I was a bit confused about the 2s prefactor. I know it comes from the spin degeneracy, but I thought that the “fermi surface” would cover that. Here (I think) the fermi surface is included in the integral, so I’m wondering, what purpose does the fermi surface have exactly, and why do we have to include it in the integral? And is it not related at all to the spin degeneracy?
The integrals in the second screenshot count all states with |\textbf{k}|<k_0. In 1D It’s all states with -k_0 < k < k_0, that’s why a factor 2 in the integrand. In 2D it’s all states in a circle with radius k_0, and in 3d it’s all states inside a sphere with radius k_0. Check if you can confirm that the integrals indeed give a length of the interval, area of a circle, or volume of a sphere.
The factor 2 for spin appears regardless of dimensionality because electrons have two possible values of spin, it is indeed not related.
