Wien's displacement law. A paradox

[lightarrow]

Wien's displacement law relates a blackbody temperature T with the wavelength l of its maximum emission: l(max) = a/T, where a is a constant. Let's calculate a.

Spectral radiance as a function of frequency v:

R(v) = [8(pi)h(v/c)^3] * 1/[exp(hv/kT)-1].

as a function of wavelength l:

R'(l) = (8(pi)hc/l^5) * 1/[exp(hc/lkT)-1].

You can obtain this last formula from the previous one, writing:

Integral(0;+infinity)R(v)dv = Integral(0;+infinity)R'(l)dl
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Now we derivate the radiance, to obtain the point of maximum:

dR(v)/dv = 0 --> 1-(1/3)hv/kT = exp(-hv/kT) -->

1-(1/3)hc/lkT = exp(-hc/lkT) (1)

dR'(l)/dl = 0 -->

1-(1/5)hc/lkT = exp(-hc/lkT) (2)
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Solving for l equations (1) and (2) gives two different values of l !

(1) gives (numerically, Mathcad): hc/lkT = 2.82 -->

l(max) = 5.11*10^-3/T

(2) gives (numerically, Mathcad): hc/lkT = 4.97 -->

l(max) = 2.90*10^-3/T
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The values taken for computation:
k = 1.38*10^-23 J/°K
h = 6.63*10^-34 J*s
c = 3.00*10^8 m/s
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Why two values for a?

Since the blackbody colour depends on l(max), this question could also be put:

which is the real blackbody's colour?

 

[lalbatros]

this is general behaviour

Hi lightarrow,

this is general behaviour
Let's consider two probability distribution functions related by a change of variable y=y(x):

f1(x) dx = f2(y) dy​

then

f2(y)=f1(x)y'​

If the maximum of f1(x) is xmax, then the maximum of f2(y) does not usually correspond to y(xmax).
Indeed:

f2' = f1' y' + f1 y''​

This equation becomes for xmax:

f2'(y(xmax)) = f1(xmax) y''(xmax)​

Therefore, f2 and f1 have correponding maxima only if y''(xmax) = 0.

This is not surprising, since the product of two functions (f1*y') has its own maximum, not necessarily at the same place as any of the two functions.

Till now, this has no physical meaning.

Now, what's the color of the black-body?
This is a physiological question. Color is a human sensation. Its analytical explanation should involve: the light source spectrum, the retina and the various detectors on the retina (3 sorts), the nervous system and the brain.
Fortunately, it is no too difficult to analyse the total result in a laboratory, simply by asking people to compare their visual sensation with different stimuli. This was done more than 100 years ago. Roughly speaking, the human eye has three different light sensible cells with each their own spectral response. Therefore, two different spectra, as measured with optical detectors, could give the same stimuli for a human. It is also clear that for high temperatures, the maximum in the energy-frequency-spectrum lies outside of the human eye sensibility. Humans will perceive it as blueish white, while the maximum (or "color'?) could be in the X-ray !
You can read more on the web about colorimetry. See a paper by http://www.fho-emden.de/~hoffmann/ciexyz29082000.pdf"for example.

Michel

 

[lightarrow]

Thank you for your reply lalbatros.

There is something I still don't understand, however:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

This relationship is called Wien's displacement law and is useful for the determining the temperatures of hot radiant objects such as stars, and indeed for a determination of the temperature of any radiant object whose temperature is far above that of its surroundings.

Which ones of the two different laws should we use for this purpose?

 

[lalbatros]

lightarrow,

It is clear that the displacement in the E(f) spectrum or in the E(L) spectrum are totally related altough they are not the same (f:frequency, L:wavelength). This is actually one physical fact seen in two different diagrams, not more. Therefore, it is mainly a matter of convention.

However, the Wien displacement refers usually to the E(L) function, and you should work with the E(L) function unless it is explicitely stated otherwise.

Note that from a laboratory point of view, there is never such a doubt: what is measured is clearly known and defined and must be compared properly with the theory without ambiguity. The ambiguity is more likely in a textbook exercise or some homework. Actually, it is not so important to remember that Wien plotted E(L) instead of E(f). What matters really are the definitions E(f) or E(L), the BB law, where it comes from, ... .

