Oom Lecture 04

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More Thermal Properties

Last time we said that c_ v={3k\over (molecular\  mass)}. 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:

 \nu _{mode}={c_ e\over 2L} \,\!

where ce 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 \left\langle s\right\rangle having frequency νn, where

 \nu _ n={c_ e\over 2L}\times n \,\!

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

 \begin{align}  P(s)& \propto e^{-s h\nu _ n\over kT}\\ & = \sum _{s=0}^\infty {e^{-s h\nu _ n\over kT}}\\ \end{align} \,\!

Then P(s)=e^{-s h\nu _ n\over kT}\left(1-e^{-h\nu _ n\over kT}\right), and so the number of phonons is given by:

 \begin{align}  \left\langle s\right\rangle & =\sum _{s=0}^\infty {s P(s)}\\ & =\left(1-e^{-h\nu _ n\over kT}\right)\sum _{s=0}^\infty {s e^{-s h\nu _ n \over kT}}\\ \left\langle s\right\rangle & = {1\over e^{h\nu \over kT} - 1}\\ \end{align} \,\!

Now these modes will tend to be well populated when \left\langle s\right\rangle \sim 1.

Now in 3D, \nu _ n={c_ e\over 2L}(n_ x^2+n_ y^2+n_ z^2)^\frac 12, 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:

 T_{Debye}={hc_ e\over ka} \,\!

Using a\sim 3\AA and c_ e\sim 3{km\over s}, we get 400K. 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 \left\langle x\right\rangle =0, but \left\langle x^2\right\rangle \ne 0. In fact we can write:
 \begin{align}  \left\langle x^2\right\rangle & \equiv {1\over N}\sum _{i=1}^ N{x_ i^2}\\ & = {1\over N}\sum _{i=1}^ N{\left[x_ i^2(m-1)+\delta ^3\pm 2\delta x_ i(m-1)\right]}\\ \end{align} \,\!

δ3 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 m\equiv {t\over \tau }, and this gives us:

 \begin{align}  \left\langle x^2(t)\right\rangle & ={t\over \tau \delta ^2}\\ & ={\delta ^2\over \tau }t\\ & =Dt\\ \end{align} \,\!

where D is the diffusivity. The diffusivity is all you care about: it can be {\delta \over \tau } or vδ, or v2τ.

  • 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:
 Flux={Net \# \over At}={-\frac12(N(x+\delta )-N(x))\over A\tau } \,\!

where A is the area of the surface. Multiplying the top and bottom by δ2, and defining C\equiv {N\over A} as a concentration, we can write the above as a derivative of concentration times the ratio of {\delta ^2\over \tau }=D:

  {Flux=-D{dC\over dx}} \,\!

this is Fick’s Law. Alternately, Flux={dC\over dt}, so we could write this:

  {{dC\over dt}=D{d^2C\over dx^2}} \,\!

we’ll simplify this equation to say that {1\over t}\sim D{1\over x^2}.

  • Back to the phonon random-walking through an insulator, we can say
 {d T \over dt}=\kappa {d T \over dx^2}  \,\!

where κ is the thermal diffusivity (D in our digression). Using that κ˜δv, we’ll say v˜ce, and δ˜λmfp, and this is typically a few lattice spacings at the Debye temperature. We’ll say \lambda \sim 3a=sim 10\AA . So  {\kappa \sim 10^{-2} {cm^2\over s}}, where we’ve divided our naive δv 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 {R^2\over t}\sim \kappa . For a 20 cm bird and \kappa =10^{-2}{cm^2\over s}, and so we get something around 10 hrs.
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