What is an approximate amplitude of a phonon?

[Karthiksrao]

I am aware that phonons are lattice vibrations - and that the amplitude of vibration would depend on the temperature. But say, at room temperature what would the order of magnitude of these lattice vibrations be ?

In particular, in continuum limit these phonons can be treated as elastic waves. So if I have take an incident planar longitudinal elastic wave at a surface, what would I need to take the amplitude of this wave to be ?

I do think that if I equate the energy stored in this wave integrated over all incident angles and frequencies to the heat capacity of the solid - I should be able to find the amplitude of this wave. Is this correct ?

Thanks!

 

[Karthiksrao]

I think I figured it out. Each mode will be populated by a number of phonons equal to the phonon occupancy number, each with energy h v, and the energy of each mode can be related to the amplitude of the lattice vibration.

 

[Henryk]

I'm glad that you answered your own question. However, this might help:
Room temperature is well above Debye's temperature for most solids. The energy per degree of freedom is then $\frac 1 2 k_B T$

 

[Karthiksrao]

Thanks for the note. I always get confused - what is the relation between a degree of freedom and a mode/state? Are they the same?

For example, in my case, I have (1/2 rho omega^2 Amplitude^2) = n * hbar omega

The left hand side is the energy contained in the elastic wave; and the right hand side is the energy contained by the total number of phonons occupying the state ('n' is occupation number for the frequency omega).

So in terms of degrees of freedom you mentioned, is it just that (1/2 rho omega^2 Amplitude^2) = 1/2 k_B T ?

Thanks

 

[Henryk]

Total energy of vibrating atoms is kinetic and potential. Kinetic is a function of momentum that has 3 components - 3 degrees of freedom. Potential is a function of displacement, again in 3 dimensions - 3 degrees of freedom. Total is 6 degrees of freedom. So, in a simple, Einstein model $\frac 1 2 \kappa x^2 = \frac 1 2 k_B T$ where $\kappa$ is the 'spring' constant.