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Ashcroft & Mermin, Solid State Physics, Chapter 16, Problem 2

  1. Mar 14, 2013 #1
    Solving Boltzmann equation

    1. The problem statement, all variables and given/known data

    Taken from Ashcroft & Mermin, Solid State Physics, Chapter 16, Problem 2:

    "A metal is perturbed by a spatially uniform electric field and temperature gradient. Making the relaxation-time-approximation (16.9) (where g0 is the local equilibrium distribution appropriate to the imposed temperature gradient), solve the Boltzmann equation (16.13) to linear order in the field and temperature gradient, and verify that the solution is identical to (13.43)."


    2. Relevant equations

    Eq. 16.9:
    [tex]{{\left( \frac{\partial g}{\partial t} \right)}_{coll}}\simeq -\frac{\left[ g\left( \mathbf{k} \right)-{{g}_{0}}\left( \mathbf{k} \right) \right]}{\tau \left( \mathbf{k} \right)}[/tex]
    Eq. 16.13:
    [tex]\frac{\partial g}{\partial t}+\mathbf{v}\cdot \frac{\partial g}{\partial \mathbf{r}}+\mathbf{F}\cdot \frac{1}{\hbar }\frac{\partial g}{\partial \mathbf{k}}={{\left( \frac{\partial g}{\partial t} \right)}_{coll}}[/tex]
    Eq. 13.43:
    [tex]g\left( \mathbf{k} \right)={{g}_{0}}\left( \mathbf{k} \right)+\tau \left( \mathrm{ }\!\!\varepsilon\!\!\text{ }\left( \mathbf{k} \right) \right)\left( -\frac{\partial f}{\partial \mathrm{ }\!\!\varepsilon\!\!\text{ }} \right)\mathbf{v}\left( \mathbf{k} \right)\cdot \left[ -e\mathbf{\varepsilon }+\frac{\mathrm{ }\!\!\varepsilon\!\!\text{ }\left( \mathbf{k} \right)-\mu }{T}\left( -\nabla T \right) \right][/tex]

    3. The attempt at a solution

    Well, I'm kinda lost. I put Eq. 16.9 into 16.13, and then what?
    Can't really see how I get from the Boltzmann equation, with the relaxation-time-approximation, to 13.43 :/

    So anyone who can help?

    Thanks in advance.
     
    Last edited: Mar 14, 2013
  2. jcsd
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