Past exam of the physics A course of the University of Cambridge / 2013 / Paper 2 / Section C / C11

One mass per cell. If $u_j$ is the displacement from equilibrium, the two springs give $m{\ddot{u}}_j=\alpha (u_{j+1}+u_{j-1}-2u_j)$. Substituting $u_j=U{e}^{i(qja-\omega t)}$ yieldsThe first Brillouin zone is $-\pi /a\le q\le \pi /a$; adding $2\pi /a$ leaves the sampled displacements unchanged. The branch is linear near zero, with sound speed $a\sqrt{\alpha /m}$, and reaches $2\sqrt{\alpha /m}$ at either zone edge.

Two masses per cell. Take a cell of length $2a$, with light displacement $u_j=U{e}^{i(2jaq-\omega t)}$ and heavy displacement $v_j=V{e}^{i[(2j+1)aq-\omega t]}$. The coupled equations reduce toA nonzero displacement requires determinant zero:Solving this quadratic in ${\omega }^2$ givesNow the first zone is $|q|\le \pi /(2a)$. The minus branch is acoustic: as $q\to 0$, both sublattices move together and ${\omega }_{-}\to 0$. Its sound speed is $a\sqrt{2\alpha /(M+m)}$. The plus branch is optical: at $q=0$ the atoms move oppositely with $mU+MV=0$, giving finite frequency $\sqrt{2\alpha (1/m+1/M)}$. This terminology describes relative motion; optical activity itself also requires coupling to the electromagnetic field.

diatomic-chain zone-edge polarizations and dimensionality. At the edge, $\cos qa=0$, so the heavy and light sublattices decouple: the acoustic edge frequency is $\sqrt{2\alpha /M}$ and the optical edge frequency is $\sqrt{2\alpha /m}$. HenceThe acoustic edge mode moves only the heavy sublattice; the optical edge mode only the light one. If $M=m$, the gap closes and the two branches are the zone-folded monatomic branch.

A three-dimensional crystal with two atoms per primitive cell has six branches: three acoustic and three optical. For propagation along a suitable symmetry direction these resolve into one longitudinal and two transverse branches of each type. The wavevector and Brillouin zone are three-dimensional; anisotropic force constants determine the dispersions, and polar crystals can have longitudinal–transverse optical splitting. A universal six-branch curve cannot be specified from masses alone.