Finally, s gives the polarization and runs from one to the number of dimensions. This is one advantage of harmonic models, see the harmonic oscillator. We get where is the quantum state of the normal mode specified by . In terms of second quantization it is also the occupation number of these modes. Hence energy is exchanged in quantized packages.
Lattice vibrations come and go in units, just like photons, and the units are called phonons .
They can be treated as quasi-particles. That interaction corresponds to the third order term in the energy expansion, where we have three vibrational coordinates. Yet we can do thermodynamics and talk about average numbers of phonons. Because the system is going to be in contact with a heat-bath which will equilibrate the sample. In real materials the phonon-phonon interaction plays an important role there.