# Acoustical phonons

Lets consider a linear chain of identical atoms of mass M spaced at a distance , the lattice constant, connected by invisible Hook's law springs. For simplicity we will consider longitudinal deformations - that is, displacements of atoms are parallel to the chain.

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:

where is a spring constant.

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 .

### Transverse acoustical standing wave

As k approaches zero (the long-wavelength limit) and we have

where is a phase velocity, which is equivalent to the velocity of a sound in the crystal.

Phonons with frequency which goes to zero in the limit of small k are known as acoustical phonons.

### "Long" wavelength acoustical vibrations

Author:Taras Kolodiaynyi; email: kolodiaynyi@chembio.uoguelph.ca
Curator: Dan Thomas email: thomas@chembio.uoguelph.ca
Last Updated: Wednesday, April 16, 1997