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Reworking of the Drude model using scattering statistics

  1. Nov 30, 2015 #1
    1. The problem statement, all variables and given/known data
    The problem I have been set is to rework the Drude model using clearly defined scattering statistics.

    2. Relevant equations
    The Drude model as we have been given it is in terms of momentum
    [itex]\vec{p}(t+dt)=(1-\frac{dt}{\tau})(\vec{p}(t)-q\vec{E}(t)dt)+(\frac{dt}{\tau})(0)[/itex]
    Where that last term represents the contribution of the scattered electrons to the total momentum of the electrons in the system.

    3. The attempt at a solution
    My attempts so far have focused on trying to use Fermi-Dirac statistics to somehow derive the momentum of a scattered electron (ie. one at thermal velocity) as a function of the average energy of an electron at thermal velocity. I have the horrible feeling I have been barking up completely the wrong tree however, so I am now making an attempt to somehow integrate classical elastic scattering off of nuclei into the equation. I don't know how well this will work though.

    If anyone has been given a similar assignment in the past or has an idea what direction sounds most right to be going in some suggestions would be greatly appreciated.
     
  2. jcsd
  3. Dec 1, 2015 #2
    Based on a question in a different thread that seems to be similar I have done the following

    Probability of scattering per unit time = [itex]\lambda[/itex]
    direction after scattering characterised by the solid angle [itex]d\Omega '[/itex]
    The probability of a given angle after scattering is given by [itex]\omega (\theta)d\Omega '[/itex]
    were [itex]\theta[/itex] is the angle between the incident and scattered momenta, [itex]\vec{p}[/itex] and [itex]\vec{p}'[/itex].

    From this I have said that
    [itex]\frac{\partial}{\partial t}f(\vec{p},t)=(1-\lambda)unscatteredthing+\lambda scatteredthing[/itex]
    which I have written as
    [itex]\frac{\partial}{\partial t}f(\vec{p},t)=f(\vec{p},t)-\lambda f(\vec{p},t)+\lambda \int \omega (\theta) f(\vec{p}',t) d\Omega'[/itex]

    What I have gotten agrees with what is in the question containing the prompt I went off except for the first term, which does not feature in the version from the hint in that question.
     
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