Concerning the cavity radiator, it contains two idealizations, which come to my mind immediately.

The first is the (thermodynamic) limit of an infinitely large cavity. This removes the descretization of the spectrum and the dependence on the geometrical shape of the cavity, whatever you have assumed it to be in the beginning. You move from the sum over e/m modes to an integration over a density of (e/m cavity) states.

The second one is thermal equilibrium (which contains the notion of temperature being well defined, i.e. the canonical ensemble). Now, strictly speaking, there can not be equilibrium if there is radiation which you can observe. But if you suppose that little hole to the infinite cavity does not significantly disturb the equilibrium state then there's no need to know the interactions that provide the equilibrium. The cavity radiator is independent of the material making the cavity and the details of the interaction processes which provide thermal equilibrium.

In most every day situations (temperature measurement of heated specimens (pyrometry) or solar observations as mentioned above) the situation is not that of thermal equilibrium. Materials properties and surface geometry/topology then matter. Many a time, though, this is sufficiently accounted for by introducing the concept of a grey radiator as mentioned in a previous contribution. The materials properties then "hide" in the proportionality factor between 'grey' and 'black'.

Magnus,

I agree with Teresa about the immateriality of a blackbody. It is quite awkward and contradictory that despite the fact that a blackbody emits energy in the form of electromagnetic radiation, which will tend to reduce its thermal energy, it is thought to be in thermal equilibrium with its surrounding environment. However, this fact is subsaned by stating that a blackbody also absorbs all radiation impinging on it.. Which I think is hiding behind this apparent contradiction in the case of an ideal blackbody is translated, into the real case, in some mechanism that internally provides that energy difference. There is also something hiding in the fact that the theory does not take into account anything related to time, while any dispense of energy involves some rate of absorption or emission. The sun, for example, that behaves much like a blackbody emitter counts with nuclear fusion to account for that difference and maintain a balance which will last several million years. However, the sun differs considerably from the ideal, theoretical, simple scheme of a blackbody being a hollow box with a small hole. You now solely need to think of a solid box at certain temperature, or even a gaseous sphere like the sun at that temperature, that will emit much in the same manner as a blackbody. Hence, the hollow core of that box was an extremely useful immaterial idealization for Planck to develop the blackbody radiation theory, during early 1900's. I think it's natural that confusions arise when dealing with these physics concepts since there is always certain amount of subjectivity when developing fundamental theories, in which radically new assumptions have to be made (energy quanta, in the case of Planck and blackbody radiation or in the case of Einstein and photoelectric effect). That subjectivity becomes objective and valid only if it's confirmed by experiments.

Hope this helps.

Regards.

I did not find your question that clear.

Planck found that he had to use a discrete sum to be able to describe the continuous emmission of a black body. That happens to be a consequence of what was later on recognized as the result of the light-matter interaction via energy quanta.

Black-bodies do not exist, or, better, cannot be measured as they are turned closed to themselves. Grey-bodies, the real things, have been studied at large. All sorts of matter-light interactions, as you mention, so that the emission is a "quasi-continuum" in nature, as is the emission of light by the Sun (an excellent grey-body by the way) a quasi-continuous spectrum that spectral analysys cannot solve.

Concerning the cavity radiator, it contains two idealizations, which come to my mind immediately.

The first is the (thermodynamic) limit of an infinitely large cavity. This removes the descretization of the spectrum and the dependence on the geometrical shape of the cavity, whatever you have assumed it to be in the beginning. You move from the sum over e/m modes to an integration over a density of (e/m cavity) states.

The second one is thermal equilibrium (which contains the notion of temperature being well defined, i.e. the canonical ensemble). Now, strictly speaking, there can not be equilibrium if there is radiation which you can observe. But if you suppose that little hole to the infinite cavity does not significantly disturb the equilibrium state then there's no need to know the interactions that provide the equilibrium. The cavity radiator is independent of the material making the cavity and the details of the interaction processes which provide thermal equilibrium.

In most every day situations (temperature measurement of heated specimens (pyrometry) or solar observations as mentioned above) the situation is not that of thermal equilibrium. Materials properties and surface geometry/topology then matter. Many a time, though, this is sufficiently accounted for by introducing the concept of a grey radiator as mentioned in a previous contribution. The materials properties then "hide" in the proportionality factor between 'grey' and 'black'.

