Formula: Dispersion Relation for a Crystal with a Monatomic Basis Angular frequency Angular wavenumber Spring constant Mass Lattice constant
$$\omega(k) ~=~ \sqrt{\frac{4 D}{\class{brown}{m}} \sin^2\left(\frac{ka}{2}\right)}$$
$$\omega(k) ~=~ \sqrt{\frac{4 D}{\class{brown}{m}} \sin^2\left(\frac{ka}{2}\right)}$$
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 $$ \frac{1}{\mathrm m} $$
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).