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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?

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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.


Reference

[1] Ziman, J. (1972). Electron-Electron Interaction. In Principles of the Theory of Solids (pp. 146-170). Cambridge: Cambridge University Press. doi:10.1017/CBO9781139644075.007

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    $\begingroup$ Thank you for the reference. Incidentally, I have found an open-access paper by Das and Green (2016) which discusses the same thing in section 4.7. $\endgroup$ May 4 '20 at 13:59

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