Lorentz oscillator model


The Lorentz oscillator model describes the optical response of bound charges. The model is named after the Dutch physicist Hendrik Antoon Lorentz. It is a classical, phenomenological model for materials with characteristic resonance frequencies for optical absorption, e.g. ionic and molecular vibrations, interband transitions, phonons, and collective excitations.

Derivation of electron motion

The model is derived by modeling an electron orbiting a massive, stationary nucleus as a spring-mass-damper system. 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 the dipole moment, polarization, susceptibility, and dielectric function.
Equation of motion for electron oscillator:
where
For time-harmonic fields:
The stationary solution of this equation of motion is:
The fact that the above solution is complex means there is a time delay between the driving electric field and the response of the electron's motion.

Dipole moment

The displacement,, induces a dipole moment,, given by
is the polarizability of single oscillator, given by
Three distinct scattering regimes can be interpreted corresponding to the dominant denominator term in the dipole moment:
RegimeConditionDispersion ScalingPhase Shift
Thomson scattering0°
Shneider-Miles scattering90°
Rayleigh scattering180°

Polarization

The polarization is the dipole moment per unit volume. For macroscopic material properties N is the density of charges per unit volume. Considering that each electron is acting with the same dipole moment we have the polarization as below

Electric displacement

The electric displacement is related to the polarization density by

Dielectric function

The complex dielectric function is given the following :
where and is the so-called plasma frequency.
In practice, the model is commonly modified to account for multiple absorption mechanisms present in a medium. The modified version is given by
where
and
  • 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,
  • and is related to the strength of the th absorption mechanism,
  • .
Separating the real and imaginary components,

Complex conductivity

The complex optical conductivity in general is related to the complex dielectric function as
Substituting the formula of in the equation above we obtain
Separating the real and imaginary components,