TheLorentz oscillator model (classical electron oscillator or CEO model) describes the optical response ofbound charges. The model is named after the Dutch physicistHendrik Antoon Lorentz. It is aclassical, phenomenological model for materials with characteristic resonance frequencies (or other characteristic energy scales) for optical absorption, e.g. ionic andmolecular vibrations, interband transitions (semiconductors),phonons, and collective excitations.^[1][2]

The model is derived by modeling an electron orbiting a massive, stationary nucleus as aspring-mass-damper system.^[2][3][4] The electron is modeled to be connected to the nucleus via a hypothetical spring and its motion is damped by via a hypothetical damper. The damping force ensures that the oscillator's response is finite at its resonance frequency. For a time-harmonic driving force which originates from the electric field,Newton's second law can be applied to the electron to obtain the motion of the electron and expressions for thedipole moment,polarization,susceptibility, anddielectric function.^[4]

Equation of motion for electron oscillator:

where

• ${\displaystyle \mathbf{r}}$ is the displacement of charge from the rest position,
• ${\displaystyle t}$ is time,
• ${\displaystyle \tau }$ is the relaxation time/scattering time,
• ${\displaystyle k}$ is a constant factor characteristic of the spring,
• ${\displaystyle m}$ is the effective mass of the electron,
• ${\omega }_0=\sqrt{k/m}$ is the resonance frequency of the oscillator,
• ${\displaystyle e}$ is the elementary charge,
• ${\displaystyle \mathbf{E}(t)}$ is the electric field.

For time-harmonic fields:$${\displaystyle \mathbf{E}(t)={\mathbf{E}}_0{e}^{-i\omega t}}$$$${\displaystyle \mathbf{r}(t)={\mathbf{r}}_0{e}^{-i\omega t}}$$

The stationary solution of this equation of motion is:$${\displaystyle \mathbf{r}(\omega )=\frac{\frac{-e}{m}}{{\omega }_0^2-{\omega }^2-i\omega /\tau }\mathbf{E}(\omega )}$$

The fact that the above solution iscomplex means there is a time delay (phase shift) between the driving electric field and the response of the electron's motion.^[4]

The displacement,${\displaystyle \mathbf{r}}$, induces a dipole moment,${\displaystyle \mathbf{p}}$, given by$${\displaystyle \mathbf{p}(\omega )=-e\mathbf{r}(\omega )=\hat{\alpha }(\omega )\mathbf{E}(\omega ).}$$

${\displaystyle \hat{\alpha }(\omega )}$ is the polarizability of single oscillator, given by$${\displaystyle \hat{\alpha }(\omega )=\frac{e^2}{m}\frac{1}{({\omega }_0^2-{\omega }^2)-i\omega /\tau }.}$$

Three distinctscattering regimes can be interpreted corresponding to the dominant denominator term in the dipole moment:^[5]

Regime	Condition	Dispersion Scaling	Phase Shift
Thomson scattering	${\displaystyle {\omega }^2\gg \frac{\omega }{\tau },{\omega }_0^2}$	${\displaystyle 1}$	0°
Shneider-Miles scattering	${\displaystyle \frac{\omega }{\tau }\gg |{\omega }_0^2-{\omega }^2|}$	${\displaystyle {\omega }^2}$	90°
Rayleigh scattering	${\displaystyle {\omega }_0^2\gg {\omega }^2,\frac{\omega }{\tau }}$	${\displaystyle {\omega }^4}$	180°

The polarization${\displaystyle \mathbf{P}}$ is the dipole moment per unit volume. For macroscopic material properties N is the density of charges (electrons) per unit volume. Considering that each electron is acting with the same dipole moment we have the polarization as below$${\displaystyle \mathbf{P}=N\mathbf{p}=N\hat{\alpha }(\omega )\mathbf{E}(\omega ).}$$

The electric displacement${\displaystyle \mathbf{D}}$ is related to the polarization density${\displaystyle \mathbf{P}}$ by$${\displaystyle \mathbf{D}=\hat{\epsilon }\mathbf{E}=\mathbf{E}+4\pi \mathbf{P}=(1+4\pi N\hat{\alpha })\mathbf{E}}$$

The complex dielectric function is given the following (inGaussian units):$${\displaystyle \hat{\epsilon }(\omega )=1+\frac{4\pi N{e}^2}{m}\frac{1}{({\omega }_0^2-{\omega }^2)-i\omega /\tau }}$$where${\displaystyle 4\pi N{e}^2/m={\omega }_p^2}$ and${\displaystyle {\omega }_p}$ is the so-calledplasma frequency.

In practice, the model is commonly modified to account for multiple absorption mechanisms present in a medium. The modified version is given by^[7]$${\displaystyle \hat{\epsilon }(\omega )={\epsilon }_{\mathrm{\infty }}+\sum _j{\chi }_j^L(\omega ;{\omega }_{0,j})}$$where$${\displaystyle {\chi }_j^L(\omega ;{\omega }_{0,j})=\frac{s_j}{{\omega }_{0,j}^2-{\omega }^2-i{\mathrm{\Gamma }}_j\omega }}$$and

• ${\displaystyle {\epsilon }_{\mathrm{\infty }}}$ is the value of the dielectric function at infinite frequency, which can be used as an adjustable parameter to account for high frequency absorption mechanisms,
• ${\displaystyle s_j={\omega }_p^2{f}_j}$ and${\displaystyle f_j}$ is related to the strength of the${\displaystyle j}$th absorption mechanism,
• ${\displaystyle {\mathrm{\Gamma }}_j=1/\tau }$.

Separating the real and imaginary components,$${\displaystyle \hat{\epsilon }(\omega )={\epsilon }_1(\omega )+i{\epsilon }_2(\omega )=\left[{\epsilon }_{\mathrm{\infty }}+\sum _j\frac{s_j({\omega }_{0,j}^2-{\omega }^2)}{{\left({\omega }_{0,j}^2-{\omega }^2\right)}^2+{\left({\mathrm{\Gamma }}_j\omega \right)}^2}\right]+i\left[\sum _j\frac{s_j({\mathrm{\Gamma }}_j\omega )}{{\left({\omega }_{0,j}^2-{\omega }^2\right)}^2+{\left({\mathrm{\Gamma }}_j\omega \right)}^2}\right]}$$

The complexoptical conductivity in general is related to the complex dielectric function (inGaussian units) as$${\displaystyle \hat{\sigma }(\omega )=\frac{\omega }{4\pi i}\left(\hat{\epsilon }(\omega )-1\right)}$$

Substituting the formula of${\displaystyle \hat{\epsilon }(\omega )}$ in the equation above we obtain$${\displaystyle \hat{\sigma }(\omega )=\frac{N{e}^2}{m}\frac{\omega }{\omega /\tau +i\left({\omega }_0^2-{\omega }^2\right)}}$$

Separating the real and imaginary components,$${\displaystyle \hat{\sigma }(\omega )={\sigma }_1(\omega )+i{\sigma }_2(\omega )=\frac{N{e}^2}{m}\frac{\frac{{\omega }^2}{\tau }}{{\left({\omega }_0^2-{\omega }^2\right)}^2+{\omega }^2/{\tau }^2}-i\frac{N{e}^2}{m}\frac{\left({\omega }_0^2-{\omega }^2\right)\omega }{{\left({\omega }_0^2-{\omega }^2\right)}^2+{\omega }^2/{\tau }^2}}$$

• Cauchy equation
• Sellmeier equation
• Forouhi–Bloomer model
• Tauc–Lorentz model
• Brendel–Bormann oscillator model

• Condensed matter physics
• Material dispersion models
• Optics
• Oscillators

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