From Time Dependence to Frequency Dependence
The easiest way to look at relaxation phenomena is to consider what happens if the driving force  the electrical field in our case  is suddenly switched off, after it has been constant for a sufficiently long time so that an equilibrium distribution of dipoles could be obtained.  
We expect then that the dipoles will randomize, i.e. their dipole moment or their polarization will go to zero.  
However, that cannot happen instantaneously. A specific dipole will have a certain orientation at the time the field will be switched off, and it will change that orientation only by some interaction with other dipoles (or, in a solid, with phonons), in other words upon collisions or other "violent" encounters. It will take a characteristic time, roughly the time between collisions, before the dipole moment will have disappeared.  
Since we are discussing statistical events in this case, the individual characteristic time for a given dipole will be small for some, and large for others. But there will be an average value which we will call the relaxation time t of the system. We thus expect a smooth change over from the polarization with field to zero within the relaxation time t, or a behavior as shown below  
 
In formulas, we expect that P decays starting at the time of the switchoff according to  
 
This simple equation describes the behavior of a simple system like our "ideal" dipoles very well. It is, however, not easy to derive from first principles, because we would have to look at the development of an ensemble of interacting particles in time, a classical task of nonclassical, i.e. statistical mechanics, but beyond our ken at this point.  
Nevertheless, we know that a relation like that comes up whenever we look at the decay of some ensemble of particles or objects, where some have more (or less) energy than required by equilibrium conditions, and the changeover from the excited state to the base state needs "help", i.e. has to overcome some energy barrier.  
All we have to assume is that the number of particles or objects decaying from the excited to the base state is proportional to the number of excited objects. In other words, we have a relation as follows:  
 
This covers for example radioactive decay, cooling of any material, and the decay of the foam or froth on top of your beer: Bubbles are an energetically excited state of beer because of the additional surface energy as compared to a droplet. If you measure the height of the head on your beer as a function of time, you will find the exponential law.  
When we turn on an electrical field, our dipole system with random distribution of orientations has too much energy relative to what it could have for a better orientation distribution.  
The "decay" to the lower (free) energy state and the concomitant builtup of polarization when we switch on the field, will follow our universal law from above, and so will the decay of the polarization when we turn it off.  
We are, however, not so interested in the time dependence
P(t) of the polarization when we apply some disturbance or input
to the system (the switching on or off of the electrical field). We rather would like to know its frequency
dependence P(w) with  
Since any signal can be expressed as a Fourier series or Fourier integral of sin functions as the one above, by knowing P(w) we can express the response to any signal just as well.  
In other words: We can switch back and forth between  
We already know the time dependence 

We thus have to consider the Fourier transform of P(t). However, while clear in principle, details can become nasty. While some details are given in an advanced module, here it must suffice to say that our Fouriertransform is given by  
 
P_{0} is the static polarization, i.e. the value of P(w) for w = 0 Hz , and i = (–1)^{1/2} is the imaginary unit (note that in electrical engineering usually the symbol j is used instead of i).  
This is an easy integral, we obtain  
 
Note that w_{0} is not 2p/t, as usual, but just 1/t. That does not mean anything except that it makes writing the formulas somewhat easier.  
The P(w) then are the Fourier coefficients if you describe the P(t) curve by a Fourier integral (or series, if you like that better, with infinitesimally closely spaced frequency intervals).  
P(w) thus is the polarization response of the system if you jiggle it with an electrical field given by E = E_{0} · exp (iwt) that contains just one frequency w.  
However, our Fourier coefficients are complex numbers, and we have to discuss what that means now.  
Usimg Complex Numbers and Functions
Using the powerful math of complex numbers, we end up with a complex polarization. That need not bother us since by convention we would only consider the real part of P when we are in need of real numbers.  
Essentially, we are done. If we know the Amplitude (= E_{0}) and (circle) frequency w of the electrical field in the material (taking into account possible "local field" effects), we know the polarization.  
However, there is a smarter way to describe that relationship than the equation above, with the added benefit that this "smart" way can be generalized to all frequency dependent polarization phenomena. Let's see how it is done:  
What we want to do, is to keep our basic equation that couples polarization and field strength for alternating fields, too. This requires that the susceptibility c becomes frequency dependent. We then have  
 
and the decisive factor, giving the amplitude of P(w), is c(w).  
The time dependence of P(w) is trivial. It is either given by exp i(wt – f), with f accounting for a possible phase shift, or simply by exp i(wt) if we include the phase shift in c(w), which means it must be complex.  
The second possibility is more powerful, so that is what we will do. If we then move from the polarization P to the more conventional electrical displacement D; the relation between D(w) and E(w) will require a complex dielectric function instead of a complex susceptibiltiy, and that is the quantity we will be after from now on.  
It goes without saying that for more complex time dependencies of the electrical field, the equation above holds for every for every sin component of the Fourier series of an arbitrary periodic function.  
Extracting a frequency dependent susceptibility c(w) from our equation for the polarization is fairly easy: Using the basic equation we have  
 
c_{s} = P_{0}/ E_{0} is the static susceptibility, i.e. the value for zero frequency.  
Presently, we are only interested in the real part of the complex susceptibility thus obtained. As any complex number, we can decompose c(w) in a real and a imaginary part, i.e. write it as  
 
with c' and c'' being the real and the imaginary part of the complex susceptibility c . We drop the (w) by now, because whenever we discuss real and imaginary parts it is clear that we discuss frequency dependence).  
All we have to do in order to obtain c' and c'' is to expand the fraction by 1 – i · w/w_{0} which gives us  
 
