# 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

If the potentials are all quadratic, then the force varies linearly with displacement. That means the superposition principle holds. (Don't confuse that with quantum superposition!) In other words the lattice waves then behave like electromagnetic waves: you put two of them together and they simply add together but don't otherwise affect one another; they don't scatter.

If the potentials are higher order, then the force does not vary linearly with position, and the superposition principle does not hold. (It holds only for linear systems.)

-

The basic reason is that the potential energy between atoms in a real crystal lattice isn't solely quadratic in the lattice spacing - there are higher order terms, so the model of a lattice as coupled harmonic oscillators breaks down.

Modeling the lattice as a series of coupled harmonic oscillators can't account for a number of observed phenomena. Thermal conductivity and thermal expansion are just two of the most familiar properties. If crystals behaved like coupled harmonic oscillators, they would have infinite thermal conductivity, and the lattice spacing would not depend on temperature - i.e. there would be no thermal expansion or contraction. Obviously that's not what is observed, and these effects are explained by the higher order (anharmonic) terms in the lattice potential energy.

There's a good additional discussion about phonons at the earlier post :

What is a phonon?

-