Integrating over the Fermi sphere

This will allow us to integrate over the vector running from the origin to kf, satisfying the given conditions for the integral. Therefore, the limits for kx should run from 0 to kf. In summary, we can solve for ky using the given equation, which will allow us to set up the integral to integrate over the vector running from the origin to kf, with the limits for kx running from 0 to kf.
  • #1
Niles
1,866
0

Homework Statement


Hi

I have the following integral over wavevectors inside the Fermi circle (we are in 2D)

[tex]
\int {dk_x \int {dk_y \sin ^2 \left( {k_x x} \right)} }
[/tex]

Ok, so I know that kx2+ky2=kf2, so ky2=kf2-kx2 - this takes care of ky. But what about kx? What should this run from in order to integrate along the vector running from the origin to kf?

Niles.
 
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  • #2
Homework Equationskx2+ky2=kf2The Attempt at a SolutionWe can use the equation kx2+ky2=kf2 to solve for kx. We can let ky be equal to the square root of kf2-kx2. So, we can set up the integral as follows: \int_{0}^{k_f}{dk_x \int_{0}^{\sqrt{k_f^2-k_x^2}}} {dk_y \sin ^2 \left( {k_x x} \right)}
 

1. What is the Fermi sphere and why is it important in integration?

The Fermi sphere is a theoretical construct in solid state physics that represents the boundary of the momentum space for electrons in a solid. It is important in integration because it helps us understand the behavior of electrons in a solid and allows us to calculate various physical properties.

2. How is integration over the Fermi sphere different from regular integration?

Integration over the Fermi sphere involves considering the entire momentum space for electrons, rather than just a specific range of values. This requires a different approach and may involve using special functions or techniques, such as the Fermi-Dirac distribution, to accurately integrate over the entire sphere.

3. What are some applications of integrating over the Fermi sphere?

Integrating over the Fermi sphere is important in many areas of solid state physics, such as calculating the electronic specific heat, electrical conductivity, and magnetic susceptibility of a material. It is also essential in understanding phenomena like superconductivity and the Hall effect.

4. How does the Fermi energy relate to integration over the Fermi sphere?

The Fermi energy is the highest occupied energy level in a solid at absolute zero temperature. When integrating over the Fermi sphere, the Fermi energy serves as a reference point and determines the upper limit of the integral. It is also used to calculate the Fermi-Dirac distribution, which is essential in accurately integrating over the sphere.

5. Are there any challenges or limitations when integrating over the Fermi sphere?

One challenge of integrating over the Fermi sphere is that it requires more complex mathematical techniques and may not have a closed-form solution for certain cases. Additionally, the accuracy of the integration may be affected by factors such as temperature and external fields. However, advancements in computational methods have made it easier to accurately integrate over the Fermi sphere.

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