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Rewriting the relation above in these variables gives = (). where we now view f as a function of k. The use of ω(k) to describe the dispersion relation has become standard because both the phase velocity ω/k and the group velocity dω/dk have convenient representations via this function. The plane waves being considered can be described by
To approximate the dispersion relation in the case of the conduction band, take the energy E n0 as the minimum conduction band energy E c0 and include in the summation only terms with energies near the valence band maximum, where the energy difference in the denominator is smallest. (These terms are the largest contributions to the summation.)
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum.
The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction: = (). The full equation is typically given as follows: [4]
Each propagates at generally different speeds determined by the important function called the dispersion relation. The use of the explicit form ω ( k ) is standard, since the phase velocity ω / k and the group velocity d ω /d k usually have convenient representations by this function.
Dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. [1] Sometimes the term chromatic dispersion is used to refer to optics specifically, as opposed to wave propagation in general. A medium having this common property may be termed a dispersive medium.
We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with electrons and ion species. We write each quantity as X = X 0 + δ ⋅ X 1 {\displaystyle X=X_{0}+\delta \cdot X_{1}} where subscript 0 denotes the "zero-order" constant equilibrium value, and 1 denotes the first-order perturbation.
This equation was studied in Benjamin, Bona, and Mahony as an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long surface gravity waves of small amplitude – propagating uni-directionally in 1+1 dimensions. They show the stability and uniqueness of solutions to the BBM equation.