Categories: Electromagnetism, Optics, Physics.

# Lorentz oscillator model

The **Lorentz oscillator model** or **dipole oscillator model**
is a classical description of light-matter interaction,
which treats the charged particles inside a solid as forming dipoles
that get pushed around by the electric field of passing light waves.
Quantitatively, it is of limited use, as it ignores quantum mechanics,
but qualitatively it captures the essential features.
It is similar to the Drude model,
but applies to insulators instead of conductors.

In insulators, the valence electrons are bound to an immobile nucleus at a certain equilibrium distance (this is a classical model, so we treat the electron as a particle). If an electric field $\vb{E}$ moves the electron, a restoring force brings it back to equilibrium, so we can pretend that it is connected to the nucleus by a spring. In other words, we treat it as a harmonic oscillator, whose spring constant $K$ should be chosen such that:

$\begin{aligned} \omega_0 = \sqrt{\frac{K}{m}} \end{aligned}$Where $m$ is the electron’s mass, and the resonance $\omega_0$ is an empirically determined transition frequency of the atom. When an electromagnetic wave travels through the material, its electric field $\vb{E}(t) = \vb{E}_0 e^{-i \omega t}$ displaces the electron by an amount $\vb{x}(t)$ governed by:

$\begin{aligned} m \dvn{2}{\vb{x}}{t} &= q \vb{E} - m \gamma \dv{\vb{x}}{t} - K \vb{x} \end{aligned}$Where $q < 0$ is the electron’s charge, and $\gamma$ represents a weak damping effect. The four terms represent Newton’s second law, the Lorentz force, the spring’s damping force, and the spring’s restoring force, respectively.

Inserting the ansatz $\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$ and isolating for the amplitude $\vb{x}_0$, we find:

$\begin{gathered} \vb{x}_0 = \frac{q \vb{E}_0}{m (\omega_0^2 - \omega^2 - i \gamma \omega)} \end{gathered}$The polarization density $\vb{P}(t)$ is therefore as shown below, where $N$ is the number of atoms per unit of volume. Note that the dipole moment vector $\vb{p}$ is defined as pointing from negative to positive, whereas the electric field $\vb{E}$ points from positive to negative:

$\begin{aligned} \vb{P}(t) \approx N \vb{p}(t) = N q \vb{x}(t) = \frac{N q^2}{m (\omega_0^2 - \omega^2 - i \gamma \omega)} \vb{E}(t) \end{aligned}$Also note that $\vb{P}$ is not equal to $N \vb{p}$; this will be clarified later. From the definition of the electric displacement field $\vb{D} = \varepsilon_0 \vb{E} + \vb{P} = \varepsilon_0 \varepsilon_r \vb{E}$, we see that the material’s dielectric function $\varepsilon_r(\omega)$ must be given by:

$\begin{aligned} \boxed{ \varepsilon_r(\omega) = 1 + \frac{N q^2}{\varepsilon_0 m (\omega_0^2 - \omega^2 - i \gamma \omega)} } \end{aligned}$You may recognize the Drude model’s plasma frequency $\omega_p$ here, but the concept of plasma oscillation does not apply, because there are no conduction electrons.

When the light’s driving frequency $\omega$ is far from the resonance $\omega_0$, we see that the “background” permittivity is higher at lower frequencies:

$\begin{aligned} \varepsilon_{\mathrm{low}} &= \, \lim_{\omega \to 0} \, \varepsilon_r(\omega) = 1 + \frac{N q^2}{\varepsilon_0 m \omega_0^2} \\ \varepsilon_{\mathrm{high}} &= \lim_{\omega \to \infty} \varepsilon_r(\omega) = 1 \end{aligned}$Using these limits, we can rewrite our previous expression for $\varepsilon_r$ as follows:

$\begin{aligned} \varepsilon_r(\omega) = \varepsilon_{\mathrm{high}} + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}}) \frac{\omega_0^2}{\omega_0^2 - \omega^2 - i \gamma \omega} \end{aligned}$In reality, atoms have multiple spectral lines, so we should treat them as if they have multiple oscillators with different resonances $\omega_n$. In that case, the relative permittivity $\varepsilon_r$ becomes:

$\begin{aligned} \boxed{ \varepsilon_r(\omega) = 1 + \frac{N q^2}{\varepsilon_0 m} \sum_{n} \frac{1}{(\omega_n^2 - \omega^2 - i \gamma_n \omega)} } \end{aligned}$This gives $\varepsilon_r$ the shape of a staircase, descending from low to high $\omega$ in clear steps at each $\omega_n$. Around each such resonance there is a distinctive “squiggle” in $\Real\{\varepsilon_r\}$ corresponding to a peak in the material’s reflectivity, and there is an absorption peak in $\Imag\{\varepsilon_r\}$. The damping from $\gamma_n$ broadens those peaks and reduces their amplitude.

Finally, recall that $\vb{P}$ was not exactly equal to $N \vb{p}$. This is because each atomic dipole generates its own electric field, affecting the response of its neighbors. There exists a formula to correct for this effect: the Clausius-Mossotti relation.

## References

- M. Fox,
*Optical properties of solids*, 2nd edition, Oxford.