Youngs Modulus (= Elastizitätsmodul) is abbreviated with an "Note:"
in this text; not with a "E" as is customary in the English literature.Y |
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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.
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As long as the potential increases quadratically with the distance from the equilibrium
position , the restoring force will be proportional to the deviation r_{o} from x = r -
r_{o}; and we have a simple harmonic oscillator.r_{o} | |||||||||||||||||||||||

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 and stop after the quadratic term. We already did that; we hadr_{o} | |||||||||||||||||||||||

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

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

The resonance frequency w of the system is known from standard mechanics; it is
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(Try it; all you have to do is to see of the solution is a solution for the differential equation above).x = x_{0}cos
wt | |||||||||||||||||||||||

While for a real oscillator there will always be some friction (or better energy dispersion);
i.e. a term , 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.k_{f} · dx/dt |
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We know the or restoring force of our system, it
is simply F_{res} | |||||||||||||||||||||||

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

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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 | |||||||||||||||||||||||

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It is easy enough to use instead of the spring constant,
we haveE | |||||||||||||||||||||||

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Which gives | |||||||||||||||||||||||

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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 | |||||||||||||||||||||||

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

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

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

What is the frequency of visible light? Easy. We know its energy ,
and we E = hnmust 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 (look it up!), we get ^{–15}
eV·sn. So our atoms are a bit slower, but _{light}
» 2 · 10^{14} Hz10 is a rather large frequency, indeed.^{13}
Hz |

© H. Föll (MaWi 1 Skript)