We thus have for the real and imaginary part of e_{0} · c(w), which is almost, but not yet quite the dielectric function that we are trying to establish:  
 
This is pretty good, because, as we will see, the real and imaginary part of the complex susceptibility contain an unexpected wealth of material properties. Not only the dielectric behavior, but also (almost) all optical properties and essentially also the conductivity of nonperfect dielectrics.  
Before we proceed to the dielectric function which is what we really want to obtain, we have to makes things a tiny bit more complicated  in three easy steps.  
1. People in general like the dielectric constant e_{r} as a material parameter far better than the susceptibility c  history just cannot be ignored, even in physics. Everything we did above for the polarization P, we could also have done for the dielectric flux density D  just replace the letter "P" by "D" and "c" by "e_{r}" and we obtain a complex frequency dependent dielectric constant e_{r}(w) = c(w) + 1 with, of course, e_{s} instead of c_{s} as the zero frequency static case.  
2. So far we assumed that at very large frequencies the polarization is essentially zero  the dipole cannot follow and c(w ® ¥) = 0. That is not necessarily true in the most general case  there might be, after all, other mechanisms that still "work" at frequencies far larger than what orientation polarization can take. If we take that into account, we have to change our consideration of relaxation somewhat and introduce the new, but simple parameter c(w >> w_{0}) = c_{¥} or, as we prefer, the same thing for the dielectric "constant", i.e. we introduce e_{r}(w >> w_{0}) = e_{¥} .  
3. Since we always have either e_{0} · c(w) or e_{0} · e(w), and the e_{0} is becoming cumbersome, we may just include it in what we now call the dielectric function e (w) of the material. This simply means that all the e_{i} are what they are as the relative dielectric "constant" and multiplied with e_{0}  
This reasoning follows Debye, who by doing this expanded our knowledge of materials in a major way. Going through the points 1.  3. (which we will not do here), produces the final result for the frequency dependence of the orientation polarization, the socalled Debye equations:  
In general notation we have pretty much the same equation as for the susceptibility c; the only real difference is the introduction of e_{¥} for the high frequency limit:  
 
The complex function e (w) is the dielectric function. In the equation above it is given in a closed form for the dipole relaxation mechanism.  
Again, we write the complex function as a sum of a real part and a complex part, i.e. as  
 
As it must be, we have  

From working with the complex notation for sin and cosinfunctions we also know that  
e', the real part of
a complex amplitude, gives the amplitude of the response that is in phase with the driving
force, e'', the imaginary part, gives the amplitude of the response that is phaseshifted by 90^{o}.  
Finally, we can ask ourselves: What does it look like? What are the graphs of e' and e''?  
Relatively simple curves, actually, They always look like the graphs shown below, the three numbers that define a particular material (e_{s}, e_{¥}, and t = 2p /w_{0}) only change the numbers on the scales.  
 
Note that w for curves like this one is always on a logarithmic scale!  
What the dielectric function for orientation polarization looks like for real systems can be tried out with the JAVA applet below  compare that with the measured curves for water. We have a theory for the frequency dependence which is pretty good!  
 
Since e_{¥} must be = 1 (or some value determined by some other mechanism that also exists) if we go to frequencies high enough, the essential parameters that characterize a material with orientation polarization are e_{s} and t (or w_{o}).  
e_{s} we can get from the polarization mechanism for the materials being considered. If we know the dipole moments of the particles and their density, the Langevin function gives the (static) polarization and thus e_{s}.  
We will not, however, obtain t from the theory of the polarization considered so far. Here we have to know more about the system; for liquids, e.g., the mean time before two dipoles collide and "loose" all their memory about their previous orientation. This will be expressed in some kind of diffusion terminology, and we have to know something about the random walk of the dipoles in the liquid. This, however, will go far beyond the scope of this course.  
Suffice it to say that typical relaxation times are around 10 ^{–11} s; this corresponds to frequencies in the GHz range , i.e. "cm waves". We must therefore expect that typical materials exhibiting orientation polarization (e.g. water), will show some peculiar behavior in the microwave range of the electromagnetic spectrum.  
In mixtures of materials, or in complicated materials with several different dipoles and several different relaxation times, things get more complicated. The smooth curves shown above may be no longer smooth, because they now result from a superposition of several smooth curves.  
Finally, it is also clear that t may vary quite a bit, depending on the material and the temperature. If heavy atoms are involved, t tends to be larger and vice versa. If movements speed up because of temperature, t will get smaller.  
© H. Föll (Advanced Materials B, part 1  script)