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 (or other characteristic energy scales) for optical absorption, e.g. ionic and molecular vibrations, interband transitions (semiconductors), phonons, and collective excitations.
[1][2] The model is derived by modeling an electron orbiting a massive, stationary nucleus as a spring-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 the dipole moment, polarization, susceptibility, and dielectric function.
[4] Equation of motion for electron oscillator:
ω
{\displaystyle {\begin{aligned}\mathbf {F} _{\text{net}}=\mathbf {F} _{\text{damping}}+\mathbf {F} _{\text{spring}}+\mathbf {F} _{\text{driving}}&=m{\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\\[1ex]{\frac {-m}{\tau }}{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}-k\mathbf {r} -{e}\mathbf {E} (t)&=m{\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\\[1ex]{\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}+{\frac {1}{\tau }}{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}+\omega _{0}^{2}\mathbf {r} \;&=\;{\frac {-e}{m}}\mathbf {E} (t)\end{aligned}}}
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 (phase shift) between the driving electric field and the response of the electron's motion.
, induces a dipole moment,
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:[5] 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
The electric displacement
is related to the polarization density
The complex dielectric function is given the following (in Gaussian units):
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[7]
and Separating the real and imaginary components,
The complex optical conductivity in general is related to the complex dielectric function
Substituting the formula of
in the equation above we obtain
Separating the real and imaginary components,
