# Talk:Wien's displacement law

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Hi, The derivation currently featured is helpful, but I think some discussion of how x is "easily" found would be useful as well. lws (talk) 17:32, 27 October 2008 (UTC)

## Alternate Formula

I have come across another way of expressing Wien's displacement law which is:

${T\lambda_{max} = \frac{1}{5} c_2}$ where c2 is second radiation constant.

Both formulas are the same but in this article is not mentioned. I'm no authority on the subject, but I would imagine Wien's displacement constant = 1/5c2. It would be nice to see where Wien's displacement constant comes from and/or other other ways of expressing Wien's displacement law —Preceding unsigned comment added by Devon Fyson (talkcontribs) 16:59, 29 January 2009 (UTC)

## Alternative form correction

Since

${\nu_{max}} \not= {c\over\lambda_{max}}$

I deleted the alternative form section (sorry, metacomet). It is possible to derive a frequency version of Wien's law, but I don't have it on hand at the moment. ...and I forgot to do an edit summary, so that's mostly the reason for this entry. ...wow, I am soo inept. --UltraHighVacuum 02:54, 23 February 2006 (UTC)

Don't apologize for fixing something that was incorrect. If it's wrong, it's wrong. I didn't write that section, I merely updated it so that it was possible to follow the logic. I didn't realize that the logic was flawed, or I would have removed it myself. -- Metacomet 10:18, 23 February 2006 (UTC)
On the other hand, it depends on what the definition of nu-max is. If it is the frequency of peak emission, then perhaps you are correct. But if the definition is the frequency that corresponds to the wavelength of peak emission, then the section that you removed was actually correct as written, although it would probably need a brief explanation of the difference in definitions. -- Metacomet 10:21, 23 February 2006 (UTC)
I have added a new, corrected section entitled "Frequency form" to the article. The source is Weisstein's World of Physics, and the information is consistent with the discussion below. -- Metacomet 11:10, 23 February 2006 (UTC)
In all my experience (and quantumm mechanics books)nu max is defined as the actual maximum frequency, so this was my motivation. It seems a definition of nu-max as being the corresponsing frequency for lambda max is confusing, and maybe also a bit arbitrary, since there is no reason for the definition not to be the other way around. But you're right, this is kind of ambiguous, so thanks for that section in the article. --UltraHighVacuum 18:04, 23 February 2006 (UTC)
Isn't that exactly what the new section says? -- Metacomet 23:07, 23 February 2006 (UTC)

${\nu_{max}} \not= {c\over\lambda_{max}}$

That is because the curves for frequency and wavelength are different. In fact, using a similar derivation it can be shown that

${ \partial u \over \partial \nu}= (3-{{h\nu_{max}}\over\ kT})e^{{{h\nu_{max}}\over\ kT}}-3=0$

This gives

${ \nu_{max}= {2.8kT \over\ h}}$

It turns out that

${C\lambda_{max} \over\ \nu_{max}} \approx 1.76$

I will go ahead and make the changes, but it would still be nice for someone to check me on this. --129.93.63.14 23:08, 9 February 2006 (UTC)

Any point is putting the frequency version in?

$f_{max} \approx 10^{11} T$

--Audiovideo 13:09, 16 May 2005 (UTC)

It's said that "This equation cannot be solved in terms of elementary functions. It can be solved in terms of Lambert's Product Log function but an exact solution is not important in this derivation. One can easily find the numerical value of x"

But this derivation is horrible. Looking back at history einstein used the wien displacement law in his paper on radiation, but in no was was the wien displacement law a derivation from plancks radiation theory. This should be noted. I suck at writing or else i would.

I put a link to adiabatic invariant which reproduces Wien's original argument, and shows its connection to Einstein-Bohr-Sommerfeld quantization. Wien didn't win the nobel prize for nothing. Likebox 18:21, 11 September 2007 (UTC)

## visual system and solar radiant power spectrum

The wavelength coincidence between the peak of the solar radiant power spectrum (at about 5800K) expressed in terms of unit wavelength interval, and the peak of the eye's spectral sensitivity function is just that - a coincidence or accident -- not something other as this article implies. If you express Planck's function in terms of unit frequency interval, you find the peak of that curve corresponds to a wavelength of about 880nm. If instead of power, you express Planck's function in terms of numbers of photons per second per unit wavelength interval (at 5800K), you find a wavelength peak at about 633nm. The problem is that Planck's function is a probability density function and thus doesn't transform simply with a change of units. The eye's spectral sensitivity function, however, is not a density function, so changing units is straightforward. But it is misleading to attempt to compare the two types of functions. This is a common error in the vision literature and a common problem when interpreting density functions.

Moreover, all visual systems plainly reflect several constraints at least as important as the solar spectrum: they can't be too sensitive to the infrared without being subjected to noise created by the warmth of the eye itself; they can't be too sensitive to uv because of molecular instability; and they most likely reflect their phylogenetic origins in water, which itself filters the solar spectrum.

All this is spelled out in "Some paradoxes,errors, and resolutions concerning the spectral optimization of human vision" Sofer and Lynch Amer. J. of Physics 67 (11) 1999

Apologies for all violations of protocol -- first time posting.

Greblams (talk) 08:15, 6 January 2008 (UTC)

So the real question is what light spectrum you use to maximize this integral, to get the maximal number of subjective lumens per watt out of your source. You can't use a laser tuned to the rhodopsin peak (which would give you the maximal efficacy)-- you're stuck using only blackbody radiation at various temperatures. Now, since this isn't a totally symmetrical curve (as the scotopic luminosity function is) you obviously cannot do this by merely using a blackbody curve with the same peak, but if the two peaks are close you'll probably be fairly close to the maximal efficiency possible for blackbodies. For example, I know that maximal photopic luminous efficacy occurs at a blackbody color temp of 6500 K, which is considerably hotter than the Sun. But the photopic luminosity function peak is 555 nm, and you're really aiming for the scotopic peak at 507 nm, which should give you a maximally efficient color temp of about 91% of that, or 5900 K. Which is very close to that of the Sun. So, faint moonlight, must be very close to maximally bright PER WATT at the color temp of our Sun, when seen by an eye using rhodopsin; if the Sun was any other temperature, the efficiency (lumens/watt) would drop, and you'd see a less bright light for the same raw total power.

I'll leave your other arguments for later, but I don't buy them. We could see very far into the infrared without being bothered by background IR, as near-IR night vision (and IR photography) shows. But there's less visual energy down there, and evolution probably had a hard time coming up with a reaction that used a single IR photon, and there wasn't much pressure to do it. As for UV, humans actually see UV when their lenses are removed, and bees see UV also, so molecules already exist that sense this. But evolution simply had too hard a time getting UV through the proteins in a lens, and why bother? Bees had a reason to need UV and they had no lenses, so they got a pigment to do it. But it only works in bright light. Humans too have a photopsin which peaks at 420 nm absorption, so it's not as though we could not have shifted our vision into the UV from that standpoint (and in fact, that’s how we see UV). If Vega was our star, we might all have compound eyes and no protein-containing lenses. Who knows? SBHarris 10:43, 6 January 2008 (UTC)

Let me try to clarify the first point. The standard luminosity efficiency function may look like a probability density function. But that is simply because it is created from instruments with finite bandwidth resolution. The data points are averages obtained from sampling within those intervals. But the function itself is not at all a density distribution. Normalized, it is defined, as you say, as a dimensionless fraction/wavelength (not fraction/wavelength interval). A probability density distribution is different. It's unit is by definition a differential interval. For example the blackbody solar spectrum tells you power/dWavelength.

