Wien's displacement law

Discovery

The law is named for Wilhelm Wien, who derived it in 1893 based on a thermodynamic argument. Wien considered adiabatic expansion of a cavity containing waves of light in thermal equilibrium. Using Doppler's principle, he showed that, under slow expansion or contraction, the energy of light reflecting off the walls changes in exactly the same way as the frequency. A general principle of thermodynamics is that a thermal equilibrium state, when expanded very slowly, stays in thermal equilibrium.

Wien himself deduced this law theoretically in 1893, following Boltzmann's thermodynamic reasoning. It had previously been observed, at least semi-quantitatively, by American astronomer Samuel Langley. This upward shift in ${\displaystyle {\nu }_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}}$ with ${\displaystyle T}$ is familiar to everyone—when an iron is heated in a fire, the first visible radiation (at around 900 K) is deep red, the lowest frequency visible light. Further increase in ${\displaystyle T}$ causes the color to change to orange then yellow, and finally blue at very high temperatures (10,000 K or more) for which the peak in radiation intensity has moved beyond the visible into the ultraviolet.

The adiabatic principle allowed Wien to conclude that for each mode, the adiabatic invariant energy/frequency is only a function of the other adiabatic invariant, the frequency/temperature. From this, he derived the "strong version" of Wien's displacement law: the statement that the blackbody spectral radiance is proportional to ${\displaystyle {\nu }^{3}F(\nu /T)}$ for some function F of a single variable. A modern variant of Wien's derivation can be found in the textbook by Gregory Wannier and in a paper by Edgar Buckingham.

The consequence is that the shape of the black-body radiation function (which was not yet understood) would shift proportionally in frequency (or inversely proportionally in wavelength) with temperature. When Max Planck later formulated the correct black-body radiation function it did not explicitly include Wien's constant ${\displaystyle b}$. Rather, the Planck constant ${\displaystyle h}$ was created and introduced into his new formula. From the Planck constant ${\displaystyle h}$ and the Boltzmann constant ${\displaystyle k}$, Wien's constant ${\displaystyle b}$ can be obtained.

Peak differs according to parameterization

The results in the tables above summarize results from other sections of this article. Percentiles are percentiles of the Planck blackbody spectrum. Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law.

Notice that for a given temperature, different parameterizations imply different maximal wavelengths. In particular, the curve of intensity per unit frequency peaks at a different wavelength than the curve of intensity per unit wavelength.

For example, using ${\displaystyle T}$ = 6,000 K (5,730 °C; 10,340 °F) and parameterization by wavelength, the wavelength for maximal spectral radiance is ${\displaystyle \lambda }$ = 482.962 nm with corresponding frequency ${\displaystyle \nu }$ = 620.737 THz. For the same temperature, but parameterizing by frequency, the frequency for maximal spectral radiance is ${\displaystyle \nu }$ = 352.735 THz with corresponding wavelength ${\displaystyle \lambda }$ = 849.907 nm.

These functions are radiance density functions, which are probability density functions scaled to give units of radiance. The density function has different shapes for different parameterizations, depending on relative stretching or compression of the abscissa, which measures the change in probability density relative to a linear change in a given parameter. Since wavelength and frequency have a reciprocal relation, they represent significantly non-linear shifts in probability density relative to one another.

The total radiance is the integral of the distribution over all positive values, and that is invariant for a given temperature under any parameterization. Additionally, for a given temperature the radiance consisting of all photons between two wavelengths must be the same regardless of which distribution you use. That is to say, integrating the wavelength distribution from ${\displaystyle {\lambda }_1}$ to ${\displaystyle {\lambda }_2}$ will result in the same value as integrating the frequency distribution between the two frequencies that correspond to ${\displaystyle {\lambda }_1}$ and ${\displaystyle {\lambda }_2}$, namely from ${\displaystyle c/{\lambda }_2}$ to ${\displaystyle c/{\lambda }_1}$. However, the distribution shape depends on the parameterization, and for a different parameterization the distribution will typically have a different peak density, as these calculations demonstrate.

The important point of Wien's law, however, is that any such wavelength marker, including the median wavelength (or, alternatively, the wavelength below which any specified percentage of the emission occurs) is proportional to the reciprocal of temperature. That is, the shape of the distribution for a given parameterization scales with and translates according to temperature, and can be calculated once for a canonical temperature, then appropriately shifted and scaled to obtain the distribution for another temperature. This is a consequence of the strong statement of Wien's law.

Frequency-dependent formulation

For spectral flux considered per unit frequency ${\displaystyle d\nu }$ (in hertz), Wien's displacement law describes a peak emission at the optical frequency ${\displaystyle {\nu }_{\text{peak}}}$ given by: $${\displaystyle {\nu }_{\text{peak}}=\frac{x}{h}kT\approx (5.879\times {10}^{10}\mathrm{H}\mathrm{z}/\mathrm{K})\cdot T}$$ or equivalently $${\displaystyle h{\nu }_{\text{peak}}=xkT\approx (2.431\times {10}^{-4}\mathrm{e}\mathrm{V}/\mathrm{K})\cdot T}$$ where ${\displaystyle x}$ = 2.821439372122078893... is a constant resulting from the maximization equation, k is the Boltzmann constant, h is the Planck constant, and T is the absolute temperature. With the emission now considered per unit frequency, this peak now corresponds to a wavelength about 76% longer than the peak considered per unit wavelength. The relevant math is detailed in the next section.

