# Dispersion relation

1. Dec 18, 2005

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

2. Dec 18, 2005

### Dr Transport

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

3. Dec 19, 2005

### Pieter Kuiper

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

4. Dec 19, 2005

### 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?

5. Dec 19, 2005

### 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 $i\vec k$.