What is the main difference between "static dielectric constant" and "high frequency dielectric constant"?

What are the underlying properties of materials in general and conjugated polymers in particular, that affect the difference in values of the static and high frequency (optical domain ~10^15Hz) dielectric constant?

Perhaps it helps to think in terms of analogies: imagine a mass m on a spring with spring constant k. - If you very slowly stretch the spring by applying a constant force F on the mass, then you will obtain a certain displacement x of that mass (when F=k*x). That picture (the resulting displacement as a function of the applied constant force) could be useful for understanding the concept of "static dielectric constant"). Notice how the mass m does not enter here at all really. - On the other hand, if the applied force becomes time-dependent, and more specifically, oscillates in time and direction (say: F(t) = F0 * sin(w*t) ) then what will be the resulting time-dependent displacement x(t) of the mass m ? ... Well, it will depend on F0 but also on w (omega), the frequency with which the force oscillates. - Then general case can be quite complicated, but we can look at two limiting situations: very small w (going to zero) and very large w (going to infinity). The former will be equivalent to the case described above, the static dielectric constant analogy (pulling the spring very slowly, until the maximum extension is reached when F(t) = F0; in fact, at all times t the extension x(t) is just given by F(t)/k if w is small enough). The latter case where w is very high then corresponds to the analogy for the high-frequency dielectric constant. But in the picture of our mass on a spring, we can see that for very high frequencies, the mass essentially stays fixed and cannot follow the rapidly changing forces. So, just as the previous commenter described, in an atom or molecule, only the lighter electrons can follow then and rapidly rearrange their "electron cloud". Permanent dipoles of molecules cannot rotate fast enough. - It is as if there is another much smaller mass attached via another spring to the big mass described above: in the static case, they are both pulled more or less equally out of their equilibrium position, but in the high-frequency case, the large mass stays fixed, while the smaller mass can oscillate rapidly.

To add to the previous answer. You can split the dipole moment of a molecule into a permanent and induced part. The permanent part is due to fixed charged distributions on a molecule. This part of the dipole moment can only contribute to the dielectric constant as the molecule can easily rotate on timescales comparable to the frequency of measurement. The induced part is due to migration of electron clouds around atoms. This part of the dielectric constant is kept until very high frequencies.

Perhaps it helps to think in terms of analogies: imagine a mass m on a spring with spring constant k. - If you very slowly stretch the spring by applying a constant force F on the mass, then you will obtain a certain displacement x of that mass (when F=k*x). That picture (the resulting displacement as a function of the applied constant force) could be useful for understanding the concept of "static dielectric constant"). Notice how the mass m does not enter here at all really. - On the other hand, if the applied force becomes time-dependent, and more specifically, oscillates in time and direction (say: F(t) = F0 * sin(w*t) ) then what will be the resulting time-dependent displacement x(t) of the mass m ? ... Well, it will depend on F0 but also on w (omega), the frequency with which the force oscillates. - Then general case can be quite complicated, but we can look at two limiting situations: very small w (going to zero) and very large w (going to infinity). The former will be equivalent to the case described above, the static dielectric constant analogy (pulling the spring very slowly, until the maximum extension is reached when F(t) = F0; in fact, at all times t the extension x(t) is just given by F(t)/k if w is small enough). The latter case where w is very high then corresponds to the analogy for the high-frequency dielectric constant. But in the picture of our mass on a spring, we can see that for very high frequencies, the mass essentially stays fixed and cannot follow the rapidly changing forces. So, just as the previous commenter described, in an atom or molecule, only the lighter electrons can follow then and rapidly rearrange their "electron cloud". Permanent dipoles of molecules cannot rotate fast enough. - It is as if there is another much smaller mass attached via another spring to the big mass described above: in the static case, they are both pulled more or less equally out of their equilibrium position, but in the high-frequency case, the large mass stays fixed, while the smaller mass can oscillate rapidly.

Extending the answer of Dr. H. Huinink, I would add the following:
The difference (Δε) between the static dielectric constant (ε0, that is, the limiting value of the real part of complex permittivity obtained at an alternating field frequency approaching zero) and its high frequency analogue (εoo), is a measure of the overall polarizability of the dielectric material.
In polymers’ case, Δε incorporates contributions from a number of mechanisms: electronic/atomic polarizations (at the high-frequency end of the EM spectrum), orientational polarizations (from motions assigned to polar side-chains, chain segments, etc.) and more complex polarization/relaxation modes (e.g., translational motion of charges that are eventually trapped at interfaces, normal mode signals, complex transitions, which become significant at the low frequency end of the EM spectrum and at relatively high temperatures).
The relative strength of the above signals (and thus the magnitude of the static dielectric constant) is mostly relevant to properties and potential applications of polymeric materials related to dielectric devices and electrical insulation, but they could also be used as complementary information for several other applications of the system.

