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Homework Help: A question regarding the number of fermions with a certain velocity component

  1. Nov 20, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that the number of electrons whose velocity's x-component is between [x,x+dx] is given by
    [tex]
    dN = \frac{4\pi V m^2 k_B T}{h^3} ln [exp(\frac{E_F -mv_x^2/2}{k_B T})+1]dv_x
    [/tex]

    2. Relevant equations
    The Fermi-Dirac Distribution function:
    [tex]
    \frac{dn}{dE}=\frac{4\pi V \sqrt{2m^3}}{h^3}\frac{\sqrt{E}}{exp \left((E-E_F)/ k_B T\right) +1}
    [/tex]

    Where E is kinetic energy,
    [tex]
    E=\frac{1}{2} m v^2.
    [/tex]

    3. The attempt at a solution
    First, I used the definition of kinetic energy above to rewrite the distribution as a function of velocity. I got
    [tex]
    dn=\frac{V m^{3/2}}{h^3}\frac{1}{exp((\frac{1}{2} m v^2 -E_F)/k_B T) +1} 4 \pi v^2 dv
    [/tex]

    Now, we know that
    [tex]
    4 \pi v^2 dv = dv_x dv_y dv_z.
    [/tex]

    Now I should be able to get the number of particles with vx in the given range by integratin out dvy and dvz. In the assignment it is suggested that I write
    [tex]
    t^2=v_y^2+v_z^2
    [/tex]
    and then I can use the formula
    [tex]
    \int_0^{\infty} (ae^x+1)^{-1}dx=ln(1+\frac{1}{a}).
    [/tex]

    But I don't know how to do that. What is the differential element dvydvz written with dt?
     
  2. jcsd
  3. Nov 20, 2008 #2
    Think of vy and vz as cartesian coordinates, and t as the radial coordinate in the plane.

    You should then be able to derive/remember the 2-D analog of the equation you have already written down:
    [tex]
    4 \pi v^2 dv = dv_x dv_y dv_z.
    [/tex]
     
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