!
 

1. What is the Drude Theory of Metals?

The Drude Theory of Metals is a classical model that describes the behavior of electrons in a metal. It assumes that the electrons are free to move within the metal and are subject to collisions with the metal ions.

2. How does the Drude Theory explain electrical conductivity in metals?

The Drude Theory explains electrical conductivity by stating that electrons are free to move within the metal and are accelerated by an applied electric field. However, they also experience collisions with the metal ions, which results in a net drift velocity and electrical current.

3. What is the Poisson Distribution Problem in relation to the Drude Theory?

The Poisson Distribution Problem in the context of the Drude Theory refers to the phenomenon where the electrons in a metal exhibit a random, discrete distribution rather than a continuous one. This can be explained by the assumption that the electrons are subject to collisions with the metal ions, causing their distribution to follow a Poisson distribution.

4. How does the Drude Theory explain the temperature dependence of electrical conductivity in metals?

The Drude Theory explains the temperature dependence of electrical conductivity by stating that as temperature increases, the number of collisions between electrons and metal ions also increases. This results in a decrease in the average drift velocity of electrons, and therefore a decrease in electrical conductivity.

5. What are the limitations of the Drude Theory?

While the Drude Theory is a useful model for understanding the behavior of electrons in metals, it has several limitations. It does not take into account quantum effects such as electron spin and the wave nature of electrons. It also does not accurately predict the behavior of electrons at very low temperatures or in highly conductive metals.

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