My teacher told us blackbody is an ideal "cavity" holding equilibrium photon gas--Bosons! That is! So I did not take a chance to think coupling or interaction!

I think it is due to the thermal equilibrium rather the materials properties and surface topology is more important . The space time geometry plays very important role in understanding the nature of interaction between matter wave and electromagnetic wave as the energy (perturbation energy) associated with such interaction is quasi-continuum in nature.

A black-body is a closed system in which radiation is in thermal equilibrium with matter.

Hi Magnus:

This is a very good question because one might really wonder how the black-body picture is entwined with the real world. Well, I have to agree with others who answered just this part of your inquiry: the black-body is an idealization of a situation that can be analytically described as Planck has shown earlier. The ingredients of this idealized situation, in particular thermal equilibrium, are virtually never encountered in real life.

Your questions basically refer to situations that are not described by a black body. The specifiic definition of the black-body were chosen exactly as to avoid the details of interactions between photons and matter. All these details are made obsolete by the assumption of thermal equilibrium.

The details of the interaction will only matter in a non-equilibirum situation as is virtually always the case in real life. This situation has been called by others in this thread the "grey-body." The sun was refered to as a notable example in this category.

In a situation, however, where the matter object is in thermal equilibrium with its surroundings, i.e., the surroundering photon gas for the black-body case, the particulars of the interaction are irrelevant. However, interactions have to be present.

Take for example the photon gas itself. The photons are to excellent approximation non-interacting and will therefore never equilibrate by themselves. But contact, i.e., interaction, with matter at a particular temperature does affect equilibration, and thus thermal equlibration within the photon gas itself.

Now you might give these arguments a twist because very close to a solid surface, for example, the photon field might not be well described by a photon gas, and Planck's famous distribution function might not apply (it probably won't apply for the photon field inside the solid either). However, this might be considered a sophistication of minor importance in most cases of interest.

To make the answer very short: all the interaction mechanisms that you mention contribute to achieve thermal equilibrium, but once thermal equilibrium has been achieved their details are unimportant.

The question concerning the nature of interaction between the photon gas and the (wall of the) black body is highly interesting from a theoretical and conceptual point of view - even now, 100 years after Einstein's derivation of the Planck formula.

To my knowledge nowadays almost all theoretical statistical physicists seem to be happy to define a "black body" just as an object which is coupled to a heat bath with constant temperature T and satisfying Kirchhoff's law "absorption = emission", not worrying about the rest of it, i.e. what kind of interactions take place. Certainly, a mathematician would first of all ask if there exists such an "object" in the vast collection of physical toy models (as a theoretician one would not care about "real life" at this stage!) - otherwise all statements about it would concern the empty set.

To be constructive and rigorous we would nowadays possibly proceed as follows:

1. First model the black body for simplicity as an harmonic crystal coupled to a heat bath (which causes phonons, i.e. lattice vibrations). Maybe one also needs to think about a special geometry, such as the usual cavity. For the harmonic crystal there won't be phonon-phonon interaction. The phonons are lattice vibrations causing the electric charges to move in the crystal and thus to generate electro-magnetic fields by the Maxwell equations. Hence, there is a phonon-electron interaction to be considered. QED-methods would allow us to handle this phonon-electron interaction (in fact, the velocities involved won't be relativistic so pedestrian approach might do).

2. Secondly, also use QED-methods to handle the photon-phonon interaction. Note, that in metals this interaction is mainly indirect via photon-electron and electron-phonon interaction but in photonic crystals it is direct. Thank heavens there is no photon-photon interaction. One may also introduce an external photon-gas source at this stage (grand canonical ensemble!)

3. The harmonic crystal needs to be specified in such a way, that photons from the external photon-gas source are neither reflected nor transmitted. This means that all incoming photons are absorbed. (To be honest, I would not know how to enforce this condition within the given model or just make it a preliminary of the harmonic crystal in consideration).

4. When all your Feynman diagrams are done you would hopefully find a steady-state situation and re-discover Kirchhoff's law as a consequence that the energy flows are balanced. And you would hopefully find also Planck's spectral distribution for a "large" system (thermodynamic limit?).

