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I have been told that phonon-phonon Interaction is an anharmonic effect so only arises if terms of third and higher order in the displacement of the ions the Hamiltonian for the nuclii is taken into account.

So how do I see the there is no phonon-phonon interaction in the harmonic approximation?

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When you solve for such Hamiltonian for the harmonic oscillator you get a set of eigenstates which by definition are orthogonal and thus you have phonons that don't interact.

When you include a non harmonic term to the Hamiltonian and treating it as a perturbation you get new eigenstates that are a mixture of those the simple harmonic oscillator. This mixture is what it's called phonon-phonon interaction.

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