1. Sep 4, 2016

### ThereIam

1. The problem statement, all variables and given/known data
Hi all,

This is Kittel 9.2. This problem has been asked about before, but the people asking found solutions.

I'm trying to find the free electron Fermi radius for a 2D metal with a rectangular primitive cell (a = 2 Ang, b = 4 Ang).

2. Relevant equations

Please forgive my lack of latex skills:

E_f = hbar^2/(2m) * k_f^2
N = 2 * "volume" of Fermi sphere / volume element size.

For 3D this is N = V / (3 pi^2) * k_f^3 = 4 pi k_f^3 / 3 *(L /2 pi) ^3

3. The attempt at a solution

Since this is a 2D problem I take the volume of the Fermi sphere to be pi*k_f^2 (the area of a Fermi circle.

I am confused about the volume element. My lattice has different periodicities in each direction. I would assume based on the derivation of the volume element for phonons that this means my volume element should reflect this. I am tempted to take it to be (4 pi^2)/(a*b) = (2 pi^2)/(a^2). This is just the area of the first Brillouin zone.

I am not sure if this makes sense. I'm requiring that the wavefunctions be periodic like traveling plane waves in the respective directions k_x, k_y, as on page 137. Do I need to modify my expression for the Fermi energy to reflect my rectangular lattice in any way?

Anyway, I then get:
N = V * 1/(volume element) = a^2*k_f^2/pi

Then I assume N = 1, for valency 1? Though I just calculated the area of the occupied states...

2. Sep 9, 2016