[Eternal Student (OP)]

Hi.  I hope everyone is well.

Blackbody radiation is usually considered (theoretically modelled and suitable equations derived) by looking at the radiation that can exist inside a cavity or an "oven".    The interior of the oven can support various modes of radiation within it,  each mode has a particular frequency,   those frequencies ultimately assumed to be caused (generated by) charged particles oscillating at that frequency in the walls of the oven.

For a cuboidal oven you can have modes of radiation within the oven with this general form:

ψ =  A.  Sin (K_x x) . Sin (K_y y) .Sin (K_z z).   

[Equation 1]

Where A = amplitude (a constant)  and  K_x,  K_y and K_z  are also constants  but not quite as arbitrary.    One of the boundary conditions that must be satisfied is that ψ = 0 at the walls of the oven   (since these waves do not continue through the walls and propagate away from the oven).  We really are looking for stationary wave solutions that fit inside the oven.   Hence  only K_x =  p.π / L_x  ,    K_y =  q π/L_y   and  K_z = r π / L_z   for some integers p,q,r   will be allowed  (L_x  being the length of the oven in the x direction etc.)

   That's a typical way in which the blackbody radiation is modelled and you can find more details in most textbooks or online resources.  I won't take up any more of your time here.

    So, taking everything at face value,  it would seem that.... if you have a big oven then you can support e-m radiation at most wavenumbers k or wavelengths λ = 2π/k.   This is just because  π/L_x ,  π/ L_y  etc. will be small numbers and so the permitted wave vectors are separated by only a small amount. 
   For a small oven, however,  the permitted wavenumbers are more widely separated.   You'd still get the overall shape given by the usual blackbody spectrum curve but you may begin to notice that you only have small discrete chunks of that spectrum present.
    I don't know... I'm asking not telling.   I've just presented some of the background so that you can hopefully see it isn't a silly question to ask.

    Taking it to a more extreme situation....  what if you don't even have a cuboidal shaped oven?   Imagine an arbitrary blob with protruding lumps shaped oven.   I haven't spent much time trying the mathematics,  it's just apparent that easy and separable solutions like [equation 1] are out of the question - that's never going to satisfy ψ=0 at the walls.  It would seem that a blob shaped oven would support considerably less modes of radiation within it.

   So,  does the spectrum of the interior of an oven actually show any changes when you alter the shape and size of that interior space?   For example, does the spectrum seem to become more discrete and less continuous with very small interior spaces?   Does a cube shaped oven have a better and cleaner match to a blackbody radiation curve than the interior of a kidney bean shaped oven?

   None of this is urgent,  I'm of the opinion that the usual way blackbody radiation is presented in the textbooks (very roughly as the above) is just far too simplified.   I am pretty sure that the shape of an oven does not affect the radiation spectrum, I'm just not sure I could explain why.

Best Wishes.

[evan_au]

One of the boundary conditions that must be satisfied is that ψ = 0 at the walls of the oven  ...since these waves do not continue through the walls

I understand that historically, black body radiation was calibrated with a block of graphite (already pretty black, visually), with a sphere carved out in the center, and a hole bored through from the outside, so you could observe the radiation inside the cavity.
- You heat up the graphite block to a certain temperature (presumably with no oxygen present), and measure the spectrum of the radiation visible at the small hole.

Because graphite is a fairly effective black body, it actually absorbs any radiation impinging on the surface of the sphere
- getting hotter in the process
- and generating black-body radiation back into the cavity
- You measure the spectrum after this has settled down to an equilibrium

Because the graphite is absorptive, you don't need "ψ = 0 at the walls", since the radiation acts as if it is propagating off to infinity
- When in fact it gets absorbed within a few mm of the wall
- So there are no standing waves in the cavity
- This is quite unlike a metal-walled microwave oven, where the E field is zero at the conductive walls, and standing waves are definitely present (causing uneven heating of the food)

So I suggest that a reflective metal oven is a much worse representation of black body radiation than a cavity carved in graphite.

See the diagram here:
https://www.physics.utoronto.ca/~phy293lab/experiments/blackbody.pdf

[Zer0]

Someone else is trying real hard to bake their cake on in here...

https://www.thenakedscientists.com/forum/index.php?topic=66414.0

(thou it's a very long cake, supposedly with no cherries atop)

[Eternal Student (OP)]

Thanks @evan_au,   I've spent a very long time thinking about what you've said and the article you linked to (teaching resource from University of Toronto) and a collection of other things.

