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Dispersion relation

  1. Dec 18, 2005 #1
    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?
  2. jcsd
  3. Dec 18, 2005 #2

    Dr Transport

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    Dispersion relations are nothing more than relating the energy to the wave-vector, [tex] E = f(k) [/tex].
  4. Dec 19, 2005 #3
    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.
  5. Dec 19, 2005 #4
    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?
  6. Dec 19, 2005 #5


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