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

## Angular frequency

`$$ \omega $$`Unit

`$$ \frac{\mathrm{rad}}{\mathrm s} $$`

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

`$$ k $$`Unit

`$$ \frac{\mathrm{rad}}{\mathrm{m}} $$`

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

## Spring constant

`$$ D $$`Unit

`$$ \frac{\mathrm{kg}}{\mathrm{s}^2} $$`

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

`$$ \class{brown}{m} $$`Unit

`$$ \mathrm{kg} $$`

Mass of an atom within the chain.

## Lattice constant

`$$ a $$`Unit

`$$ \mathrm{m} $$`

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