# Calculating maximum amplitude of atomic vibrations

1. Apr 15, 2010

### nandugopan

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?

2. Apr 16, 2010

### Mr.Miyagi

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:
$$\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}$$

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
$$\frac{3}{8}\hbar\omega_D+\frac{3\hbar}{\omega_D^3}\int_0^{\omega_D}\frac{\omega^3}{e^{\frac{\hbar\omega}{kT}}-1}$$
with
$$\omega_D^3=6\pi^2\frac{N}{V}\overline{v}^3$$.
$$\overline{v}$$ 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.

Last edited by a moderator: May 4, 2017
3. Apr 16, 2010

### nandugopan

Thank you Mr.Miyagi for your reply, but I am interested in calculating the maximum possible amplitude of atomic vibrations.

4. Apr 16, 2010

### Mr.Miyagi

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. Apr 17, 2010

### nandugopan

Thank you Mr.Miyagi. So there can be no 'maximum' amplitude as all amplitudes are statistically possible. Is that right?

6. Apr 17, 2010

### Mr.Miyagi

That is correct.

7. Mar 10, 2011

### metin_mehel

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?