The article you linked to includes this statement:

Classical physics suggested that all modes had an equal chance of being produced, and that the
number of modes went up proportional to the square of the frequency.            (From page 2, just above fig.1)

   I would have to ask what were they considering as a "mode"?    If it was a particular frequency of radiation that could exist within the oven,  then EVERY frequency is possible and no special conditions or restrictions apply at any frequency  (assuming as they state, you don't need standing waves, or ψ = 0 at the walls).   The second half of the sentence then seems to have no explanation - why would the number of modes available increase with the square of the frequency?

Compare with these diagrams (and brief explanations):

http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html#c2      (LATE EDITING: I think that link goes exactly where needed):
   That idea shown in the diagram is NOT the only way people think about fitting different waves in a box so that ψ = 0 at the walls   -  but it's an online resource that I know people can access so that's why I've given it.

and then this shows how the number of modes is determined:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html#c2
   Notice that the number of modes varies ~  1/λ³.   Invert to obtain relationship to frequency ~f³ and then instead of considering all modes from f=0 to f=our value,   we usually consider a density  (the number of modes per unit frequency at f=our value)   so that is ~ f ²  as mentioned in the article you linked to.   (They were a bit sloppy not mentioning that it was a density).   Anyway, it seems more common to consider the density of modes in terms of wavelength so it is a density per unit wavelength at a given wavelength which has the form   ~ 1/λ⁴ .

In all the derivations of blackbody radiation that I have seen (classical based on Rayleigh-Jeans  OR   quantum based on Planck etc.)  the density of modes permitted is ALWAYS of that form   ~ 1/λ⁴  (it seems more common to use wavelength instead of frequency).     It is only the allocation of Energy in each mode that changes.   

   So, there just really ought to be some reason why the density of modes  varies with wavelength.   In most texts (I can find multiple references if required) that is explained purely on the basis of requiring standing waves inside the oven.   

    None-the-less, I agree with the general idea presented in the article you sighted.   I've been thinking for quite a while that it isn't the  electromagnetic waves inside the oven that are the explanation for the density of modes.   It does seem reasonable to consider a situation much as you have outlined with the graphite oven -  waves of any frequency can be generated, they won't be stationary, they will travel but they will be dealt with at the walls regardless.  To say this another way, why on earth should we require standing waves? 

   Under that assumption, the explanation for the density of modes must be something else......   I'll keep the speculation to a minimum but the most obvious candidate is the particles in the walls of the oven, they are the thing that would generate the radiation of each frequency.   They could have more modes (of oscillation) when their oscillation frequency is higher - but I've tried and still can't see a good reason to explain why.

   Overall, I'm seriously wondering if we should just abandon the notion of there being "modes" (of radiation in the oven) entirely.   If you don't require standing waves then there doesn't seem to be any sensible way of defining what a "mode" should be.   I'm going to have another good look through various bits of literature about Blackbody radiation and see  what's what.

   For the moment, if anyone has any information and especially any relevant practical investigations showing that the spectrum of radiation inside an oven is affected by the shape of the oven, please let me know.   Otherwise I'm going to assume that everyone is of the opinion that the shape of the oven makes no difference at all.

So I suggest that a reflective metal oven is a much worse representation of black body radiation than a cavity carved in graphite.

   Yes, that seems absolutely certain.   Obviously the walls were not blackbodies, they reflected something but by definition a blackbody must absorb all radiation that hits it.   So it is clearly just the cavity and not the walls that we are talking about here.   Such a metal oven really should have a spectrum that is very different to a blackbody spectrum -  there was nothing in the walls (an atom) that absorbed the radiation you are describing as being reflected  and so there were no particles in the walls that could possibly emit such radiation into the cavity.  In short there should be a clear deficit in radiation of the frequency (frequencies) that the walls reflect.   The only way radiation of those frequencies could have got into the cavity would be when the oven door was opened.   
     If we remove the part about the walls being reflective,  so we just have an oven with (dull) conductive metal walls then your comment is more interesting.   That is precisely the sort of thing I would like to know about,  especially if anyone has done a suitable experiment and analysed the spectrum inside a metal oven compared to a graphite oven.
    Microwave ovens are complicated and seem only partially related to this discussion - it's not the walls of the oven that are being raised to some temperature to heat that oven  - but yes, I see some of your point about the standing waves.     

