Radiation from solid black bodies

[Philip Koeck]

TL;DR

How can one show that Planck's distribution is also valid for solid black bodies?

I understand that Planck's law is derived for a cavity with a hole in it.
I haven't found a clear argument that the same result and all results that follow from it also apply to solid surfaces that are black.
Can anybody point me to a text that shows this?

 

[Spinnor]

Most real bodies are not black bodies. See this thread,

Getting the emissivity from scratch ,​

https://www.physicsforums.com/threa.../#:~:text=2.,the emissivity can be determined.

Material	Emissivity
Human Skin	0.98
Water	0.95
Aluminum (Polished)	0.1
Aluminum (Anodized)	0.65
Plastic	0.93
Ceramic	0.94
Glass	0.87
Rubber	0.9
Cloth	0.95

Material	Emissivity
Aluminium foil	0.03
Aluminium, anodized	0.9[19]
Aluminium, smooth, polished	0.04
Aluminium, rough, oxidized	0.2
Asphalt	0.88
Brick	0.90
Concrete, rough	0.91
Copper, polished	0.04
Copper, oxidized	0.87
Glass, smooth uncoated	0.95
Ice	0.97-0.99
Iron, polished	0.06
Limestone	0.92
Marble, polished	0.89–0.92
Nitrogen or Oxygen gas layer, pure	~0[20][21]
Paint, including white	0.9
Paper, roofing or white	0.88–0.86
Plaster, rough	0.89
Silver, polished	0.02
Silver, oxidized	0.04
Skin, human	0.97–0.999
Snow	0.8–0.9
Polytetrafluoroethylene (Teflon)	0.85
Transition metal disilicides (e.g. MoSi2 or WSi2)	0.86–0.93[8]
Vegetation	0.92-0.96
Water, pure	0.96

From the Wiki on emissivity, note the white and black colored sides of a cube look the same in a thermal image.

 

[PeterDonis]

I understand that Planck's law is derived for a cavity with a hole in it.

Planck's original derivation made use of this as a heuristic for deriving the density of states for EM radiation. However, I'm not sure that modern derivations of the Planck distribution have this limitation. Certainly it is not required to obtain the Bose-Einstein formula, which is the key new element of the Planck distribution as compared to the Boltzmann distribution.

Note also that the composition of the emitting or absorbing body does not appear anywhere in the derivation; the only density of states that is used is that of the EM field itself. Basically the assumption is that the only constraint on whether radiation of a given frequency can be emitted or absorbed is the energy associated with that frequency. Ultimately such an assumption would have to be based on a model of the material of which the emitter or absorber is made. And, as @Spinnor has said, most materials are not perfect black bodies so we would not expect to be able to find a general material model that would predict black body radiation for any solid.

 

[Philip Koeck]

And, as @Spinnor has said, most materials are not perfect black bodies so we would not expect to be able to find a general material model that would predict black body radiation for any solid.

I was thinking of an ideal black body, not necessarily anything that could be realized.

My main conceptual problem is how to apply a statement about EM-waves in a cavity to a situation where EM-waves are emitted from the surface of a solid, no matter what the solid is made of.

Derivations that don't build on a "photon gas" the way Planck's does would clearly remove this problem.
Do you have some link to such a derivation or a textbook?

 

[PeterDonis]

a statement about EM-waves in a cavity

Again, this was a heuristic used in Planck's time to derive the density of states for the EM field (note that it was not limited to the derivation of Planck's formula, the same density of states appears in the previous derivations that led to the ultraviolet catastrophe). I believe modern derivations make use of coherent states of the quantum EM field (since those are the states that typical EM radiation from an emitting body will be in), which does not rely on any assumptions about "putting particles in a box" or other heuristics. I think we had a thread on this general topic a while back.

 

[Vanadium 50]

A black body is an idealization. If your question is "how do I show a real body exactly matches that for a perfect black body" the answer is "you don't".

If you question is "how do I derive the (most common) improvement?" you postulate emissivity, and then consider your object in thermal equilibrium with an ideal black body.

If you then ask "how close to reality is this new approximation" the answer is "experiment tells you".

 

[Charles Link]

The reason the small aperture is placed in the box with the cavity is that then we have very nearly unity emissivity by Kirchhoff's law, basically independent of the emissivity of the walls of the cavity. The derivation with the cavity thereby makes perfect sense. See also https://www.physicsforums.com/threa...-density-of-a-black-body.956343/#post-6063569

and

https://www.physicsforums.com/threa...erivation-of-plancks-law.958747/#post-6079472

 

[hutchphd]

I think this topic is covered in every introductory sophomore "Modern Physics" text (like Eisberg or Tipler). One problem is to provide interaction sufficient to temperature equilibrate the "photon gas": hence the "black body" cavity walls. The rest follows from energy equipartition and counting states in the quantum way as Planck did by Ansatz. The resuts are then independent of the details (other than a belief in thermodynamics).