We frequently omit the differential piece when referring to the power spectrum (as you did), but this is what makes the difference when you transform the function from wavelength to frequency, using wavelength=c/frequency. (Sorry, I haven't figured out a more convenient notation). In that case, you have to also supply the Jacobian weighting factor -- here equal to c/(frequency)2 (ignoring the sign since that only affects the order of integration.) And this is what changes the shape of the curve! You don't have to do this with the luminosity function because it doesn't involve differentials. If you plot a luminosity function in terms of frequency or wavelength, the shape of the curve looks the same. That is, the wavelength associated with the peak of the curve remains unchanged. When you plot the solar spectrum in terms of dwavelength or dfrequency, while the information it contains is the same(!) the shape of the curve changes as does the wavelength associated with its peak. This is a property of probability density distributions generally: they may change shape radically with transformation of variables. (Very narrow ones do not do so, however.)

That's why it's not legitimate in principle to attempt to infer anything at all from the peak of the solar spectrum plotted against a certain bandwith and, in this case, the peak of a luminosity function, as this article attempts to do, albeit parenthetically. It's a case of "comparing apples and oranges." Essentially, I can transform the shape of the solar spectrum (and the wavelength associated with its peak) any way I like by some combination of a change of variables or plotting against a different bandwidth. We usually see the solar spectrum plotted against wavelength and people often draw significance from the closeness of its peak to the peak of a luminosity function. My main point was that even apart from the cogency of any particular evolutionary argument, the closeness of the two peaks is itself misleading and essentially meaningless. When plotted against frequency the wavelength associated with the peak of the solar spectrum is about 880nm. So what exactly is the "closeness" to the peak of the luminosity function that we are trying to explain via evolution? It's not that too much is made of this -- it's that it's meaningless to talk about. This misunderstanding crops up a lot in both the technical and popular literature about vision.

While it isn't legitimate to relate the peak of the solar spectrum to the luminosity function, it certainly IS legitimate to multiply the two functions as you describe. What you get is indeed then a density distribution. And yes the integrals will all come out the same for different units, as you say --as long as you mind the Jacobians. And it is also legitimate (and most interesting!) to ask of that integral whether it is in some sense optimal. Your comments about source blackbody temperature are intriguing. That at least strikes me as one plausible way to begin to think about how/if the visual system may be optimal at least in a global sense. You have to be careful about what the limits of integration are taken to be, though. For example, they have to take into account the atmospheric cutoff at about 320 nm.

As for my comments about other constraints, some of your points are well taken. For the infrared, there certainly are plenty of photons in the near infrared for a visual system to potentially sop up. During the day, there is plenty of sunlight above 700nm and at night, I believe there is the OH air glow as well. Why visual systems generally don't exploit these, I'm not sure. You are probably right that it has to do with the low energy of those photons. A photoisomerization system would need ever longer molecules to make the low energy transitions to detect these photons, but such molecules are increasingly unstable in aqueous solutions. There is also a temperature of the eye however that will fog a retina attempting to detect in the infrared. I'm not sure what that temperature is though. As for UV, it is true that losing your cornea or lens will enhance your ability to detect UV. But it can't be simply the case that "evolution simply had too hard a time getting UV through the proteins in a lens." Folks without a cornea or lens also suffer increased rates of UV-related ocular damage. I'm not sure what drives a visual system to flirt with UV radiation at all since UV is hell on organic molecules. I'm also not sure what drove the insect-flower communication system to move into the uv in the first place. And I wonder how many bees and other insects are going blind by the time they die. I believe older Brown pelicans typically suffer from glaucoma as a result of accumulated eye damage from all those high impact dives for fish. Maybe bees, like pelicans, simply gotta do what they gotta do to get as far as they do. I'm a great believer in the power of natural selection to climb adaptive peaks, but sometimes living things reflect quite sub-optimal adaptations.

I think the really under-rated factor is water. If you compare the sunlight transmission curve in water at 1 meter and a luminous efficiency curve (you can do this because the transmission curve is not a density distribution) -- it looks like that that is what the visual system evolved to track. My hunch is that we have been tinkering on the margins with an essentially water-evolved visual system ever since.

