I am trying to understand how normal modes, optical and acoustical cases and longitudinal/transversal propagation of waves are all together related.
Lets say we are have a clystal chain (for simplicity) in 3D and we have more then 2 different atoms per unit cell.
A normal mode would be, said in a simplistic way, a wave pattern + nodes that we can "see" along the chain. We have as many normal modes are we have atoms. Max normal mode would be for max wavenumber of $\pi/a$ (a lattice constant). Basically we would have a standing wave.
Now each mode (wavenumber value) can change it's energy somehow, which will result in the same pattern but with higher energy, correct?
And depending one their motion in relation to the propagation of the wave in the crystal we can say that we are having a transversal or longitudinal wave.
That much I understand, but what i don't get is how do we differ between the optical and acoustic case? (i am not using the word mode here to not create confusion with the normal modes).
In Wikipedia for the optical case it says : " Atoms ( 2 or more different per unit cell) move in opposite direction. Isn't that the case only for when the wave number is maxima? Meaning for the last normal mode, which is characterized for the max. value of wavenumber. As I said above, only for this wavenumber value the particles move in an opposite direction, which results into a standing wave, and for the optical case the atoms must move in the opposite direction? How can we have an optical case if the wavenumber isn't in it's maximal value?