## Blackbody Thermal Absorption and Emission

A blackbody is defined as an idealized material that absorbs all radiation incident on it, irrespective of the wavelength and of the angle of incidence. Reflection and transmission are therefore absent. As will be discussed here, the absorption and emission properties of a blackbody can be derived from one another, so that no new definition of the blackbody will be needed to describe its emission.

The expression "blackbody"—which has historical roots — is somewhat inaccurate, because, as is clear from the definition, almost always only the surface of the body is important. Thus, if there is heat flowing by conduction to the surface, there will be a distribution of temperature in the body different from the surface temperature. Yet, conventionally, one still refers to the latter in this case as "blackbody temperature."

There are no materials that match exactly the definition of a blackbody, although some — such as black paint in the visible range of wavelengths— come very close. Ideal blackbodies, however, are an important standard with which real surfaces can be compared.

The following thought experiment, illustrated in Fig. 7.1, allows one to draw some simple conclusions from purely thermodynamic arguments: One considers an evacuated enclosure at a uniform temperature and of arbitrary shape. The exterior is thermally insulated. The interior surface emits and absorbs as an ideal blackbody. Also, there is a test body (assumed for simplicity to be convex so that radiation from it strikes only the enclosure) which is also a black body.

Initially, the temperatures of enclosure and test body are arranged to be the same. It then follows from the first and second laws of thermodynamics

Figure 7.1 Schematic illustrating derivation of Lambert's law.

Figure 7.1 Schematic illustrating derivation of Lambert's law.

that neither temperature can either increase or decrease and that the test body's emission equals its absorption. Because the argument is independent of the test body's position or orientation, the radiation in the space between enclosure and test body must be homogeneous (i.e., has constant radiation density) and isotropic.

In Fig. 7.1a let dS be a surface element of the enclosure and p the angle from the normal n of dS to some chosen direction of emitted radiation. Also let dco be the solid angle bounding this direction.

Whereas in Fig. 7.1a all rays emitted from a point of dS that fall within the solid angle dco are shown, in Fig. 7.1b the rays from all points of dS are shown that converge to an aperture with area dB (located at point P at distance R.) Similarly, a second surface element dS' is shown, with rays converging to an aperture with the same area dB at P and at distance R'. The area dB can be chosen arbitrarily small compared with dS and dS'. Also, for convenience, dS and dS' are chosen such that the solid angles dQ extended from the apertures to dS and dS' are the same.

The radiation from a point of dS or dS' striking the respective aperture therefore has directions bounded by the solid angles dco = dB/R2 and dco' = dB/R'2

Another such geometrical relation, evident from Fig. 7.1.b, is dQ, = dS cos p/R2 = dS' cos p'/R'2

Let e\ypx dS dco dX be the black body radiant power from dS, emitted within a cone with solid angle dco centered about ft and within a wavelength interval dX centered about X. The e^p, is referred to as the directional, monochromatic emitted power (per unit area and unit wavelength interval; in the present case for a blackbody). It is a function of p, X, and T, and its magnitude is usually expressed in units of W/(m2 /xm). Analogous statements hold for e^u-

Isotropy of the radiation at point P then requires that the two radiant powers be equal, hence e^px dS dco dX = e\,p>x dS' dco' dX

Substitution of the geometrical relations found for dco, dco', and dQ yields

In particular, when expressing the power emitted in a direction given by p with the power emitted along the normal, one obtains by setting p' = 0 the important relation ebpk = iwcos P (7.1')

The subscript notation used here and in what follows is more or less standard in the theory of thermal radiation. Thus n refers to the normal direction. Omitting among the subscripts the letter b means that a surface more general than a blackbody is considered. Omitting the angle p or n means that the radiant power has been integrated over all pertinent angles

(usually a half-space). Omitting the wavelength X means that the radiant power has been integrated over all wavelengths.

Equation (7.1') is known as Lambert's cosine law (Lambert, astronomer and physicist, 1728-1777). The dependence on p only through the cosine has a simple representation: As is easily shown, if the radiant power of a surface element is drawn as the lengths of vectors pointing along the directions of propagation of the radiation, the end points are located on a sphere that is tangent to the radiating surface.

The radiant power, e bx, per unit area and unit wavelength interval, emitted into the half-space above dS, is known as the hemispherical, monochromatic emitted power (per unit area and unit wavelength interval) and is readily obtained by integrating ebpX over the half-space:

rn/2

It was first shown by Planck (1858-1947), based on an electromagnetic calculation of the emission and treating the photons in the enclosure as a perfect gas, that

where Cx = he2 = 0.59544 10~16 W m2 and C2 = hc/k = 1.4388 104 pm K (.h = Planck's constant = 6.6252 10"34 ]s;k = Boltzmann's constant = 1.3806 10"23 kg m2/(s2K); c = speed of light in vacuum = 2.99792 10® m/s). Common units for ebx are W/(m2 //m), therefore the same as for ebpk.

Equation (7.3) is known as Planck's radiation law. Its derivation became the basis for the later development of quantum mechanics.

In Fig. 7.2, the hemispherical monochromatic emitted power ebx is plotted for several temperatures that are important in space technology. The curve labeled 5760 K corresponds to the effective blackbody temperature of the solar photosphere. Its radiation, to a large extent, determines the skin temperature of spacecraft. The curve labeled 300 K is representative of the majority of spacecraft components. They are designed for, and work best, in a fairly narrow temperature range around room temperature (e.g., storage batteries in the range of 5 to 30°C). The third curve applies to the boiling temperature (77 K) of nitrogen. This temperature is typical of many scientific instruments that work at infrared wavelengths. Liquid nitrogen is also used in spacecraft thermal testing, where it cools the test chamber walls, thereby suppressing unwanted thermal radiation from the chamber.

With increasing temperature, the maximum emitted power shifts toward shorter wavelengths. If Amax designates the wavelength for which at a chosen temperature the hemispherical monochromatic emitted power has a maximum, then

where the constant C3 = 2.8978 103 ¡im K. First established experimentally,

Figure 7.2 Planck's radiation law (Eq. 7.3) for three temperatures significant in space technology: (1) effective temperature of sun's photosphere, (2) ambient temperature, (3) liquid nitrogen temperature.

but also an easily derived consequence of (7.3), the equation is known as Wien's displacement law (Wien, 1864-1928).

The factor e i-,^ of the directional, monochromatic emitted power, expressed by ebx fromPlanck's radiationlaw, becomes, byusing (7.1) and (7.2), ebßX=7t~le bxcosyß (7.5)

The hemispherical emitted power (per unit area), e^, is defined as the integral of eh, over all wavelengths. The integration over Planck's relation can be carried out in closed terms (although this is not immediately obvious). The result is the Stefan-Boltzmann law, originally found by experiment, eb = aTi (7.6)

From this, the factor et,/; of the directional emitted power, that is, the power emitted per unit solid angle in the direction defined by the angle ß, when integrated over all wavelengths is ehß = cosß er T4 (7.7)

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