Why is the black body radiation independent of composition and incident radiation?

Question

There have been questions similar to this, but most of them do not explain the mechanism responsible for the phenomena but instead explain through contradiction of second law of thermodynamics, for example this answer https://physics.stackexchange.com/a/130901/324947 can anyone explain qualitatively the mechanism responsible so that on imposing that an object absorb all the incident radiation in thermal equilibrium must lead to emission of radiations irrespective of its composition and the nature of incident radiation?

Answer 1

explain qualitatively the mechanism responsible so that on imposing that an object absorb all the incident radiation in thermal equilibrium must lead to emission of radiations irrespective of its composition and the nature of incident radiation?

First, emission independence of the nature of incident radiation.

This nice simple property of the black body is part of the definition of the black body as used in physics considerations. It is assumed intentionally as part of the definition, because this is an hypothetical limit of the real behaviour that is never attained in reality but some bodies get close, and assuming it simplifies the model and mathematics in calculations.

If the emission characteristics of the body depended on incident radiation, we would not call that body a black body. We would call it a real body, or gray body. Because that behaviour is complicated and what real bodies manifest; and we often find some more details as reasons for this behaviour, e.g. we can often ascribe that kind of emission to reflection from the body surface, or to internal conversion of energy from one frequency to some other frequency (fluorescence). Black body is not capable of these complicated processes because that is the desideratum in the definition - to get a simple model.

Then, there is the independence of the composition of the body. This is also effectively part of the definition, because the idea was to have a concept of body whose emission does not depend on these details. Again, as a sort of idealized limiting behaviour of the real bodies, where while the emission does depend on the composition, often the dependence can be neglected and this simplifies things.

But here, we can find another reason: the compatibility with the concept of thermodynamic equilibrium. In short, if we had two bodies that absorb everything incident on them, of same temperature but with different composition, and assuming their emissions had different frequency distribution, this would prevent establishment of thermodynamic equilibrium between them in a reflective box. One of the bodies would radiate more and one of them less, and since both absorb everything coming their way, this would lead to transfer of energy from one body to the other, which would lead to diverging temperatures of the two bodies, a behaviour not compatible with the so-called "minus first" law of thermodynamics: bodies in closed system tend to thermodynamic equilibrium.

With real bodies, whose emission does depend on composition, this hypothetical incompatibility with thermodynamics does not arise, because different compositions causing different emissions are accompanied by different reflecting behaviour, and presence of these different reflecting behaviours breaks the above argument: even if more energy is radiated by one body than the other body at some frequency interval, this need not be completely absorbed by the other body, so there is no implication of impossibility of equilibrium. On the contrary, the equilibrium can happen because any body absorbs only as much as it also radiates, at every frequency, and this is only possible because the body reflects the surplus in the radiation coming its way, i.e. the body is not a black body.

Answer 2

That is must be radiating is clear from the fact that it is absorbing energy from the environment it is in thermal equilibrium with while not heating up.

So the question becomes why this is independent of material type and why it had that particular distribution.

It is important to remember that a black body is an idealation and is only an approximation of the real world. In this sense it is not actually materially independent. But it remains a good approximation. The key word here is black. To be a good approximation the material needs to be able to absorb any wavelength of light.

For a material to absorb light it needs to interact with light in some way. in normal matter this is caused by electrons in the material being able to absorb a wife range of wavelengths into one of many internal degrees of freedom. The electron is then excited and can step down to ground.

Thermodynamics is reversible so if an object can absorb any wavelength is must be possible for the electrons in that object to Emit that wavelength.

But why is it this distribution? This can be obtained from the equipartion theorem which states that given many possible degrees of freedom the energy is spent equally on each degree of freedom. There are only finitely many possible high energy wavelengths because the shortest possible is a plank length. The peak wavelength changes with temperature because energy depends on the square of the velocity of the electrons.

Answer 3

This may be a semantic point, but parsing your question: Blackbody radiation is not independent of the composition and incident radiation.

We use the term black body radiation both is a a more theoretical sense and a practical sense. In the more theoretical sense we consider a cavity with a small hole and look at the allowed modes. That can be done classically, or quantum mechanically, and the quantum mechanical way gives us what we observe in nature. Hyperphysics has a nice explanation.

If you buy a blackbody source it actually will look like a bar of something that can handle high temperatures like Silicon Carbide in an insulated box with a hole. If I want to calibrate an optical thermometer or pyrometer I would point it into the cavity and I would be more confident would be getting the right temperature.

One reason people bring up thermodynamics to explain blackbodies is that in a equilibrium situation we know know the temperatures are constant and there is a "principle of detailed balance" that lets use use Stefan Boltzmann's Law to find the temperature of the two objects if only radiation is being exchanged. For example we know the temperature of the sun from its black body radiation and we can see that the temperature of the earth is about right using the Stephan Boltzmann's equations. Even though the sun is emitting in the visible, and the earth is emitting in the far infrared the equations are in balance.

So how does the black body depend on the material or incident radiation? Well if it is not a cavity and just flat surfaces, then you may have a spectral dependence of the material. It might be reflective or absorptive over a wavelength range. Or the material might be polished smooth or be textured.

So for real materials as you change the composition you need to consider the emissivity a value of 0 is a perfect reflector and a value of 1 is a perfect absorber. But the emissivity is also a function of wavelength. Something that is absorbing in one part of the spectrum might be reflective in another part of the spectrum.

So if you take three different materials say copper aluminum and whatever, pure and with a nice polished surface and heat them to the same temperature and look at them with an optical thermometer you will get three different temperatures, because the black body spectrum is modified by the emissivity.

If I take three blocks of the same material but instead of polishing them all, and I polish one, scratch one up or have a rough surface and then maybe pattern the third block periodically on the micron size scale. With them all at the same temperature, and I point an optical thermometer at them I would get three different temperatures. This is becasue the emissivity is different. Similarly if one block is oxidized and one isn't I would measure two different temperatures.

Note: When I say optical thermometer, it could have a very expensive optical thermometer that would measure the spectrum of the radiation from the blocks as a function of wavelength. The I might fit that spectrum to a black body curve by using the equation varying the temperature. The Temperature that fits the best is what I would call the temperature.

So the composition actually matters a lot. Sometimes you will also see terms like albedo or absorptivity when talking about blackbody radiation.

Similarly, in detail we say the sun is has a blackbody temperature around 5800 degrees. That is true, but if you look at the spectrum more closely there are Fraunhofer lines where some of the spectrum is absorbed, or on the earth the atmosphere might be absorbing portions of the spectrum. However in many cases it is very useful to ignore the additional physics of the Fraunhofer lines or atmosphere absorption when doing back of the envelop calculations.

If you have a materials like a gas of solid, you might shine light on it an excite the electronic transitions, those electronic transitions between discrete energy levels would not be part of the blackbody spectrum. However, if you looked at the spectrum corresponding to the temperature material you would find that in addition to the electronic transition a broader blackbody spectrum. It might be that the sides of the box or cavity is very far away, but in equilibrium the allowed electromagnetic modes of the box would be correspond to the blackbody spectrum. Again assuming you are in equilibrium.