Energy of 3D free electron gas.

[leoneri]

Hi, I have two questions,

1. I am reading the book 'Solid State Physics', 1976, by Aschcroft and Mermin. I am reading chapter II about 'The Sommerfeld Theory of Metals'. (I hope anyone here have the same book..)

I found it hard to figure out one integral on the equation (2.30) on calculation of the 3D electron gas energy density 'E/V' when putting limit V -> infinity.

My question is why the integration (1/(4*Pi^3)) * integral(dk * (h_bar^2*k^2)/(2*m)) is equal to (1/Pi^2) * (h_bar^2*k_Fermi^5)/(10*m) ? I have tried it but I know that integration of k^2 is (1/3)*k^3, so I don't get it on how to get the result...

Sorry that I do not know LaTex..

2. I read on wikipedia. They calculate the total energy by doing integration of Fermi energy over N, which is the total number of electron. My question is, why is it like that? As far as know, not all electrons sit in the same energy level equal to Energy Fermi due to Pauli principle. So it does not make sense to me while the result is correct the the total energy is equal to 3/5 of Energy Fermi times the number of electron. Can anybody here explain it? Source: http://en.wikipedia.org/wiki/Fermi_energy#The_three-dimensional_case

Many Thanks in advance.

 

[alxm]

Skipping 1 because I don't have the book...

2: The Pauli principle is neglected. Note that you can have several particles with the same energy if they have different quantum numbers; you're basically approximating/assuming there's an infinite number of available states. More generally you're basically assuming a homogeneous gas of non-interacting electrons.

In a real situation, you have the Pauli principle, electrons interact, and materials are not homogeneous. This is basically the simplest, crudest model you could have. But it does give meaningful results for electrons in a metal crystal. (OTOH this kind of model fails pretty badly - as one would expect - for a molecule or single atom, where the system is not at all homogenous and the number of states is quite small)

 

[kanato]

On 1, I believe the integration factor should be d^3k, since you are looking at a 3 dimensional electron gas. Then you go to spherical coordinates, since the kinetic energy depends only on the magnitude of k, and use the transformation d^3k = 4*pi*k^2*dk.

 

[leoneri]

@kanato
Thanks .. you are right .. it is clear for me now. I was puzzled because of it for two days until now. xD

@alxm
So the Pauli principle is neglected by saying that we are assuming there are infinite number of states that can be occupied by electrons? Is it because of the distance between the allowed k values is too small?

So, this means that it is allowed for an electron to have energy equal to energy Fermi when T = 0 ? I always thought that it is the limit, so electrons' energy are always smaller than Fermi Energy...