Kohn singularity of static Lindhard screening ($F(1+u)=\frac{1}{2}+\frac{u}{2}\ln |u|+O(u)$)

The three-dimensional static screening function is finite at $q=2k_F$, but its derivative diverges logarithmically. Expanding the explicit static Lindhard dielectric function gives the title's nonanalytic term. The singularity comes from particle–hole excitations connecting opposite parts of the Fermi surface. It produces a Friedel density tail from a derivative singularity; analogous electronic response can cause a Kohn anomaly in a coupled lattice vibration.