# Lindhard RPA Dielectric Function Electron Gas

The longitudinal dielectric function of a gas of free electrons (+ homogeneous positive background) is often described in the Lindhard- or Random Phase Approximation (RPA).
The dielectric function depends on both frequency omega and wavevector k. However, it is non-analytic at the point omega=0, k=0. Namely its value depends on how the constant ratio of omega/k is chosen in the limit omega to 0. What is the physics behind this behaviour?

Physics Monkey
Homework Helper
The singular structure comes from the existence of gapless excitations at the Fermi surface. Is this answer too brief/trivial for what you were looking for?

Dear Physics Monkey,

Too brief yes, too trivial no. I was thinking the following: when taking the limit omega to 0 before k to 0 (static screening) the electrons have all the time of the world to adjust to the field. In the other limit ( k to 0 before omega to 0) they would have to move with too high velocity over too large a distance.
I also think that the latter limit changes drastically if scattering/band structure is to be included.

Physics Monkey