In applications, some additional practical concepts are needed too. Like (directional) intensity, emissive power, emissivity, absorptivity, reflectivity, transmissivity, view factors, and much more sometimes.

Michel

 

[lightarrow]

Thank you again for your answer, lalbatros.

It is clear that the displacement in the E(f) spectrum or in the E(L) spectrum are totally related altough they are not the same (f:frequency, L:wavelength). This is actually one physical fact seen in two different diagrams, not more. Therefore, it is mainly a matter of convention.

It was not this to worry me.

However, the Wien displacement refers usually to the E(L) function, and you should work with the E(L) function unless it is explicitely stated otherwise.

Said in this terms it could seem that you take E(L) function because from this comes: l(max) = 2.90*10^-3/T and you know (and me too) that this one only fits the experimental results, so, not because it's arbitrary.
Why is: l(max) = 5.11*10^-3/T wrong? It was derived correctly or not?

 

[lalbatros]

lightarrow,

Said in this terms it could seem that you take E(L) function because from this comes: l(max) = 2.90*10^-3/T and you know (and me too) that this one only fits the experimental results, so, not because it's arbitrary.
Why is: l(max) = 5.11*10^-3/T wrong? It was derived correctly or not?

Both may be correct. None are wrong with the information available.
It all depends on the details.

What is incorrect is to say "the maximum of the spectrum is given by L(max) = 5.11*10^-3/T".
One should be more precise and say 'the maximum of the E_L(L) is ..." or "the maximum of E_f_(f) is ..." ...

http://scienceworld.wolfram.com/physics/WiensDisplacementLaw.html" .
You will see two results, because you can seek the maximum of two functions (even more if you want).

In summary: you need to define:

the function you explore to find a maximum

E_F: density per frequency range
E_L: density per per wavelength range)​

the variable you use to express your result

f: frequency
L: wavelength​

Therefore, you can define four ways to give a Wien displacement:

E_F(F) is maximum defines the maximum Fmax_E_F
E_F(F(L)) is maximum defines the maximum Lmax_E_F
E_L(L) is maximum defines the maximum Lmax_E_L
E_L(L(F)) is maximum defines the maximum Fmax_E_F
​

This problem should not trouble you if you imagine you measured a spectrum in the laboratory. Depending on your instruments and/or you preferences, you might represent your experimental results as E_F or as E_L on the vertical axis. Similarly, you can choose F or L as your horizontal axis. There are therefore 4 different ways for looking at a maximum in the spectrum. The Wien displacement is a physical fact that can be represented in different ways (or coordinates). It is necessary to be precise and tell people the details needed so that they can figure out what you are talking about: which function exactly, which variable.

Note however, that, historically, Wien probably made a particular choice to represent its experimental results. But this does not change physics in any way, of course.

Michel

PS:

You remember maybe that thermodynamics needs a strong discipline in this respect.
The specific heat for example can be defined as a derivative of internal energy or a derivative of enthalpy.

In practical applications there is also a need to develop such disciple and precision.

Take for example an industrial plant. You will often ear questions like "what happens if the temperature there is increased". Such questions are generally meaningless and require further discussions to receive a meaning before they can be answered. For example, you have to determine if some control loop will be running or not. You may also need to define how exactly you will increase the temperature: will material flows (like fuels or raw materials or combustion gas) be kept constant or will some increase. etc etc

 

[lightarrow]

Thank you for your detailed answer, lalbatros.

Forgive me if I keep asking about the same subject: so, if I say (as it is written in many books) that the L of maximum intensity in the solar light's spectrum is ~ 540 nm (yellow-green), as it comes from the relation L = 2.90*10^-3/T, assuming a solar surface temperature of 5400 °K, it's a completely arbitrary statements, with these informations only, because, using the other relation we have: L ~ 950 nm (infrared)?

 

[lalbatros]

lightarrow,

You are totally right.

However, the context in the book should be checked. Maybe they refer to a graph of the spectrum where their point of view can be seen.

You have seen on the Wolfram page that they made the derivation for two intensity functions: either B(f/Hz) ot B(L/cm).

Writing books with precision may degrade the style, but for physics most of the message lies in precision and there is no reason to speak loosely.



Michel

 

[lightarrow]

Thank you very much lalbatros.

Alberto.