My goal in this section is to keep the math as simple as possible. Those interested in a fuller treatment should consult some of the sources that I list. In spite of my goal, there is some unavoidable math in this discussion, and some of the equations are not simple. I try to explain what the equations mean in words. I realize that some readers will simply skip the equations: that's okay; I hope that the explanations still make some intuitive sense.

**What is a Black Body?**

Black objects are black because they absorb all colors of visible light and only a negligible amount of light is reflected. Imagine the ideal case of this phenomenon. Imagine a body that absorbs all wavelengths of light, regardless of the wavelength with 100% efficiency so that no light is reflected or transmitted. Also, imagine that this behavior extends beyond the visible into all relevant regions of the spectrum. It turns out that such a body is also a perfect emitter of radiation, if heated.

Such a body is a black-body. It absorbs and emits radiation, but it does not reflect or transmit radiation.

A room temperature black-body emits radiation in the infrared. As it is heated, it emits more radiation. The previous section on infrared radiation discussed the fact that as the temperature of a body changes, the wavelength shifts as well. The reader of that section will remember that hotter bodies emit shorter wavelength (higher frequency) radiation. So as bodies are heated, they emit more radiation and the wavelengths of the distribution shifts. These two relationships can be quantified.

**Quantitative Relationships for a Black-body**

These two relationships between temperature and IR were noticed a long time ago. The first relationship, the fact that heated objects radiate more can be expressed as Stefan's Law:

M = σT

^{4}

M is the excitance of the radiation, or power per square meter in Watts per square meter. In the graph shown (red line), the excitance is expressed in kilowatts (kW/m

^{2}). T is the temperature in Kelvin. Kelvin scale is the Celsius temperature scale, shifted so that zero is absolute zero. Absolute zero is -273.15 degrees Celsius, so to get the temperature in K, simply add 273.14 to the temperature in Celsius. Notice that temperature is taken to the fourth power. This fact indicates a very strong relationship, a slight increase in temperature leads to a large increase in the excitance of the radiation.

σ is the constant of proportionality. This constant is called the Stefan-Boltzmann constant:

σ = 5.67 x 10

^{-8}W/(m

^{2}K

^{4})

The second relationship, the fact that the wavelength changes as Temperature changes is can be quantified as Wien's Displacement Law:

Tλ

_{max}= B

In this equation λmax is the wavelength at which the maximum radiation is emitted. T is the temperature in Kelvin, and B is a constant, whose value is 2.9 x 10

^{-3}in meters * Kelvin. This equation can also be written:

λ

_{max}= B/T

It says that the wavelength at which the maximum radiance occurs is inversely proportional to the Temperature. Remember that the frequency is also inversely proportional to the wavelength:

λ

_{max}= c/ ν

_{max}

We can combine these two expressions to get an expression for the maximum frequency:

ν

_{max }= cT/B

As we increase the temperature of a black-body, the frequency at which the maximum radiance occurs also increases.

**Planck's Law**

I have characterized the relationship between the total radiance as a function of temperature and also the frequency dependence of the frequency of maximum radiance, but what is really needed is a full expression that relates the amount of radiation emitted at any given frequency (or wavelength) as a function of temperature. The equation that describes this relationship is called Planck's Law. This equation is somewhat complicated and its derivation is even more complicated. I am not going to derive it here; the interested reader should refer to some of the sources I cite (each of which derives it in a slightly different manner).

The main point to understand here about Planck's Law is that it provides a relationship between the radiance emitted at a given frequency from a black-body to the temperature of that body:

I = 2 h ν

^{3}/c

^{2}(exp(hν/kT)-1)

I warned you that the equation is complicated. I is the spectral radiance. It is a measure of how much radiation is emitted. The other quantities except k should be familiar to this who have read this far. The constant h is Planck's constant, the proportionality constant between energy and frequency that was visited in the previous post. ν is the frequency. c is the speed of light. The expression "exp()" is the natural exponential function. exp(x) could also be written e

^{x}. T is the absolute temperature in Kelvin.

The constant k is called Boltzmann's constant. It is a constant that arises out of the field of thermodynamics, a field that I am only touch on here. Some readers may be familiar with the universal gas constant, R, and Avogadro's number N. Boltzmann's constant is simply R divided by N.

From Planck's Law, it is possible to draw curves of spectral radiance as a function of frequency or wavelength for a given temperature.

As expected the curves of higher temperature shift to shorter wavelength (higher frequency).