Derivation from Planck's law

Parameterization by wavelength

Planck's law for the spectrum of black-body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization. Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is: $${\displaystyle u_{\lambda }(\lambda ,T)=\frac{2h{c}^2}{{\lambda }^5}\frac{1}{e^{hc/\lambda kT}-1}.}$$

Differentiating ${\displaystyle u(\lambda ,T)}$ with respect to ${\displaystyle \lambda }$ and setting the derivative equal to zero gives: $${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\lambda }=2h{c}^2\left(\frac{hc}{kT{\lambda }^7}\frac{e^{hc/\lambda kT}}{{\left(e^{hc/\lambda kT}-1\right)}^2}-\frac{1}{{\lambda }^6}\frac{5}{e^{hc/\lambda kT}-1}\right)=0,}$$ which can be simplified to give: $${\displaystyle \frac{hc}{\lambda kT}\frac{e^{hc/\lambda kT}}{e^{hc/\lambda kT}-1}-5=0.}$$

By defining: $${\displaystyle x\equiv \frac{hc}{\lambda kT},}$$ the equation becomes one in the single variable x: $${\displaystyle \frac{x{e}^x}{e^x-1}-5=0.}$$ which is equivalent to: $${\displaystyle x=5(1-e^{-x}).}$$

This equation is solved by $${\displaystyle x=5+W_0(-5e^{-5})}$$ where ${\displaystyle W_0}$ is the principal branch of the Lambert W function, and gives ${\displaystyle x=}$ 4.965114231744276303.... Solving for the wavelength ${\displaystyle \lambda }$ in millimetres, and using kelvins for the temperature yields:

${\displaystyle {\lambda }_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}=hc/xkT=}$(2.897771955185172661... mm⋅K)${\displaystyle /T}$.

Parameterization by frequency

Another common parameterization is by frequency. The derivation yielding peak parameter value is similar, but starts with the form of Planck's law as a function of frequency ${\displaystyle \nu }$: $${\displaystyle u_{\nu }(\nu ,T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{h\nu /kT}-1}.}$$

The preceding process using this equation yields: $${\displaystyle -\frac{h\nu }{kT}\frac{e^{h\nu /kT}}{e^{h\nu /kT}-1}+3=0.}$$ The net result is: $${\displaystyle x=3(1-e^{-x}).}$$ This is similarly solved with the Lambert W function: $${\displaystyle x=3+W_0(-3e^{-3})}$$ giving ${\displaystyle x}$ = 2.821439372122078893....

Solving for ${\displaystyle \nu }$ produces:

${\displaystyle {\nu }_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}=xkT/h=}$ (0.05878925757646824946... THz⋅K⁻¹)${\displaystyle \cdot T}$.

Parameterization by the logarithm of wavelength or frequency

Using the implicit equation ${\displaystyle x=4(1-e^{-x})}$ yields the peak in the spectral radiance density function expressed in the parameter radiance per proportional bandwidth. (That is, the density of irradiance per frequency bandwidth proportional to the frequency itself, which can be calculated by considering infinitesimal intervals of ${\displaystyle \ln \nu }$ (or equivalently ${\displaystyle \ln \lambda }$) rather of frequency itself.) This is perhaps a more intuitive way of presenting "wavelength of peak emission". That yields ${\displaystyle x}$ = 3.920690394872886343....

Mean photon energy as an alternate characterization

Another way of characterizing the radiance distribution is via the mean photon energy $${\displaystyle ⟨E_{\text{phot}}⟩=\frac{{\pi }^4}{30\zeta (3)}kT\approx (3.7294\times {10}^{-23}\mathrm{J}/\mathrm{K})\cdot T,}$$ where ${\displaystyle \zeta }$ is the Riemann zeta function. The wavelength corresponding to the mean photon energy is given by $${\displaystyle {\lambda }_{⟨E⟩}\approx (0.53265\mathrm{c}\mathrm{m}\cdot \mathrm{K})/T.}$$

Criticism

Marr and Wilkin (2012) contend that the widespread teaching of Wien's displacement law in introductory courses is undesirable, and it would be better replaced by alternate material. They argue that teaching the law is problematic because:

1. the Planck curve is too broad for the peak to stand out or be regarded as significant;
2. the location of the peak depends on the parameterization, and they cite several sources as concurring that "the designation of any peak of the function is not meaningful and should, therefore, be de-emphasized";
3. the law is not used for determining temperatures in actual practice, direct use of the Planck function being relied upon instead.

They suggest that the average photon energy be presented in place of Wien's displacement law, as being a more physically meaningful indicator of changes that occur with changing temperature. In connection with this, they recommend that the average number of photons per second be discussed in connection with the Stefan–Boltzmann law. They recommend that the Planck spectrum be plotted as a "spectral energy density per fractional bandwidth distribution," using a logarithmic scale for the wavelength or frequency.