# Phase velocity in monatomic chain

When considering a one-dimensional monatomic chain of atoms (identical masses $$m$$ & spring constant $$\kappa$$), one finds the following dispersion: $$\omega(k) = \sqrt\frac{\kappa}{m}\cdot\left|\sin\left(\frac{ka}{2}\right)\right|\, ,$$

which is $$\frac{2\mathrm{\pi}}{a}$$-periodic. So wavewectors higher than $$\mathrm{\pi}/a$$ do not provide new physical behaviour.

However, when computing the phase velocity, one finds: $$v_p = \frac{\omega}{k} = \frac{1}{k}\sqrt\frac{\kappa}{m}\cdot\left|\sin\left(\frac{ka}{2}\right)\right|\, .$$ This means that the phase velocity goes like a sinc, which is not periodic; wavevectors outside the first Brioullin zone yield a much lower phase velocity.

How is this possible? Is there a good reason to consider only the first Brioullin zone for the phase velocity? Or are there other errors my calculation?

• Why do you care about phase velocity? Group velocity is much more important and meaningful Aug 21, 2020 at 0:41

While you can define a continuous function for the displacement of the atoms from their equilibrium position $$u\left(x, t\right)$$ for the wave, that doesn't mean that the wave is really continuous; the wave only has a meaningful displacement at the $$x$$ positions where there are atoms. So, some of the intuition coming from waves in a continuous medium doesn't really apply.
• You are right. I had a hard time thinking about increasingly small $v_p$ for large values of $k$. But in this simple case, wavevectors should be truncated to the first BZ in the first place. So maybe one should redefine $v_p$ to be $\omega / (k\,\mathrm{mod}\, (\pi/a))$. This yields a much more sensible graph. Aug 21, 2020 at 15:36