Long wavelength dispersion relation in a solid

Long wavelength dispersion relation in a solid

In solid state physics, perhaps the most visited two elementary excitations are phonons (vibration) and magnons (spin wave). It is interesting to notice that the long wavelength dispersion relation are different for them, i.e.
phonon:
magnon: ,
where is the frequency and is the wave vector.

One can go back to the derivation of those two excitations and find the mathematical reason. On the other hand, the origin can be resorted to the fundamental difference between the two equation of motions.
phonon:
magnon: ,
where x is the displacement, is the angular momentum, is the spring constant and is the torque.

In solid, the restoring force for an oscillator comes from the imbalance of the interaction between neighbors. For long sinusoidal wave, it is always propotional to .

Hence, the final dispersion relation will be decided by the equation of motion. Since for phonon, it involves second derivative, one gets ; for magnon, the first derivative gives .