Past exam of the condensed matter physics course of the University of Oxford / 2013 / 3 / a

Let $a$ be the equilibrium spacing of this monatomic chain and $u_j$ the displacement from $ja$. The right nearest-neighbour bond exerts $K(u_{j+1}-u_j)$ and the left bond $K(u_{j-1}-u_j)$. The two next-nearest-neighbour bonds similarly connect $j$ to $j\pm 2$. ThusEvery uniform displacement gives zero force, as translation symmetry requires. There is one displacement coordinate per primitive cell. Adding further-neighbour springs does not introduce a second atom or turn this into a diatomic-chain problem.