
Note: Youngs Modulus (= Elastizitätsmodul)
is abbreviated with an "E" in this text; not with a
"Y" as is customary in the English literature. 



We have
already employed the picture of an
atom or particle oscillating (or vibrating) in its potential well. Now we shall
compute the vibration frequency w = 2pn from the binding
potential. 


As long as the potential
increases quadratically with the distance from the equilibrium position
r_{o}, the restoring force will be proportional to the
deviation x = r  r_{o} from
r_{o}; and we have a simple harmonic oscillator. 


The harmonic approximation is good enough for getting an
order of magnitude estimate of the vibration frequency; i.e. we simply replace
the proper potential by its Taylor expansion around r_{o}
and stop after the quadratic term. We already did that;
we had 


U 
= 
U_{0} +
1/2U_{0}'' · x^{2} 




and






U''(r_{0}) 
= 
U_{0} ·
(nm/r_{0}^{2}) 




The basic equation for oscillations
in this potential that we have to solve is 


m_{a} · 
d^{2}x
dt^{2} 
+ k_{s} · x =
0 




with m_{a} =
mass of the vibrating particle (we use the symbol m_{a}
instead of m to avoid confusion with the exponent m
in the potential equation). In this formulation we also used a "spring constant"
k_{s} in order to be able to compare the solutions with
standard formulations of classical mechanics. 


The resonance frequency w of the
system is known from standard mechanics; it is 


w 
= 
æ
ç
è 
k_{s}
m_{a} 
ö
÷
ø 
1/2 




(Try it; all you have to do is to see
of the solution x = x_{0}cos wt is a solution for the differential equation
above). 


While for a real oscillator there
will always be some friction (or better energy dispersion); i.e. a term
k_{f} · dx/dt, we do not have to worry
about that because friction does not change the resonance frequency. If you
want to know more about this, use the
link. 

We know the or restoring force
F_{res} of our system, it is simply



F_{res } = – 
dU
dx 
= – U_{0}'' ·
x = U_{0} · (nm/r_{0}^{2})
· x 




The spring constant thus is simply
k_{s} = U_{0} ·
(nm/r_{0}^{2}), and the resonance frequency is 


w = 
æ
ç
è 
U_{0} ·
(nm/r_{0}^{2})
m_{a} 
ö
÷
ø 
½ 
= 
1
r_{0} 
æ
ç
è 
U_{0} ·
n · m
m_{a} 
ö
÷
ø 
½ 



While this is good enough, we
remember that we had the second derivative of the potential at some other
occasion: When we found a formula for
Youngs modulus E. 


What we had was 


E = 
1

· 
d^{2}U

= 
n · m ·U_{0}

r_{0} 
dr^{2} 
r_{0}^{3} 




It is easy enough to
use E instead of the spring constant, we have 


k_{s} 
= 
U_{0} · 
n · m
r_{0}^{2} 
= 
E · r_{0} 




Which gives 


w 
= 
æ
ç
è 
E · r_{0}
m_{a} 
ö
÷
ø 
1/2 



The vibration frequency of an atom in
a lattice thus will be determined  approximately  by the easily obtainable
quantities Youngs modulus, lattice constant and mass of the atom. Lets see what
we get for some examples 


Lets take Silicon. We have 


E = 150 GPa = 1,5 ·
10^{11} N/m^{2} 

w =
8,4 · 10^{13} Hz
n = 1,34 · 10^{13} Hz 
m_{a} = 31 · 1,67
· 10^{–27} kg 
Þ 
r_{0} = 0,31 nm = 3,1 ·
10^{–10} m 




That is very satisfactory because it
gives us the common result, always just claimed without justification, that the
vibration frequency of atoms in a lattice is in the order of 10^{13}
Hz. 


That the vibration
frequency of atoms in a solid is in the order of n » 10^{13} Hz
is a number we will commit to memory now, and which we will never forget! 

Is a frequency of 10^{13}
Hz large or small? Dumb question, you always have to add "In relation
to what"? 


In electrical engineering, the
highest frequencies "commonly" employed are in the (1  100) GHz =
10^{9} Hz  10^{11} Hz "Microwave" range.
However, there is a lot of excitement about novel devices in the
"Terahertz" (= THz = 10^{12} Hz) region. Our atoms,
however, vibrate still faster  but not much. 


What is the frequency of visible
light? Easy. We know its energy E = hn, and we must
know that the energy of visible light is in the 1 eV region. It's
actually a bit higher, 1 eV is still infrared, but it is good enough for
our purpose. With h = 4.13 · 10^{–15} eV·s (look it up!),
we get n_{light} » 2 · 10^{14} Hz. So our atoms are a
bit slower, but 10^{13} Hz is a rather large frequency,
indeed. 