# How to derive the divergence leading to Kohn anomalies?

I'm trying to understand the mathematical derivation given in the book "A Quantum Approach to Condensed Matter Physics" on page 215 (see 1), for explaining how the phonon-energy perturbed by phonon-electron interaction, namely

$$\hbar\omega_q^{(p)} = \hbar\omega_q - \sum\limits_k 2 | M_{kk'}|^2 \langle n_k \rangle (\varepsilon_{k'} - \varepsilon_{k})^{-1}$$

with $k' = k - q$ gives rise to the Kohn anomaly by making $\partial \hbar\omega_q^{(p)} / \partial q_x$ diverge when $q \sim 2k_F$.

The book explains that

Let us suppose $q$ to be in the $x$-direction and of magnitude $2k_F$, and evaluate $\partial \hbar\omega_q^{(p)} / \partial q_x$. If we neglect the variation of $M_{kk'}$ with $q$ the electron-phonon interaction contributes an amount $$2 \sum\limits_k | M_{kk'}|^2 \langle n_k \rangle (\varepsilon_{k'} - \varepsilon_{k})^{-2} {\partial \varepsilon_{k-q} \over \partial q_x} .$$ On substituting for $\varepsilon_{k-q}$ one finds the summation to contain the factors $\langle n_k \rangle (k_x - k_F)^{-2}$. These cause a logarithmic divergence when the summation is performed [...]

What I don't grasp yet is the step of substituting for $\varepsilon_{k-q}$ (what exactly does one substitute with? $\varepsilon_k = \hbar^2 k^2 / 2m$?) and what the summation then looks like concretely to cause a logarithmic divergence.

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There is either something weird in their derivation or I am stupid, but I will try to figure it out. – BebopButUnsteady Nov 19 '12 at 18:25
Ah I see. That's silly – BebopButUnsteady Nov 19 '12 at 18:33