Let Un =displacement of atom n from its equilibrium position
Un-1=displacement of atom n-1 from its equilibrium position
Un+1=displacement of atom n+1 from its equilibrium position
The force on atom n will be given by its displacement and the displacement of its nearest neighbors :
The equation of motion is:
The above equation is not obviously a wave equation, but let us assume a traveling wave solution, namely,
Let Uno = Uo since if it is a wave, it has to have a definite amplitude.
If we substitute our wave solution into equation of motion we find a phonon's dispersion relation for linear monatomic chain as follows:
The dispersion curve is shown below.
One important feature of the dispersion curve is the periodicity of the function. For unit cell length , the repeat period is , which is equal to the unit cell length in the reciprocal lattice. Therefore the useful information is contained in the waves with wave vectors lying between the limits
This range of wave vectors is called the first Brillouin zone. At the Brillouin zone boundaries the nearest atoms of the chain vibrate in the opposite directions and the wave becomes a standing wave .
As k approaches zero (the long-wavelength limit) and we have