Calculating maximum amplitude of atomic vibrations

In summary: The kinetic energy is due to the motion of the atoms. The amplitude is the size of the vibration. The velocity is the speed of the vibration.
  • #1
nandugopan
3
0
Is it possible to predict theoretically the maximum amplitude of vibration that atoms of a metallic species, like Copper, will exhibit at a given temperature?
 
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  • #2
Well, I'm not sure about maximum amplitude of vibration. It is possible to calculate average energy per atom and thus average amplitude of vibration.

Einstein used a simplified model, in which each atom represents three harmonic oscillators (for the three dimensions). All the oscillators are considered independent. This independence can't be correct of course, but the model still gives rather good predictions, except for very low temperatures. The average energy (that is per atom, per dimension) turns out to be:
[tex]\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}[/tex]

Debye improved on this model by taking into account that the oscillators affect each other. Now oscillation of any of the atoms will propagate through the whole thing. Debye made some clever assumptions which allowed him to solve the equations that arise in this situation. His result for the average energy (again per atom, per dimension) was
[tex]\frac{3}{8}\hbar\omega_D+\frac{3\hbar}{\omega_D^3}\int_0^{\omega_D}\frac{\omega^3}{e^{\frac{\hbar\omega}{kT}}-1}[/tex]
with
[tex]\omega_D^3=6\pi^2\frac{N}{V}\overline{v}^3[/tex].
[tex]\overline{v}[/tex] is the average velocity of the propagating waves in the crystal, N the number of atoms and V the volume.

As you may notice Debye's method is a little harder to use. So, when you're not dealing with very low temperatures, you might as well use Einstein's results. I'll leave it to you to calculate the amplitude from these results.

Also, see http://en.wikipedia.org/wiki/Debye_model" [Broken] for some more information about both models.
 
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  • #3
Thank you Mr.Miyagi for your reply, but I am interested in calculating the maximum possible amplitude of atomic vibrations.
 
  • #4
Well, it is unbounded. The fact is that the energy of the crystal fluctuates at a constant temperature, as does any system in a heat bath. Noticeable fluctuations almost never occur, but they can happen.

So, worst case scenario: the copper samples disintegrates right in your face, because a huge energy fluctuation occurred. But for this to happen you have to wait for a long, long, long time (on average!).
 
  • #5
Thank you Mr.Miyagi. So there can be no 'maximum' amplitude as all amplitudes are statistically possible. Is that right?
 
  • #6
That is correct.
 
  • #7
why? why atoms vibrate?
by vibration the atoms are moving. so there must be a kinetic energy. so there must be dissipated heat? what is the velocity of an atom when it is vibrating? what is the amplitude? or displacemet?
 

1. What is the maximum amplitude of atomic vibrations?

The maximum amplitude of atomic vibrations refers to the maximum displacement of an atom from its average position during a vibration. It is a measure of the extent to which an atom is able to move from its equilibrium position.

2. How is the maximum amplitude of atomic vibrations calculated?

The maximum amplitude of atomic vibrations can be calculated using the equation A = Δx/2, where A is the maximum amplitude, and Δx is the total displacement of the atom from its equilibrium position.

3. What factors affect the maximum amplitude of atomic vibrations?

The maximum amplitude of atomic vibrations is affected by the atomic mass, bond strength, and temperature. Heavier atoms and stronger bonds have lower maximum amplitudes, while higher temperatures result in larger maximum amplitudes.

4. Why is it important to calculate the maximum amplitude of atomic vibrations?

Calculating the maximum amplitude of atomic vibrations is important in understanding the behavior of atoms in a solid. It helps in predicting the thermal and mechanical properties of materials, as well as in designing and optimizing materials for specific applications.

5. Can the maximum amplitude of atomic vibrations be measured experimentally?

Yes, the maximum amplitude of atomic vibrations can be measured experimentally using techniques such as X-ray diffraction, neutron scattering, and optical spectroscopy. These methods allow for the analysis of atomic motion and can provide valuable information about the maximum amplitude of atomic vibrations.

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