Greblams (talk) 23:35, 6 January 2008 (UTC)

Actually, the primary factor would probably be that the system (like everything else about living things) is basically made of water. Ben Standeven (talk) 00:29, 12 January 2009 (UTC)

## Wien's Real Law

This page is mostly about what textbooks call Wien's displacement law, which is an observation about the Planck distribution, not what Wien actually discovered. The derivation should derive the real law, which tells you how to get the blackbody curve at any temperature from the blackbody curve at any other temperature. The relationship for the "peak" frequency comes out right, but it's not the fundamental thing.Likebox (talk) 14:11, 23 April 2009 (UTC)

Just to clarify--- the reason the peak is no good is because the peak is where the derivative of the Planck distribution is zero. But the full Planck distribution includes the geometric mode-counting factor in front, it is not the fundamental quantity. This mode counting part is just a power, so the peak ends up inversely proportional to the wavelength anyway, because of the peculiar properties of powers, but the physically fundamental quantity is the thermal energy in each mode. This is what Wien is talking about.

The full displacement law says that the average energy per mode divided by T is a function of omega/T only, in Planck's constant terms, it's a function of hf/T. so that the Wien correspondence is the correct quantum relationship between frequency and energy. Leaving this out makes Wien look like a hack.Likebox (talk) 14:35, 23 April 2009 (UTC)

The textbook version is not as totally moronic as it first appears. The full Wien law is equivalent to the statement that the peak location is proportional to the temperature when the blackbody distribution is expressed as a function of $\lambda^\alpha$ for any possible exponent \alpha. So if you have an appreciation for the arbitrariness of $\lambda$ as a thermodynamic coordinate, you can deduce the full Wien law from the textbook version. But it's hard to see. It certainly doesn't make the beautiful adiabatic relationship between different temperatures obvious.Likebox (talk) 16:35, 23 April 2009 (UTC)

## Wien's "b"

Wien's constant "b", as I remember it, was exactly equal to k/h. It didn't have the ridiculous transcendental factor that appears in modern textbooks, as far as I remember, because Wein formulated his law in terms of the energy per mode, not in terms of the peak of the distribution. The constant b appears in Wien's work as the dimensional coefficient inside the function F,

$u/\nu = F(\nu/bT)$

which immediately shows that b is just Boltzmann's constant divided by Planck's constant. The "b" in textbooks is a nonsense constant, it depends on whether you express the distribution in terms of frequency or in terms of wavenumber. The statement here that Wien did not consider his constant as a constant of nature is also suspect. What did he think it was then?Likebox (talk) 16:47, 23 April 2009 (UTC)

## Reflection from the moon

In the "familiar approximate examples" section, under light from the sun and moon, the article states "Even nocturnal and twilight-hunting animals must sense light from the waning day and from the moon, which is reflected sunlight with this same wavelength distribution.". Is this actually true? Does this not require the moon be a perfect mirror? Obviously spectra from the moon will show some of that original shape, but accounting for the fact that the moon is a non-ideal blackbody itself, it seems fair to point out that light will be coloured. --144.32.126.12 (talk) 15:26, 27 April 2009 (UTC)

## Value for b incorrect

Now, it says "b = 28,977,685(51) nm·K." This is off by a factor of 10. I'd fix it, but don't know the proper formatting. 128.252.78.82 (talk) 14:55, 4 May 2009 (UTC) There seems to be a problem of units in the values listed for b. one value has units of nm*K while the other has nm*eV. The latter is wrong and should be eV*K? May 26 2011 — Preceding unsigned comment added by 129.240.152.169 (talk) 07:49, 26 May 2011 (UTC)

## Article Rot

Why does this article rot? The description of the law in the first paragraph decays to the textbook explanation with a time constant of about 1 month. If you don't know what Wein did, please don't edit the article.Likebox (talk) 15:51, 7 September 2009 (UTC)

## The lead has some issues of style

While I have no right to impose myself as an authority on how article leads should be written, I would like to appeal to some common sense. This is an encyclopedia, and should be written in a style that is accessible to anyone that has the prerequisite mental faculties using a vocabulary common to a wide interested audience. Yet the article began referring to a "blackbody curve" with a link to an article about "black body" that does not contain the expression "blackbody curve". I just replaced that term with its definition. It further talks about "each mode" when it is anyone's guess what modes it is talking about. In fact, you have to know that we are talking about resonant radiation inside a cavity, and that such resonances have modes corresponding to the geometry of the cavity. This makes the lead read as a paragraph from the middle of some quite long story.