Best Wishes.

[Eternal Student (OP)]

...and thanks @Zer0 ,  I must have been still writing when you posted.

[evan_au]

The only way radiation of those frequencies could have got into the cavity would be when the oven door was opened

If you are looking at the spectrum excited by the cavity magnetron, it spans a range of about 10MHz (it jumps around between different modes inside the magnetron).
- So there are a number of modes that might be excited by the magnetron
- To minimise uneven heating due to standing waves, microwave ovens often have a rotating platform for the food, or else a mode stirrer (like a metal fan, but usually out of sight)

https://www.researchgate.net/figure/Magnetron-spectrum-without-PLL-control-and-with-full-heater-power-68-W_fig4_3074637

[evan_au]

Lasers have a 1-dimensional cavity, for which there are several modes.
- Cheap semiconductor laser pointers continually jump between these modes
- More expensive lasers for telecommunications often have a second optical cavity coupled to the first; only one mode is able to resonate in both cavities, reducing interference due to the different speed of light of different modes when traversing an optical fiber.

This video shows mode changes every few seconds in a green laser.

This page shows a typical laser spectrum, averaged over time - but in practice, not all of these resonances are there at the same time.
https://www.researchgate.net/figure/The-laser-spectrum-at-t-6-nsec-Axial-cavity-modes-are-visible-in-the-laser-spectrum_fig4_321257044

[Bored chemist]

I'm not sure it's possible to tell.
I think we are looking at the idea that wavelengths comparable with the length of the side of the box will be perturbed by the size and shape of the box.


If I get a metal box (like a microwave oven) and put a tungsten light bulb in it. the light doesn't get out.
If I make a small (say 1mm) hole in the wall, the light gets out and I can measure its spectrum. The box hardly makes a difference to that.
I can, also measure the infra red from the lamp.

In principle, the bulb also emits microwave and radio wave radiation.
But I can't see them- not because the box changes the spectrum of the lamp, but because the long wavelengths will not pass through the small hole.
The longer than about 1mm wavelengths are filtered out

So, I can only observe the spectrum of what's happening in the box through a filter which I know removes anything with a wavelength longer than 1mm.
If I make the hole bigger (comparable with the size of the box) in order to look at wavelengths comparable with the size of the box then...
it isn't a box any more.

[alancalverd]

A waveguide is effectively a long, thin oven designed to support one particular mode and frequency. It can obviously support harmonics but these are usually filtered out or not generated in the first instance.

Much the same applies to wind instruments. A flute extracts a fairly pure sine wave from the exciting edge tone, whereas a conical-bore instrument like a saxophone carries a lot more overtones.

[Bored chemist]

To a good approximation*, an electron in the wall of the oven which drops to a slightly lower energy level (and emits a long wave photon as a consequence) does not know what shape and size the oven is.

Waveguides are often silver plated or even lined with superconductors to improve the sharpness of their frequency response.
A black body is  (pretty much definitively) not.

The analogy isn't  getting a resonance from a flute; it is trying to play a tune on an exhaust silencer. (I might pay good money to watch someone try to do that.)

If the wall opposite absorbs all the radiation which hits it, there's no way for the radiation to "know" how far away it is.

It's not a good approximation if one of the walls is a good mirror.
I did some pointless research on this as a student. You can change the half life of fluorescence by putting the emitter near a mirror.
But you don't make black bodies out of mirrors so it's beside the point.

[Eternal Student (OP)]

Hi.

Thank you to everyone who has spent some time here.   I am busy reading and thinking about what has been said.

For the moment, if anyone has any information and especially any relevant practical investigations showing that the spectrum of radiation inside an oven is affected by the shape of the oven, please let me know.

   Overall,  especially with the comments from @evan_au about lasers,  it seems that the shape of an oven / cavity almost certainly can influence the spectrum of radiation within it.

(sampling the spectrum is a problem....) because the long wavelengths will not pass through the small hole.

