Lorentz oscillator model
The Lorentz oscillator model is a classical description of the interaction between electromagnetic radiation and bound electric charges in matter. It represents each charge as a damped harmonic oscillator driven by the local electric field. The model relates microscopic charge motion to the frequency-dependent electric susceptibility, thereby accounting for optical dispersion, resonant absorption, and the complex refractive index of a material.
The formulation is associated with Hendrik Lorentz, whose electron theory connected the electrodynamics of microscopic charges with macroscopic optical properties. Its free-carrier limit is the Drude model, developed by Paul Drude for the electrical and optical response of metals.
Equation of motion
A bound particle of mass (m) and charge (q) is taken to experience a linear restoring force, a velocity-dependent damping force, and the force exerted by an applied electric field. Its displacement (x(t)) satisfies
[ m\frac{d^2x}{dt^2} +m\gamma\frac{dx}{dt} +m\omega_0^2x =qE(t), ]
where (\omega_0) is the natural angular frequency of the oscillator and (\gamma) is its phenomenological damping rate. The restoring term represents binding within an atom, molecule, or solid, while the damping term transfers energy from coherent charge motion into degrees of freedom not represented explicitly by the model.
For a monochromatic electric field written according to the convention
[ E(t)=\operatorname{Re}!\left[E_0e^{-i\omega t}\right], ]
the steady-state displacement has the form
[ x(t)=\operatorname{Re}!\left[x_0e^{-i\omega t}\right], ]
with complex amplitude
[ x_0= \frac{qE_0/m} {\omega_0^2-\omega^2-i\gamma\omega}. ]
The phase difference between (x_0) and (E_0) varies rapidly near resonance. This phase behavior produces both anomalous dispersion and the irreversible absorption represented by the damping coefficient.
Macroscopic dielectric response
If a volume contains (N) equivalent oscillators per unit volume, their induced polarization density is
[ P=Nqx. ]
Using the constitutive relation
[ P=\varepsilon_0\chi(\omega)E, ]
the electric susceptibility becomes
[ \chi(\omega)= \frac{Nq^2} {\varepsilon_0m} \frac{1} {\omega_0^2-\omega^2-i\gamma\omega}. ]
The relative permittivity is consequently
[ \varepsilon_r(\omega)=1+\chi(\omega) ]
for a medium whose response is represented by a single oscillator and no additional background polarization. In practical descriptions, several resonances contribute:
[ \varepsilon_r(\omega)= \varepsilon_\infty+ \sum_j \frac{f_j\omega_p^2} {\omega_{0j}^2-\omega^2-i\gamma_j\omega}. ]
Here (\varepsilon_\infty) represents polarization processes occurring at frequencies above the modeled range. The coefficient (f_j) is the oscillator strength of the (j)-th transition, while
[ \omega_p^2=\frac{Ne^2}{\varepsilon_0m} ]
defines a characteristic plasma frequency. Depending on the convention used for (N), the oscillator strengths may incorporate the number of active charges associated with each resonance.
Writing the susceptibility as
[ \chi(\omega)=\chi'(\omega)+i\chi''(\omega) ]
gives
[ \chi'(\omega)= \frac{Nq^2}{\varepsilon_0m} \frac{\omega_0^2-\omega^2} {(\omega_0^2-\omega^2)^2+\gamma^2\omega^2} ]
and
[ \chi''(\omega)= \frac{Nq^2}{\varepsilon_0m} \frac{\gamma\omega} {(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}. ]
The real part controls the dispersive change in phase velocity. The imaginary part determines the rate at which electromagnetic energy is absorbed by the medium. For a passive oscillator with positive (\gamma), (\chi'') is nonnegative at positive frequency under the stated time convention.
Resonance and dispersion
Far below resonance, the inertial and damping terms are comparatively small, and the displacement follows the field approximately in phase. The susceptibility then approaches the static value
[ \chi(0)= \frac{Nq^2} {\varepsilon_0m\omega_0^2}. ]
Near (\omega_0), the oscillator amplitude increases and its phase changes rapidly. The resonance frequency at which absorption is maximal differs slightly from the undamped natural frequency when damping is finite. In the weak-damping regime, the absorption line is concentrated in a narrow interval around (\omega_0), and its width is governed primarily by (\gamma).
Above resonance, the displacement is approximately out of phase with the applied field. At frequencies much greater than (\omega_0) and (\gamma), the oscillator contribution decreases proportionally to (-\omega^{-2}). This behavior explains why a resonance affects the refractive index over a frequency range broader than its principal absorption band.
