Drude-Lorentz Model for Dispersion in Dielectrics

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.