- #1
Jopi
- 14
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Homework Statement
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]
Homework 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]
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?