I have noticed that Ag, Au, Cu’s Fermi Surfaces are very nearly nice spheres. Is this due to the fact that the outer shell filling is one electron? But if so, I would expect Cr to also be a nearly sphere-like structure. What is the reason behind these shape similarities then? I guess it has something to do with the BZ interactions on the gaps, but I don’t see it.
Another question: Q1.4 asks: " What Fermi surface shape would you expect the [NaCl crystal] to have?". I can only surmise it is not spherical, but how do you know what shape it is?
The dispersion is well described by the nearly free electron model whenever the electrostatic potential is sufficiently weak. Most metals have many electrons on the outer shell. Gold electronic structure, for example, is [Xe] 6s¹4f¹⁴5d¹⁰, so quite a lot of electrons on the outer shell.
The Fermi surface being similar to a single sphere, however, isn’t a criterion for the nearly free electron model. Rather, the NFEM Fermi surface is made out of pieces of a sphere moved and glued together. The best example I think is aluminium.
To answer this question let’s first consider if NaCl is a metal or an insulator.
“Gold electronic structure, for example, is [Xe] 6s¹4f¹⁴5d¹⁰, so quite a lot of electrons on the outer shell.”
So due to the many electrons in the outer shell, the single 6s1 electron ‘feels’ a net low charge, as the nucleus is very postitivly charged but the f and d orbitals are very negatively charged. Then this would lead to a low electrostatic potential?
To answer this question let’s first consider if NaCl is a metal or an insulator.
It is an insulator, optically transparent - large bandgap, even number of electrons/unit cell. Since there is no interaction, there are no BZ gaps/crossings, and thus the entire Wigner-Seitz unit cell is exactly the Fermi Surface?
I’m actually unsure how to determine whether lattice potential is weak or strong in a given material. I don’t think there’s a simple argument that would prove that in advance.
Rather the Fermi surface is an empty set: no k-points at all.