Phonon spectrum I had a question regarding phonon spectrum in condensed matter.
Consider a cubic lattice with '$p$' atoms per primitive cell. Consider the lattice plane used for derivation of the phonon spectrum to be the face of the cube and consider only nearest neighbour interactions. Consider the force on the plane to be directly proportional to the nearest neighbour distance with that of the neighbouring planes. Using this prescription (described in Introduction to Solid State Physics by Charles Kittel for example) we get $\omega$ versus $k$ curves depending on the value of $p$ . My question is how do you classify these curves into 'acoustic' and 'optical' modes? On what basis is the nomenclature of 'optical' and 'acoustic' used? Further, why are there only 3 acoustic modes be it any '$p$' value?
 A: Citing the wikipedia article:

Acoustic phonons are coherent movements of atoms of the lattice out of their equilibrium positions. If the displacement is in the direction of propagation, then in some areas the atoms will be closer, in others farther apart, as in a sound wave in air (hence the name acoustic). Displacement perpendicular to the propagation direction is comparable to waves in water.
Optical phonons are out-of-phase movement of the atoms in the lattice, one atom moving to the left, and its neighbour to the right. This occurs if the lattice is made of atoms of different charge or mass. They are called optical because in ionic crystals, such as sodium chloride, they are excited by infrared radiation.

Bottom line:

*

*acoustic phonons are comparable by their mode of movement to sound waves

*optical phonons couple to electromagnetic field (part of the spectrum of which is studied by optics)

A: The classification into acoustic and optical is to do with the behaviour of the Dispersion relation $\omega(k)$ at the centre of the zone, ie as $k$ tends to 0. For the acoustic branches the dispersion relation is approximately linear there, so that $\omega(k)=v_{sound}k$ which is equivalent to the usual frequency/wavelength relationship. 
In addition, taking the one-dimensional two mass setup of this problem as an example, there are two modes of vibration, one where the two masses are vibrating in phase with each other, and one where they are vibrating in anti-phase. The first of these resembles "normal" sound waves, where the vibrations are straightforwardly passed down the crystal, and as such this is another factor. 
Lastly I believe there is the fact that the wave speed for the acoustic branches, the limit of $\omega/k$ as $k$ goes to zero, is of the same order as sound speeds in e.g. air. 
