Radius of Fermi Sphere

  1. 1. The problem statement, all variables and given/known data
    Problem 9.2(B) from Kittel Solid State Physics.

    A two-dimensional metal has one atom of valence one in a simple rectangular primitive cell of a1 = 2Å and a2 = 4Å. Calculate the radius of the free electron Fermi sphere and draw this sphere to scale on the drawing of the Brillouin zones.

    A 2D solution that seems to be correct is posted here . Can anyone tell me what is wrong with my approach? Also, some equations weren't working for the latex. Sorry.

    3. The attempt at a solution

    First I find the electron concentration in terms of [itex]k_{F}[\latex]

    [itex]V=(4/3) \pi k^{3}[/itex]

    [itex]N=2*(4/3)*\frac{\pi k^{3}}{V_{k}}[/itex]

    where

    V_{k}=\frac{2 \pi}{a}*\frac{2 \pi}{a}*\frac{2 \pi}{b}=\frac{4 \pi^{3}}{a^{3}} (latex code didn't work for this)

    which is the k-space volume. The factor of 2 is the electron spin degeneracy.

    The electron concentration, N, is then:

    [itex]N=\frac{8}{3}\frac{\pi k^{3}a^{3}}{4\pi^{3}}=\frac{2k^{3}a^{3}}{3\pi^{2}}[/itex]

    which gives

    [itex]k=(\frac{3\pi^{2}N}{2a^{3}})^{1/3}[/itex]

    using N=1 and plugging in the values for a and b I get.

    k=(3\pi^{2})/16 (the latex code is screwing up for some reason)

    This gives me 1.23 A^-1.

    Where did I go wrong?
     
  2. jcsd
  3. Why is your momentum space 3d when the problem is 2d?
     
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