# Screened Coulomb potential in metals

Physics Asked by curio on November 4, 2020

One of the reasons why we can neglect electron-electron interactions in metals is the fact that their coulomb interaction is screened. I’m confused about the nature of this screening. In the literature the process is usually described like this:

If we bring an additional charge inside a neutral metal, the coulomb potential of this charge will be screened due to electrons accumulating (or rejecting) at the charge. This leads to the Thomas-Fermi potential

$$phi(r) =frac{e}{r},e^{-r/r_{TF}}.$$

But if we consider the potential between electrons in the neutral metal, the screening must have a different origin because the electron density is homogeneous. The above mentioned approach can’t be the reason can it?

I think the only way for the electron-electron interaction to be screened in the neutral metal is by the presence of the positively charged ions. If we consider a small volume in the metal, the total charge inside will be zero so the long range potential falls off faster then $$frac{1}{r}$$. It appears the literature claims that the other electrons are responsible for the screening but I can’t see how this is possible.

Edit:
Maybe I should rephrase the question. In the literature, it seems the mechanism behind the screening is the dynamics of the electrons, rather then the presence of the ions, so my question is

Q: How can this possibly be true, given that the electron distribution is homogeneous?

Maybe not very useful regarding you question, but other way to see why e-e interactions in a metal can be neglected is by comparing typical kinetic to potential energy.

$$E_F = hbar^2k_F^2/2m$$ and $$U_{e-e} = e^2/{4piepsilon_0d}$$.

Using the free electron relation $$k_F = (3pi^2n)^{1/3}$$ and $$d^{-1} approx n^{1/3}$$ you get

$$E_F/U_{e-e} approx 5 a_B n^{1/3}$$

where $$a_B$$ is the Bohr radius in SI units. When the potential energy is equal to or larger than the kinetic energy it is also known as the Mott criterion. The insight of appx is that counter-intuitively, as the density increases e-e interactions are negligible.

Answered by Cooper_pair on November 4, 2020

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