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 vibration A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz ...

s, interband transitions (semiconductors),

phonon In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...

s, 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 Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...

, susceptibility, and dielectric function. Equation of motion for electron oscillator: :

\mathbf F_  =  m\frac

\mathbf F_ + \mathbf F_ + \mathbf F_  =  m\frac

\frac \frac - k \mathbf r -   \mathbf E(t) = m\frac

\frac + \frac \frac +  \omega_0^2 \mathbf r\; = \; \frac  \mathbf E(t)

where *

\mathbf r

is the displacement of charge from the rest position, *

t

is time, *

\mathrm \tau

is the relaxation time/scattering time, *

k

is a constant factor characteristic of the spring, *

m

is the effective mass of the electron, *

\omega_0 = \left(k / m \right)^

\omega_0

is the resonance frequency of the oscillator, *

e

is the elementary charge, *

\mathbf E(t)

is the electric field. For time-harmonic fields: :

\mathbf E(t) = \mathbf E_0 e^

\mathbf r(t) = \mathbf r_0 e^

The stationary solution of this equation of motion is: :

\mathbf r(\omega) = \frac  \mathbf E(\omega)

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.

Dipole moment

The displacement,

\mathbf r

, induces a dipole moment,

\mathbf p

, given by :

\mathbf p(\omega) = -e \mathbf r(\omega) = \hat\alpha(\omega)  \mathbf E(\omega)

\hat \alpha(\omega)

is the polarizability of single oscillator, given by :

\hat \alpha(\omega)=  \frac \frac

Polarization

The polarization

\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 :

\mathbf P = N \mathbf p = N \hat \alpha(\omega) \mathbf E(\omega)

Electric displacement

The electric displacement

\mathbf D

is related to the polarization density

\mathbf P

by :

\mathbf D = \hat\varepsilon \mathbf E = \mathbf E + 4\pi \mathbf P = (1 + 4\pi N \hat \alpha) \mathbf E

Dielectric function

The complex dielectric function is given by :

\hat \varepsilon(\omega) = 1 +  \frac \frac

where

4\pi N e^2/m = \omega_p^2

and

\omega_p

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 :

\hat \varepsilon(\omega) = \varepsilon_ + \sum_ \chi_^(\omega; \omega_)

where :

\chi_^(\omega; \omega_) = \frac

and *

\varepsilon_

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, *

s_ = \omega_p^ f_

and

f_

is related to the strength of the

j

th absorption mechanism, *

\Gamma_=1/\tau

. Separating the real and imaginary components, :

\hat \varepsilon(\omega) = \varepsilon_1(\omega) + \mathrm i \varepsilon_2(\omega) = \left \varepsilon_ + \sum_ \frac \right + \mathrm i \left \sum_ \frac \right /math>

Complex conductivity

The complex

optical conductivity Optical conductivity is the property of a material which gives the relationship between the induced current density in the material and the magnitude of the inducing electric field for arbitrary frequencies. This linear response function is a gene ...

in general is related to the complex dielectric function :

\hat \sigma(\omega) = \frac (\hat\varepsilon(\omega) - 1)

Substituting the formula of

\hat\varepsilon(\omega)

in the equation above we obtain :

\hat \sigma(\omega) =  \frac \frac

Separating the real and imaginary components, :

\hat \sigma(\omega) =  \sigma_1(\omega) + \mathrm i \sigma_2(\omega) =
\frac \frac -\mathrm i \frac \frac

References

{{reflist , refs = {{cite book , last1 = Lorentz , first1 = Hendrik Antoon , year = 1909 , title = The theory of electrons and its applications to the phenomena of light and radiant heat , language = en , volume = Bd. XXIX;Bd. 29 , location = New York; Leipzig , publisher = B.G. Teubner , oclc = 535812 {{cite book , last1 = Dressel , first1 = Martin , last2 = Grüner , first2 = George , year = 2002 , title = Electrodynamics of Solids: Optical Properties of Electrons in Matter , chapter = Semiconductors , pages = 136–172 , language = en , location = Cambridge , doi = 10.1017/CBO9780511606168.008 , isbn = 9780521592536 {{cite web , last = Colton , first = John , title = Lorentz Oscillator Model , year = 2020 , website = Brigham Young University, Department of Physics & Astronomy , publisher = Brigham Young University , url = https://physics.byu.edu/faculty/colton/docs/phy442-resources/Lorentz-oscillator-model.pdf , access-date = 2021-11-18 {{cite web , last1 = Almog , first1 = I. F. , last2 = Bradley , first2 = M. S. , last3 = Bulovic , first3 = V. , title = The Lorentz Oscillator and its Applications , year = 2011 , website = Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science , publisher = Massachusetts Institute of Technology , url = https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/readings/MIT6_007S11_lorentz.pdf , access-date = 2021-11-24 {{cite journal , last1 = Spitzer , first1 = W. G. , last2 = Kleinman , first2 = D. , last3 = Walsh , first3 = D. , title = Infrared Properties of Hexagonal Silicon Carbide , journal = Physical Review , volume = 113 , issue = 1 , pages = 127–132 , year = 1959 , url = https://doi.org/10.1103/PhysRev.113.127 , doi = 10.1103/PhysRev.113.127 , bibcode = 1959PhRv..113..127S , access-date = 2021-11-24 {{cite journal , last1 = Zhang , first1 = Z. M. , last2 = Lefever-Button , first2 = G. , last3 = Powell , first3 = F. R. , title = Infrared Refractive Index and Extinction Coefficient of Polyimide Films , journal = International Journal of Thermophysics , volume = 19 , issue = 3 , pages = 905–916 , year = 1998 , url = https://doi.org/10.1023/A:1022655309574 , doi = 10.1023/A:1022655309574 , s2cid = 116271335 , access-date = 2021-11-24