


This is simply the force equilibrium
and the only nontrivial force in this equation is the term
k_{F} · m · dx/dt 


This is the damping or friction term, we simply
assume that it is proportional to the mass m and its velocity
dx/dt. The proportionality constant is our damping
constant k_{F} times the
mass. 


Often the friction term is just written as
k*_{F} · dx/dt, i.e. the mass is
included in k*_{F}, but our approach has a certain
advantage as we will see below. 


While all other terms come from ironclad first
principle physical law (always assuming harmonic potentials), the friction term
is a bit arbitrary; its exact formulation depends on the specific problem.



However, if you have a system where the amplitude
"decays" exponentially after the driving force is switched off, you
must have a damping term as given. Essentially you are back to the very general
model of relaxation into the ground state as employed for the
frequency dependence of the
orientation polarization. 

We are now stuck with solving a
linear second order differential equation  and we know how that is done. 


Usually, we would move step by step, first
looking at a simplified system without damping and driving forces, and then
adding the complications. 


What we would find for the simplified system is
that there is a special frequency w_{0} called the
resonance
frequency or "Eigenfrequency", which is the simply the
frequency with which the system will oscillate by itself if started once. The
resonance frequency without damping we call w_{0}'; it is given by 


w_{0}' 
= 
æ
ç
è 
k_{S}
m_{ } 
ö
÷
ø 
1/2 




With damping added, the resonance frequency
changes somewhat, and the amplitude will decrease with time after some initial
push started an oscillation. This is described by the following equations 





x(t) 
= 
x_{0} · cos(w_{0}t) · exp – 
k_{F}
2_{ } 
· t 
w_{0} 
= 
æ
ç
è 
k_{S}^{ }
m_{ } 
– 
k_{F}^{2}
4_{ } 
ö
÷
ø 
1/2 




If, for a moment, we apply these
equations to an ion sitting in a lattice, we will notice two interesting
points: 

1. The "spring
constant" follows from the binding potential. It is  of course  related
to Youngs modulus Y which tells us how much the length of a
specimen changes under an applied force, or more precisely, how stress applied to a material creates (elastic)
strain. For a homogeneous isotropic
material we actually
have 





With a_{0} = bond length
» lattice constant. In other words, we
know a lot about the spring constant for the systems we are treating here. 


What that means is that we also have a good idea
for the order of magnitude of the resonance frequency.
It will come
out to be roughly 10^{13} Hz. 

2. The
damping or friction constant k_{F} for a single atom,
which is coupled by "bond springs" to some other atoms, which are
coupled by bond springs ... and so on, is far more difficult to assess. Off
hand, most of us probably do not have the faintest idea about a possible
numerical value, or if k_{F} relates somehow to some
quantities we already know, like the spring constant. 


However, realizing that the dimension of the
damping constant is [k_{F}] = 1/s, and that it takes just
a few reciprocal k_{F}'s before the oscillation dies out,
we can make an educated guess: 


If you "snap" just one atom of a huge collection of more or less
identical atoms, all connected by more or less identical springs, pretty soon
all atoms will oscillate. And the original energy, initially contained in the
amplitude of the "snapped" atom, is now spread out over all atoms 
which means that their amplitudes will be far smaller than the original one. To
get the idea, just look at the picture. 





In other words: There is no doubt that it will
just take a few  say 5 or maybe 50  oscillations of the primary
atom, before the orderly energy contained
in the oscillation of that one atom will have spread and became diluted and disordered. 


In yet other words: excess energy contained in
the oscillations of one atom will turn into thermal energy (= random vibrations
of all the atoms); it becomes thermalized rather quickly  in the time it takes to
oscillate back and forth a few times. 


k_{Fw} is thus tied to w_{0}, we expect it to be very roughly in the
order of 5w_{0} .... 50w_{0}. 

So far so good. But now we must go
all the way and switch on "driving", in our example an electrical
field that pulls at the charged mass with a force that oscillates with some
arbitrary frequency w 


However, we will not even try to write down the
solution the full differential equation given above in "straight"
terms  it is too complicated, and there is a better way. We will, however,
consider the solution qualitatively. 


We (should) know that the mass oscillates with
the frequency of the driving force and an amplitude that depends on the
frequency (and the damping constant and so on), and that there will be a phase
shift between the driving force and the oscillating mass that also depends on
the frequency, and so on. 


We also (should) know what all of this looks like
 qualitatively. Here it is: 




What we are going to do, of course,
is to describe the driven damped harmonic oscillator in
complex
notation. The basic equation than is 


m · 
d^{2}x
dt^{2} 
+ k_{F} ·
m · 
d x
d t 
+ k_{s} ·
x 
= 
q · E_{0} ·
exp(iwt) 



The solutions are most easily
obtained for the inphase amplitude x_{0}' and the
outofphase amplitude x_{0}''. 


The total amplitude x_{0}
and the phase shift f are contained in these
amplitudes. If we want to have them, we simply calculate them as
outlined above. 

The solution we will obtain is 


x(w,
t) 
= 
x(w) · exp (iwt)

x(w)
= 
q · E_{0}
m 
æ
ç
è 
æ
è 
w_{0}^{2} – w^{2}
(w_{0}^{2} –
w^{2})^{2} +
k_{F}^{2} w^{2} 
ö
ø 
– i · 
æ
è 
k_{F} w
(w_{0}^{2} –
w^{2})^{2} +
k_{F}^{2} w^{2}

ö
ø 
ö
÷
ø 

x'(w)
= 
q · E_{0}
m 
æ
ç
è 
w_{0}^{2} – w^{2}
(w_{0}^{2} –
w^{2})^{2} +
k_{F}^{2} w^{2} 
ö
÷
ø 

x''(w)
= 
q · E_{0}
m 
æ
ç
è 
k_{F} w
(w_{0}^{2} –
w^{2})^{2} +
k_{F}^{2} w^{2}

ö
÷
ø 




This looks complicated, but is, in fact, far more
elegant than the description without complex numbers. If we plot
x'(w) and x''(w), we obtain the following curves 




These curves are purely
qualitative. A quantitative rendering can be obtained by the JAVA module
below 


Instead of the spring constant, you
may enter Youngs modulus directly. Typical numbers (in GPa) are:
 Diamond: 1000
 Carbides, Oxides, Nitrides: » 300  600
 Glas: 70
 Quartz: 100
 Alkali halides: 15  70
 Wood: 10
 Polymers: 1  10
 Rubber: 0.001  0.1



The damping constant enters with its
reciprocal value normalized to w, i.e.
roughly the number of cycles it takes to dampen out an oscillation. 


You can compare two sets of
parameters, because the last curve will always be shown with the new
curve. 


You can also enlarge any portion of
the diagram by simply drawing a window on the part you like to see
enlarged. 





