Error incurred from approximating fermi surfaces to be a sphere

[tut_einstein]

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!

 

[sokrates]

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.