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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.
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We expect then that the dipoles will randomize, i.e. their dipole moment or their polarization
will go to zero. |
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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 |
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In formulas, we expect that P decays starting at the time of the switch-off according
to |
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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 non-classical, i.e. statistical mechanics, but beyond
our ken at this point. |
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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 change-over
from the excited state to the base state needs "help", i.e. has to overcome some energy barrier. |
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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: |
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dn
dt |
µ n |
= | – |
1
t |
· n |
| | |
| |
| n | = |
n0 · exp – |
t
t |
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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. |
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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. |
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The "decay" to the lower (free) energy state and the concomitant built-up 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. |
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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 w = 2pn =
angular frequency, i.e. the output to a periodic harmonic input, i.e. to a field like E = Eo
· sinwt. |
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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. |
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In other words: We can switch back and forth between P(t)
and P(w) via a Fourier
transformation. |
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We already know the time dependence P(t) for a switch-on
/ switch-off signal, and from that we can - in principle - derive P(w). |
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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 |
|
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| P(w) = |
¥
ó
õ
0 |
P 0 · exp – |
t
t |
· exp – (iwt ) · dt |
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P0 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). |
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This is an easy integral, we obtain |
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| P(w) = |
P0
w0 + i · w |
| | |
| w0 = |
1
t |
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Note that w0 is not
2p/t, as usual, but just 1/t.
That does not mean anything except that it makes writing the formulas somewhat easier. |
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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). |
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P(w) thus is the polarization response of the system if you jiggle
it with an electrical field given by E = E0 · exp (iwt) that contains
just one frequency w. |
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However, our Fourier coefficients are complex numbers, and we have
to discuss what that means now. |
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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. |
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Essentially, we are done. If we know the Amplitude (= E0) and (circle)
frequency w of the electrical field in the material (taking into account possible "local field" effects), we know the polarization. |
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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: |
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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 |
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and the decisive factor, giving the amplitude of P(w), is c(w). |
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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. |
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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. |
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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. |
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Extracting a frequency dependent susceptibility
c(w) from our equation for the polarization is fairly easy:
Using the basic equation we have |
|
|
| e0 · c(w)
| = |
P(w)
E(w) | = |
P0
E0 |
· | 1
w0 + i · w |
= |
cs · |
1
1 + i · w/w0 |
|
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cs = P0/ E0
is the static susceptibility, i.e. the value for zero frequency. |
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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 |
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| c(w) |
= |
c'(w) + i · c''(w) |
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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). |
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All we have to do in order to obtain c' and
c'' is to expand the fraction by 1 – i · w/w0 which gives us |
| |
| e0 · c(w)
| = | cs
1 + (w/w0)2
| – i · |
cs · (w/w0)
1 + (w/w0)2
|
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We thus have for the real and imaginary part of e0
· c(w), which is almost, but not yet quite the dielectric
function that we are trying to establish: |
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| e0 · c' |
= | cs
1 + (w/w0)2 |
| | | |
| – e0 · c'' |
= |
cs · (w/w0)
1 + (w/w0)2 |
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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 non-perfect dielectrics. |
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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. |
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1. People in general like the dielectric constant er 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 "er"
and we obtain a complex frequency dependent dielectric constant er(w) = c(w) + 1 with, of course, es
instead of cs as the zero frequency static case. |
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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
>> w0) = c¥
or, as we prefer, the same thing for the dielectric "constant", i.e. we introduce er(w
>> w0) = e¥
. |
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3. Since we always have either e0
· c(w) or e0 ·
e(w), and the e0
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 ei are what they are as the relative dielectric "constant" and multiplied
with e0 |
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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 so-called Debye equations: |
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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: |
| |
| D(w) |
= |
e (w) · E(w) = |
æ
ç
è |
es – e¥
1 + i (w/w0)
|
+ e¥
| ö
÷
ø |
· E(w) |
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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. |
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Again, we write the complex function as a sum of a real part and a complex part, i.e. as e(w) = e'(w) – i · e''(w). We use
a "–" sign, as a matter of taste; it makes some follow-up equations
easier. But you may just as well define it with a + sign and in some books that is what
you will find. For the dielectric function from above we now obtain |
|
|
|
e' = e¥
+ |
es – e¥
1 + (w/w0)2
|
| e'' = |
(w/w0)(es
– e¥)
1 + (w/ w0)2 |
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As it must be, we have |
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| e'(w = 0) |
= | es |
|
e'(w = 0) |
= | 0 | | |
| | |
| | |
| e'(w ®
¥) | = |
e¥ |
| | |
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From working with the complex notation
for sin- and cosin-functions we also know that |
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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 phase-shifted by 90o. |
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Finally, we can ask ourselves: What does it look like? What are the graphs of e'
and e''? |
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Relatively simple curves, actually, They always look like the graphs shown below, the three
numbers that define a particular material (es, e¥, and t = 2p /w0)
only change the numbers on the scales. |
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Note that w
for curves like this one is always on a logarithmic scale! |
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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! |
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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
es and t (or wo). |
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es 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 es.
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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. |
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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. |
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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. |
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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. |
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© H. Föll (Advanced Materials B, part 1 - script)
