# Self consistent equation by FT

1. Mar 7, 2012

1. The problem statement, all variables and given/known data
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 eletronic 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

2. Relevant 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) )

3. 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
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution