Talk:Planck's law/Archive 2

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Planck Law linked to the Fine Structure Constant

The fine structure constant α = e² /ħc ≈ 1/137.036 and the blackbody radiation constant α_R = e² (a/k⁴ )^1/3 ≈ 1/157.555 are two dimensionless constants, derived respectively from a discrete atomic spectra and a continuous radiation spectra and linked by an infinite prime product. The blackbody radiation constant governs large density matter where oscillating charges emit or absorb photons that obey the Bose-Einstein statistics. The new derivations of Planck’s law, the Stefan-Boltzmann law, and Wein’s displacement law are based on the fine structure constant and a simple 3D interface model. Planck’s original formula is a experimental fitting result. There are many derivations to explain the blackbody radiation law, including Planck in 1901, Einstein in 1917, Bose in 1924, Pauli in 1955. This is not reinventing the blackbody radiation law, but instead pointing out that the surface charge is Planck’s oscillator, and it is related to the fine structure constant. Planck’s radiation law is derived as the result of the 3D interface interaction between photons and charged particles. Using the 3D surface charge model and the energy quanta ε = hν, Planck’s law in terms of the spectral energy density in [erg · cm^-3  · sr^-1  · Hz^-1 ] can be rewritten as

$I(\nu ,\,T)={\frac {c}{4}}\cdot u(\nu ,\,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}=(hc)({\frac {1}{2\pi }})^{2}({\frac {\alpha h\nu }{e^{2}}})^{3}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$

where α = 1/137.036, and Stefan–Boltzmann law

$J=\sigma T^{4}={\frac {c}{4}}\cdot ({\frac {\alpha _{R}}{e^{2}}})^{3}(kT)^{4}$

where α_R =1/157.555

^ Xiao, K. "Dimensionless Constants and Blackbody Radiation Laws". Electronic Journal of Theoretical Physics EJTP 8, No. 25 (2011) 379–388. http://www.ejtp.com/articles/ejtpv8i25p379.pdf

— Preceding unsigned comment added by Happyboy2011 (talk • contribs) 03:30, 28 May 2011 (UTC)

The removed changes added information and dealt with the illogicality of the previous wording; as for the language, that is perhaps a matter of taste.

The change from "A black body is an ideal surface that absorbs completely, with no reflection or transmission" was intended to deal with the illogicality of its saying that a body (a three-dimensional object in space) is a surface (a two-dimensional imaginary mathematical object that cannot absorb or emit, as noted carefully by Planck); it was also intended to deal with the point carefully made by Planck, that it is necessary in defining a black body to require that it not scatter, which was ignored by the previous wording.

The change from "Black bodies are Lambertian objects" was intended to deal with the illogicality of trying to apply a law about surfaces (Lambert's cosine law) to three-dimensional objects (bodies). Moreover, Lambert has two laws named after him, the cosine law referred to here, which relates to interfaces, and the attenuation law with relates to non-black three-dimensional media. As noted by Kirchhoff and Planck, black media have infinite optical thickness even in infinitesimally thin layers; the notion of attenuation hardly applies.

Dear Q Science, Perhaps you will very kindly be willing to deal with these problems in ways that are in accord with Planck's treatment of the matter and that meet your own standards of non-confusing language. For me, it is confusing to be asked to identify a two-dimensional mathematical object with a three-dimensional physical object, and to ignore the possibility of scattering that Planck found significant.Chjoaygame (talk) 03:21, 20 January 2011 (UTC)

I have no problem with adding the word "scatter" though I think that "absorbs completely" covers it. My problem is in changing

A black body is an ideal surface that absorbs completely ...

to

Ideally, a black body absorbs completely ...

which has a very different meaning. In addition

Black bodies are Lambertian objects ...

is pretty straight forward, but I have no idea what

Black bodies have Lambertian interfaces with their contiguous media ...

is supposed to mean. (What is "contiguous media"?)

As far as 2- vs 3-dimensional treatment, I don't see a difference when dealing with the surface of a 3-dimensional object. The math should be identical. The problems are related to optically thin gases which, by definition, are not blackbodies simply because they ARE optically thin.

I am a bit curious what you think "infinite optical thickness" means. From your text, it appears that we may have different understandings of that phrase. At any rate, I agree that the current wording should be improved. Q Science (talk) 07:09, 20 January 2011 (UTC)

Dear Q Science, Thank you for your thoughtful reply. I accept that my wording could have been better.

"I have no problem with adding the word "scatter" though I think that "absorbs completely" covers it."

Yes, "absorbs completely" nearly covers it; but thinking so would perhaps lead to deletion of ", with no reflection or transmission"; according to Kirchhoff, a perfectly black body doesn't just absorb completely; it does so within a infinitesimal layer of the surface of the body. Planck mostly seems to consider bodies that are not perfectly black; that is the reason he requires that a (non-perfect) black body is not permitted to scatter.

We agree on this. Q Science (talk) 20:41, 20 January 2011 (UTC)

"what Black bodies have Lambertian interfaces with their contiguous media ... is supposed to mean. (What is "contiguous media"?) "

It is not possible to make direct measurements of radiation within an ideal or perfect black body. One can only measure its radiation by making it interface with another medium, not black; a vacuum would be ideal for the purpose if it did not destroy the black body. 'Contiguous' means 'touching or interfacing immediately'. One wants that other medium to be contiguous with the black body. According to Planck (1914, section 10), "First the body must have a black surface in order to allow the incident rays to enter without reflection. Since, in general, the properties of a surface depend on both of the bodies which are in contact, this condition shows that the property of blackness as applied to a body depends not only on the nature of the body, but also on that of the contiguous medium. A body which is black relatively to air need not be so relative to glass, and vice versa. Second, the black body must have a certain minimum thickness depending on its absorbing power, in order to insure that the rays after passing into the body shall not be able to leave it again at a different point of the surface. The more absorbing a body is, the smaller the value of this minimum thickness, while in the case of bodies with vanishingly small absorbing power only a layer of infinite thickness may be regarded as black. Third, the black body must have a vanishingly small coefficient of scattering (Sec.8). Otherwise the rays received by it would be partly scattered in the interior and might leave it again through the surface."

Well, that makes sense, but your text was not clear enough to get that meaning. At any rate, this is way too much detail for the intro for this article. Q Science (talk) 20:41, 20 January 2011 (UTC)

I think this means that Planck distinguishes ideally or perfectly black bodies from effectively black bodies. Kirchhoff 1860 makes the "supposition that bodies can be imagined which, for infinitely small thicknesses, completely absorb all incident rays, and neither reflect nor transmit any. I shall call such bodies perfectly black, or more briefly, black bodies."

"I am a bit curious what you think "infinite optical thickness" means."

For me, a body has infinite optical thickness just when a beam that enters it is entirely absorbed. The 'optical thickness' is the negative of the logarithm of the attenuation ratio, the ratio of transmitted to entering power. For example the ocean is practically opaque to infrared radiation because it is very deep, and practically all the radiation from the atmosphere that enters the ocean is absorbed before it reaches the bottom. In effect, therefore, the ocean has practically infinite optical depth. This means that the body of the ocean (as distinct from its interface with the atmosphere) is effectively (as per Planck) black to infrared radiation that enters it, though not perfectly black as defined by Kirchhoff. But its surface is not perfectly black, because some physically significant amount of infrared radiation is reflected at the ocean/atmosphere interface. Thus the ocean, as observed looking downwards within itself, is a (non-perfect as per Planck) black body, because it does not immediately extinguish entering radiation, which can penetrate a finite distance into it; a finite skin depth if you like. But as observed from the atmosphere, the ocean does not have a black (as per Planck) interface with the atmosphere, and thus does not appear to be black.

We agree on the definition. However, it appears that the ocean absorbs longwave IR radiation in only a millimeter or two. Q Science (talk) 20:41, 20 January 2011 (UTC)

"As far as 2- vs 3-dimensional treatment, I don't see a difference when dealing with the surface of a 3-dimensional object. The math should be identical."

Personally I see a logical difference between having the math identical and having the physics identical; I am reading that by "the math" you mean the formulas that put the physics in numerical terms.

Perhaps some thought may lead to a better wording for the article.Chjoaygame (talk) 11:07, 20 January 2011 (UTC)

To be more concrete, perhaps we could say that a "blackbody is an ideal object that absorbs...". Actually, the absorption is always carried out by the three-dimensional body (whether it is a black object or a cavity), but we are interested in what enters and exits its geomertical surface. In this sense, "scattering" is superfluous, as it is already included in "reflection", which can be specular or diffuse (i.e. the result of multiple scattering events), and here indicates all the light the surface gives back into the source hemispace. Also, with the aim of simplifying, I think that the subsequent phrase on the "Fresnel-Stokes-Helmholtz-Stewart reversion-reciprocity principle", which has a name more complicated than the principle itself, should be reworded or omitted. It may be interesting for non-black bodies and for BRDFs, but its usefulness here is not clear. --GianniG46 (talk) 14:12, 20 January 2011 (UTC)

The purpose of this article is to present the equations that describe blackbody emissions, not to define a blackbody. The details on what a blackbody is, and is not, should be in the blackbody article, and not in the introduction to this article. That said, please keep either the word "ideal" or perhaps "theoretical" in the definition since these both imply that blackbodies don't actually exist. Q Science (talk) 20:41, 20 January 2011 (UTC)

Dear Q Science, Thank you for your advice. It is not normally considered logical to include in the definition of something a statement about its existence. In normal logic, the existence question is tackled after the definition has been made. How can one investigate the existence of something if its existence has already been specified in its definition? The present wording of this article explicitly notes the non-existence of perfectly black materials. Planck clearly thinks of effectively black bodies that really exist but are not perfect.

This is a special case where that distinction is part of the definition. Q Science (talk) 06:31, 21 January 2011 (UTC)

The article on the Black body omits most of Planck's careful and important distinction between the surface and the bulk of the body, and does not note that Kirchhoff's law needs something such as a local thermodynamic equilibrium condition to make it work, and does not recognise Planck's concern that the cavity must be in a relatively non-reflective opaque material, focusing instead on the rigidity of the wall.Chjoaygame (talk) 03:36, 21 January 2011 (UTC)

Then add that information there, not here. Plank's equation is just one of the properties of a blackbody. If a piece of information is not needed to understand the equation, then it does not belong here. Q Science (talk) 06:31, 21 January 2011 (UTC)

Dear Q Science, I am not intending to add that information anywhere. I just thought you might like to know about it.Chjoaygame (talk) 17:30, 21 January 2011 (UTC)

The factor 2 in the Planck Law

It is true that the factor 2 does not occur in equation (274) nor in equation (276) on page 168 of Planck 1914. The reason for the apparent discrepancy is that Planck states his equations for the intensity of polarized rays of thermal radiation. The equation in the Wiki article is for unpolarized rays, that have both polarizations and thus twice the intensity, which calls for the factor 2 in the equation. Planck notes on page 6 that the thermodynamic equilibrium cavity radiation is unpolarized. Planck includes the factor 2 in his statements of energy density, for example in equation (275) on page 168 and equation (301) on page 176..Chjoaygame (talk) 17:58, 21 January 2011 (UTC)

Lost electrons of the black body

Planck's law of black body radiation describes distribution of energy in spectrum of the black body radiation. This law is a special case of Bose-Einstein statistics. According to present representations radiation is the effect of the electrons transition between energy levels. Hence the ideal black body radiation must be described also by the Fermi-Dirac statistics.

${\frac {h\nu }{e^{h\nu /kT}-1}}={\frac {2kT}{e^{(h\nu -\mu )/kT}+1}}$

$\mu$ - chemical potential. If $h\nu >>kT$

$\mu =kT\ln \left({\frac {h\nu }{2kT}}\right)$

It is interesting that during the existence of Fermi-Dirac statistics has not been attempted to apply these statistics to calculate the energy of the black body. Leonid 2 (talk) 07:18, 27 April 2011 (UTC)

Dear Leonid, this is indeed an interesting question. I think that the physics of Planck's law is that it is determined by just the temperature, regardless of the particular material that sustains it. The temperature is a character of thermodynamic equilibrium of material (ponderable matter), or of local thermodynamic equilibrium (and perhaps of some other states) of material. In a cavity with perfectly reflecting walls and no material content, any state of the electromagnetic field will persist unchanged, as noted by Planck 1914 and by Landsberg 1978. It is not the electromagnetism that sets the spectrum, it is the material (Planck talks of a "small particle of carbon"), in this case in thermodynamic equilibrium or local thermodynamic equilibrium. The material must be a poor reflector, so that the radiation in the cavity can thoroughly interact with the interior of the material, not just be reflected from its surface. It is the thorough interaction with the equilibrium material that sets the spectrum. The material must have a special balance of excited states to do the job. Planck's oscillators are not enduring physical objects. They are very briefly transient evanescent processes within microscopic atoms, molecules or crystals or whatever (let us call them 'molecules'). These molecules must be exchanging energy according to the equipartition law of equilibrium. Within each molecule, there are two kinds of process: a stationary process in which no radiative change is occurring; and an evanescent oscillatory process in which radiative change is occurring. This is the meaning of the quantum theory. At reasonable temperatures, most of the time is spent in the stationary state, and occasionally an evanescent process occurs. There are three kinds of evanescent process. One in which an excited stationary state spontaneously decays with the emission of a quantum of radiation. This is what thermodynamics recognizes as emission. One in which an excited stationary state is triggered by resonance with the radiation field to decay with the emission of a quantum of radiation that is consequently identical in all respects and coherent with the triggering radiation. This is what thermodynamics recognizes as negative absorption, though it is sometimes given the misleading, for thermodynamics, name of stimulated "emission". One in which a stationary state is triggered by the electromagnetic field to resonate and thus to lead to an excited state of the molecule. This is what thermodynamics recognizes as positive absorption. Thermodynamic absorption is the algebraic sum of negative and positive absorption. At thermodynamic equilibrium, the rate of emission is equal to the rate of absorption, for every kind of radiation. The temperature of the matter determines the balance of processes which determine the numbers of molecules in excited states ready to participate in the many various evanescent processes. In a gas, this balance of processes and excited states is governed by the Maxwell-Boltzmann distribution, combined with the ability of intermolecular collisions to affect the distribution of stationary excited states of the molecules. For the emitted and (negatively and positively) absorbed quanta of radiation, the frequencies of the radiation are governed by the dynamically possible resonances that link the relevant pairs of stationary states of molecules, according to the Bohr rule ΔE = hν, where ΔE denotes the energy difference between the relevant pairs of stationary states; at present, no one has the slightest idea why or how these resonances work; it is part of the doctrine of the quantum theory that no one will ever find out how or why. The derivation in the article as it stands makes no mention of these dynamical physical factors: it just pulls the mathematical partition function out of a hat, talking about "particles obeying Bose-Einstein statistics". In a nutshell, the Planck distribution is determined by the quantitative degrees of availability of material molecules in various stationary excited states ready to provide the transient creation-annihilations of just the right numbers of the relevant transient evanescent Planck oscillators. As Planck notes: no material (carbon particle) able to set the temperature by reaching thermodynamic equilibrium in the cavity (or, nowadays, material stably driven away from thermodynamic equilibrium as in a laser) means no tendency to form a Planck distribution. Something along these lines may be useful in the article. Chjoaygame (talk) 14:39, 27 April 2011 (UTC)

Planck's views on this subject are worthy of respect

Planck's views on this subject are worthy of respect. GianniG46, I suggest you read Planck before making more edits. It is clear that you have not done so already, or that if you have read him, you have not taken aboard what he wrote.Chjoaygame (talk) 07:03, 10 December 2010 (UTC)

The statement that perfect black bodies do not exist is not a Planck's "view", but is a trivial thing if you take the word "perfect" literally, or is an observation of the experimental physicists before Planck, if you take it as meaning "enough good to give reliable measurements". Anyway, to show you my respect for Planck, I have not only restored the citation, but have added a link to a web page containing the complete book The Theory of Heat Radiation, so that interested users can read it. Concerning whether I have read or not Planck, maybe you have not read the "Philosophiæ Naturalis Principia Mathematica", but this does not imply that you ignore Newton's laws. --GianniG46 (talk) 22:06, 10 December 2010 (UTC)

It is great that you give the link to the webpage for Planck's book. Good work.

Planck is a good enough authority for me about this. The non-existence of ideal black bodies is not too trivial for Planck to mention and give a reference for. Planck on page 42 notes that the cavity walls are not permitted to be totally reflecting, but for every frequency must absorb at least slightly somewhere; the slightness of absorption influences the time required to establish stable stationary radiation. Planck adds that it is essential that the walls of the cavity transmit no radiation to the exterior. You prefer to think primarily in terms of an approximation, instead of deriving the approximation from the ideal case. You don't like even to use Planck's word 'ideal', you seem to want to improve on it by writing of 'perfect' black bodies. You seem to want to avoid citing Planck in favour of a better authority in a more recent book. I don't see why Planck's statements about black bodies and cavity radiation are not good enough for you. You seem to be implying that you haven't read Planck. (I have read only parts of Philosophiae Naturalis Principia Mathematica, and that only in translation, and I am not wikiing about Newton's laws.)

I think isotropic radiation can be purely right circularly polarized in every ray? Or not? Thermal radiation is unpolarized. Chjoaygame (talk) 17:38, 11 December 2010 (UTC)

No answer to my question? I think it is necessary to state that thermal radiation is polarized unpolarized(Chjoaygame (talk) 10:44, 12 October 2011 (UTC)), as this is not necessarily implied by its being isotropic. At least Planck goes to the trouble of making the unpolarized character explicit.Chjoaygame (talk) 10:39, 18 December 2010 (UTC)

Yes, strictly speaking you are right: circularly polarized radiation may be considered isotropic, at least if one has equal probabiities of having right- and left-polarized wawes. But usually when one says "polarized", the average reader thinks to linear polarization. Including me and our friend Max, who at page 6 of his "The Theory of Heat Radiation" says "Since the medium was assumed to be isotropic the emitted rays are unpolarized".

Joking aside, the main reason I removed the expression "unpolarized" is simplicity and ease of reading (you know, I love simplicity). The reader would stop on this word and think "Why 'unpolarized'? Who would ever expect radiation to be polarized if there is no particular reason for this? When one says "radiation" wihout attributes, one means unpolarized radiation, unless there are particular conditions. Which are these particular conditions?". This would uselessly make a bit more difficult to follow the text. To make an example, the phrase "With a well tuned cello you can play Bach's Suites" is correct, I hope you will never play them with an untuned instrument. But in reading this phrase people think there is a particular reason to add "well tuned", and may erroneously imagine there is a reason similar to the one for which one says "Well-tempered clavier". Instead, "isotropic and homogeneous" are the necessary and qualifying attributes of the radiation in the cavity (not of BB emission, as it was written before your edit)--GianniG46 (talk) 14:55, 18 December 2010 (UTC)

It seems that you are a mind-reader, able to read the reader's mind in some considerable detail. We are talking about cavity radiation that is emitted by the walls of the cavity, not by the medium inside the cavity. Surely if one has equal probabilities of left- and right-polarized rays one has unpolarized radiation? I am asking is it possible to have purely-right polarized isotropic homogeneous radiation, that is to say every ray is right circularly polarized? I think it is topologically impossible to have isotropic linearly polarized radiation?Chjoaygame (talk) 19:02, 18 December 2010 (UTC)

Please, take apart the absolutely irrelevant phrase on left and right, which comes from an extemporary thought of mine on symmetries. I have deleted it, and, if you want so, I can delete all the first part of my reply (I wrote it was a joke), so we can discuss on the second part. Concerning mind reading, of course I am reading mine. --GianniG46 (talk) 08:37, 19 December 2010 (UTC)

Being a rather ignorant character, when first reading about this, I found I needed to be reminded about the requirement for two orthogonal polarizations to be specified, to get a factor of 2 in the result, which was not obvious to me.Chjoaygame (talk) 09:36, 19 December 2010 (UTC)

Can't understand statement

"This function represents the emitted power per unit area of emitting surface in the normal direction, per unit solid angle, per unit frequency."

Since it appears to make no sense to measure surface area in any particular direction, I assume this means "(emitted power per unit area of emitting surface) in the normal direction..." and not "emitted power per (unit area of emitting surface in the normal direction)..."? But how can it be "per unit solid angle" if it only measures power emitted in one direction, and, in any case, wouldn't the emission in exactly one direction be zero? I don't get it. Is it maybe talking about emission over a conical steradian whose axis is normal to the surface, or something like that? 86.179.2.89 (talk) 03:14, 14 February 2011 (UTC)

Oh, sorry, hold on ... is it talking about a cone pointing in the normal direction, and then taking the limit of emission/(cone size in steradians) as the cone gets (in principle) infinitesimally small? 86.179.2.89 (talk) 03:29, 14 February 2011 (UTC)

Yes, that's it, and I agree, the wording could be better. Any suggestions? PAR (talk) 20:47, 14 February 2011 (UTC)

How about This function represents the emitted power per unit area of emitting surface, per unit solid angle around the normal to the surface, per unit frequency.?Chjoaygame (talk) 01:06, 15 February 2011 (UTC)

Hmm. I'm not sure this is any clearer. In fact, I think it's marginally worse since it sounds even more like the formula gives the total energy over a conical steradian centred on the normal, which is incorrect. Instead, the formula gives the energy that would be emitted over a steradian if the emission was at the same rate as along the normal (as I understand it). 86.181.174.29 (talk) 12:48, 15 February 2011 (UTC)

On that fragmentary reading, the total energy over a conical steradian centred on the normal is zero; a point emits zero energy; the energy must be emitted from a non-zero area. The energy is emitted from infinitely many infinitesimal areas each emitting infinitesimal conical solid angles around the normals to their respective areas; the statement has to be read as a whole; it should not be read fragmentarily, just picking out one phrase as in that fragmentary reading. The specific radiative intensity is a limit with respect jointly both to area and to solid angle; its dimensions are power per unit area per unit solid angle per unit frequency; its meaning is not intuitively obvious at first glance; at first encounter with this quantity, one has to think carefully to grasp it.

Mihalas and Mihalas 1984 on page 311 express it like this:

"The radiation field is, in general, a function of position and time, and at any given position has a distribution in both angle and frequency. We define the specific intensity I (x, t ; n, ν) of radiation at position x and time t, traveling in direction n with frequency ν, to be such that the amount of energy transported by radiation of frequencies (ν, ν + dν) across a surface element dS, in a time dt, into a solid angle dω around n is

dE = I (x, t ; n, ν) dS cos α dω dν dt ,

where α is the angle between n and the normal to dS."

If one felt like it, one could complain that this fails to tell the reader that the point x lies in the surface element dS.

Pehaps it would be better to delete the attempt to say in this article what a specific radiative intensity is, and just write "This quantity is a specific radiative intensity" and let the reader work it out for himself by looking at the article on specific radiative intensity, which has a diagram that helps partly, and I emphasize partly, to explain it.

How about a suggestion from you?Chjoaygame (talk) 01:56, 16 February 2011 (UTC)

I see no way to properly explain this without devoting quite a few words to it. If the explanation takes up too much space in the lead section, perhaps it could be relegated to a footnote? My attempt follows. I think "per unit solid angle" and "per unit frequency" are potentially equally difficult, so it's worth covering both at the same time. "per unit area of emitting surface" seems much easier to grasp.

"Since frequency and direction of emission are continuous quantities, the power emitted in exactly the normal direction or at exactly a given frequency ($\nu$) is theoretically zero. Instead, the formula gives the limit of emitted power divided by solid angle over a narrow cone centred on the normal, as that cone becomes indefinitely small. Similarly, it gives the limit of emitted power divided by bandwidth (frequency spread) over a small range of frequencies centred on v, as this range becomes indefinitely small. This can be visualised as the power that would be emitted if the intensity found in a very small region around the normal were continued over one steradian, and the intensity found in a very small region around $\nu$ were continued over over one hertz of bandwidth."

81.159.109.69 (talk) 14:03, 16 February 2011 (UTC)

This article is about Planck's law. A few words about specific radiative intensity aka spectral radiance are appropriate, but not a thorough explanation of the concept; instead of a footnote, which is not appropriate, the reader is already directed by a link to the Wikipedia article on it. It is necessary to write that the law as stated by Planck referred to the specific intensity evaluated at the normal to the emitting black surface (Planck 1914, pages 168-169). I think the proper course is either to delete the attempt to say in the article what specific radiative intensity is and to indicate only that it is to be evaluated for the normal to the emitting black surface, and so to write just This quantity is a specific radiative intensity evaluated at the normal to the emitting black surface , or else to leave it at a few words, as for example in This function represents the emitted power per unit area of emitting surface, per unit solid angle around the normal to the surface, per unit frequency.Chjoaygame (talk) 00:04, 17 February 2011 (UTC)

I don't really agree with you. Specific radiative intensity is daunting and seems like more information than is necessary if all one wants to do is understand what the units of this formula mean. Also, I read the lead section several times without once imagining that clicking on that link would answer my question. 86.160.218.188 (talk) 12:28, 17 February 2011 (UTC) Sorry, I changed my mind. I think you're right that the place to explain this in full is Specific radiative intensity. If that article is daunting then it needs to be made more accessible there, not compensated for here. I think the units should be given in this article, though. Maybe there's some way to make it clearer that to understand the units you should click on the "Specific radiative intensity" link. Perhaps even just putting it first might help:

"This quantity is a specific radiative intensity: it represents the emitted power per unit area of emitting surface in the normal direction, per unit solid angle, per unit frequency."

86.160.218.188 (talk) 12:51, 17 February 2011 (UTC)

It is hard to see the difference between understanding what the units mean and understanding what specific radiative intensity means.

I am not happy with the wording "the emitted power per unit area of emitting surface in the normal direction". I think it is verging on being contradictory to the next phrase "per unit solid angle"; it keeps the very wording that you found confusing in the present text; the word "direction" is the problem. The wording This quantity is a specific radiative intensity evaluated at the normal to the emitting surface. It tells the emitted power per unit area of emitting surface, per unit solid angle around the normal to the surface, per unit frequency. is more conventional and safer, I think.Chjoaygame (talk) 14:44, 17 February 2011 (UTC)

If you already know that "specific radiative intensity" is the concept that you need to understand in order to understand the units, then of course there is no difference. However, coming to this article knowing nothing about the subject, and wanting a quick understanding of what the law is saying, one just sees the usual forest of blue links, some of which might be useful to read, some of which might not. As I have mentioned before, I think "per unit solid angle around the normal to the surface" is marginally worse than what we already have. It seems to be referring to the power emitted over an actual conical steradian centred on the normal. 86.160.218.188 (talk) 15:06, 17 February 2011 (UTC).

Ok. I accept that logically speaking, around the normal to the surface is redundant (it has already been stated that the specific radiative intensity is to be evaluated at the normal to the surface), and is intended only to provide possible benefits of redundancy of expression, which seem in this case to be negative, to judge from your personal reaction. So, taking this into account I propose This quantity is a specific radiative intensity evaluated at the normal to the emitting surface. It has the dimensions of emitted power per unit area of emitting surface, per unit solid angle, per unit frequency.Chjoaygame (talk) 07:05, 18 February 2011 (UTC)

I agree that some ways of wording the relevant geometrical (as distinct from spectral) concepts can be difficult to assimilate at first. To this end I've tried a different approach, based on how Rybicki and Lightman explain the concept on pages 7-8 of their book, which may or may not work for this article. Instead of talking about infinitesimal areas dA and solid angles dΩ, how about considering one square meter of emitting surface and one square meter of receiving surface, with the line between them normal to both surfaces, and with the distance r between them much much greater than one meter? Taking into account that the power through the receiving surface is attenuated by a factor of r², in the limit as r tends to infinity we get the same effect, numerically speaking, as with infinitesimal dA and dΩ. This approach simplifies things to the extent that it replaces two infinitesimal quantities dA and dΩ by one large quantity r. --Vaughan Pratt (talk) 07:45, 12 October 2011 (UTC)

The article on specific radiative intensity has a diagram not far from that of Rybicki and Lightman to which you refer. It seems odd that the lead on Planck's law should spend so much space on the definition of a term that is the subject of two other entire articles. If a detailed definition is needed in the Planck article, it might be better to put it in the body of the article somewhere.Chjoaygame (talk) 10:35, 12 October 2011 (UTC)

Fair enough. So how about we get the body to the point where we're all happy with it and then return to the question of the most appropriate way of satisfying WP:LEDE? --Vaughan Pratt (talk) 15:49, 12 October 2011 (UTC)

Suggest conforming notation to Rybicki and Lightman

Rybicki and Lightman write $I_{\nu }(T)$ rather than $I(\nu ,T)$. This has the advantage of allowing $I'(\lambda ,T)$ to be simplified to $I_{\lambda }(T)$. With this change the whole paragraph starting "This is perhaps more clearly stated" can be deleted, since the other points in that paragraph are made elsewhere in the article.

It should be pointed out that these two functions depend on T in the same way, and differ only in how they depend on wavelength/frequency, the difference being expressible in terms of the relation $\lambda I_{\lambda }=\nu I_{\nu }$. Put differently, whereas $I_{\nu }(T)$ is a density with respect to frequency, $I_{\lambda }(T)$ is a density with respect to wavelength. Once this relationship is clear, one can mix $\nu$ and λ in the same formula without confusion, for example as in the current lead which uses both $\nu$ and λ, bearing in mind that $\nu$ = c/λ and λ = c/$\nu$.

It may also be worth mentioning density with respect to log of frequency, needed for semilog plotting of multiple copies of Planck's law in the same graph over say 100 to 50,000 cm⁻¹, as with solar vs. terrestrial radiation. This requires its own function $I_{\log(\nu )}$ distinct from both these other two, definable via $I_{\log(\nu )}=\ln(2)\nu I_{\nu }$, with units of W/m²/sr/octave (without the ln(2) it's per 1.4427 octaves). $I_{\log(\nu )}$ peaks at the wavelength-frequency-neutral peak given by $h\nu /kT=\rho =3.920690394872886...$. --Vaughan Pratt (talk) 21:10, 9 October 2011 (UTC)

I agree. $I_{\nu }(T)$ is better notation than $I(\nu ,T)$, because the latter form could lead a reader to conclude that $I(x,T)$ is a single function, where x can be either wavelength or frequency.--Srleffler (talk) 04:42, 11 October 2011 (UTC)

I think the present notation of $I_{\nu }(\nu ,T)$ should be changed to $I_{\nu }(T)$ as per Rybicki and Lightman and as per the above suggestion. PAR (talk) 04:33, 12 October 2011 (UTC)

Dear PAR, the present notation has the advantage that it distinguishes the form of the function from the value of its first argument. It is hard or impractical or confusing to put the value of an argument as a subscript. The preachy section (that troubled Vaughan Pratt) that I removed made this point in a sort of way. The proposal that $I_{\lambda }(T)$ is simpler than $I'(\lambda ,T)$ seems a stretch. I do not think we have an obligation to conform to the notation of any particular textbook.Chjoaygame (talk) 05:14, 12 October 2011 (UTC)

edit by Headbomb

For Planck's law, my copy of Rybicki and Lightman (1979/2004, page 22 [1]) uses $B_{\nu }(T)$ and $B_{\lambda }(T)$. I put this into the article following the above suggestions. Besides Rybicki and Lightman 2004, other texts that use $B$ for the Planck quantities are Mihalas and Mihalas 1984, Chandrasekhar 1950, Goody and Yung 1989, Liou 2002. Although Bohren and Clothiaux 2006 use $P_{e}$, not $I$, they note that $B$ is sometimes used. I do not have by me a text that uses $I$. Planck uses K as a general symbol for specific intensity, and also for his own law. Perhaps Headbomb has a reason why $I$ is preferable, that he will very kindly tell us?Chjoaygame (talk) 23:34, 12 October 2011 (UTC)

I prefer $B_{\nu }(T)$ over $I_{\nu }(T)$ because very often a general radiation field $I_{\nu }(T)$ is being compared to the black body field $B_{\nu }(T)$, and then the notation gets ad hoc. $B_{\nu }(T)$ is a very special (actually, the most special) flux density and should have a special name. I agree, we need a good reason not to use B. PAR (talk) 00:17, 13 October 2011 (UTC)

Because I is the most-used version (pretty sure it's the one Eisberg and Resnick used, and pretty much all other modern-physics texts I've seen also use I instead of B), and it was the one in place for years. There's no reason to change it. Headbomb {talk / contribs / physics / books} 01:13, 13 October 2011 (UTC)

According to Chandra (Radiative Transfer - 1950, p 290),

$B(T)=\int _{0}^{\infty }B_{\nu }(T)d\nu ={\frac {\sigma }{\pi }}T^{4}$

The same text also uses the $I_{\nu }(T)$ notation, but not for blackbody radiation. Q Science (talk) 01:38, 13 October 2011 (UTC)

Chandrasekhar is the best reference for this subject that I know of, with Rybicki and Lightman high on the list. Also, the use of e.g. B_ν(T) rather than $B_{\nu }(T)$ is inferior. I will revert to the B notation. PAR (talk) 03:37, 13 October 2011 (UTC)

Dear Headbomb, you say that I is the most-used version and that you are pretty sure that it's the one Eisberg and Resnick (and pretty much all other modern-phyiscs texts you've seen) used, and that it was the one in place for years. My reading of Eisberg and Resnick is that they do not have a special symbol for Planck spectral quantities as distinct from general spectral quantities. For example, page 3 of Eisberg and Resnick reads: "The spectral distribution of blackbody radiation is specified by the quantity $R_{T}(\nu )$, called the spectral radiancy, which is defined so that $R_{T}(\nu )d\nu$ is equal to the energy emitted per unit time in radiation of frequency in the interval $\nu$ to $d\nu$ from a unit area of the surface at absolute temperature $T$."? Eisberg and Resnick also use $\rho _{T}(\nu )$ for energy density for various distributions. Perhaps you will very kindly give more detail of what you are pretty sure of?Chjoaygame (talk) 06:28, 13 October 2011 (UTC)

I agree with Headbomb that I is the symbol used most frequently in radiometry, which concerns the Intensity of radiation. However Planck's law is about radiation emitted specifically by a black body, for which many authors use the symbol B (albeit usually for Brightness rather than Blackness) when referring to radiation from something, as opposed simply to radiation casually encountered in space. Since the Planck's law article is about radiation from something it should follow the convention of authors who use B for radiation so generated. --Vaughan Pratt (talk) 07:48, 13 October 2011 (UTC)

Turned out I had a copy of Halliday and Resnick (Part II, 2nd edition) on my shelf. They use I for intensity as the square of the electric field E, and a cursive R (not quite ${\mathcal {R}}$ or $\Re$) for emitted radiation (what we're writing as B). Since they use B for the magnetic component orthogonal to the electric field E they presumably object to overloading it. --Vaughan Pratt (talk) 19:58, 13 October 2011 (UTC)

Vaughan Pratt's new edits

Vaughan Pratt is concerned to eradicate preachiness but it seems that his new lead uses notation that may be, or is verging on, own research with no citation of reliable source, without comment on the talk page. I have recently seen the Wikipedia thought-police overwrite a lead with a very short new lead with no citation of reliable source. Now Vaughan Pratt is offering his own rather technical and detailed viewpoint as to how to make a short summary less short. Personally I would prefer the previous lead.Chjoaygame (talk) 23:28, 9 October 2011 (UTC)

The problem with writing $I(\nu ,T)$ and $I(\lambda ,T)$ is that the "I" can be taken as specifying a single particular function $I(\cdot ,\cdot )$ which is not the case. If this notation is used, then every occurrence of $I(\nu ,T)$ should include the differential, e.g. $I(\nu ,T)d\nu$ and $I(\lambda ,T)d\lambda$ in order to avoid confusion. Writing $I_{\nu }(T)$ and $I_{\lambda }(T)$ or $I(\nu ,T)$ and $I'(\lambda ,T)$ avoids this problem, making it very clear that two different functions are involved. I prefer the $I_{\nu }(T)$ notation, although both ways are common in the literature. Whatever the choice, we should use consistent notation throughout, which is not presently the case. In the first paragraph, the equation given mixes wavelength and frequency, which although not wrong, is not helpful in an introduction. I have rewritten the first paragraph to give a clear definition of the meaning of the spectral radiance, leaving the exact mathematical statement to the first section. PAR (talk) 01:07, 10 October 2011 (UTC)

Thanks, PAR. I'd been considering writing dA, dΩ, and d$\nu$ myself but decided against mentioning infinitesimals in the lead. Glad to see someone else take that initiative so that I don't get blamed for it.  ;)

The varied notation in the literature makes it hard to know which to prefer. In the preceding section I picked Rybicki and Lightman's notation partly on the ground that theirs was the first non-Planck citation in the article, partly that they'd clearly thought it through carefully. However they use $I_{\nu }(T)$ for spectral radiance independently of Planck's law, and the article's actual citation is to p.22 of R&L where they write BB radiation as $B_{\nu }(T)$, drawing a sensible distinction between I for intensity broadly construed (what computer graphics people have in mind by radiance) and B specifically for black body radiation. This distinction is observed elsewhere in Wikipedia, e.g. Rayleigh-Jeans law though not Wien approximation. Any chance of making Wikipedia notationally more consistent here? --Vaughan Pratt (talk) 02:00, 10 October 2011 (UTC)

Incidentally, on the off-chance that some future editors may decide the formula isn't "problematic" after all, here for the record is what I wrote. "$I_{\nu }(T)={\frac {\rho }{e^{\rho }-1}}\cdot {\frac {2kT}{\lambda ^{2}}}$ where $\rho ={\frac {h\nu }{kT}}$ is a dimensionless ratio relating photon energy $h\nu$ at frequency $\nu$ to particle-level thermodynamic energy $kT$ at temperature T, and $\lambda ^{2}$ can be interpreted as the area a photon of wavelength $\lambda =c/\nu$ interacts with." While I'm flattered that anyone would think this was my "original research," it's not my idea at all but merely what "Just granpa" proposed a year ago (thanks gramps!), which is simply a reorganization of the terms of the formula that makes it much easier to conceptualize and work with. (But I'll gladly take credit for the "where $\rho ={\frac {h\nu }{kT}}$" innovation, which should prove my IQ is over 80.) --Vaughan Pratt (talk) 02:16, 10 October 2011 (UTC)

I did not realize the intent of that equation. At any rate, if it is included, it should probably go in further down, not in the introduction, with a clearer explanation and, strictly speaking, a source. PAR (talk) 03:39, 10 October 2011 (UTC)

Is Vaughan Pratt saying that the suggestion by "Just granpa" was not original research, or that "Just granpa" is a reliable source, or that Vaughan Pratt's "innovation" is not original research?

With care to define in some detail the conceptually difficult term spectral radiance in the lead, is there also need to warn the reader not to look up the Wikipedia article to clarify the term "power density", which in the lead here seems redundant anyway? In the Wikipedia article definition of power density, it is definitely a wrong term to use to express Planck's law. Is this an example of the need for consistent terminology? Would it be appropriate to mention that the intensities referred to are energy flux spectral densities with respect to several arguments, or that flux density is one of several quantities that are sometimes used to state Planck's law?

Vaughan Pratt's suggested way of rewriting Planck's law has some merit as a teaching aid or mnemonic, and perhaps deserves a place somewhere in the article if it can be reliably sourced, but is perhaps not ideal for the lead, for the reasons that PAR has pointed out.Chjoaygame (talk) 03:17, 10 October 2011 (UTC)

I agree, power density is not the right term. I always check out the Wikipedia articles on e.g. radiance if I want to define terms exactly, (or at least consistent with Wikipedia). An editor named User:Srleffler seems to be very knowledgeable in this area. PAR (talk) 03:39, 10 October 2011 (UTC)

There are two separate articles, one on volume power density (called "power density", confusingly I agree), one on surface power density. Since it's the latter that's the appropriate one here I linked to that. I also added "radiance" and "spectral radiance" as specific applications of surface power density to that article. Hopefully that's enough to avoid the problem. I'm thinking there ought to be a dab page for "power density" given the breadth of meanings of "density" arising in conjunction with "power". --Vaughan Pratt (talk) 04:05, 10 October 2011 (UTC)

or that Vaughan Pratt's "innovation" is not original research Well, I'm inclined to view x+y and y+x as the same function, but WP:OR only allows arithmetical manipulations of source material (provided other editors agree that the arithmetic is correct) and (to date) not algebraic manipulations. So I'm happy to abide by WP:OR and to consider changing x+y to y+x as original research. If WP:OR ever broadens arithmetic manipulations to include algebraic manipulations, e.g. on the ground that it's the function that matters and not how you arrange the terms of a representation of the function (which is my interpretation, applied to numbers instead of functions, of why they allow arithmetic manipulation), then we could revisit this. --Vaughan Pratt (talk) 04:15, 10 October 2011 (UTC)

If Vaughan Pratt is keen to keep the term power density, linking it to the article on surface power density, would he also consider directly and explicitly letting the reader of this present article know that he means surface power density? Only one extra word would save the reader following a link. I still think that the two words 'power density' in the first sentence are redundant clutter, especially as some statements of Planck's law are not in terms of power density of any kind. Is there something very evil about the usual term flux density?Chjoaygame (talk) 05:55, 10 October 2011 (UTC)

No problem. I changed it to "surface flux density," how does that look? It's easy to change to whatever seems best. The only reason I wrote "power density" was because I was thinking of it as the least specialized term that indicated the concept, catering for readers that might not be familiar with the jargon of radiation physics such as "spectral radiance." University boards of trustees are forever complaining about overly specialized jargon in tenure cases. I was figuring on everyone knowing what "power" and "density" meant individually, and that they'd naturally ask what was in the density denominator. --Vaughan Pratt (talk) 06:59, 10 October 2011 (UTC)

Come to think of it, I suppose "surface flux" is redundant, like "free gift." Is "flux density" best there? (Trying to keep everyone happy.) --Vaughan Pratt (talk) 07:02, 10 October 2011 (UTC)

The new lead spends four sentences explaining the formalism of one form of expression of Planck's law. Then it mentions other forms of expression. Then it mentions the asymptotic forms. Then it mentions Planck's progress. The most important things about the law are that it describes equilibrium radiation from a black body, and that it can be explained by the postulation of quantal emission and absorption(Chjoaygame (talk) 19:46, 10 October 2011 (UTC)) from countably many unspecified discrete resonators.Chjoaygame (talk) 12:10, 10 October 2011 (UTC)

Spoken like a fan of the journal Foundations of Physics. :) To philosophers of physics (the main audience for that journal) the properties you mention would be important. To historians of physics the fact that it resolved the glaring discrepancy between the Rayleigh-Jeans law and the Wien approximation by unifying them is important. To users of the law such as myself, the law itself is what's important, which is why I felt it deserved a place in the lead. Since there seems to be no support for that point of view it makes me wonder whether I'm the only editor of this article who depends on this law for their research (in my case atmospheric physics). It's extremely unusual for an article about a law not to state the law in the lead, in fact now that the law has been deleted from the lead (for the second time!) I suspect this must be almost the only such article. --Vaughan Pratt (talk) 16:46, 10 October 2011 (UTC)

Fair comment, Vaughan Pratt. The law is important from many points of view. As it happens I don't like the new part that says "In terms of photons, it describes the equilibrium energy distribution of photons" and I would like to see that deleted. My reason, some time ago, for taking the statement out of the lead was that it was too complicated to fit comfortably. I compensated for that by devoting the immediately following section just to statements of the law, and for that reason making the lead extremely short. I just felt that it was too hard to put more information in the lead with an appropriate level of detail. The new lead seems to be growing like Topsy. Perhaps a one-size-fits-all policy is not ideal.Chjoaygame (talk) 19:46, 10 October 2011 (UTC)

I agree. PAR, why did you insert that comparison with Maxwell-Boltzmann when there's nothing in the article about it? It's covered in Gas in a box and Bose gas, so a "see also" to those articles would be appropriate, but a lead sentence with nothing backing it up in this article seems wrong. And why did you delete the law when it's the central object of interest for this article? I'm not religious when it comes to choice of expression of the law, if you have a preferred one by all means use it instead. They can all easily be seen, at least by us editors if not every reader, to denote exactly the same function, making it purely a matter of taste which to pick. (Regarding User:Just granpa's elegant expression, rephrasing "where λ = c/ν" as "where λ abbreviates c/ν" might remove the confusion you raised; if not then 2kT/λ² can simply be spelled out as 2kT/(c/ν)².) --Vaughan Pratt (talk) 21:25, 10 October 2011 (UTC)

Well, make it unanimous then, the equation needs to be put back in. I favor the frequency statement. Regarding the Maxwell-Boltzmann statement, I think the point needs to be made that Planck's law is not an isolated concept. Viewed as a photon energy distribution, it is a member of a family of thermal equilibrium distributions, which include massive bosons and fermions, massless fermions, and "Maxwell-Boltzons" (I made that up) which is the distribution for massive bosons and fermions in the limit of low density. It is already alluded to in the statement "Max Planck ... showed that, expressed as an energy distribution, it is the unique stable distribution for radiation inthermodynamic equilibrium". I was trying to bring a little more content and clarification to that statement by bringing it into the family. PAR (talk) 03:57, 11 October 2011 (UTC)

I prefer the old 2 sentence introduction. The equations are better left in the body of the text where there is room to explain them. I am also fairly certain that the word "normal" (meaning "at right angles to a surface") is not part of the definition. (Actually, I would like to see a better explanation of how Lambert's cosine law works in this case.) At any rate, the current intro contains way too many details. Q Science (talk) 05:35, 11 October 2011 (UTC)

For an article with this many sections, two sentences is nowhere near in line with WP:LEDE.

Regarding "normal," you may be thinking of spectral irradiance which integrates spectral radiance over the whole 2π-steradian hemisphere, thereby introducing a factor of π (the 2 disappears on account of the cos(θ) factor). Spectral radiance is radiation in a specific direction. Planck's law can indeed be stated for spectral irradiance simply by including the factor of π, but as standardly defined (Rybicki and Lightman, Goody and Yung, etc.) it governs spectral radiance for radiation normal to the radiating surface, from which all other directions are easily obtained using Lambert's cosine law, namely a factor of cos(θ) where θ is the zenith angle, that is, the angle between the specified direction and the normal. --Vaughan Pratt (talk) 14:03, 11 October 2011 (UTC)

For reference, the article on quantum optics has the following lead: "Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter." This does not conform to a one-size-fits-all lead policy. Probably the quantum optics local editors have their reasons for this, and perhaps we are entitled to ours for this article.

It was a little disingenuous of PAR to write about making it "unanimous" to put the equation back in. I wrote above, at the start of this section: "Personally I would prefer the previous lead."

It is true that Planck's law is not an isolated concept, but is part of an account of many phenomena in terms of statistical mechanics. But this is not mentioned in the body of the article as it stands at present, and so it is hardly right to state it in what is intended to be a summary of the article.Chjoaygame (talk) 12:58, 12 October 2011 (UTC)

I wouldn't expect an article on a field of research in physics and an article on a law of that field to follow the same format in their respective leads. In particular I would expect the latter kind of lead to state the law, as it almost invariably does with this article being one of the rare exceptions. In the lead of an article about a field, what would be the counterpart of stating the law?

My point about the three functions of frequency/wavelength (in both senses of "/" in the case of the third function) and temperature is that (for any fixed value of T) there are precisely three such functions, as depicted in Goody and Yung's figure 2.3, the middle one of which they've drawn dotted, perhaps to suggest that there are two primary functions (which arguably is an accident of history but we can discuss that elsewhere). As far as "one size fits all," are you suggesting that different sources disagree on the number of functions (to within the question of whether to count the third one)? It seems to me that every source should be in agreement that there are two primary functions, whether or not they mention the third.

What distinguishes these functions is not the watts, or the geometry (whether in terms of area or solid angle), but the manner in which they divide up the spectrum, namely into equal sized blocks of either frequency, wavelength, or fractions of an octave (or decade, or neper, where the distinction between frequency and wavelength disappears).

