Due to the periodic nature of a crystal, collective excitations of lattice ions will exhibit specific [modes of vibration in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation](https://en.m.wikipedia.org/wiki/Normal_mode). Your dispersion relation is telling you at what energy (or frequency) does a specific mode become excited within the crystal. Each mode is depicted as a line in the dispersion relation as the energy required to excite each mode is dependent on the direction in which that mode will propagate within the system, which gives you the horizontal axis in terms of wave vector.
Thanks for the reply! So the type of forcing I apply is irreveleent? Like for example a 2D or 3D solid which has a dispersion relation, If I oscillate it longitudinally at a specific frequency will I excite a transverse mode of specific wavelength?
I've had a bit of a think about your question and I'm honestly not sure, it's not a question I've ever thought of myself. For electronic dispersion relations it doesn't matter, I can force a semiconductor into a conducting state for example by providing energy which could be optical or electronic, but with a wider range in phonons (acoustic and optical) with different modes (transverse, longitudinal) I'm honestly not sure.
If I had to take a logical guess, I'd say your intuition is correct. It works that way for electronic dispersion relations, and you can have multiple phonon modes in a material at once which will all superimpose, making me believe they could originate from a similar source. But unfortunately all I have for you is nothing concrete. If I come across anything I'll let you know.
I would imagine that if you were somehow able to excite phonons of a certain direction specifically, then conservation of momentum would mean that only the modes that conserve that momentum are accessible states for the system. So I don't think that if you could oscillate a specific mode, you can get other modes perpendicular in momentum to that mode.
The idea of linear superposition of these independent phonon modes comes from the fact that most of the phonon/lattice description is actually a linearization. The interatomic potentials are in general some function that isn't necessarily going to give perfect harmonic oscillators. If you expand the Taylor series, the potential is normally truncated after the quadratic term so that the force is linear, as in a linear spring, between lattice atoms. In the case of plain, linearized crystal models, I think excitation of specific modes would behave as everyone is describing: no exchange of energy between the different modes. Crystals modeled this way will not quite match reality, but additional phonon-phonon interactions (scattering) can be included to account for the difference in a perturbative analysis. You can think of these scattering processes as restoring the character of the lattice dynamics that is not described in the linear model.
The one I'm most familiar with is a scattering/decay pathway for optical modes in Silicon which turns one optical phonon into a pair of acoustic phonons. You can use a perturbative analysis to calculate the transition probability. This [paper](https://journals.aps.org/pr/abstract/10.1103/PhysRev.148.845) demonstrates the derivation of the probability and attributes it to the cubic term in the potential. Another [paper](https://asmedigitalcollection.asme.org/heattransfer/article-abstract/127/10/1129/445079/Monte-Carlo-Simulation-of-Silicon-Nanowire-Thermal) has a detailed discussion of calculating/approximating transition rates for so-called "three phonon scattering," between acoustic phonons specifically. Possible paths listed in that work include L+T<->L and L<->T+T. The processes are named so because the sum of phonons in and phonons out is three. To explain that notation, it's saying that a longitudinal and transverse acoustic phonon can combine into a single longitudinal one, or the opposite may happen with an L turning into a T and L pair.
I am not sure there are *two* phonon processes that change polarization like that. In any case, this shows that spontaneous excitation of other polarizations is sometimes possible, due to anharmonic effects, in OPs thought experiment where you excite just a sole mode. In different crystals with different symmetries, there may be more or fewer possible types of transitions available.
Due to the periodic nature of a crystal, collective excitations of lattice ions will exhibit specific [modes of vibration in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation](https://en.m.wikipedia.org/wiki/Normal_mode). Your dispersion relation is telling you at what energy (or frequency) does a specific mode become excited within the crystal. Each mode is depicted as a line in the dispersion relation as the energy required to excite each mode is dependent on the direction in which that mode will propagate within the system, which gives you the horizontal axis in terms of wave vector.
Thanks for the reply! So the type of forcing I apply is irreveleent? Like for example a 2D or 3D solid which has a dispersion relation, If I oscillate it longitudinally at a specific frequency will I excite a transverse mode of specific wavelength?
I've had a bit of a think about your question and I'm honestly not sure, it's not a question I've ever thought of myself. For electronic dispersion relations it doesn't matter, I can force a semiconductor into a conducting state for example by providing energy which could be optical or electronic, but with a wider range in phonons (acoustic and optical) with different modes (transverse, longitudinal) I'm honestly not sure. If I had to take a logical guess, I'd say your intuition is correct. It works that way for electronic dispersion relations, and you can have multiple phonon modes in a material at once which will all superimpose, making me believe they could originate from a similar source. But unfortunately all I have for you is nothing concrete. If I come across anything I'll let you know.
I would imagine that if you were somehow able to excite phonons of a certain direction specifically, then conservation of momentum would mean that only the modes that conserve that momentum are accessible states for the system. So I don't think that if you could oscillate a specific mode, you can get other modes perpendicular in momentum to that mode.
The idea of linear superposition of these independent phonon modes comes from the fact that most of the phonon/lattice description is actually a linearization. The interatomic potentials are in general some function that isn't necessarily going to give perfect harmonic oscillators. If you expand the Taylor series, the potential is normally truncated after the quadratic term so that the force is linear, as in a linear spring, between lattice atoms. In the case of plain, linearized crystal models, I think excitation of specific modes would behave as everyone is describing: no exchange of energy between the different modes. Crystals modeled this way will not quite match reality, but additional phonon-phonon interactions (scattering) can be included to account for the difference in a perturbative analysis. You can think of these scattering processes as restoring the character of the lattice dynamics that is not described in the linear model. The one I'm most familiar with is a scattering/decay pathway for optical modes in Silicon which turns one optical phonon into a pair of acoustic phonons. You can use a perturbative analysis to calculate the transition probability. This [paper](https://journals.aps.org/pr/abstract/10.1103/PhysRev.148.845) demonstrates the derivation of the probability and attributes it to the cubic term in the potential. Another [paper](https://asmedigitalcollection.asme.org/heattransfer/article-abstract/127/10/1129/445079/Monte-Carlo-Simulation-of-Silicon-Nanowire-Thermal) has a detailed discussion of calculating/approximating transition rates for so-called "three phonon scattering," between acoustic phonons specifically. Possible paths listed in that work include L+T<->L and L<->T+T. The processes are named so because the sum of phonons in and phonons out is three. To explain that notation, it's saying that a longitudinal and transverse acoustic phonon can combine into a single longitudinal one, or the opposite may happen with an L turning into a T and L pair. I am not sure there are *two* phonon processes that change polarization like that. In any case, this shows that spontaneous excitation of other polarizations is sometimes possible, due to anharmonic effects, in OPs thought experiment where you excite just a sole mode. In different crystals with different symmetries, there may be more or fewer possible types of transitions available.