Drude model relaxation time approximation

In summary, the Drude model explains the conduction of a metal by using the relaxation time approximation, which states that the electron has a collision in an infinitesimal time interval of dt. The mean time between collisions is τ, and the average time since the last collision and the average time to the next collision are both τ. However, the average time between the last collision and the next collision can be shown to be 2τ. This can be explained by considering intervals along the particle's path and the fact that the electron is more likely to be in certain intervals than others. Ultimately, the time delay between two collisions is defined by τ.
  • #1
CAF123
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In the Drude model of the free electron gas to explain the conduction of a metal, the relaxation time approximation that the electron has a collision in an infinitesimal time interval ##dt##is ##dt/\tau##. It can be shown that the mean time between collisions is ##tau##. If we choose an electron at random, the average time since the last collision is ##\tau## and the average time to the next collision is ##\tau##. The average time between the last collision and the next collision can then be shown to be ##2\tau## .
My question is how does this agree with the fact that the mean time between collisions is in fact ##\tau##?
So in my mind, ##\tau## is that time where we consider some interval ##[0,L]## say and divide the total number of collisions in that interval by the time taken for the particle to travel from ##0## to ##L##.
I am thinking the 2##\tau## comes about by considering the fact that we might have a collision at ##0## and then at ##0 + \epsilon, |\epsilon| \ll L## and then maybe the next one is not until ##3L/4## and then again at ##3L/4 + \epsilon##. At random, it is more likely to see the electron in the intervals ##[0+\epsilon, 3L/4]## than it is in the intervals ##[0, 0+\epsilon]## or ##[3L/4, 3L/4 + \epsilon]##. If we apply this to a more symmetric set up then maybe this could explain the ##2\tau##, but I am not sure if this observation is helpful at all.
Many thanks.
 
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  • #2
Can anybody provide any help?
 
  • #3
No. The time delay between two collisions is defined by τ.
"Time before next collision" - "time since last collision" = τ. It's not more complex than that...
 

1. What is the Drude model relaxation time approximation?

The Drude model relaxation time approximation is a theoretical model used to describe the behavior of electrons in a metal. It assumes that the electrons behave like a gas, moving freely through the metal and colliding with the metal ions, resulting in a net flow of electric current.

2. How does the Drude model relaxation time approximation differ from other models?

The Drude model relaxation time approximation differs from other models in that it does not take into account the quantum nature of electrons. It is a classical model that simplifies the behavior of electrons in a metal to a gas-like behavior, while other models, such as the quantum mechanical model, consider the wave-like nature of electrons.

3. What is the relaxation time in the Drude model relaxation time approximation?

The relaxation time in the Drude model relaxation time approximation is a measure of how long it takes for an electron to return to its equilibrium state after a collision with a metal ion. It is a key parameter in the model and is used to calculate the electrical conductivity of the metal.

4. What are the limitations of the Drude model relaxation time approximation?

The Drude model relaxation time approximation has several limitations. It does not consider the effects of quantum mechanics, such as the wave-like nature of electrons and their discrete energy levels. It also does not take into account the interactions between electrons, which can significantly affect the behavior of a metal. Additionally, it does not account for the temperature dependence of electrical conductivity.

5. How is the Drude model relaxation time approximation used in practical applications?

The Drude model relaxation time approximation is widely used in practical applications, such as in the design of electronic devices and materials. It is also used in the study of electrical and thermal conductivity in metals and other conductive materials. However, its limitations must be considered when applying it to real-world scenarios.

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