The Independent-Particle Approximation

[ratphysicsfun]

Homework Statement

The IPA potential-energy function U(r) is the potential energy "felt" by an atomic electron in the average field of the other Z - 1 electrons plus the nucleus. If one knew the average charge distribution rho(r) of the Z - 1 other electrons, it would be a fairly simple matter to find U(r). The calculation of an accurate distribution rho(r) is very hard, but it is easy to make a fairly realistic guess. For example, one might guess that rho(r) is spherically symmetric and given by rho(r)=(rho.naught)exp(-r/R) where R is some sort of mean atomic radius. (a) Given that rho(r) is the average charge distribution of Z - 1 electrons, find rho.naught in terms of Z, e, and R. (b) Use Gauss's law to find the electric field E at a point r due to the nucleus and the charge distribution rho. (c) Verify that as r goes to 0 and r goes to infinity, E bahaves as required by F= Zkexp2/(r^2) for r inside all other electrons, and F=kexp2/(r^2) for r outside all other electrons.

Homework Equations

F=Zke^2/(r^2), F=ke^2/(r^2), U(r)= -ke^2/r for r outside other electrons, U(r)=~-Zke^2/r as r goes to zero (inside other electrons)

The Attempt at a Solution

I'm not sure how rho fits into my equations. I know it's related to e, and Z.

 

[BigJDubs]

I'm not sure if I need to use U(r) or F=Zke^2/(r^2). (a) rho.naught= Z*e/R(b) E(r)= (Z*e/4piEpsilon.naught)*exp(-r/R)/r^2 (c) As r goes to zero, E(r)= Z*ke^2/r^2 and as r goes to infinity, E(r)= ke^2/r^2, so both equations are satisfied.