I think that it is perfectly admissible to have less accessible stuff later in the article, but the lead should be a little more inviting. I urge the editors to imagine the non-expert, intelligent reader when they write the lead.

Turn the reference to the CODATA source into a reference, to get it where it belongs, at the end of the article. (I shall see what I can do with that.) —Preceding unsigned comment added by Cacadril (talkcontribs) 04:33, 9 March 2010 (UTC)

## pičinks?

According to the ångström article, "the use of the ångström is officially discouraged by the tohle je ončo vončo lvončo for Weights and Measures." Anyone have any objection to sticking to SI units in this article? --Vaughan Pratt (talk) 10:32, 16 May 2011 (UTC)

I didn't know what Wien's Displacement Law was before reading this article (I was aware of the general relationship, but not the specifics). This article needs a good cleanup to be more reader friendly, and removing the darned angstroms from the article would be a good first step. Who uses angstroms anymore? Only those above the age of 40, that's who:P. If we're going to keep the angstroms in this article, maybe we should change all the temperatures from kelvin to Rankine, just for the old fogies who refuse to go SI. Gopher65talk 16:22, 6 October 2011 (UTC)

## Wien's approximation

Somewhere in the opening I would like to see mention that Wien's "law" only applies to a finite range of frequencies.
At the very least, mention it is also known as Wien_approximation, providing the link.

This would improve understanding of Wien's with regard to physical reality and historical relevance.

Andrew Church (talk) 08:10, 15 November 2014 (UTC)

Wien's displacement law (the relationship between spectra at different temperatures) is generally true. Wien's distribution law (which attempts to predict the form of the spectrum) is only right for high frequencies, which is why it's generally referred to as the Wien approximation. There is a link to the latter in the lead section of this article, but it should be clear that they are different things. Djr32 (talk) 12:28, 15 November 2014 (UTC)

Thank you Dj,
I was in the process of differentiating the two without much luck. Even my preferred text on topic is ambiguous, and only softly implies that:
his discovery that the wavelength of the light frequencey emitted with the greatest intensity, multiplied by the temperature is contsant - as displacement law, and predictions of other frequencies/wavelengths for that same K, as distribution law (that breaks down for all wavelengths infrared and beyond - my Young and Freedman University text doesn't mention this).

After my exams, I'd like to attempt a more artful recraft of the opening paragraph, lauding the work of Wien (1911 Nobel -Physics) in mid 1890's, sure but some how expressing the following: then at the Feb 2 1900 meeting of the German Physical society, Lummer and Pringsheim revealed discrepancies between predictions of Wien's law in the infrared and experiment. Then at the October 19 meeting, Plank announced "an improvement" to the formula. It was trying to explain the new formula's success on firm thoretical ground, that forced "an act of desperation" ultimately leading to the discovery of the quantum. [1]

In 1896 Plank thought Wien had solved the Blackbody problem and set about deriving the law from first principles. When it turn out that not only did Rayleigh's model break down at high frequencies (Ultraviolet catastrophe) but now also Wien's law misbehaved badly for all frequencies infrared and beyond, Plank was the proverbial "right man in the right place at the right time", with the capability, integrity and resolve to play his part.
Andrew Church (talk) 23:44, 15 November 2014 (UTC)

Your input is very welcome, but these historical issues do not belong (in any detail) on this page. Rather you should see if you can usefully contribute to pages such as Planck's law#History and Wilhelm Wien where such history is discussed. This page should concentrate on the content of Wien's displacement law which is a significant relationship in and of itself (more so than Wien's approximation aka Wien-Planck law which of course wasn't correct but was an historically important step in the course of understanding black body radiation).