# 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 U_{n }=displacement of atom n from its equilibrium position

U_{n-1}=displacement of atom n-1 from its equilibrium position

U_{n+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 U_{no} = U_{o} 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