During my MSc graduation study I studied the molecular backgrounds of the dielectric constant. At that moment I read the two volumes (books) of Bottcher. I can advise these books to everyone. See links below.

Finally a remark on polymers connecting to the answer of Ioannis. In my lab we have done some dielectric measurements on nylon-6 containing different amounts of water (equilibrated with RH). Extremely dry nylon-6 has a dielectric constant of about 3 and 4. That is the value that you expect on the basis of only induced polarizability that is limited to distances of atomic scale. The polymer is in the dry state glassy and permanent dipoles (amide part of the molecule) can not rotate and contribute to the polarization. At high humidities the dielectric constant reaches values around 10 and higher. This has two causes: water enters and that has a high dipole moment. However this cannot explain completely the rise. At higher water contents the polymer is plasticized and the permanent dipoles in the polymer can also rotate and contribute to the dielectric constant.

all properties of materials, beside their names, including dielectric constant depend on frequency of irradiation. It is due to possibility to employ a quasi-static approximation in low frequency, which became invalid at higher frequency. For example below 1kHz in many cases the inductive component of Maxwell equations might be neglected, while it have to be accounted in high frequency range. What is even more disturbing for beginners in this area is to realize that physical parameters including dielectric constant became not only became a frequency dependent values, they also but appear to be a complex numbers! The exact relationship between epsilon at zero and high frequency dielectric constant is also the property of material.
So you could not use the zero frequency dielectric constant in explanation/calculation of HF epsilon.

I find the comment of Scheicher to be quite instructive. To take it a bit further, consider the classical equation of an electron with position vector r, mass m and charge -e in a harmonic potential with force constant k, subject to a uniform vector electric field oscillating in time t with angular frequency w : E sin (w t) The equation of motion is

m d^2 r / dt^2 + k r = -e E sin (w t)

which has the driven solution

r(t) = E sin (w t) (-e/m) / ( w0^2 -w^2 ) where w0^2 = k/m. The time-dependent electric dipole moment is

-e r(t) = alpha(w) E sin (w t) where alpha(w) = (e^2 / m) / (w0^2 - w^2) is the frequency-dependent polarizability.

For w < w0, alpha is positive, and for w > w0, alpha is negative. One gets the same result in quantum mechanics using time-dependent perturbation theory. So you see, in the high-frequency limit the polarizability of this system is negative and is equal to the polarizability of a free electron.

I think this is also true for the ground states of real atoms and molecules: the static polarizability is always positive, and the high-frequency polarizability tends to -N e^2/( m w^2) where N is the total number of electrons (there is also a small correction due to finite masses of the nuclei).

This question is related to dispersion of the dielectric constant of the medium due to the electromagnetic field of high frequency compare the frequencies that are characteristic for process of electric and magnetic polarization of material. For such high frequency we can not describe the field of the medium from the macroscopic point of view due to the dispersion effects.
However, there is an interval of the frequencies (called also optical frequencies) for which we can still use the macroscopic of the electromagnetic field in the medium (i.e. use Maxwell equations) even though the dispersion effects are present.
To understand the optical frequency: think about the fact that in the electric and magnetic polarization the main role is played by the electronic mechanism. For wavelengths of the field (and so equivalently frequencies) that correspond to the characteristic time of relaxation of the electron we can still describe the field in the medium using macroscopic description (i.e. wavelength >> dimensions of atom).

For this high frequency, in the case of dispersion, we can write for electric filed in the medium
D = \epsilon_0 \epsilon(\omega) E
where \omega is the frequency of the field. \epsilon(omega) gives the law of dispersion of the dielectric constant, which now is a complex number.

For omega->0 we have that \epsilon(omega) --> static dielectric constant.

