So far we have considered only the ground state of some condensed matter systems. If we excite such objects many interesting things can happen. One of the simplest and most interesting is that we can excite wave-like motions of the atoms and molecules. If we hit a solid with a hammer, we know that sound waves will move through the body. These acoustic waves are part of a class of motions called lattice vibrations which also can be excited by thermal effects. If temperatures are low or the wave amplitude is small, quantum mechanics must be used. The quantized version of a lattice vibration is made up of phonons.

Less familiar is the fact that if we wiggle a spin in a magnet it will cause its neighbors to move, and the motion will be transmitted in a spin wave. Once more there is a quantum version, the magnon.

Lattice vibrations and phonons

We first look at the vibrations of atoms away from their equilibrium positions in the crystal. We do this in three ways. First we briefly allude to the macroscopic theory of elasticity which does not use microscopic details at all. Then we look at classical atoms bound together by forces. For vibrations near the equilibrium positions the forces become an effective set of springs that bind the atoms. Then we quantize the motions.