There may well be thousands of different ways of packaging this information, but I'm not aware that there are any differences between these sources on the core facts about the law that I've just listed here. I'm of the opinion that the body of the article should present those facts clearly, and that the lead should summarize them as per WP:LEDE. I do not agree that the lead has to be limited to two sentences, which is very much at odds with WP:LEDE. --Vaughan Pratt (talk) 17:02, 12 October 2011 (UTC)

break

According to WP:LEAD, "... It should define the topic, establish context,...". I would put this under the "establish context" heading. PAR (talk) 16:14, 12 October 2011 (UTC)

Agreed. It should also "summarize the most important points". For an article on a law, I would consider the law itself an "important point." Other articles on laws seem to interpret "summarize the law" as "give the law." I'm not sure how else one would "summarize" a law, nor why one would omit it if it's easily given as a formula. --Vaughan Pratt (talk) 17:12, 12 October 2011 (UTC)

Looking over it, I can say that the new lead is quite good. Well done, chaps.Chjoaygame (talk) 20:13, 12 October 2011 (UTC)

I don't agree. There is no way this law is "easily given as a formula". For instance, the current formula is not complete (which is one reason there is a special section on various forms of the equation). I have written computer programs using this law and "easy" is not a word I would use to explain what I now know. Q Science (talk)

Lucky I wrote "quite good". I didn't write "ultimately perfect". That gives me leeway to ask for more from Q Science about his thoughts on this?Chjoaygame (talk) 06:33, 13 October 2011 (UTC)

For starters

$B(T)=\int _{0}^{\infty }B_{\nu }(T)d\nu =\int _{0}^{\infty }B_{\lambda }(T)d\lambda ={\frac {\sigma }{\pi }}T^{4}$

though I'm not sure about the π. Then there is the question of a surface or a volume. And of course, Lambert's cosine law. All of this is covered in the body of the article, as it should, but why should we put a confusing partial equation in the introduction? At any rate, one of the best references I know is http://www.spectralcalc.com/blackbody/blackbody.html. In spectroscopy (and my programs), the most useful version is based on Wavenumbers, but that version is missing from the wikipedia article. In summary, there is nothing simple about that equation or how it is used. Q Science (talk) 07:51, 13 October 2011 (UTC)

I agree with the π. --Vaughan Pratt (talk) 20:05, 13 October 2011 (UTC)

Is that a proposal to remove the formula from the lead?Chjoaygame (talk) 08:40, 13 October 2011 (UTC)

I have put in a spectroscopists' wavenumber formula.Chjoaygame (talk) 09:21, 13 October 2011 (UTC)

Sources

User talk:Chjoaygame's point about WP:OR is a good one, and it behooves us to look at what we have in the way of sources.

The article cites p.22 of Rybicki & Lightman, but that's after 20 pages of discussion of radiative flux, all of which is also available for citation. Another good source is Goody and Yung's book, pages 27-30 of which treats black body radiation leading up to Planck's law. Both use I for radiation in general and B for radiation emitted from matter (where Planck's law enters), making the two books reasonable authorities for the B convention in this article, which Wikipedia already follows in the Rayleigh-Jeans law article.

R & L also have a nice way of looking at solid angle in terms of two small surfaces at distance r obeying the inverse r² law. Since surfaces of unit area are small relative to r when the latter is much greater than unity, this suggests the following way of presenting the concepts. The use of $\nu B_{\nu }$ and $\lambda B_{\lambda }$ when a log-scaled spectrum is desired is from Goody and Yung, and various papers in the literature use that method, for example Bakan and Raschke in Der natürliche Treibhauseffekt. Figure 2.3 of Goody and Yung gives the respective shapes of the three functions referred to in the following.

Planck's law is either of two functions $B_{\nu }(T)$ and $B_{\lambda }(T)$ giving r² times the power in watts per unit spectrum, respectively Hz or meters, of radiation at frequency $\nu$ and wavelength $\lambda =c/\nu$ emitted from unit area of an ideal black body at absolute temperature T into unit area at distance r ≫ 1, where both areas are normal to the direction of radiation. The functions may be defined simultaneously via a third function $\nu B_{\nu }(T)=\lambda B_{\lambda }(T)=2h(\nu /\lambda )^{2}/(e^{h\nu /kT}-1)$, which itself constitutes a form of Planck's law giving the power per neper of spectrum (1.4427 octaves or 0.4343 decades) suitable for plotting Planck's law on a logarithmic scale. The shape of these three functions is that of the curve $x^{n}/(e^{x}-1)$ for n respectively 3, 5, and 4. --Vaughan Pratt (talk) 15:59, 11 October 2011‎ (UTC)

The foregoing part of this section seems to be from Vaughan Pratt?Chjoaygame (talk) 18:55, 11 October 2011 (UTC)

Oops, sorry about that. --Vaughan Pratt (talk) 19:19, 11 October 2011 (UTC)

comment on sources

Planck 1914 on page 168 states it first for specific intensity as a function of frequency, then for spectral energy density per unit volume as a function of frequency, then as specific intensity as a function of wavelength.

Atmospheric sciences. Paltridge and Platt 1976 state the law on page 43 first as spectral energy density per unit volume of black body radiation as a function of frequency, then spectral radiance as a function of frequency, then as a function of wavelength, then as a function of wavenumber. Goody and Yung 1989 state it as specific intensity as a function of frequency, and as a function of wavelength; Liou 2002 does the same.

Astrophysics and others. Chandrasekhar 1950 states it just for specific intensity as a function of frequency. Mihalas and Mihalas 1984 state it first for spectral energy density per unit volume as a function of frequency and then for specific intensity as a function of frequency and then as the spectral total flux per unit area from a black body as a function of frequency. Rybicki and Lightman 2004 state it first for spectral energy density per unit volume as a function of frequency, then for specific intensity as a function of frequency and then as a function of wavelength. Mandel and Wolf state it just as a spectral density as a function of frequency. Loudon 2000 states it just for spectral energy density per unit volume as a function of angular frequency; Gerry and Knight 2005 do the same. There are thousands of others.

Who will select from this the preferred one? That is a reason for following the suggestion of not putting the statement into the lead, but for keeping the lead short, and making the various statements in the first section. I do not think the one-size-fits-all lead policy is suitable here.Chjoaygame (talk) 20:17, 11 October 2011 (UTC)

Chjoaygame, could you please clarify the point you're making here? Are you saying that these authors use $B_{\nu }(T)$ and $B_{\lambda }(T)$ to denote different functions? If so then that's a very serious matter that is more relevant to the body of the article than to the lead, since the latter should be no more or less than a summary of the former. If not then I don't understand your concern. There are thousands of different presentations in the literature of the concept of hyperbola. By your logic the lead of that article should therefore say nothing about what a hyperbola is but just be two noncommittal sentences, because no one is allowed to select from among those thousands of presentations. --Vaughan Pratt (talk) 05:02, 12 October 2011 (UTC)

Just to clarify, I believe that R&L use, $B_{\nu }(T)$ to denote a function of $\nu$ and T, and they use $B_{\lambda }(T)$ to denote a different function of $\lambda$ and T. PAR (talk) 05:38, 12 October 2011 (UTC)

Quite right. Actually it occurs to me, given the way Planck's function is standardly drawn, that $\nu$ and T are the wrong way round. The typical graph fixes T and plots a function of $\nu$. This would suggest a family of functions $B_{T}(\nu )$ indexed by T, with the independent variable $\nu$ increasing linearly along the abscissa or x-axis.

But we have three such functions families. For the second we have $\lambda$ in place of $\nu$, while for the third we have the log of either one (since the only difference between the log of a quantity and its reciprocal is one of sign, which is a linear transformation that has no bearing on shape of functions). Each of these three functions could take either $\lambda$ or $\nu$ as input, or even both as long as $\lambda \nu =c$ was observed, without changing its meaning as long as the function was notified as to which one it was receiving.

This seems to me the heart of the matter. --Vaughan Pratt (talk) 06:51, 12 October 2011 (UTC)

Thank you for this comment, Vaughan Pratt. The point that I am making is that I do not think the one-size-fits-all lead policy is binding, and that I think it is unsuitable for the present article. I agree with Q Science's above remark "I prefer the old 2 sentence introduction." I don't see why the article on the hyperbola should set the pace for an article on Planck's law. Planck's law is in a sense more complicated than a hyperbola and can be treated differently. Yes I am saying that $B_{\nu }(T)$ and $B_{\lambda }(T)$ denote different functions; that is the reason for those textbook notations. Perhaps there is some confusion about the meaning of the term 'function'? As for the present new lead being a summary of the article - well!Chjoaygame (talk) 05:52, 12 October 2011 (UTC)

(Feel free to call me by my given name, Vaughan.) Ah, good, we agree on one thing at least: the two functions $B_{\nu }(T)$ and $B_{\lambda }(T)$ are different, as is the third function $\nu B_{\nu }(T)=\lambda B_{\lambda }(T)$.

As to the meaning of 'function,' the customary notion in mathematics is that two functions are equal if and only if they have the same graph. If you have a different meaning of "function" then this could be an interesting topic for discussion. If not, then by inspection of the three graphs in Figure 2.3 of Goody and Yung, which are spaced well apart, one would be inclined to say that these three functions are distinct.