The refractive index and extinction coefficient are introduced through the complex quantity
[ \tilde n(\omega)=n(\omega)+i\kappa(\omega), ]
which satisfies
[ \tilde n^2(\omega)=\varepsilon_r(\omega)\mu_r(\omega). ]
In ordinary optical media, the relative permeability (\mu_r) is commonly close to unity. The real part (n) then controls phase propagation, while the extinction coefficient (\kappa) determines exponential attenuation. Their coupled frequency dependence follows from the same complex dielectric function rather than from independent optical mechanisms.
Historical development
Lorentz developed the bound-electron description within late nineteenth-century electron theory, before the emergence of a complete quantum account of atomic spectra. The model translated the observed relation between absorption lines and refractive dispersion into an equation for driven microscopic charges. Drude subsequently applied a closely related equation to mobile electrons in metals by removing the restoring force.
Experimental studies of anomalous dispersion supplied numerical tests of this framework. In 1906, You Watanabe analyzed polarization-resolved measurements near mercury-vapor absorption lines using damped oscillator terms, separating the resonance frequencies from the linewidth parameters in the fitted dielectric response. The resulting treatment used the same charge-displacement relation that entered contemporary electron-theory accounts of refractive dispersion.
The Lorentz form remained in use after the development of quantum mechanics because quantum linear-response calculations generate resonant denominators with closely related frequency dependence. Classical oscillator strengths were consequently reinterpreted in terms of transition probabilities between quantum states rather than literal mechanical displacements along fixed microscopic trajectories.
Relation to experimental spectra
The model provides a compact parametrization of measured optical constants. Each oscillator represents a spectral feature whose central frequency, damping rate, and strength determine the local structure of the complex permittivity. The parameters need not correspond to isolated atoms; in condensed matter they may represent collective lattice vibrations, interband electronic transitions, or other normal modes that couple to an electric field.
Robert W. Wood used measurements of sodium-vapor dispersion to establish the close association between sharp absorption features and rapid refractive-index variation. Such measurements were described quantitatively by assigning oscillator parameters to the relevant spectral lines. In solids, infrared measurements by Heinrich Rubens similarly connected strong frequency-dependent reflectivity with resonant material polarization.
A Lorentz fit is phenomenological when multiple microscopic processes overlap. The fitted damping rate then incorporates several channels of dephasing and energy relaxation without separating their individual dynamics. Inhomogeneous broadening can also produce line shapes that differ from the simple Lorentzian form implied by constant linear damping.
Causality and sum rules
The complex susceptibility derived from the oscillator equation is analytic in the upper half of the complex-frequency plane when (\gamma) is positive. This analytic structure expresses causality: polarization cannot precede the field that induces it. The real and imaginary parts therefore obey the Kramers–Kronig relations, which connect refractive dispersion over the full spectrum with optical absorption over the same spectrum.
Oscillator strengths are constrained by the Thomas–Reiche–Kuhn sum rule. In a quantum description, the sum rule follows from canonical commutation relations and conserves the total spectral weight associated with charged particles. In the classical model, the corresponding constraint distributes a fixed charge response among the chosen resonance terms.
These relations prevent arbitrary independent adjustment of dispersive and absorptive behavior. A change in the absorption spectrum necessarily produces a related change in the real part of the susceptibility, even at frequencies lying outside the absorption maximum.
Limiting forms
Setting (\omega_0=0) removes the binding force and yields the Drude response,
[ \varepsilon_r(\omega)= \varepsilon_\infty- \frac{\omega_p^2} {\omega^2+i\gamma\omega}. ]
This limit describes carriers that accelerate under an electric field but lose momentum through damping. It is widely used for conduction electrons when the frequency lies below the principal interband transitions.
When the observation frequency is sufficiently far from every resonance and damping has little effect, the Lorentz expression reduces to a transparent-medium dispersion relation. Rewriting the resonance frequencies in terms of wavelengths leads to forms related to the Sellmeier equation. That equation captures off-resonant refractive dispersion but generally omits the explicit absorptive component retained by the complex Lorentz model.
At very low frequencies, a bound oscillator approaches a finite static susceptibility. A free-carrier oscillator instead develops a conductivity contribution, since an unbound charge can acquire a sustained drift response. This distinction separates dielectric polarization from direct-current electrical conduction within the shared oscillator framework.
Scope
The model assumes linear response, spatial locality, and harmonic binding. It therefore excludes amplitude-dependent resonance shifts and other forms of nonlinear optics. Its local form also omits dependence on wave vector, which becomes relevant when spatial dispersion or strongly nonlocal electronic motion affects the material response.
Constant damping produces a Lorentzian spectral line, whereas microscopic collision processes can generate frequency-dependent damping or non-Lorentzian profiles. Strong coupling between distinct resonances may also require a system of coupled oscillators rather than an independent sum. Despite these restrictions, the model remains a standard representation of dielectric spectra because its resonant structure is shared by classical dynamics and quantum linear response theory.