# Oom Lecture 04

### From AstroBaki

## Contents |

### More Thermal Properties

Last time we said that . If you work this out for Al (A=13), we get something a bit larger than what we measure at 300K. The reason for this is that this temperature is below the **Debye Temperature**, which is another way of saying that not all the vibrational modes of Al are populated. We can calculate the mode frequency of a cube of Al (length L on a side) as the wave speed over the wavelength. If we are in resonance, this is:

where *c*_{e} is the elastic wave speed. Now we say the amplitude of the mode is equivalent to the energy in the mode, which is the number of **phonons** having frequency ν_{n}, where

and n is the number of nodes. Boltzmann statistics says that the probability of having some number of phonons is:

Then , and so the number of phonons is given by:

Now these modes will tend to be well populated when .

Now in 3D, , so the above graph will become a 3D plot with a threshold representing a spherical surface.

The minimum wavelength of a phonon (set by the lattice spacing) sets a maximum ν, which is the maximum frequency where we have to worry about not all states being populated. We can estimate this temperature as:

Using and , we get 400*K*. So the take-away point is that **Debye temperatures are similar to room temperature**.

### Thermal Diffusivity (Conductivity) in insulators

- Heat flows by diffusive propagation of elastic waves (phonons). It’s sort of like shaking one end of a net–the wave scatters off into heating all the nodes. Phonons do a random walk (scatter) down an insulator because of inhomogeneities in the lattice, and because of phonon-phonon interference (phonons are collisional).
- Digression on Random Walks (Diffusing) taken from Howard Berg,
*Random Walks in Biology*. You can get somewhere by going nowhere. This is the drunkards walk: you have a 50/50 chance of moving one direction or another. You say that a step width is δ, and each step is taken in time τ. If you have a bunch of drunkards, then you have the property that , but . In fact we can write:

δ^{3} is a term which is always positive, and the 3rd term above averages to 0 in the limit of large N. Now we’ll say that , and this gives us:

where *D* is the **diffusivity**. The diffusivity is all you care about: it can be or *v*δ, or *v*^{2}τ.

- You can get a flux from diffusion if you have more drunkards on one side of a surface than on the other. We can write this as:

where A is the area of the surface. Multiplying the top and bottom by δ^{2}, and defining as a concentration, we can write the above as a derivative of concentration times the ratio of :

this is **Fick’s Law**. Alternately, , so we could write this:

we’ll simplify this equation to say that .

- Back to the phonon random-walking through an insulator, we can say

where κ is the thermal diffusivity (D in our digression). Using that κ˜δ*v*, we’ll say *v*˜*c*_{e}, and δ˜λ_{mfp}, and this is typically a few lattice spacings at the Debye temperature. We’ll say . So , where we’ve divided our naive δ*v* by 3 for 3 dimensions (which slows our naive linear diffusion down).

- We can estimate the time it takes to cook a turkey: We’ll just by dimensional analysis say . For a 20 cm bird and , and so we get something around 10 hrs.