Error incurred from approximating fermi surfaces to be a sphere

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SUMMARY

The discussion centers on the error incurred from approximating the Fermi surface as a sphere in k-space, which is established to be inversely proportional to the number of electrons (N), typically around 10^23. The participants clarify that in two dimensions, the actual shape of the electron region in k-space is more squarish than spherical. The area discrepancy between the Fermi circle and the squarish region leads to a missing area of Kf^2*(4-π), which is linked to the electron density and ultimately results in the error being proportional to 1/N. The relationship kf = sqrt(2πN) is highlighted as a key factor in this analysis.

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tut_einstein
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I read somewhere that the error incurred from approximating the Fermi surface to be a sphere in k-space goes as 1/N where N is the number of electrons. So, N is generally of the order 10^23.
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!
 
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In 2D- as you correctly pointed out the error is proportional to kf^2 ...

But N is the electron number as you point out.

in 2D at low temperatures

kf = sqrt ( 2 pi N )

that is

kf ^ 2 is proportional to Electron Density ...

Then the error indeed goes as 1/N.
 

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