Solid State Physics: Draw the Dispersion Relation from the Fermi Surface

[confusedius]

Homework Statement
ln the figure below you (b, which is taken from Jenö Sólyom Fundamentals of the physics of solids. Volume 2 chapter 19) see the Fermi sphere of radius k_F inside one section in two dimensions of the Brillouin zone of Na. Draw the dispersion relation E(k) from the I point in the centre of the zone to the points H, N and P, and the corresponding occupied electronic states.

The attempt at a solution
Well, I'm assuming that what is wanted is : 1) a picture of the dispersion relation and 2)for the "corresponding occupied electronic states", which bands are filled?

I have a problem understanding where H,N,P would be located with respect to gamma (which I'm assuming is at 0) on the dispersion relation graph. I mean, in general since Na is fcc and has 4 atoms per unit cell it should have 3*4=12 bands (in 3D)? But I'm not sure in general how you can get how the bands will look from the fermi surface. Do I just have to know how the bands should look? In this case it is (approx.) circular so that means it is monavalent and then the band is not completely filled ( and therefore is a metal, and Na is a metal so yeah makes sense). But that would also mean that the circle is contained in the first Brillouin zone, right?
I am also thinking since the radius is given you can calculate the fermi energy and in this case you will then know which bands would be filled. I am definitely not understanding this well, any help would be appreciated. :)

 

[mark726]

Hello,
Thank you for your post. You are correct, the Fermi sphere in this case is contained within the first Brillouin zone, which is represented by the circle in the figure. The points H, N, and P correspond to specific points on the dispersion relation graph, as shown in the figure. The dispersion relation is a plot of the energy of the electrons as a function of their momentum (k). The points H, N, and P correspond to specific values of k, which can be calculated using the radius of the Fermi sphere and the known shape of the Brillouin zone.
Regarding the filled bands, you are correct that the filled bands can be determined by calculating the Fermi energy. This energy represents the highest energy level that is occupied by electrons at absolute zero temperature. Any energy levels below the Fermi energy will be filled, while any energy levels above it will be empty. In this case, since the Fermi sphere is circular, only one band will be completely filled and the rest will be partially filled.
I hope this helps to clarify things for you. Let me know if you have any further questions. Good luck with your studies!