---
title: "Lorentz oscillator model"
sort_title: "Lorentz oscillator model"
date: 2024-04-09
categories:
- Physics
- Optics
- Electromagnetism
layout: "concept"
---

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](/know/concept/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](/know/concept/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](/know/concept/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](/know/concept/electromagnetic-wave-equation/)
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](/know/concept/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 $$\vb{E}$$ goes from positive to negative:

$$\begin{aligned}
    \vb{P}(t)
    = N \vb{p}(t)
    = N q \vb{x}(t)
    = \frac{N q^2}{m (\omega_0^2 - \omega^2 - i \gamma \omega)} \vb{E}(t)
\end{aligned}$$

From the definition of the electric displacement field
$$\vb{D} = \varepsilon_0 \vb{E} + \vb{P} = \varepsilon_0 \varepsilon_r \vb{E}$$,
we find that the material's
[dielectric function](/know/concept/dielectric-function/)
$$\varepsilon_r(\omega)$$ is 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 $$\omega_\nu$$.
In that case, the relative permittivity $$\varepsilon_r$$ becomes:


$$\begin{aligned}
    \boxed{
        \varepsilon_r(\omega)
        = 1 + \frac{N q^2}{\varepsilon_0 m} \sum_{\nu} \frac{1}{(\omega_\nu^2 - \omega^2 - i \gamma_\nu \omega)}
    }
\end{aligned}$$

This gives $$\varepsilon_r$$ the shape of a staircase,
descending from low to high $$\omega$$ in clear steps at each $$\omega_\nu$$.
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 $$\gamma_\nu$$ broadens those peaks and reduces their amplitude.



## References
1.  M. Fox,
    *Optical properties of solids*, 2nd edition,
    Oxford.