The Lorentz oscillator model describes the optical response of bound charges. The model is named after the Dutch physicist
Hendrik Antoon Lorentz
Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
. It is a
classical, phenomenological model for materials with characteristic resonance frequencies (or other characteristic energy scales) for optical absorption, e.g. ionic and
molecular vibrations, interband transitions (semiconductors),
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
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
.
Equation of motion for electron oscillator:
:
:
:
:
where
*
is the displacement of charge from the rest position,
*
is time,
*
is the relaxation time/scattering time,
*
is a constant factor characteristic of the spring,
*
is the effective mass of the electron,
*
*
is the resonance frequency of the oscillator,
*
is the elementary charge,
*
is the electric field.
For time-harmonic fields:
:
:
The stationary solution of this equation of motion is:
:
The fact that the above solution is
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
means there is a time delay (phase shift) 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
:
Polarization
The polarization
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
:
Electric displacement
The electric displacement
is related to the polarization density
by
:
Dielectric function
The complex dielectric function is given by
:
where
and
is the so called
plasma frequency Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability i ...
.
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,
: