Understanding the Six Degrees of Freedom in Crystal Structures

[Zarquon]

It seems to me that there should only be three degrees of freedom for each atom in a crystal, one for each direction of vibration; but apparently there are six? Can someone explain?

Thanks.

 

[DrDu]

Are you referring to the heat capacity being 3Nk in the high temperature limit?
That's not that there are 3 degrees of freedom but 3 harmonic oscillators per atom, each having a heat capacity of k. As a mnemonic one sometimes says that there is one degree of freedom for kinetic and potential energy in an oscillator.

 

[Zarquon]

Thanks, that clarifies things a bit! But now I'm a bit confused with the equipartition principle: according to what I've been told, a degree of freedom is the same as an independent variable that contributes an amount to the energy proportional to its square, and each degree of freedom contributes 1/2 k T to the mean energy. So according to this a 1-dimensional harmonic oscillator only has one degree of freedom? (That is, its energy is proportional to the square of its amplitude)

 

[DrDu]

I would have to think about seriously to give you details, but maybe you want to work this out yourself. The point is that the equipartition theorem is a theorem from classical statistical mechanics. Hence you have to consider the degrees of freedom in phase space.

 

[Darwin123]

Thanks, that clarifies things a bit! But now I'm a bit confused with the equipartition principle: according to what I've been told, a degree of freedom is the same as an independent variable that contributes an amount to the energy proportional to its square, and each degree of freedom contributes 1/2 k T to the mean energy. So according to this a 1-dimensional harmonic oscillator only has one degree of freedom? (That is, its energy is proportional to the square of its amplitude)

The total energy of the 1-D harmonic oscillator is proportional to the square of the amplitude. However, the amplitude is the maximum displacement of the harmonic oscillator.

At any specific time, the square of the displacement is less than or equal to the square of the amplitude. This "less than or equal to" means there is a degree of freedom in the displacement.

The total energy of the harmonic oscillator is constant in time when there is no damping. However, at any moment of time the total energy has two components. The total energy is the sum of the kinetic energy and the potential energy. The ratio of kinetic energy to total energy changes with time.

Therefore, there are actually two degrees of freedom. There is a degree of freedom corresponding to the kinetic energy. There is a degree of freedom corresponding to the potential energy.

Suppose you have a harmonic oscillator of known frequency. At any moment of time, the amplitude can be calculated only by knowing BOTH the kinetic energy and the potential energy.

There is no way to determine the amplitude from the potential energy alone because the potential energy is proportional to the square of the instantaneous displacement. There is no way to determine the amplitude from the kinetic energy because the kinetic energy is proportional to the square of the velocity. One can determine the amplitude knowing both the kinetic energy and the potential energy at anyone time. Therefore, there are two degrees of freedom.

Amplitude is not just the displacement. Amplitude is the maximum displacement. Any displacement less than maximum is not the amplitude.

 

[Zarquon]

Alright, it's beginning to make sense to me now. Thanks a lot, guys!