**Experimental Black-Bodies**

The black-body is a theoretical entities. Real bodies behave somewhat differently. In fact, no real physical object will exactly exhibit the black-body curve. There are devices, however, that very closely approximate black-bodies. These experimental black-bodies can be very expensive, costing 10s of thousand of dollars depending upon their size. They are sold by companies such as Elecro-Optics Incorporated (EOI).

Figure source.

**Real Bodies and Kirchoff's Radiation Law**

Black-bodies are important to understand because they provide a theoretical limit to the ability of real materials to emit and absorb infrared radiation. To address the fact that real bodies do not emit according to Planck's Law a factor called the emissivity, ε

**.**The emissivity is a correction factor for the fact that real bodies do not emit according to Planck's Law:

I

_{emitted}(ν) = ε(ν)I

_{BB}(ν)

The the spectral radiance of a real body is its emissivity multiplied by the radiance expected from an ideal black-body. In general the emissivity depends upon the frequency. The Planck curve is also frequency dependent; so the radiance as a function of frequency of a real body is the product of two frequency dependent functions: the emissivity and the Planck's Law curve. I can rearrange this equation to obtain the emissivity:

ε(ν) = I

_{emitted}(ν)/I

_{BB}(ν)

Real bodies also absorb radiation. Let I

_{incident}be the incident radiance on body. The radiation that will be absorbed by that body is:

I

_{absorbed}(ν) = α(ν)I

_{incident}(ν)

Like the emissivity, the absorptivity, α, is a function of frequency. In the case of a blackbody, α=1, by definition, but for real bodies less than 100% of the incident radiance will be absorbed. I can also rearrange this equation:

α(ν) = I

_{absorbed}(ν)/I

_{incident}(ν)

Kirchoff was interested in the case of an enclosed chamber made out of real material

*at thermal equilibrium*, where no radiance can escape from the chamber. The energy absorbed must be equal to the energy radiated. There is a further consideration that is a little more complex to understand, but it is a consequence of the second law of thermodynamics that the radiation distribution inside the chamber must be distributed according to Planck's Law. A result of Kirchoff's Law is:

α(ν) = ε(ν)

The absorptivity of a body is equal to its emissivity. A direct consequence of this law is that a real body that is

*at thermal equilibrium*cannot emit more than a black-body at that same temperature. Remember that a black-body absorbs all of the incident radiation. A body cannot absorb more radiation than is incident upon it, and therefore real bodies at best absorb as much as a black-body. According to Kirchoff's Law as long as the body is in thermal equilibrium, it cannot emit more than a black-body either.

**Brightness Temperature**

It is possible to remotely observe bodies from their IR spectra, but one does not necessarily know what the emissivity of the objects being observed. One can assign a temperature to the IR spectra observed based upon Planck's Law. The radiance observed at a given wavelength is converted to the Temperature at which a black-body would emit the same radiance at the same wavelength. This temperature is, in general, wavelength dependent, i.e., the same object will emit at different brightness temperatures in different parts of the spectrum. Also note, that the brightness temperature is in general, not equal to the thermodynamic temperature, although it may be close for some materials. If one is observing the emission spectrum of a body in the infrared, and the body is at equilibrium, or nearly so, then the object is at least as hot as the brightness temerature and probably hotter.

**Other Considerations**

Not all matter absorbs in the infrared. In the next post, I explain why atmospheric molecules such as N

_{2}, and O

_{2}do not absorb or emit infrared radiation at all. Additionally, many materials absorb in some parts of the infrared, but not others. Many materials, have specific band structures in the absorption spectra that can be understood by the interaction of radiation with matter on a molecular or inter-atomic level. To understand climate science and global warming, it is necessary to delve into these interactions for some of the gases of interest,such as water vapor, carbon dioxide, methane, nitrous oxide, and others. The next post starts the discussion of the interaction of radiation with molecules.

**Sources**

- Atkins, P. W.
*Physical Chemistry*, New York: W. H. Freeman and Company, New York, 3rd edition, 1986 - McQuarrie, Donal d A.,
*Statistical Thermodynamics*, University Science Books, Mill Valley, CA, 1973 - http://en.wikipedia.org/wiki/Planck%27s_law
- Steinfeld, Jeffrey I,
*Molecules and Radiation*, The MIT Press, Cambridge, MA, 2nd edition, 1985 - Struve, Walter S.,
*Fundamentals of Molecular Spectroscopy*, John Wiley & Sons, New York, 1989 - Hecht, Eugene,
*Optics*, 4th edition, Addison Wesley, San Francisco, CA, 2002 - http://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation

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