I am having trouble understanding this concept.
The frequency ω is plotted against the wave vector k, but how do I actually read it? Do I search for a frequency and look which modes are "(co)existing" at that frequency? Or do I pick a wave vector (a direction) and look which frequencies are allowed for these values of k? I can probably read it both ways, but where is cause and effect exactly?
Here's what I know: Let's assume a 2D case with a simple Brillouin Zone Γ-X-Y-Γ. The sections of the dispersion relation correspond to values of k, where Γ denotes the point where k is very small and the wavelength λ is very large. Traveling along the x-axis is basically like traversing the edges of the Brillouin Zone, covering all possible directions of the wave vector.
Dear Devanjith,
the dispersion relations only describe the relation between frequency and wave vector. From this relation the velocity of sound can be derived.
If you excite a mode experimentally, for instance by using a quartz generator than you define the frequency experimentally and the associated wave vector can be determined by the dispersion relation. If you have a musical instrument - lets say a string - than the wavelength (wavevector) is fixed and the frequency spectrum relates from the velocity of sound.
All materials have a temperature T. Therefore, the energy states of lattice vibrations are occupied using the laws of quantum statistics. Phonons obey Bose-Einstein statistics. For a constant speed of sound and applying the Bose statistics, the Debye model of specific heat was derived. The Einstein model used the dispersion relation of the optical modes.
As a consequence, it must be known which parameter are fix - frequency, wavelength, energy. The dispersion relation gives the counterpart.
With Regards
R. Mitdank
Dear Devanjith,
the dispersion relations only describe the relation between frequency and wave vector. From this relation the velocity of sound can be derived.
If you excite a mode experimentally, for instance by using a quartz generator than you define the frequency experimentally and the associated wave vector can be determined by the dispersion relation. If you have a musical instrument - lets say a string - than the wavelength (wavevector) is fixed and the frequency spectrum relates from the velocity of sound.
All materials have a temperature T. Therefore, the energy states of lattice vibrations are occupied using the laws of quantum statistics. Phonons obey Bose-Einstein statistics. For a constant speed of sound and applying the Bose statistics, the Debye model of specific heat was derived. The Einstein model used the dispersion relation of the optical modes.
As a consequence, it must be known which parameter are fix - frequency, wavelength, energy. The dispersion relation gives the counterpart.
With Regards
R. Mitdank
The dispersion curves indicate several modes with several velocities where v=f*wavelength, so at a specific frequency you can find several modes with different velocities and wavelength or k. both f and k are related together, with the same manner at a spesific k you can find several modes with different frequencies.
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