## Saturday, January 01, 2011

### Quantum mechanics of thermal radiation

UPDATE: (Jan 17th 20100).

I posted the question below at the Physics StackExchange and received some excellent answers here. They add to my post below by factoring in something I had ignored: Raman scattering events.

How do objects emit thermal radiation?

This is an example of a simple query for which Google fails to provide a simple answer. You are inexorably drawn into endless websites which describe:

* Planck's Law
* Wien's displacement law
* The Stefan-Bolzmann law.

In the process you will learn a great deal about cavities, naive application of the equipartition theorem and its consequent ultraviolet catastrophe, electromagnetic standing waves, Planck's introduction of the quantisation of radiated energy. Here's a good overview. What you will NOT learn is how objects get to emit/absorb thermal radiation in the first place.

You will be particularly confused by some of the non-sequiturs in the articles Google turns up. You will read about the discrete frequencies emitted and absorbed by atoms as their electrons jump between orbitals at different energy levels. And you will wonder what on earth this has to do with the continuous blackbody spectrum. (At room temperature, nothing).

The truth of the matter is well explained in this powerpoint presentation. At ordinary temperatures on earth thermal radiation is in the infrared. This frequency band is associated with energy transitions where molecular links vary their stretch, bend and spin in different modes of oscillation (recall inter-molecular links are modelled by techniques such as LCAO, with more detail on coupling to the electromagnetic field here).

It turns out that molecules have a very large number of energy levels when doing these contortions and so the emission and absorption spectrum looks like a comb with an uneven height profile (see slide 33 and following). However, we have to factor in two other phenomena (as described in slide 32 in the context of the atmosphere):

"Doppler broadening: random translational motions of individual molecules in any gas leads to Doppler shift of absorption and emission wavelengths (important in upper atmosphere).

"Pressure broadening: collisions between molecules randomly disrupt natural transitions between energy states, so that absorption and emission occur at wavelengths that deviate from the natural line position (important in troposphere and lower stratosphere)

"Line broadening closes gaps between closely spaced absorption lines, so that the atmosphere becomes opaque over a continuous wavelength range."

Similar phenomena smear out the quantised lattice vibrations in insulating solids. In metals the free conduction electrons have a very fine-grained hierarchy of energy levels determined by the wave-numbers which fit into a macrosized metal object.

So there you are. You get a smooth blackbody spectrum of thermal radiation because of the extremely large number of finely-spaced energy levels involved in molecular/atomic/electron motions in the bulk (degrees of freedom as we say). These generate a wide, dense comb-like quantised structure of emission/absorption frequencies which is then smeared into a continuous spectrum by doppler and bulk kinetic effects.

Other keyphrase: Quantum mechanisms of thermal radiation.