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

[Jopi]

Homework Statement

Prove that the number of electrons whose velocity's x-component is between [x,x+dx] is given by
<br /> 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<br />

Homework Equations

The Fermi-Dirac Distribution function:
<br /> \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}<br />

Where E is kinetic energy,
<br /> E=\frac{1}{2} m v^2.<br />

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
<br /> 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<br />

Now, we know that
<br /> 4 \pi v^2 dv = dv_x dv_y dv_z.<br />

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

But I don't know how to do that. What is the differential element dv_ydv_z written with dt?

 

[naresh]

But I don't know how to do that. What is the differential element dv_ydv_z written with dt?

Think of v_y and v_z 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:
<br /> 4 \pi v^2 dv = dv_x dv_y dv_z.<br />