Error incurred from approximating fermi surfaces to be a sphere

In summary, the error incurred from approximating the Fermi surface to be a sphere in k-space is proportional to 1/N where N is the number of electrons. This is because the actual shape of the region filled with electrons in k-space is squarish, and the area of this square region is approximately 4*Kf^2, while the area of the Fermi circle is ∏*Kf^2. This results in a missing area of Kf^2*(4-∏) ≈ Kf^2, leading to the error being proportional to 1/N. At low temperatures in 2D, Kf^2 is proportional to the electron density, hence the error being proportional to 1
  • #1
tut_einstein
31
0
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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  • #2
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.
 

1. What is the reason for approximating fermi surfaces to be a sphere?

The Fermi surface is a theoretical concept that represents the boundary between occupied and unoccupied energy levels in a material. In reality, the Fermi surface can have complex shapes and is difficult to visualize. Therefore, it is often approximated as a sphere for simplicity and ease of calculation.

2. How accurate is the approximation of fermi surfaces as a sphere?

The accuracy of the approximation depends on the material and the level of precision needed. In some cases, the spherical approximation may be sufficient, while in others it may lead to significant errors. It is important to consider the limitations of the approximation and its impact on the results.

3. What are the consequences of using the spherical approximation for fermi surfaces?

Using the spherical approximation can lead to errors in calculations and predictions, particularly in systems with complex Fermi surfaces. This can affect the accuracy of physical properties such as electrical conductivity and magnetic behavior. Therefore, it is important to carefully consider the consequences of using this approximation.

4. Are there any alternative methods for approximating fermi surfaces?

Yes, there are various methods for approximating fermi surfaces, such as using multiple spherical shells or using a more complex mathematical model. However, these methods can be more computationally intensive and may not always be necessary depending on the specific research or application.

5. How can I determine if the spherical approximation is appropriate for my research/application?

The suitability of the spherical approximation depends on the specific material and the level of accuracy needed for the research or application. It is recommended to consult with experts in the field and carefully evaluate the limitations and potential errors of the approximation before using it in your work.

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