Thus we would have demonstrated that certain harmonic lattices (satisfying point 3.) are candidates for Planck's black body and in addition described the desired interactions.

I don't know whether or not this project has been carried out in full rigor and beauty. However, Albert Einstein has done it in 1916 in a less formal and more pedestrian way, using only Bohr's atom model and his own theory of photons. Later, in the 1980s-90s Robert F. O'Connell from Louisiana State University started some analysis on how photons react with a heat bath (modelled as vibrating harmonic lattice).

Einstein, A. (1916). "Zur Quantentheorie der Strahlung". Mitteilungen der Physikalischen Gesellschaft Zürich 18: 47–62.

Markus

I think your examples are not quite realistic. A black body can rather be seen as a harmonic crystal with all frequencies present with the same probability. Kirchhoff's law is an older idea and is not of interest one you use the exchange of photons and the idealisation that the matter is not grey but black meaning all incoming radiation is fully absorbed at all frequencies. Einstein's emission law which is derived from the absorption then suffices to know the emission (rate)

A system emits black body spectrum every time that some conditions are met: i) the optical depth of the region is higher than one, ie the radiation must not be free to escape the region, or the free mean path of photons must be much smallet than the volume containing the gas, ii) a thermodynamical equilibrium (at least locally) does exist. The resulting spectrum is indipendent by the emission and absorption processes and depends only on the temperature of the emitting region (ie observing a black body tells nothing about the processes of emission but only on the T of the source). The BB emission is stable and self regulated: it too many photons are in, absorption tends to reduce them, if too few, stimulated emissions supply the missing photons. As a result, a system in thermal equilibrium can not emit more radiation than corresponding BB at the same T, if the two conditions are met. This is the key point

Emanuele

Correction

The optical depth must be much higher than one. At one (and assuming log-base 10)

it is a grey body with absoptivity of 90%. Also temperature is not in the definition of BB. BB means complete absorptivity at all frequencies and all frequencies

No real material nor a model of a material can be a black body. If a material absorbs, then by the Kramers Kronig relation there is a refractive index differing from 1 and, on top of that, dispersion. Therefore, reflectivity is nonzero and frequency dependent.

The probably best approximation to the BB is tha cavity radiator with a small hole. What goes in there has a very low probability of coming out again, hence an absorbtion probability of nearly 1 (whatever the details are of what happens inside). Whatever comes out there is a very small fraction of the radiation energy inside, hence the existence of almost equilibrium.

We then fall back on what Markus stated, that the processes do not matter for a black body. They do, of course, determine the behavior of any real material -- which never is and never will be an ideal black body.

Dispersion does not per se mean reflection as in a gaseous meium, but for a realstic absorber/emitter the gas mus be confined by a material that would reflect

In a gas you have a pretty small density (compared to solids). Then of course the polarizability (\chi(\omega)) is very small and the dielectric function will be close to one - and so will be the refractive index etc. But then also - at some point - we have left the definition range of a homogeneous medium at the length scale of radiation wavelength. (See the corresponding discussion on classical fields and local fields in J.J. Jackson, "Electrodynamics").

I guess that in a gas, all length scales scale accordingly, and so do time scales (e.g. of reaching equilibrium)?

Kai

You are right that the scale is not that of the shortest wavelngth, say hundreds of nm corresponding to a BB with a T of that of the sun; at the longer ones it will do

The early Universe is actually a great play ground for this question. Photons and electrons+baryons interact through various processes. To redistribute photons over frequency Compton scattering is crucial. This usually means that a large number of scatterings is needed to achieve kinetic equilibrium. To restore full equilibrium also the photon number has to be adjusted. In the early Universe this is mainly done by double Compton scattering (the number of photons is 1.6x10^9 time larger than the number of baryons) and only at later times by Bremsstrahlung.

It is really interesting to think about the CMB (Cosmic Microwave Background) photon field. It could in principle be produced from a hot electron-proton plasma with no photons to begin with if you start the evolution early enough!

It is interesting that the materials with the highest imaginary parts of the dielectric constants (absorption) are often bent chain hydrocarbons.  These seem to provide a natural coupling between photons and the phonons (where the vast majority of thermal energy is stored).