   Yes.   I was going to say they would need to put a spectrum analyser inside the oven  -  but I'm fairly sure a "spectrum analyser" was actually the name for a piece of hi-fi audio kit that we all wanted back in the 1980's.   Anyway, it's that sort of equipment that would be wanted regardless of the name it may have.   I'm aware that analysing the spectrum of all e-m radiation is actually extremely difficult.   Radio waves aren't a problem, an aerial and a simple electrical circuit can catch those but shorter wavelengths are a problem and electrical equipment is not fast enough to rectify a ray of light shone onto an aerial.   Spectrum analysers in the lab are often based on refracting the e-m radiation (say through a prism) and looking at the rainbow of rays that emerge.   Such pieces of equipment can be too big to get in the oven and very likely to melt  etc.     Obviously the equipment must be such that it doesn't significantly interfere with the properties of the oven (e.g. it had better be a blackbody itself, very small and avoid removing too much of the radiation during the sampling process).   You may be unable to control the spectrum anlayser remotely or get data from it, radio won't get through a typical oven wall and you must not flood the cavity with e-m radiation of that range because that will obviously interfere with the results, so the device may need to operate automatically and record the data for later retrieval.  Overall quite a demanding set of criteria for the spectrum analyser.   I'm not much of a practical scientist but I'll guess that such equipment isn't available for standard issue from the storeroom?   (question mark because you've got to be hopeful).

Best Wishes.

LATE EDITING:  Someone added the tag "microwave oven" to this thread.  I have reduced that to "oven".   Microwave ovens are not a good example of blackbody radiation existing inside the cavity and are only relevant when properties like standing waves were being discussed.

[alancalverd]

Part of your practical problem would be the amplitude range as well as the frequency range of your black body radiation. Evan's lasers at least restrict the frequency range to near-negligible so it's not inconceivable that you could look for missing lines in their uniformly high intensity spectrum but if we look inside an ideal carbon spherical shell* at red heat, the tiniest surface imperfection will alter the Planck resonance spectrum by adding or subtracting wavelengths equal to arbitrary multiples of the height of the imperfection, and all against the background of every other surface element doing its bit!


*mathematicians and theoretical physicists may like cubes, but engineers ask awkward questions about what happens in the corners, so physicists prefer to use spherical shells.

[evan_au]

Multiple modes can be put to good use by making the second cavity tunable.
- This allows selection of one mode amongst many possibilities
- By selecting different modes in different devices, you can have many different devices transmitting on the same optical fiber, increasing the fiber capacity by a factor of 10 to 100: "Wavelength Division Multiplexing"
- If the independent wavelengths are closer together, then better control over wavelength accuracy and stability is needed (ie more cost)
https://en.wikipedia.org/wiki/Wavelength-division_multiplexing

Today, Erbium-Doped Fiber Amplifiers (EDFA) are used to amplify a wide range of wavelengths (other exotic elements like Thulium, Praseodymium and Ytterbium are also usable).
- This is effectively a laser without a cavity, and it is able to amplify all wavelengths in a Wavelength Division Multiplexing system.
- See EDFA at https://en.wikipedia.org/wiki/Optical_amplifier

[Bored chemist]

if we look inside an ideal carbon spherical shell* at red heat, the tiniest surface imperfection will alter the Planck resonance spectrum by adding or subtracting wavelengths equal to arbitrary multiples of the height of the imperfection,

No, it won't.
That's why (or because, depending on how you look at it) they use graphite; it's black.

You can not get cavity resonances if the walls are black.

[alancalverd]

So Planck was wrong after all.

[Zer0]

Best Wishes.

LATE EDITING:  Someone added the tag "microwave oven" to this thread.  I have reduced that to "oven".   Microwave ovens are not a good example of blackbody radiation existing inside the cavity and are only relevant when properties like standing waves were being discussed.

Thanks for Correcting it & providing an Explanation for the same.

[Bored chemist]

So Planck was wrong after all.

No, he was right, which is why he didn't say what shape or size the cavity was; he knew it didn't matter as long as it was black.

[hamdani yusuf]

Blackbody radiation is usually considered (theoretically modelled and suitable equations derived) by looking at the radiation that can exist inside a cavity or an "oven".    The interior of the oven can support various modes of radiation within it,  each mode has a particular frequency,   those frequencies ultimately assumed to be caused (generated by) charged particles oscillating at that frequency in the walls of the oven.

The effect of standing wave to radiation spectrum is only significant when the oven walls are highly reflective. Otherwise, the effect of standing wave, and shape of the oven, would be miniscule.
Candle soot is highly absorbtive/emissive, even when it's only applied to a flat surface. It absorbs light and reemit electromagnetic radiation close to black body spectrum.

