1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: A question regarding the number of fermions with a certain velocity component
Loading...