# Drude-Lorentz Model for Dispersion in Dielectrics

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Drude and Lorentz (ca. 1900) developed a classical theory to account for the complex index of refraction and dielectric constant of materials, as well as their variations with the frequency of light. The model is based on treating electrons as damped harmonically bound particles subject to external electric fields. A highly simplified version of the model is given in this Demonstration, with results limited to a qualitative level. Still, the phenomena of normal and anomalous dispersion and their relation to the absorption of radiation can be quite reasonably accounted for. The classical parameters of the theory transform simply to their quantum analogs, so that the results remain valid in modern theories of materials science.

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Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

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The equation of motion for a harmonically bound classical electron interacting with an electric field is given by the Drude–Lorentz equation , where is the natural frequency of the oscillator and is the damping constant. For an incident electromagnetic field of frequency , at the point can be conveniently represented by a complex exponential . The steady-state solution, in complex form, is given by . The electric dipole moment of the electron, , corresponds to the macroscopic relation for the polarizability , where is the complex electric susceptibility , being the number of polarizable electrons per unit volume. It is sufficient to approximate , with transition frequencies lying in the optical region rad/s. Also, . The real part of the susceptibility gives the frequency dependence (dispersion) of the dielectric constant and index of refraction . The imaginary part represents the absorption coefficient. This function has the form of a Lorentzian.

In the more accurate quantum theory of dispersion, the frequency is replaced by a sum over several atomic transition frequencies and the damping parameters are determined by excited-state lifetimes.

The real and imaginary parts of the susceptibility are connected by the Kramers–Kronig relations: and , where signifies the Cauchy principal value of the integral.

Reference

[1] L. Rosenfeld, *Theory of Electrons*, New York: Dover Publications, 1965, pp. 68 ff.

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