Past exam of the condensed matter physics course of the University of Oxford / 2015 / 3 / b / Solution

Let $u_p$ be the longitudinal displacement of atom $p$. Translational invariance and a symmetric harmonic pair force give the long-range harmonic-chain dispersion equationsUse a traveling normal mode $u_p=u_0{e}^{i(kpa-\omega t)}$. Dividing by its nonzero amplitude givesHencePositive pair constants give a nonnegative squared frequency and a uniform-translation zero mode. More general signed force constants require checking stability of the entire sum. The frequency is even in $k$ and periodic with $2\pi /a$, reflecting reciprocal-lattice equivalence. The force sum must converge in the model. A finite ordinary acoustic slope additionally requires convergence of $\sum n^2{C}_n$, in which case $v_s=a\sqrt{\sum n^2{C}_n/M}$. Both finite-range examples below satisfy this condition.