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+---
+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.