 

[Philip Koeck]

A black body is an idealization. If your question is "how do I show a real body exactly matches that for a perfect black body" the answer is "you don't".

If you question is "how do I derive the (most common) improvement?" you postulate emissivity, and then consider your object in thermal equilibrium with an ideal black body.

If you then ask "how close to reality is this new approximation" the answer is "experiment tells you".

Actually I wasn't asking about real objects. My question was how we can know that a formula that was derived for the radiation emitted from a hole in a cavity is also valid for a hypothetical, solid black body, for example a lump of metal painted with ideal black paint.
My conceptual problem is that the cavity derivation discusses a "photon gas" inside the cavity, but I can't see how such a "gas" could exist inside a solid body.
Maybe the more recent derivations mentioned by PeterDonis are what I'm looking for.

 

[pines-demon]

Actually I wasn't asking about real objects. My question was how we can know that a formula that was derived for the radiation emitted from a hole in a cavity is also valid for a hypothetical, solid black body, for example a lump of metal painted with ideal black paint.
My conceptual problem is that the cavity derivation discusses a "photon gas" inside the cavity, but I can't see how such a "gas" could exist inside a solid body.
Maybe the more recent derivations mentioned by PeterDonis are what I'm looking for.

How comfortable are you with statistical field theory? The exact full quantum calculation can get as ugly as you can imagine... (However that is an overkill)

 

[pines-demon]

Maybe what you need is a good reference on how to derive heat transfer between two objects, then you will see that Planck's law gives you the maximum heat transfer between two bodies at different temperature separated by vacuum and large distances.

 

[Philip Koeck]

How comfortable are you with statistical field theory? The exact full quantum calculation can get as ugly as you can imagine... (However that is an overkill)

That wouldn't work for me, I'm afraid.

 

[hutchphd]

My conceptual problem is that the cavity derivation discusses a "photon gas" inside the cavity, but I can't see how such a "gas" could exist inside a solid body.

Thephoton that exists in your head and the photon that is the fundamental excitation of the quantized EM field are rather different creatures. Some of the processes seem isolated in space but the photon energy exchange is facilitated somehow by the presence of charge and this allows a steady state temperature to be associated with the EM field. I cannot describe the detailed interactions but so long as one can define a Temperature for the isolated system then the rest has to be true. Thats how I deal with my ignorance on this!

 

[pines-demon]

That wouldn't work for me, I'm afraid.

Ok, I do not know how useful is this but let me try to explain it.

The problem of an object emitting radiation without being in thermal equilibrium should be hard because it is a nonequilibrium problem (it means that somewhere far away there is another object with another temperature for example).

However, one can deal with it in linear response theory (given vacuum, easy geometries and large distances) by writing the electric and magnetic fields as products of the fluctuating thermal currents/dipoles in the material (it's is the inverse of Kubo formula). Then you can relate the radiation to the imaginary (dissipative) part of the dielectric function of the material using the quantum fluctuation–dissipation theorem. You can find a simpler treatment in Novotny&Hetch Principles of Nanoptics (in the chapter of fluctuation–dissipation intereaction). The whole idea is that for a metal this imaginary part depends on the frequency, while for a blackbody is independent of the frequency. By blackening the material (that is by adding layers of disordered nonhomogenous materials you remove the specific absorption at a given frequency) thus approaching a Planck's law.

This calculation can be done exactly rigorous by using quantum thermal field theory. Again even with all that effort, one finds that if we took Planck carelessly as working with the equilibrium formula (and some emissivity factor that you can measure) and apply it to non equilibrium situations, it works more than perfect. Deviations appear at nanoscopic distances but at that point the linear response theory above works fine.

 

[Lord Jestocost]

Actually I wasn't asking about real objects. My question was how we can know that a formula that was derived for the radiation emitted from a hole in a cavity is also valid for a hypothetical, solid black body, for example a lump of metal painted with ideal black paint.
My conceptual problem is that the cavity derivation discusses a "photon gas" inside the cavity, but I can't see how such a "gas" could exist inside a solid body.
Maybe the more recent derivations mentioned by PeterDonis are what I'm looking for.

The spectral radiant exitance at the aperture in a cavity (the Planck blackbody radiant exitance) can only approximately be “mimicked” by a solid surface covered, for example, with carbon black.