Consider a lattice with two kinds of atoms - that is lattice with a basis of two atoms in the primitive cell. Now we have to write two solutions for the displacement corresponding to the two masses m and M. The equations of motion are:
and we assume solutions:
The diatomic case has two solutions of the dispersion relation:
These solutions are plotted in the figure below.
The allowed frequencies of propagation wave are split into an upper branch known as the optical branch, and a lower branch called the acoustical branch. There is a band of frequencies between the two
branches that cannot propagate. The width of this forbidden band depends on the difference of the
masses. If the two masses are equal, the two branches join (become degenerate) at 
. Note that the first
Brillouin zone goes from 
to 
just as in the monatomic case if we use the lattice constant
instead of the
interatomic spacing 
. The acoustical branch is qualitatively similar to the dispersion relation for a monatomic lattice, but the optical branch represents a completely
different form of wave motion.
What is the difference physically between the acoustical and optical branches? It can be found that for optical branch (in the long wavelength limit) the two atoms in the unit cell move opposite to each other and the light mass amplitude is greater. For acoustical branch (in the long wavelength limit) the displacement of both atoms has the same amplitude, direction and phase. Click here to see the difference between acoustical and optical modes .