If you still feel there is some confusion about the meaning of the term 'function' I'd be more than happy to discuss it with you. It's certainly a very interesting question, and you may well have some good ideas on how it bears on physics. --Vaughan Pratt (talk) 06:18, 12 October 2011 (UTC)

Again, just to clarify, $B_{\nu }(T)$ and $B_{\lambda }(T)$ are not different functions of T, they are different functions of the variable in the subscript. Sorry if this is already understood. PAR (talk) 06:23, 12 October 2011 (UTC)

Thanks for the clarification, PAR. Meanwhile I just noticed an ambiguity in my earlier question "Are you saying that these authors use $B_{\nu }(T)$ and $B_{\lambda }(T)$ to denote different functions?" What I meant there was "Are you saying that these authors use $B_{\nu }(T)$ to denote different functions?" and separately the same question for $B_{\lambda }(T)$. Sorry not to have spotted that ambiguity sooner, it's the sort of ambiguity that quickly leads to talking at cross purposes. --Vaughan Pratt (talk) 06:30, 12 October 2011 (UTC)

new section Introduction to Planck's law

This new section seems to me like an account of some of the mathematical properties of the Planck distribution functions.Chjoaygame (talk) 06:57, 13 October 2011 (UTC)

Or it could be viewed as an explanation of the ultimate purpose of Planck's law from the perspective of its users, which has been notably absent from the rest of the article. If you have a way of explaining that purpose that involves no mathematical properties of the law, you have my full attention. --Vaughan Pratt (talk) 07:55, 13 October 2011 (UTC)

The explanation does however appear to have confused Headbomb. I'd be very interested to know which parts he finds confusing. --Vaughan Pratt (talk) 08:44, 13 October 2011 (UTC)

Michael Price, would you kindly please explain (a) what you find confusing about the sentence "An ideal radiator or black body at temperature T degrees Kelvin radiates equal amounts of thermal radiation above and below frequency ν_T = 72.995T gigahertz" and (b) why your difficulty with that sentence constitutes grounds for deleting an entire section! Are you claiming the sentence is false, unclear, or something else? --Vaughan Pratt (talk) 09:32, 13 October 2011 (UTC)

Do you have an explicit source for your idiosyncratic presentation? I don't see one. Note the other sections are sourced. -- cheers, Michael C. Price ^talk 20:09, 13 October 2011 (UTC)

Well, this is unusual. A true statement on Wikipedia confuses a reader, nothing unusual about that, it must surely happen more than once a second. What's unusual is that instead of rephrasing the statement to fix whatever the confusion was, the reader deletes the entire section. No explanation beyond the bald statement that the reader is confused. I immediately ask what the problem is, no reply (that was eight hours ago).

And not just once but with two readers, both apparently knowledgeable about the subject, ruling out vandalism.

Were the section indeed confusing, other editors would by now have leapt into the breech to either validate the deletion or revert it. Instead, total silence. Most unlike Wikipedia.  :)

While waiting for someone else besides the silent deleters to engage, I should perhaps say why this article is badly in need of some sort of introduction. The body of the existing article starts out with the bald statement that Planck's law is written

$B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}.$

That certainly makes sense mathematically, but in that case the lead should start something like "In mathematics Planck's law is a function..." If you're going to start with "In physics Planck's law is..." then logically the body of the article should approach the subject from a physical rather than mathematical perspective.

Mathematically one has equally valid choices of defining position as the integral of velocity or velocity as the derivative of position. Physically however the natural starting point is position, with velocity then being understood differentially in terms of position. (With flux vs. flux density the opposite is the case, as Maxwell himself saw clearly early on, because the intrinsic geometry of flux is that of the light field, a delicate four-dimensional notion if one hopes for any sort of conservation law. No such complication arises with the spectrum, which has only one dimension.)

In the case of radiation, Planck's law governs spectral flux density emitted by a surface, namely the derivative of flux density at that surface with respect to position in the spectrum in the same way that velocity is the derivative of position with respect to time. As pointed out yesterday by User talk:Chjoaygame, Planck's law is independent of the geometry of radiant flux, whence the buck for handling that can be passed to the appropriate articles.

What the law does is to govern the dependence of flux (or flux density but why bother with density?) on position in spectrum, analogously to the treatment of velocity as the dependence of position on time, with the spectral energy of the whole band below the spectral position playing the role of the distance from the origin to the moving point whose velocity is being considered. That is, one should understand flux in watts as the physical quantity, and spectral flux in watts per "variation of position in the spectrum" as its derivative. Since position in the spectrum can be specified in at least three useful but nonlinearly related ways, namely frequency, wavelength, and percent increase of a band, the usual mathematical techniques for handling such dependencies apply.

If you read my suggested introduction in that light, you will notice the following things about it.

1. It works with flux independently of geometrical considerations of the light field, per User talk:Chjoaygame's suggestion but without even the geometrical distraction of flux density.

2. The spectrum enters by way of the concept of a band, namely the whole spectrum above or below a given point, whose total flux can be taken to be watts, very simple conceptually, by analogy with position. (Think of the origin of the spectrum as one end of it.)

3. It maintains neutrality between frequency, wavelength, and percent change in band until the distinction is needed.

Any proper understanding of Planck's law must arrive at this point of view at least implicitly, or the concept has not been properly grasped. (This is how I would diagnose many of the misunderstandings on this discussion page, both those depending on the geometry of the light field and the different representations of spectral position.) The only question is whether the article should make this point of view explicit. It seems to me that doing so is sound pedagogy. --Vaughan Pratt (talk) 18:13, 13 October 2011 (UTC)

On the question of flux vs. flux density, a straightforward approach to the former is to work with the total flux $\Phi$ emitted by an arbitrary black Lambertian surface of unit area and arbitrary shape (e.g. a sphere), whose derivative with respect to frequency is given by $d\Phi /d\nu =\pi B_{\nu }(T)$. The Lambertian assumption can then be removed by passing to normally emitted radiance $L_{e}$ whose derivative satisfies $dL_{e}/d\nu =B_{\nu }(T)$ (aka ${L_{e}}_{\nu })$. Putting the π back when going the other way requires reinstating the Lambertian assumption. (This should satisfy User talk:Q Science who wanted it done this way from the beginning but didn't mention the assumption of a Lambertian surface.) --Vaughan Pratt (talk) 19:08, 13 October 2011 (UTC)

Oops, ignore the bit about Lambertian, black bodies are Lambertian by definition. Sorry to doubt you, Q Science. (But you still need the factor of π.) --Vaughan Pratt (talk) 19:21, 13 October 2011 (UTC)

Other notable facts currently not covered in the article

I don't know a lot about the history of physics as most of the current editors here :) , but I have read about the following interesting historic discussion. In the early days of quantum physics, it was thought that the agreeement with the theoretical derivation of the Planck distribution with observations was proof that the photon was massless. Simply put, if the photon is massive, there should be a longitudunal mode in addition to the two transversal modes for each wavevector, so a black body radiator should emit a factor 3/2 more.

But, it think it was Bohr who pointed out that this argument is bogus. As the let the photon mass tend to zero, the longitudonal component interacts less and less with matter, and it completely decouples in the limit of zero mass. So, you can't distinguish between a theory that has a finite but almost infinitessimal photon mass and a photon mass that is srictly zero by observing black body radiation.

Another issue is that we should point out that the Planck distribution is a psecial case of the Bose-Einstein distribution at zero chemical potential. We should briefly mention the Fermi-Dirac distribution, and explain that this is relevant at very hight temperatures larger than a few MeV when electrons and postrons will be present in the photon gas. This is also relevant for the present day relic background, despite the temperature being very low. The fact that the photons used to be in thermal equilibrium with an electron-postron plasma which have now disappeared, means that the photon sector has undergone re-heating relative to the neutrino secto, due to the electrons and postrons vanishing in the cooling universe. You can very easily compute the ratio of the present day neutrino temperature and the photon temperature. Count Iblis (talk) 16:04, 13 October 2011 (UTC)

Maxwell understood in the 1870s that light should exert pressure, although it was only measured experimentally in 1901 by Nichols and Hull. (William Crookes thought he'd observed it in 1873 but it turned out he was observing a delicate exchange of momentum between the vanes of his radiometer, the bulb, and the residual air in the apparatus, with the incident radiation serving only to heat the vanes.) Bohr reconciled Maxwell's understanding with Planck's law.

I was hoping PAR would say something about the connection with the Maxwell-Boltzmann distribution in the article in order to justify his reference to it at the end of the lead. Since that and related connections are covered in the articles Gas in a box and Bose gas, pointers to those articles should suffice. --Vaughan Pratt (talk) 18:34, 13 October 2011 (UTC)

notation

We have seen various notations in the recent spate of edits.

The present (as of the picosecond when I last glanced at the career of change) notation is/was/used to be: $B_{\nu }(T)\$, $B_{\lambda }(T)\$, $B_{k}(T)\$, $u_{\nu }(T)\$.

Shortly before that it was: $B(\nu ,T)\$, $B(\lambda ,T)\$, $B(k,T)\$, $u(\nu ,T)\$.

One text uses an equivalent of: $B_{T}(\nu )\$, $B_{T}(\lambda )\$, $B_{T}(k)\$, $u_{T}(\nu )\$.

The different forms are densities with respect to different arguments, for functions appropriate to their respective arguments. That means they can have different units.

The subscripted form shows that different functions are intended. The brackets are to some extent a relic of an older notation. They indicate the values of the arguments and can be seen as values of the functions for those argument values. It is a little unorthodox to differentiate with respect to a subscript when one might expect differentiation with respect to an argument.

The above use of wavenumber k is contrary to the way it is defined in the article on wavenumber, and is likely to confuse. The previous use of the phrase "spectroscopists' wavenumber" was a flag to warn of this, along with an explicit definition, precautions now missing from the present article.

I do not think we are strictly bound to follow any one particular textbook.Chjoaygame (talk) 20:28, 13 October 2011 (UTC)

forms of expression

The present edit has cut some of the forms on the apparent ground that they are not "common". I don't see why a collection of forms should be restricted to "common" ones, or who will be the arbiter of commonness. One of the current common forms is new to the article, and is, to work from the link it gives, wrong; moreover, its latest manifestation is notationally inconsistent within itself.Chjoaygame (talk) 20:38, 13 October 2011 (UTC)

Chjoaygame, I think what we might be seeing here is an instance of well-intentioned vandalism, if there is such a thing on Wikipedia. I would like to echo PAR's comment in one of his edits: "Revert to last by Chjoaygame - Restore proper notation, deleted table. Headbomb, please slow down and discuss changes. Valid changes may be lost by this revert, but would not be if edits were done more slowly and with consensus." One of Headbomb's edits completely removed the (badly needed) introduction I had just spent several hours writing, with no explanation as to why he removed it, and no response to my question as to why. He is also in the process of making a great number of other edits some of which I find sufficiently objectionable that I am inclined to undo most of them as inappropriate. For example he has arbitrarily decided that "every article should begin with a history section" without consulting anyone, a major change for this article. It would count as an edit war if there were such a thing as a one-editor edit war. Edit bombing might be the appropriate term.

