Many commonly encountered light sources, including the sun and incandescent light bulbs, are closely modelled as "blackbody" emitters. A blackbody absorbs all radiation incident on its surface and emits radiation based on its temperature. Blackbodies derive their name from the fact that, if they do not emit radiation in the visible range, they appear black due to the complete absorption of all wavelengths. The blackbody sources which are of interest to photovoltaics, emit light in the visible region. The spectral irradiance from a blackbody is given by Planck's1 radiation law, shown in the following equation:

## Planck's Radiation Law

where:

*λ *is the wavelength of light;

*T* is the temperature of the blackbody (K);

*F* is the spectral irradiance in Wm^{-}^{2}µm^{-1}; and

*h,c* and k are constants.

Getting the correct result requires care with the units. The simplest is to use SI units so that *c* is in m/s, *h* is in joule·seconds, *T* is in kelvin,* k* is in joule/kelvin, and *λ* is in meters. This will give units of spectral irradiance in Wm^{-3}. Dividing by 10^{6} gives the conventional units of spectral irradiance in Wm^{-2}µm^{-1}. The notation of F(λ) denotes that the spectral irradiance changes with wavelength.

The total power density from a blackbody is determined by integrating the spectral irradiance over all wavelengths which gives:

## Blackbody Power Density

where σ is the Stefan-Boltzmann constant and T is the temperature of the blackbody in kelvin.

An additional important parameter of a blackbody source is the wavelength where the spectral irradiance is the highest, or, in other words the wavelength where most of the power is emitted. The peak wavelength of the spectral irradiance is determined by differentiating the spectral irradiance and solving the derivative when it equals 0. The result is known as Wien's Law and is shown in the following equation:

## Wien's Law

where λ_{p} is the wavelength where the peak spectral irradiance is emitted and

T is the temperature of the blackbody (K).

## Peak Wavelength, Radiation Intensity Calculator

Drag the slider at the bottom of the graph to see the change in the blackbody radiation spectrum as the temperature is increased from 1000 to 6000 K. MATLAB/Octave Code.

The above equations and animation show that as the temperature of a blackbody increases, the spectral distribution and power of light emitted change. For example, near room temperature, a blackbody emitter (such as a human body or light bulb which is turned off) will emit low power radiation at wavelengths predominantly greater than 1µm, well outside the visual range of human observation. If the blackbody is heated to 3000 K, it will glow red because the spectrum of emitted light shifts to higher energies and into the visible spectrum. If the temperature of the filament is further increased to 6000 K, radiation is emitted at wavelengths across the visible spectrum from red to violet and the light appears white. The graphs below compare the spectral irradiance of a blackbody at these three temperatures. The room temperature case of 300K (the black dotted line) has essentially no power emitted in the visible and near infrared portions of the spectrum shown on the graph. Because of the huge variation in both emitted power and the range of wavelengths over which the power is emitted, the log graph below demonstrates more clearly the variation in the emitted blackbody spectrum as a function of temperature.

- 1. , “Distribution of energy in the spectrum”, Annalen der Physik, vol. 4, pp. 553-563, 1901.

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