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Formula Dispersion Relation for a Crystal with a Diatomic Basis Angular frequency    Angular wavenumber    Spring constant    Mass    Lattice constant

Formula
Formula: Dispersion Relation for a Crystal with a Diatomic Basis
Lattice vibration inside a crystal with diatomic basis
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Angular frequency

Unit
This dispersion relation \(\omega_{\pm}(k)\) describes the relation between the frequency (energy) and the wavenumber (wavelength) of a diatomic chains of a crystal. The oscillation is purely longitudinal (or transversal) and only the interaction between the neighboring chains is considered here.

In a crystal with a diatomic basis there are two vibrational frequencies: \(\omega_{+}(k)\) optical branch and \(\omega_{-}(k)\) acoustic branch.

The angular frequency is related to the frequency \(f\) via \(\omega = 2\pi \, f \).

Angular wavenumber

Unit
Wavenumber is related to wavelength \(\lambda\) via \(k = 2\pi / \lambda \).

Spring constant

Unit
Spring constant (or coupling constant) comes from the Hooke spring law and describes how strongly a diatomic lattice plane is coupled to its neighboring lattice planes.

Mass

Unit
The two masses of a diatomic basis.

Lattice constant

Unit
Lattice constant is the distance between two adjacent lattice planes when they are in equilibrium (i.e. not deflected).