Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 33A / Solution

Label the two displacements in cell $n$ by $u_n,v_n$, with cell length $2a$. Choose the $K$ spring within a cell and the $G$ spring between cells. The equations areA Bloch wave $(u_n,v_n)=(u,v)e^{i(2aqn-\omega t)}$ gives a two-by-two eigenvalue problem, whose determinant isThus the alternating-spring chain dispersion relation isPeriodic boundaries require $e^{2iaqN}=1$, so $q=\pi j/(Na)$ modulo $\pi /a$. There are $N$ distinct wavenumbers in the Brillouin zone $-\pi /(2a)\le q<\pi /(2a)$ and two branches, accounting for $2N$ modes. The minus branch is acoustic, with zero frequency and in-phase motion at the centre; the plus branch is optical, with opposite cell displacements.

Expanding the square root near zero givesAt the zone boundary the square root is $K-G$, so the acoustic maximum is $\sqrt{2G/m}$ and the optical minimum is $\sqrt{2K/m}$. The frequency gap is thereforeQuantizing each harmonic mode gives phonons, bosonic excitations of energy $\hslash {\omega }_{\pm }(q)$ and crystal momentum $\hslash q$ defined modulo a reciprocal lattice vector. Their group velocity is $d\omega /dq$; acoustic phonons carry long-wavelength sound, while optical phonons have a nonzero centre frequency. There is one longitudinal polarization per branch here, and phonon number is not conserved in thermal equilibrium.