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Could you please explain what exactly is the relevant information that is conveyed through a dispersion relation?

Edit 1: Sorry about being vague. I am currently trying to understand the dispersion relations obtained in the one-dimensional monoatomic and diatomic lattice and its relation to the optical and acoustic phonons. I understand how we arrived at the results, but I cannot fully visualize its implications from a physical point of view. I was hoping may be I can get some insight into the relations from a physical point of view.

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  • $\begingroup$ They're many. For example $dE/dk$ can be useful to know the density of states in statistical mechanics, from which you calculate many things. It's also relevant to describe semiconductor devices... they're so many things $\endgroup$ – FGSUZ Feb 17 '19 at 0:15
  • $\begingroup$ This is a very broad question. Maybe you could specify more. $\endgroup$ – noah Feb 17 '19 at 0:23
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Dispersion relations connect the energy to wavelength (or momentum) of a particle/wave.

For example:

$$\hbar \omega=\hbar c k=\hbar c \frac{2\pi}{\lambda}$$

Would be the dispersion relation of light, and it shows that energy and momentum are linearly proportional. Waves with zero momentum have zero energy.

Compare this to:

$$\hbar \omega=\hbar \omega_0 + a k^2$$

Now, zero momentum does not imply zero energy, and there is a non-linear relation between energy and momentum.

(Answering OP's edited question) key property that can be obtained from the dispersion relation is the speed of propagation, or group velocity. This is given by the slope $d\omega/dk$, which for acoustic phonons gives the speed of sound, and for optical phonons is typically quite small.

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Acoustical phonons have an approximately linear dispersion, where $\omega/k$ of the longitudinal modes gives the velocity of sound. Up to a cutoff. This gives the Debye expression of the specific heat as a function of temperature ($c_{\rm v} \propto T^3$ at low temperatures).

The dispersion of the optical branches is much less. In ionic solids, this explains the infrared spectrum (${\it Reststrahlen}$).

And the dispersion curves can be measured experimentally with inelastic neutron diffraction or with inelastic x-ray scattering. The details of the phonon structure can be involved in superconductivity or in structural phase transitions or in non-linear optical effects.

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