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 E\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 KK should be chosen such that:

ω0=Km\begin{aligned} \omega_0 = \sqrt{\frac{K}{m}} \end{aligned}

Where mm is the electron’s mass, and the resonance ω0\omega_0 is an empirically determined transition frequency of the atom. When an electromagnetic wave travels through the material, its electric field E(t)=E0e−iωt\vb{E}(t) = \vb{E}_0 e^{-i \omega t} displaces the electron by an amount x(t)\vb{x}(t) governed by:

md2xdt2=qE−mγdxdt−Kx\begin{aligned} m \dvn{2}{\vb{x}}{t} &= q \vb{E} - m \gamma \dv{\vb{x}}{t} - K \vb{x} \end{aligned}

Where q<0q < 0 is the electron’s charge, and γ\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 x(t)=x0e−iωt\vb{x}(t) = \vb{x}_0 e^{- i \omega t} and isolating for the amplitude x0\vb{x}_0, we find:

x0=qE0m(ω02−ω2−iγω)\begin{gathered} \vb{x}_0 = \frac{q \vb{E}_0}{m (\omega_0^2 - \omega^2 - i \gamma \omega)} \end{gathered}

The polarization density P(t)\vb{P}(t) is therefore as shown below, where NN is the number of atoms per unit of volume. Note that the dipole moment vector p\vb{p} is defined as pointing from negative to positive, whereas the electric field E\vb{E} points from positive to negative:

P(t)≈Np(t)=Nqx(t)=Nq2m(ω02−ω2−iγω)E(t)\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 P\vb{P} is not equal to NpN \vb{p}; this will be clarified later. From the definition of the electric displacement field D=ε0E+P=ε0εrE\vb{D} = \varepsilon_0 \vb{E} + \vb{P} = \varepsilon_0 \varepsilon_r \vb{E}, we see that the material’s dielectric function εr(ω)\varepsilon_r(\omega) must be given by:

εr(ω)=1+Nq2ε0m(ω02−ω2−iγω)\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 ωp\omega_p here, but the concept of plasma oscillation does not apply, because there are no conduction electrons.

When the light’s driving frequency ω\omega is far from the resonance ω0\omega_0, we see that the “background” permittivity is higher at lower frequencies:

εlow= lim⁡ω→0 εr(ω)=1+Nq2ε0mω02εhigh=lim⁡ω→∞εr(ω)=1\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 εr\varepsilon_r as follows:

εr(ω)=εhigh+(εlow−εhigh)ω02ω02−ω2−iγω\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 ωn\omega_n. In that case, the relative permittivity εr\varepsilon_r becomes:

εr(ω)=1+Nq2ε0m∑n1(ωn2−ω2−iγnω)\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 εr\varepsilon_r the shape of a staircase, descending from low to high ω\omega in clear steps at each ωn\omega_n. Around each such resonance there is a distinctive “squiggle” in Re{εr}\Real\{\varepsilon_r\} corresponding to a peak in the material’s reflectivity, and there is an absorption peak in Im{εr}\Imag\{\varepsilon_r\}. The damping from γn\gamma_n broadens those peaks and reduces their amplitude.

Finally, recall that P\vb{P} was not exactly equal to NpN \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

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