Self consistent equation by FT

[saiaditya]

Homework Statement

Magnesium atom is introduced into a Copper crystal where electrons are free to move in a fermi sphere until equilibrium is reached.
Assuming the mean field (slowly varying) description applies with potential energy eψ(r)
the energy given by ε(r,p) where the dist of electrons is given by fermi function f(r,p)
for change in electronic density due to impurity we have δn(r).
Mean field is related to its source (charge distribution) through poisson equation -Δψ(r) .
Solve for the self consistent equation by Fourier Transform

Homework Equations

ε(r,p)=ε(p)+eψ(r)

f(r,p)=1/( e^(ε(p)+eψ(r)-εf/kt) +1 )

δn(r)=2∫dp/(2∏h)^3 (f(r,p)-f(p)

-Δψ(r)=e/ε (δn(r)-δ(r) )

The Attempt at a Solution

Am not sure how to solve for f(r,p) but by equating δn(r) to Aψ and using FT i hope to arrive
at a solution, any help will be helpful. Thanks

 

[mark726]

for reading and any insights you have on this problem.

Hello, thank you for your forum post. Solving for the self-consistent equation in this scenario involves using the Poisson equation, which relates the mean field (potential energy) to its source (charge distribution). In this case, the charge distribution is affected by the impurity (magnesium atom) and can be represented by δn(r). Using the Fourier transform, we can rewrite the Poisson equation as -Δψ(r) = e/ε (δn(r)-δ(r) ).

To solve for f(r,p), we can use the fact that the distribution of electrons is given by the Fermi function, which takes into account the energy of the electrons and the Fermi energy (εf). We can rewrite this as f(r,p) = 1/( e^(ε(p)+eψ(r)-εf/kt) +1 ).

To solve for the self-consistent equation, we can equate δn(r) to Aψ, where A is a constant. Then, using the Fourier transform, we can solve for ψ(r) and substitute it back into the equation for f(r,p) to solve for the electronic distribution.

I hope this helps! Let me know if you have any further questions or if you need clarification on any of the steps. Good luck with your research.