Dispersion Relations: Confusing Me & What Information is Gained?

[Quasi Particle]

Dispersion relations have the tendency to confuse me.
In general, I know what dispersion is, but trying to apply it to crystals, I just "can't see the forest among all those trees". :uhh:
In phonon dispersion, acoustical and optical phonons have quite a different dispersion behaviour. Why is that? I do know the difference between acoustical and optical phonons, but I don't see the physical meaning.
Electron dispersion "creates" the energy bands. But again, I don't really have a concept of the physical meaning.
Can anyone depict this, please?
And also: what exactly do you need dispersion relations for? What information do you get from them?

 

[Dr Transport]

Dispersion relations are nothing more than relating the energy to the wave-vector, $$E = f(k)$$.

 

[Pieter Kuiper]

Electron dispersion "creates" the energy bands. But again, I don't really have a concept of the physical meaning.
Can anyone depict this, please?

As Dr. Transport says, it is the energy as a function of wave vector. This the same relation that links frequency to wavelength. The relation is very simple when the propagation velocity is independent of frequency (as for EM-waves, and for sound in air), but is more interesting in other cases (light in glass, waves on water).
For free electrons it is the relation between De-Broglie wavelength and kinetic energy.

And also: what exactly do you need dispersion relations for? What information do you get from them?

In the case of electrons the derivative gives effective masses.
In the case of phonons, it gives the velocity of sound.

 

[Quasi Particle]

Yes I know it sounds a pretty stupid question, but it just seems to be opaque to my understanding.

Thanks for your answers so far (*notes* effective mass, velocity of sound)

If, say, I had an exam about solid state physics and plasma physics and I were asked to draw and explain dispersion relations of electrons, phonons and different kinds of plasma waves, is there a quick and simple way to deduce them?

 

[Tide]

Yes. For one dimensional waves, write out the wave equation and replace derivatives wrt x by ik and derivatives wrt to t by $i\omega$. The rest is just algebra. In higher dimensions, replace the grad operator with $i\vec k$.