Wien's Displacement and Distribution Laws

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In 1893 Wilhelm Carl Werner Otto Fritz Franz Wien (1864 Jan 13 - 1928 Aug 30) devised his well-known displacement law and in 1896 he devised his less-well-known distribution law, which Max Planck used in his formal derivation of the correct formula describing blackbody radiation. Now I want to show you how to deduce those laws much as Wien did.

Imagine that we have a spherical cavity of radius R and volume V and that we have lined it with a perfectly elastic, perfectly reflecting film. And imagine that we have filled the cavity with blackbody radiation that has an absolute temperature T. Now imagine that we change the radius of the cavity and, thus, change the cavity's volume. What happens to the temperature of the radiation inside the cavity and what happens to the wavelengths of the rays that comprise the radiation?

We know right away, as did Wien, that the adiabatic expansion or compression of light does not change the light's entropy,

(Eq'n 1)

so we also know that

(Eq'n 2)

Replacing the volume in that equation with its description as a function of the cavity's radius and extracting the cube root of the result yields

(Eq'n 3)

We also know, thanks to Kirchhoff, that blackbody radiation must conform to a description in which the spectral density follows a function, u(λ,T), of the wavelength and the absolute temperature only. That function describes how much of the radiation's total energy goes into the rays that have wavelengths around λ, so we need to know how the wavelength of any given ray changes as the radius of the cavity changes.

Assume that the cavity expands and that the expansion occurs quasi-statically; that is, that the wall of the cavity moves at a speed v that is vanishingly small relative to the speed of light. Any given ray bounces off the cavity wall, traces a chord across the cavity, bounces off the wall again, and continues on in that fashion indefinitely. At any given bounce point the trajectory of the ray meets the cavity's radius at an angle θ, so we calculate the length of the chord as 2RCosθ. Any arbitrarily chosen wavefront in the ray traverses the chord in the interval

(Eq'n 4)

On each bounce the wavelength of the ray changes, in accordance with the Doppler shift, by the amount

(Eq'n 5)

But we know that v=dR/dt, so we can rewrite that equation as

(Eq'n 6)

If we divide that equation by λ and integrate it, we get

(Eq'n 7)

which necessitates that

(Eq'n 8)

for all wavelengths comprising the radiation.

When we combine that result with Equation 3 we get

(Eq'n 9)

in which w represents a pseudo-constant. If we examine the spectrum of blackbody radiation at some given temperature, then w changes with the wavelength. But if we look at the part of the radiation that has a particular wavelength while we subject the radiation to an adiabatic expansion or compression, then w remains constant and the wavelength has an inverse relationship with the radiation's temperature. That fact necessitates that we properly describe the spectral density of the radiation with a function, not of wavelength and temperature, but of wavelength multiplied by temperature; that is,

(Eq'n 10)

That equation expresses Wien's displacement law.

But most of us did not learn Wien's displacement law in that form. The more familiar form, most often found in astronomy courses, comes from assuming (or deducing) that the spectral density function has a maximum when the wavelength has a certain value. We thus expect w to take on a unique value that would allow us to calculate the temperature at which the maximum wavelength takes on a given value. Otto Lummer and Ernst Pringsheim first determined that value in 1900 from their observations of cavity radiation. Further measurements have refined the value so that we have Wien's displacement law appearing in textbooks as

(Eq'n 11)

Thus, if we determine that space is filled with blackbody radiation that has its peak wavelength at 1059.5 microns, then we can infer that the radiation and, therefore, space itself has a temperature of 2.735 Kelvin. Likewise, we can calculate that the human body (T=98.6 F = 37 C = 310.15 K) radiates heat in a spectrum that has its peak at 9.343 microns, at the border between the near and far infrared regions of the electromagnetic spectrum.

So in the wild melee of electromagnetic waves flittering about within the cavity he imagined Wien discovered a simple rule. But then he re-examined his imaginary experiment and advanced the theory of blackbody radiation even further toward the goal that Kirchhoff had set.

Adiabatic expansion or compression of light does not change the entropy of the light, but it certainly changes the light's energy. We know that at any given instant the energy contained in the radiation in the cavity equals the product of the volume of the cavity and the uniform density of the energy in the cavity: E=uV. Also we know that light in an enclosure exerts a pressure p=u/3, so in a spherical cavity of radius R the light exerts upon the wall a total force of

(Eq'n 12)

If the radius of the wall changes by the minuscule increment of dR, then the force does upon the wall or upon the light an increment of work

(Eq'n 13)

But u=E/V, so we can rewrite that equation as

(Eq'n 14)

Dividing that equation by E and integrating the result gives us

(Eq'n 15)

which corresponds to

(Eq'n 16)

Finally, we want to look again at the energy density of the radiation in the cavity. We have u=E/V, so we also have, as a consequence of Equation 16, the statement that

(Eq'n 17)

Combining that equation with Equation 3 tells us that

(Eq'n 18)

which just gives us back the Stefan-Boltzmann law and verifies our work so far (insofar as the Stefan-Boltzmann law accurately describes Reality).

But that result merely gives us total energy density. We actually want to obtain a description of the spectral energy density, u(λ,T)dλ, the energy density in that part of the radiation whose wavelengths lie between λ and λ+dλ. To that end we note that the total energy density,

(Eq'n 19)

We should properly express that equation as a function of w and note that we carry out the integration with the understanding that the temperature has the same value at all wavelengths of the blackbody radiation, so we have, combining Equations 18 and 19,

(Eq'n 20)

If we multiply that equation by T and remove the limits from the integral, we get

(Eq'n 21)

in which the constant that I have represented by the Greek letter kappa stands revealed as the number coming from the evaluation of a function of w in coordination with the evaluation of the integral. But now we can simply differentiate that integral with respect to w and obtain

(Eq'n 22)

And at last we invoke Equation 10 to get Wien's distribution law in the form in which Max Planck used it in his 1901 paper on his derivation of the spectral density of blackbody radiation

(Eq'n 23)

However, some sources offer a different version of that equation,

(Eq'n 24)

Clearly we can obtain that equation from Equation 23 by making the appropriate substitution from T=w/λ and then absorbing the fifth power of w into the unspecified function of λT. We can see that form of Wien's distribution law reflected in Planck's law in the form

(Eq'n 25)

in which the first radiation constant

(Eq'n 26)