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I couldn't figure out how they came up with the value. I was trying to say that the actual shape of the region filled with electrons in the k-space will be squarish. If we look at it in 2 dimensions (so Fermi sphere -> Fermi circle), the area of this square region is approximately

4* Kf^2, where Kf is the Fermi wave vector. And the area of the circle is ∏*Kf^2. So we're missing points in the area Kf^2*(4-∏) ≈ KF^2. I don't know how to get 1/N from here.

Thanks!