#
Formula: **Dispersion Relation for a Crystal with a Diatomic Basis**

$$\begin{align}

\omega_{\pm}^2 ~&=~ D \, \left( \frac{1}{\class{brown}{m_1}} + \frac{1}{\class{brown}{m_2}} \right) \\\\

~&\pm~ D \, \sqrt{\left(\frac{1}{\class{brown}{m_1}} + \frac{1}{\class{brown}{m_2}}\right)^2 ~-~ \frac{4}{\class{brown}{m_1} \, \class{brown}{m_2}}\,\sin^2\left(\frac{k\,a}{2}\right) }

\end{align}$$

\omega_{\pm}^2 ~&=~ D \, \left( \frac{1}{\class{brown}{m_1}} + \frac{1}{\class{brown}{m_2}} \right) \\\\

~&\pm~ D \, \sqrt{\left(\frac{1}{\class{brown}{m_1}} + \frac{1}{\class{brown}{m_2}}\right)^2 ~-~ \frac{4}{\class{brown}{m_1} \, \class{brown}{m_2}}\,\sin^2\left(\frac{k\,a}{2}\right) }

\end{align}$$

## Angular frequency

`$$ \omega_{\pm} $$`Unit

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

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

`$$ k $$`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 spring law and describes how strongly a diatomic lattice plane is coupled to its neighboring lattice planes.

## Mass

`$$ \class{brown}{m_1}, \class{brown}{m_2} $$`Unit

`$$ \mathrm{kg} $$`

The two masses of a diatomic basis.

## Lattice constant

`$$ a $$`Unit

`$$ \mathrm{m} $$`

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