The Shift in Black-Body Radiation

Wien’s displacement law, often called “Wien’s law,” states that the peak wavelength of black-body radiation is inversely proportional to temperature. As temperature increases, the peak wavelength shifts to shorter values, while lower temperatures result in longer peak wavelengths. This inverse relationship between wavelength and temperature is a direct consequence of Planck’s radiation law, which describes the spectral intensity of black-body radiation as a function of wavelength for a given temperature. In simpler terms, hotter objects emit radiation at shorter wavelengths, while cooler objects emit radiation at longer wavelengths.

Wien’s displacement law specifies that the spectral radiance of black-body radiation reaches its maximum at a particular wavelength [math]\lambda _{peak}[/math], which is determined by the black body’s temperature [math]T[/math].

[math]\lambda _{peak}=\frac{b}{T}[/math]

The constant b is a constant of proportionality called Wien’s displacement constant. 

[math]b=2.897\ \cdot \ 10^{-3}\ m\ K[/math]

Figure 1 depicts black-body radiation as a function of wavelength for different temperatures. Each temperature curve reaches its peak at a distinct wavelength, with Wien’s law describing the shift in peak position as temperature changes. 

Wien’s Displacement Law does not accurately predict long wavelengths and low temperatures. A continuous curve cannot be obtained from Wien’s Law at low temperatures. As seen in Figure 1, the graph of peak wavelength begins to asymptote as the wavelength increases and the temperature decreases. Moreover, Planck’s Law broadens as the temperature increases.

Wien’s displacement law can be derived and approximated from Planck’s law. To do so, Planck’s spectral radiance function [math]B_{\lambda}\left(\lambda,T\right)[/math]  is differentiated for wavelength, [math]\lambda[/math], and the result is zero for calculating the peak’s position.

[math]\frac{d}{d\lambda}\cdot B_{\lambda}\left(\lambda,T\right)=\frac{hc}{\lambda\cdot k_{B}\cdot T}\frac{e\frac{hc}{\lambda\cdot k_{B}\cdot T}}{e\frac{hc}{\lambda\cdot k_{B}\cdot T}-1}-5=0[/math] 

Defining [math]x\ =\ \frac{hc}{\lambda k_B T}[/math], the derived Planck’s function can be simplified as it becomes one in the single variable. 

[math]\frac{x\ e^x}{e^x-1}-5=\ …\ =\ e^x\ (x-5)\ +5=0[/math] 

This equation must be solved numerically using Newton’s Method, resulting in [math]x\ =\ 4.965[/math]. Using this for the relationship for peak wavelength, [math]\lambda _{peak}[/math], the Wien’s constant can be derived.

[math]\ \lambda _{peak}\ =\ \frac{h\ c}{x\ k_BT}\ =\ \frac{h\ c}{4.965\ k_B}\ \frac{1}{T}=\ \frac{b}{T}\ [/math]

About 25% of the total infrared energy lies at wavelengths shorter than the peak, and the other 75% lies at wavelengths longer than the peak [4]. There, infrared radiometry and long wavelength sensors can most often be tuned to measure high temperatures, but short wavelength sensors have challenges measuring small temperatures when the wavelength peak is still in the long wavelength domain.