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

Formula
Formula: Dispersion Relation for a Crystal with a Monatomic Basis
Single-Atom Longitudinal Lattice Vibration (1d)
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Angular frequency

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

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's law and describes how much an atomic chain is coupled to its neighboring chains.

Mass

Unit
Mass of an atom within the chain.

Lattice constant

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