Frequency dependence of the dielectric constant

 

Viewed from a microscopic perspective we know that the molecular polarizability is frequency dependent. Electronic polarizability is present in all molecules and has a response time that is rapid (> 1014 s-1). The high frequency response can follow the undulations of electromagnetic radiation in the visible region and hence this response gives rise to refraction of light. Sometimes it is said that this contribution is the high frequency or optical dielectric constant, e¥. There is a nuclear polarizability in polar molecules due to their tendency to align in an applied electric field. The motion of the molecules is a rotation due to the torque of the applied field. The frequency range for these motions is 106 - 1010 s-1. These motions give rise to absorption and dispersion in the microwave region. They also contribute to the low frequency or static dielectric constant, e0. The static dielectric constant is not really static, but rather is due to changes in the electrical response due to dipolar reorientation.

We shall dissect the relative permittivity, er into real and imaginary parts.

These two contributions represent the in-phase (er') and out-of-phase (er'') components of the frequency response of the medium. The in-phase component results in dispersion. Physically this means refraction of the electromagnetic radiation as it passes through the medium. The out-of-phase component gives rise to absorption. Absorption occurs in the visible (electronic state transitions), infrared (vibrational transitions), and microwave (rotational transitions). The real and imaginary parts of the frequency dependence dielectric response are related to one another by Kramers-Kronig relations:

The importance of these equations for spectroscopy is that we can obtain information on absorptive processes by measuring dispersion. For example, diffuse reflectance spectra from crystals can be transformed into absorption spectra.

The refractive index can also be represented as a complex quantity.

The high frequency part of the dielectric response is equal to the square of the index of refraction, er(w) = n(w). Equating real and imaginary parts leads to

er'(w) = n2(w) - k2(w)

er''(w) = 2n(w)k(w)

The real part of the index of refraction, n r(w) is the factor by which the speed of light is reduced as it traverses a medium. The imaginary part of the index of refraction, k(w) is an absorption coefficient. To understand the effects of these two terms, consider an electric field

The wavevector in vacuum is

and in a dielectric medium it is

Considering both the real and imaginary parts of the index of refraction we have

The exponentially decaying term represents the attenuation of radiation as it passes through an absorptive medium. Since the intensity is proportional to the square of the amplitude of the electric field

The absorption coefficient g is

The absorption coefficient can be related to the molar absorptivity Î (units of L mol-1 cm-1) by comparing Beer's law to the above expression

In the above expression x is the pathlength and C is the concentration. Using these relations we can establish the connection between the imaginary part of the dielectric constant and the molar absorptivity

where N is the number of absorbing molecules per cm3. This can also be expressed as the number of moles per L.

Using the relation c = w/k and k = 2p/l this expression can also be recast as