# What are soft phonon modes?

In chapter 4 of Charles Kittel's Introduction to Solid State Physics, a problem titled "soft phonon modes" asks us to derive the dispersion relation for the following setup: a 1d crystal consists of a line of point ions equal in mass but alternating in charge $$e_p = (-1)^p e$$ separated by distance $$a$$, with force constant $$\gamma$$ between neighbouring ions and Coulomb interaction between all ions.

Then it can be shown that the dispersion relation is $$\omega^2 = \frac{4\gamma}{M}\left[ \sin^2(Ka/2) + \frac{e^2}{4\pi\epsilon_0 a^3\gamma} \sum^\infty_{p=1} \frac{(-1)^p}{p^3}\left(1-\cos{pKa}\right) \right] ,$$

and various criteria for stability can be derived from here. My question is, what exactly does "soft phonon modes" mean here, and what does it have to do with the setup described in the problem?

• soft quantum particles are low energy, i.e. low $\omega$, which in turn should be low momentum, low $K$. This makes sense in this context too, because the derivation is really only good for long wavelength phonons. Almost-bulk motion, far from probing the interatomic spacing. Commented Nov 15, 2023 at 4:37
• How many modes does this have? If it has a lower branch, then that is probably the soft mode, and the higher branches are not. Commented Nov 16, 2023 at 0:47