# Derivation of logarithmic singularity that causes the Kohn anomaly

The basis of the Kohn anomaly is that a logarithmic singularity at $$q=2k_\rm F$$ (where $$k_\rm F$$ is the radius of the Fermi sphere) causes extreme fluctuations of the dielectric function about that point.

I'm interested in how that is derived, but I could not find anything in Lindhard theory which the Wikipedia article (linked above) points to.

Is there literature that details a proof of this result?

A derivation of the Lindhard formula and the emerging Kohn anomaly can be found in J. M. Ziman's Principles of the Theory of Solids (Chapter 5: Electron-Electron Interaction1). Equation (5.16) from the cited book gives a general formula for the dielectric constant. The Kohn anomaly (shown in equation (5.36) in Ziman) is a consequence of the free-electrons model approximating (5.16) at $$0$$ Kelvin. Here is what Ziman writes in Chapter 5.4:
If we wish to study screening at short distances we need to evaluate the sum (5.16) for large values of $$\bf q$$. This depends on the detailed structure of the energy surfaces $$\mathcal E(k)$$. For our free-electron model at absolute zero it is not very difficult to evaluate the sum by a straightforward integration over $$\bf k$$-space. The result is as follows: $$\epsilon({\bf q},0)=1+\frac{4\pi{\boldsymbol e}^2}{q^2}\frac n{\frac23\mathcal E_F}\left\{\frac12+\frac{4k_F^2-q^2}{8k_Fq^2}\ln\left|\frac{2k_F+q}{2k_F-q}\right|\right\}\tag{5.36},$$ where $$k_F$$ is the radius of the Fermi sphere.