Exploring Waves in Crystals: Acoustical & Optical Branches

[aaaa202]

My book considers waves in crystals. Especially it develops the idea of acoustical and optical branches of the waves. From what I can see these are different ways the dispersion of our phonon may look like. It is developed when the crystal contains more than one atom per unit cell, but I am not sure why this is. Do we not have acoustical and optical branches for a wave in any crystal regardless of how many atoms we have per unit cell and what their mass ratio might be?
Also I am not sure what dispersion even means. It is a relationship between the phase velocity and the wavelength (or wavevector k), but what is the physical meaning of this quantity. When a physicist describes different kinds of dispersion in a crystal is he then describing different forms of wavemotion that can occur in the crystal?
Maybe my question is kind of confusing but I just want to clear up some things that my books explains quite poorly.

 

[UltrafastPED]

For acoustic phonons all of the motions are in phase ... just as with classical sound. Everything else is an optical phonon.

Dispersion is a property of waves in matter: if the velocity varies with the wavelength, then the relationship which describes this is the dispersion relation. In optics this is connected to the index of refraction; in a vacuum (n=1) there is no dispersion. Likewise for ordinary acoustics there is almost no dispersion in air, but it can be very important for ultrasound in biological systems.

For a crystal the speed of sound can vary according to the direction of travel ...

For more details see: http://en.wikipedia.org/wiki/Phonon#Dispersion_relation

Kittel's "Introduction to Solid State Physics" is a great reference whenever you get confused - very easy to follow.