Clearly Chjoaygame and I are having trouble with this. PAR, what's your opinion on Headbomb's editing? --Vaughan Pratt (talk) 20:40, 13 October 2011 (UTC)

I think Headbomb is making positive contributions and some negative ones. He's machine-gun editing, aggravating and not interested in discussion. However, the bottom line is a better article, not a struggle over who can win an edit war or revenge for aggravation. I'm in favor of letting him finish up, then look at what has been done, then keep the good, add the good that has been deleted, and throw out the bad. Revert to a better version if necessary. That's the time to be adamant about things that are wrong. If I or anyone else is right and Headbomb is wrong, then most interested editors will agree, and the article will be improved even more, with no edit war. PAR (talk) 22:07, 13 October 2011 (UTC)

My position exactly. --Vaughan Pratt (talk) 16:33, 14 October 2011 (UTC)

Some of the present common forms are stated as mathematical formulas without indication of their physical meanings. The law can be expressed in diverse physical ways. For example, Eisberg and Resnick express it terms of spectral radiant exitance/radiosity (SI) which they call "spectral radiancy".Chjoaygame (talk) 21:24, 13 October 2011 (UTC)

derivation

The present derivation is written as if Bose-Einstein statisitics is a given from which it is appropriate to derive Planck's law. It assumes that the equilibrium radiation forms standing waves. The possibility of creation and destruction of photons is expressed in the Bose-Einstein statistics.Chjoaygame (talk) 20:46, 13 October 2011 (UTC)

Attempting to edit during this "bombing run" may be counterproductive. I attempted to undo the mess Headbomb made of the k notation but he simply reverted it back to the inconsistent, cluttered, and nonstandard notation. He seems eager to start an edit war. I suggest people wait until Headbomb has finished editing the article to his satisfaction before attempting to assess the resulting damage. --Vaughan Pratt (talk) 20:51, 13 October 2011 (UTC)

I wrote the current derivation a long time ago. It doesn't invoke the Bose-Einstein distribution, we simply work in the canonical ensemble. Where we say that this is a special case of the Bose-Einstein distribution, we say that mu = 0, because there are no restrictions on the number of phtons, but it would be better to reverse the logic here. Since we obtain the distribution with mu = 0, this is fact a computation showing that mu = 0 and that therefore there are no restrictions on the number of photons. Count Iblis (talk) 21:14, 13 October 2011 (UTC)

flimsy sources

I am not keen on sourcing from the article by Helge Kragh (2000), nor from the article by Ribaric and Sustaric (2008). They may or may not be reliable sources, but seem not to be.Chjoaygame (talk) 21:04, 13 October 2011 (UTC)

notes

I do not like footnotes such as the newly placed one. I think that note should stand in the body of the text.Chjoaygame (talk) 21:06, 13 October 2011 (UTC)

As I said, I think the best procedure is just to wait this out. He's not listening to anyone. --Vaughan Pratt (talk) 21:10, 13 October 2011 (UTC)

Restored the Appendix

I don't mind if it is moved, but note that there are links from this and another article to this Appendix. Count Iblis (talk) 20:54, 13 October 2011 (UTC)

He reverted you too? He's doing it to everyone who tries to stop him, and has stopped engaging with the discussion page. Does that count as an edit war, or is this behavior covered under some other WP guideline? --Vaughan Pratt (talk) 21:09, 13 October 2011 (UTC)

Well, I don't mind a big editing drive by one editor, but we do need to make sure that the different articles are consistent with each other. If we don't want the Appendix here, we can combine the Appendix here and the Appendix in Stephan-boltzmann law and move that to the article Methods of contour integration. But then the math editors there would have to agree. Count Iblis (talk) 21:21, 13 October 2011 (UTC)

Such appendices don't belong anywhere on Wikipedia. We shouldn't be in the business of teaching people how to integrate per WP:TEXTBOOK. Headbomb {talk / contribs / physics / books} 21:30, 13 October 2011 (UTC)

I disagree, the derivation is quite notable, and there are many articles on Wikipedia that do similar things. Count Iblis (talk) 22:25, 13 October 2011 (UTC)

There is zero reason to have an entire section devoted to one integral.

$J=\int _{0}^{\infty }{\frac {x^{3}}{e^{x}-1}}\,dx={\frac {\pi ^{4}}{15}}.$

If the reader doesn't know how to do this, all they have to do is look up an integral table. This is not worth 4KB of text (or roughly 1.5 page of text if you print it), nor does it warrant a deviation from WP:TEXTBOOK. Headbomb {talk / contribs / physics / books} 23:26, 13 October 2011 (UTC)

It can be moved out of here, but these types of integrals are covered in text books about this subject; it's notable not as an integration exercise for math studens, but more in relation to this physics subject.

To save space here and in the Stephan-Boltzmann article (which has a different, longer derivation), I propose to move the deleted Appendix to the the Appendix of the Stephan-Boltzmann article and move the Appendix from the Stephan-Boltzmann article to the article Methods of contour integration right below the section about the integral of the Cauchy distribution. The section can be named "Integral of the Planck distribution", which is at least as notable as the integral of the Cauchy distribution. This will save a lot of space in both physics articles without any content being lost. Count Iblis (talk) 23:49, 13 October 2011 (UTC)

Note that the general Bose-Einstein integrals are covered (but not derived) in the polylogarithm article. In this particular case, its $6\,\operatorname {Li} _{4}(1)=\pi ^{4}/15\,$. PAR (talk) 01:38, 14 October 2011 (UTC)

It could have a place at in some mathematics articles I suppose, although I'm not really sure it's particularly topical at methods of contour integration. But if the math crowd is fine with it, I'm probably fine with it too.Headbomb {talk / contribs / physics / books} 01:51, 14 October 2011 (UTC)

Headbomb has an opinion about the place of the derivation of the integral. So do others. I am for Count Iblis putting the derivation of the integral in this article as he thinks fit.Chjoaygame (talk) 04:27, 14 October 2011 (UTC)

The derivation belongs in this article and should stay here. The WP:TEXTBOOK argument is interesting, but does not prohibit useful information. Q Science (talk) 16:16, 14 October 2011 (UTC)

It's an interesting question whether science (including mathematics) consists of its propositions or its methods. Since both viewpoints are well represented in academia and in the literature, it would seem preferable that Wikipedia present both. It's also an interesting question as to whether Wikipedia has any obligation to go the extra mile in making its content clearer. If we followed all the arguments made above we'd have to delete one third of the Bubble sort article on the ground that it's covered in text books, one third because it treats a method, and the remaining third on account of its tutorial flavor. --Vaughan Pratt (talk) 17:36, 14 October 2011 (UTC)

The bubble sort article doesn't look like a tutorial to me. It is not suprising that an article about a method shows the alogrithm and an optimized form. IRWolfie- (talk) 20:56, 16 October 2011 (UTC)

Notability doesn't apply to article content. The derivation adds absolutely nothing to the article at all in any shape or form. It's also not referenced. How to solve a particular integral step by step may be of interest to a maths article but certainly not in a physics one. IRWolfie- (talk) 18:15, 16 October 2011 (UTC)

Solving certain integrals can be of interest to a physics article, otherwise high level physics textbooks would not bother to devote so much space on the computation of integrals that frequently appear in the theory. Students at that level know how to compute integrals and they can always be referred to math texts, integral tables etc. etc. However, certain techniques are frequently used in certain fields of physics and are notable for that reason only. Also typical math sources are not good sources to look up the evaluations of such integrals. The computations and approximations of integrals relating to Bose-Einstein and Fermi-Dirac distributions is such a notable physics subject. Similarly, the evaluation of certain Feynman integrals by introducing so-called "Feynman parameters", see e.g. here is also primarily a physics subject.

So, perhaps the best thing to do is to compile a new article on the subject of Bose-Einstein and Fermi-Dirac integrals, but then as a physics article, not a math article, because it's not so notable as far as pure math is concerned. Count Iblis (talk) 20:27, 16 October 2011 (UTC)

You keep mentioning notability, notability is not a criteria for inclusion in articles. You mention physics textbooks have the derivations, wikipedia is not a textbook WP:NOTTEXTBOOK. An article on the mathematical solutions to Bose-Einstein and Fermi-Dirac integrals sounds like a maths article to me. Unsuprisingly a theoretical physicist as part of his job will have to solve mathematical problems. IRWolfie- (talk) 20:48, 16 October 2011 (UTC)

I'm in favor of a math article, because Bose-Einstein statistics are not limited to physics. They occur in network theory and information retrieval. The Bose-Einstein distribution should be viewed as a pure probability density function, separate from any particular application. The Bose-Einstein integral is a special case of a polylogarithm, but could use its own article which is why I just linked the solution of that integral to the "Bose-Einstein integral" article, which is a redirect to the polylogarithm article. PAR (talk) 20:51, 16 October 2011 (UTC)

Ok, then we can move both the contour integration method given in the appendix of Stephan-boltzmann law and the derivation that was in this article to your article. Count Iblis (talk) 21:16, 16 October 2011 (UTC)

Notability is the criteria for the creation of an article, if you feel a method meets the WP:NOTABILITY test you can create a separate article on it. IRWolfie- (talk) 21:34, 16 October 2011 (UTC)