Dispersion Relations
The kinetic energy of a particle depends on the magnitude and direction of the wave vector k, the properties of the particle and the environment in which the particle is moving. For example, the kinetic energy of an electron in a Fermi gas is given by
where m is the electron mass. The dispersion relation is a spherically symmetric parabola and it is continuously rising so the DOS can be calculated easily.
For longitudinal phonons in a string of atoms the dispersion relation of the kinetic energy in a 1-dimensional k-space, as shown in Figure 2, is given by
where is the oscillator frequency, the mass of the atoms, the inter-atomic force constant and inter-atomic spacing. For small values of the dispersion relation is rather linear:
When the energy is
With the transformation and small this relation can be transformed to
Read more about this topic: Density Of States
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