BLACKBODY RADIATION

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Blackbody radiation physics is used extensively in infrared imaging system design and performance analyses. Blackbody radiation formalism is used often in this document. Blackbody radiation units and terminology can be confusing. When dealing with the blackbody radiation laws, care must be taken to avoid common errors by factors of 2, [[pi]], or 10. This appendix provides a brief review of Planck's radiation laws and terminology. The terminology has been adopted from Vincent [VINCENT90].

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What is a blackbody?* A blackbody is defined as an ideal body that completely absorbs all radiant energy striking it and, therefore, appears perfectly black at all wavelengths. A blackbody is also a perfect emitter of radiation; one that, for a given temperature, radiates the maximum number of photons possible per unit time per unit area in a specific spectral interval.

From the temperature T of a blackbody, Planck's radiation law determines the quantity of radiation Mq([[lambda]],T) emitted by a blackbody, per cm^(2) of blackbody area, as a function of wavelength [[lambda]] according to:

photons/(cm^(2)*sec*um) (A-1)

where c = 3 x 10^(10) cm/sec (speed of light), h = 6.626 x 10^(-34) J*sec (Planck's constant), k = 1.381 x 10^(-23) J/degrees K (Boltzmann's constant), [[lambda]]= wavelength in cm, T = temperature in degrees K.

Mq is called *spectral photon exitance*. (Exitance is the ANSI standard name. Others refer to this quantity as emittance.) The word *spectral* indicates that the quantity is expressed per unit wavelength. If an integration is performed over a wavelength band, or over all wavelengths, the term spectral is dropped and we obtain *photon exitance*.

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Spectral radiant exitance* Me([[lambda]],T) has units of W/(cm^(2)*um) and is related to the spectral photon exitance Mq by:

W/(cm^(2)*um). (A-2)

We are oftentimes interested in the amount of radiation emitted in a certain direction per unit solid angle. *Photon sterance,* Lq, or *radiant sterance,* Le, are the terms and symbols used for quantities normalized to unit solid angle. Figure A-1 (Solid Angle from Blackbody to Detector Aperture) illustrates a detector observing a blackbody from a distance. The aperture of the detector subtends a solid angle [[Omega]] at the differential area dA of the blackbody.

Spectral photon sterance is obtained by dividing the spectral photon exitance by [[pi]]:

photons/(cm^(2)*sec*um*sr) (A-3)

It is tempting to divide by 2[[pi]] , but the correct factor is [[pi]]. If the detector views the differential area at an angle, [[theta]] ,the differential area will appear reduced by a factor of cos([[theta]] ). The factor of [[pi]] accounts for this cos([[theta]] ) area reduction averaged over a hemisphere. Spectral photon sterance is the rate at which photons are emitted per unit area, per unit wavelength, per unit solid angle.

Spectral radiant sterance curves for several temperatures are shown in Figure A-2 (Blackbody Spectral Radiant Sterance). For a fixed wavelength, as the temperature increases the spectral radiant sterance increases. Also, as the temperature increases, the peak of the curves shift toward shorter wavelengths. The wavelength of the maximum [[lambda]]m (in um) is determined from Wien's displacement law:

(A-4)

If the spectral radiant exitance is integrated over all wavelengths, the result is the power per unit area (called *radiant exitance* Me [with no arguments [[lambda]],T]). This is the area under the spectral radiant exitance curve (or [[pi]] times the area under a spectral radiant sterance curve of Figure A-2). This is the classic Stefan-Boltzmann law:

W/cm^ (2 ) (A-5)

where s = 5.670 x 10^(-12) W/(cm^(2)*K^(4)); the Stefan-Boltzmann constant. (If Mq were integrated over all wavelengths the result would be in photons/sec/cm^(2) and the Stefan-Boltzmann constant is 1.52 x 10^(11) photons/(sec*cm^(2)*K^(3)).

When only a wavelength band, from [[lambda]]1 to [[lambda]]2 is integrated, we will use the notation:

or (A-6)

We also will have occasion to use the temperature derivative .

A blackbody is an ideal radiator. The spectral emissivity e(l) of a graybody is defined as the ratio of the spectral photon (or radiant) exitance of the graybody to that of a blackbody:

. (A-7)

The inverse Planck function is used to derive the temperature of a blackbody from its *measured* photon exitance *M*. Solving equation A-1 for T gives:

. (A-8)

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