Past exam of the condensed matter physics course of the University of Oxford / 2018 / 3 / c

The mass per lead atom isLet $u_n$ be its longitudinal displacement. Newton's second law gives the harmonic chain equationInsert the normal mode $u_n=u{e}^{i(kna-\omega t)}$ to obtainAt long wavelength, $\omega \simeq a\sqrt{\kappa /m}|k|$. The sound speed is the corresponding low-frequency group velocity:The relevant inertia in this chain equation is one atomic mass. The reduced mass $m/2$ belongs to the relative vibration of an isolated pair and should not be substituted for $m$ in the chain's dispersion relation.