# Why must phonon-phonon interaction be anharmonic?

I'm learning about phonons and this is really interesting. Why can't I describe two phonons colliding as a harmonic oscillation? Why does it have to be anharmonic for the phonon model to work?

• Harmonic interaction would mean that you could decouple the interaction by defining appropriately chosen linear combinations. The interaction could have been included in the matrix that describes the potential energy up to quadratic terms and you should have diagonalized that matrix. – Count Iblis Jun 15 '14 at 16:10
• @CountIblis Thanks for the answer. Could you elaborate a little bit about defining interactions as linear combinations? What does that imply? Also if you could elaborate your second sentence a little bit. What happens when I diagonalize the matrix? Isn't that ok? – bodacydo Jun 15 '14 at 16:56
• @CountIblis I can't quite understand why if I defined phonons as harmonic linear combinations, then the interactions woudn't happen. I'd greatly appreciate some clues. Thanks :) – bodacydo Jun 15 '14 at 17:06
• It's just like the case where you have two coupled oscillators, you can decouple them by doing a suitable coordinate transform (e.g. if you take the sum and differences of the two coordinates in case the oscillators are identical). This will always work as long as you only have quadratic terms. – Count Iblis Jun 15 '14 at 17:14