[Eternal Student (OP)]

Hi.    Thank you to everyone who has spent some time here.

@evan_au,  your point about lasers and fibre optic cables is understood and appreciated.   I have glanced at the various links you provided but not spent hours on it - it seems slightly tangent.  Overall, I can appreciate that they (laser and fibre experts) make a lot of use of resonance effects from cavities and chambers.

@alancalverd and @Bored chemist ,    the effect of imperfections on the surface of a chamber is not all that clear to me.   For a graphite wall it may not be important but I don't know,  that's half the point of this thread and the original question.

   

So Planck was wrong after all.

   Yes, it looks like it,  although possibly not directly about the imperfections in the graphite wall but over some things, yes.
    Based on the information I have found recently Planck made a number of assumptions which were later revised,  some by Planck himself,  others by people who came much later.     The modern presentation of the derivation of blackbody radiation that appears in textbooks is,  to my best knowledge and belief,  often an awkward mesh of the original ideas and some new ones to produce something based on more reasonable and often much more simply explained assumptions.

    Examples:
1.   Planck assumed radiation of (angular) frequency ω would only be generated by harmonic oscillators in the walls that oscillate at the same same frequency.     
2.   Most sources of information state:  He also assumed the radiation was generated by transitions between energy levels of those oscillators.
3.   One of the key assumptions, which every source of information agrees on, is that the energy levels (of the oscillators) were quantised and came in packets of size ħω.

    Those assumptions get re-worked and re-phrased in many modern textbooks.    With modern science it is possible to understand that classical electromagnetism is not always the best way to go.   While you would have wanted an oscillator to be a charged particle oscillating at frequency ω to generate e-m radiation of (angular) frequency ω,  if you understand that the things generating the radiation were atoms and not just arbitrary charged particle oscillators, then assumption 2 can point you in a different direction.     Radiation of a given frequency ω could be generated by a transition of energy levels in many different ways:  It could have been an electron jumping just one electron orbital,  or a different atom having an electron jumping two orbitals all in one go,  or a different atom jumping 3,  etc.  etc.    The requirement for there to be a thing in the walls that is an oscillator with frequency ω starts to look a bit unnecessary or abstract.   You can do reasonably well replacing the density of modes (of radiation supported in the oven) with the notion of there being more ways that radiation of frequency ω can be generated when ω increases.
    Another common variation on the assumptions made is that you just assume the radiation (in the oven) must have quantised packets of energy,  ħf.   (Just to be clear, this is different from assuming the oscillators in the walls were quantised).  Then the Boltzmann distribution can be applied directly to each mode of radiation instead of applying it to the oscillators in the walls.   This seems to be the way many of the modern texts have decided to go and it has some advantages,  in particular you don't need many (if any) assumptions about oscillators existing in the walls.    However, to apply the Boltzmann distribution you do really want a "mode" of radiation to be a thing that exists.   You will want to treat a "mode" of radiation as being a small system all of its own that is in thermal equilibrium with a larger system at temperature T   (the larger system being the rest of the oven including all other modes of radiation EXCEPT the one mode you are working with at the time).

The effect of standing wave to radiation spectrum is only significant when the oven walls are highly reflective. Otherwise, the effect of standing wave, and shape of the oven, would be miniscule.

     Possibly correct and almost exactly what I had though originally.   However, the following two things must be noted:
1.    There is some need to define what a "mode" of radiation should be, especially with the assumptions made in most modern textbook derivations.   A "mode" is usually explained as being a frequency of radiation that can be sustained within the oven,  i.e.  a standing wave.
2.      Comments made by  @evan_au  et.al.   there do seem to be examples where standing waves become very important and do effectively eliminate other frequencies of radiation  (lasers, fibre optics etc.)

Best Wishes.

[Bored chemist]

As (almost) always, the answer is "it depends".
The emission line from a laser can be subject to "mode hopping" because, as its temperature changes, the wavelengths reinforced by resonance also chance. The Fabry Perot resonator depends critically on the size and shape of the cavity.

With a black body cavity, the light can't know how far way the other side is, and the spectrum depends on the temperature.

If I get a box with mirror walls and put a police car in it with the lights flashing, that light is still blue.