Black body and continuous spectrum

[jaumzaum]

When we see the black body radiation graphic (intensity vs frequency), we could say the spectrum emitted by a black body is not discrete, it's continuous (as all the frequencies are emitted). How can this be possible? If we have a black body made of Sodium, wouldn't the spectrum have only certain frequencies?

http://img69.imageshack.us/img69/6078/94400250.png

 

[Bill_K]

There's a lot going on that doesn't involve the atomic energy levels. Atoms collide with each other thermally, producing photons, which collide with other atoms, etc.

 

[Darwin123]

When we see the black body radiation graphic (intensity vs frequency), we could say the spectrum emitted by a black body is not discrete, it's continuous (as all the frequencies are emitted). How can this be possible? If we have a black body made of Sodium, wouldn't the spectrum have only certain frequencies?

http://img69.imageshack.us/img69/6078/94400250.png

It is not a black body unless the energy is reabsorbed and remitted many times before leaving the cavity. If the walls of the black body cavity completely reflect electromagnetic radiation, then the electromagnetic radiation has to be reabsorbed and re-emitted many times by the sodium atoms. So consider the absorption and emission of radiation by the sodium atoms.

I'll explain just one broadening mechanism, Doppler shift. There are others. However, Doppler shift is fundamental and unavoidable.

Consider a cavity with 100% reflecting walls containing sodium gas. There is a small hole in the cavity walls from which the spectrum can be measured. However, the hole is so small that if there was no sodium in the cavity, a typical photon would have to reflect millions of times from the walls to get out of the cavity. Suppose the sodium gas is dense enough so a photon is absorbed by about once every 1000 reflections. So a typical photon leaving the cavity represents energy that has been absorbed and remitted about 1000 times.

An atom of sodium gas has quantized energy levels. Therefore, in the inertial frame of the sodium atom photons are absorbed or emitted that match differences in the energy levels. So one could naively think that the light in the cavity could show the sodium spectrum even though the energy has been absorbed and emitted one thousand times. However, there are processes that broaden the spectrum.

In the inertial frame of the reflecting walls, the sodium does not absorb or emit photons of fixed energy. The sodium atoms are moving around. The velocities of the sodium atoms are distributed in a thermal probability distribution. The sodium gas and the photons are in thermal equilibrium, so they have the same temperature. So the kinetic energies of the sodium atoms are spread out. Each sodium atom has a different

Suppose that a sodium atom emits a photon of fixed energy. The emitted photon matches one of the spectral lines of sodium in the inertial frame of the sodium atom. However, the sodium atom is moving fast in the inertial frame of the cavity. So the frequency of the emitted photon is Doppler shifted. Depending on the direction of the photon relative to the velocity of the atoms, the frequency can increase or decrease by a small amount. All the sodium atoms emit light which has a frequency equal to the frequency of a spectral line plus the frequency shift caused by the Doppler effect. So the spectrum of the radiation emitted by all the sodium atoms is broadened. The emitted energy is not just a sodium line spectrum.

Doppler effect also works on the absorbed radiation. In the inertial frame of a sodium atom, the sodium atom only absorbs photons with energy equal to the differences in energy levels. However, the light headed toward the atom is Doppler shifted. Every atom has a different Doppler shift. So absorption spectrum of the sodium gas is broadened. It isn't just a line spectrum. After a thousand times of absorption and re-emission by different sodium atoms, the spectrum of the radiation is not going to look anything like a line spectrum. In fact, it looks like a black body spectrum. The sum of individual Doppler shifts randomize the energy of the photon.

Suppose the initial condition of the cavity isn't in thermal equilibrium. Suppose that this non equilibrium condition has a distribution that looks like a line-spectrum. After a thousand absorption and re-emissions, a thermal equilibrium is reached. The final distribution will look like a black body spectrum.

The spectrum can also be broadened by nonlinear optical processes such as nondegenerate two-photon absorption and emission. However, that gets a little fancy.

A good example is the sun. The sun is mostly hydrogen plus helium. The spectrum of the sun looks like a black body spectrum superimposed on a line spectrum. The black body component comes from the deep layers of the sun, where every photon has been absorbed and re-emitted millions of times per second for one hundred thousand years. However, the line spectra come from the very sufrace levels of the sun where a photon can escape into outer space almost immediately.

 

[Bill_K]

Darwin123, Nice explanation, but you're making it sound like black body radiation is entirely a result of the broadening of emission lines. Black body radiation does not depend on emission lines. Black body radiation would exist, and have the same characteristics, even in a gas of particles that had no excited states at all.