Consider a 1-dimensional chain of $2 N$ atoms of mass $m$ (with $N$ large and with periodic boundary conditions). The interactions between neighbouring atoms are modelled by springs with alternating spring constants $K$ and $G$, with $K>G$.

In equilibrium, the separation of the atoms is $a$, the natural length of the springs.

Find the frequencies of the longitudinal modes of vibration for this system, and show that they are labelled by a wavenumber $q$ that is restricted to a Brillouin zone. Identify the acoustic and optical bands of the vibration spectrum, and determine approximations for the frequencies near the centre of the Brillouin zone. What is the frequency gap between the acoustic and optical bands at the zone boundary?

Describe briefly the properties of the phonons in this system.