Quantum Mechanics- statistical physics fermi-dirac distribution.

[jcharles513]

Homework Statement

Consider a free-electron gas at a temperature T such that kT << E_f Write down the expression for the electron number desnity N/V for electrons that have an energy in excess of of E_f. Show by making the change of variables (E-E_f)/kT = x. that the number desnity is proportional to T. Calculate an expression for N/V under these circumstances, making the use of that the fact that the ∫ from 0 to infinity of dx/(exp(x)+1) = ln 2 [Hint: In working out the integral over E the integrand is such that (x+E_f)^1/2 = E_f^(1/2)

Homework Equations

I'm not sure where to start with this one. I know N is the Fermi Dirac distribution N(E) = 2/(exp( (E-E_f)/kT) + 1) but after that I'm not sure.

The Attempt at a Solution

I'm a bit confused about what the question is asking. So I just looked up the equation for a Fermi Dirac distribution and tried to relate it to what I need. I'm not sure where the integral comes into play in this.

 

[JesseC]

N is not the Fermi-Dirac distribution, because that would solve your problem immediately. The Fermi-Dirac distribution describes the average occupancy of an energy state E (more accurately written E_k because states are discrete, but for a large number of states it is effectively continuous).

To find N you need to integrate over the Fermi Dirac distribution multiplied by the density of states. This is a horrible integral, which is why your question has given you some help with that. First of course you need to find the density of states, which could be worked out assuming free particles in a box (as your question explicitly states).

Britney Spears has a good explanation for finding the density of states if you're unsure about that: http://britneyspears.ac/physics/dos/dos.htm

 

[jcharles513]

Using the results from Britney Spears, 3d density of states = \frac{1}{2{\pi}^{2}}\frac{2m}{{{\hbar}^{2}}}^{3/2}E^{1/2}

and integrating from 0 to ∞ of the density of states multiplied by Fermi-Dirac distribution The result is:

\frac{1}{{\pi}^{2}} {\frac{2m}{{\hbar}^{2}}}^{3/2}*E_f * ln(2)

I appreciate the help. I'm trying to get ready for a final and this was thrown at us with no real explanation. It seems rushed so I have the weekend to learn.

There is no dependence on temperature in my answer so it doesn't make sense. I'm not sure where I messed up or how close I am. Thanks,

 

[JesseC]

I can guess where you might have gone wrong on the integral, and I'm sorry that I don't have time to write a mathematical reply (tex is a pain) but I'll try to explain in words.

If you make the substitution suggested in your question, then x = (E - E_f)/kT and you should find that dx = dE/kT --> dE = kTdx. So there is a factor of kT in there, which you can remove from the integral. The limits on the integral are from E_f to infinity, thus using the substitution that becomes zero to infinity. The integral itself is ultimately just a constant, so N/V is proportional to T.

This question needs you to know the density of states, and if you've got an exam coming up then it seems surprising that you don't know what it is...

PS that density of states doesn't seem to include a factor for the spin degeneracy (electrons can be spin up/down).

 

[jcharles513]

That makes sense. I hate when the mistake I make is a simple integral mistake. Thanks again. You would think I should know this if I have a final soon. He taught it to us yesterday and testing monday on the final on it. So I've been working on learning it ever since.

Thanks,
James