Some good references:
1) Classical electrodynamics, Jackson
2) Intr. Electrodynamics, Griffiths
3) Foundations of electrodynamics, H. Sykja

## Popular Answers

Ralph H. Scheicher· Uppsala University## All Answers (8)

H. P. Huinink· Technische Universiteit EindhovenRalph H. Scheicher· Uppsala UniversityIoannis M. Kalogeras· National and Kapodistrian University of AthensThe difference (Δε) between the static dielectric constant (ε0, that is, the limiting value of the real part of complex permittivity obtained at an alternating field frequency approaching zero) and its high frequency analogue (εoo), is a measure of the overall polarizability of the dielectric material.

In polymers’ case, Δε incorporates contributions from a number of mechanisms: electronic/atomic polarizations (at the high-frequency end of the EM spectrum), orientational polarizations (from motions assigned to polar side-chains, chain segments, etc.) and more complex polarization/relaxation modes (e.g., translational motion of charges that are eventually trapped at interfaces, normal mode signals, complex transitions, which become significant at the low frequency end of the EM spectrum and at relatively high temperatures).

The relative strength of the above signals (and thus the magnitude of the static dielectric constant) is mostly relevant to properties and potential applications of polymeric materials related to dielectric devices and electrical insulation, but they could also be used as complementary information for several other applications of the system.

H. P. Huinink· Technische Universiteit EindhovenVolume 1 about statics:

http://www.amazon.com/Theory-Electric-Polarization-Vol-Dielectrics/dp/0444410198/ref=sr_1_1?s=books&ie=UTF8&qid=1377716434&sr=1-1

Volume 2 about dynamics:

http://www.amazon.com/Theory-Electric-Polarization-Vol-Time-Dependent/dp/0444415793

Finally a remark on polymers connecting to the answer of Ioannis. In my lab we have done some dielectric measurements on nylon-6 containing different amounts of water (equilibrated with RH). Extremely dry nylon-6 has a dielectric constant of about 3 and 4. That is the value that you expect on the basis of only induced polarizability that is limited to distances of atomic scale. The polymer is in the dry state glassy and permanent dipoles (amide part of the molecule) can not rotate and contribute to the polarization. At high humidities the dielectric constant reaches values around 10 and higher. This has two causes: water enters and that has a high dipole moment. However this cannot explain completely the rise. At higher water contents the polymer is plasticized and the permanent dipoles in the polymer can also rotate and contribute to the dielectric constant.

Vladimir Burtman· University of UtahSo you could not use the zero frequency dielectric constant in explanation/calculation of HF epsilon.

Charles Winthrop Clark· National Institute of Standards and Technologym d^2 r / dt^2 + k r = -e E sin (w t)

which has the driven solution

r(t) = E sin (w t) (-e/m) / ( w0^2 -w^2 ) where w0^2 = k/m. The time-dependent electric dipole moment is

-e r(t) = alpha(w) E sin (w t) where alpha(w) = (e^2 / m) / (w0^2 - w^2) is the frequency-dependent polarizability.

For w < w0, alpha is positive, and for w > w0, alpha is negative. One gets the same result in quantum mechanics using time-dependent perturbation theory. So you see, in the high-frequency limit the polarizability of this system is negative and is equal to the polarizability of a free electron.

I think this is also true for the ground states of real atoms and molecules: the static polarizability is always positive, and the high-frequency polarizability tends to -N e^2/( m w^2) where N is the total number of electrons (there is also a small correction due to finite masses of the nuclei).

Hiqmet Kamberaj· International Balkan UniversityHowever, there is an interval of the frequencies (called also optical frequencies) for which we can still use the macroscopic of the electromagnetic field in the medium (i.e. use Maxwell equations) even though the dispersion effects are present.

To understand the optical frequency: think about the fact that in the electric and magnetic polarization the main role is played by the electronic mechanism. For wavelengths of the field (and so equivalently frequencies) that correspond to the characteristic time of relaxation of the electron we can still describe the field in the medium using macroscopic description (i.e. wavelength >> dimensions of atom).

For this high frequency, in the case of dispersion, we can write for electric filed in the medium

D = \epsilon_0 \epsilon(\omega) E

where \omega is the frequency of the field. \epsilon(omega) gives the law of dispersion of the dielectric constant, which now is a complex number.

For omega->0 we have that \epsilon(omega) --> static dielectric constant.

Some good references:

1) Classical electrodynamics, Jackson

2) Intr. Electrodynamics, Griffiths

3) Foundations of electrodynamics, H. Sykja

Can you help by adding an answer?