It is usually completely unnoticed by the senses that all material bodies are constantly emitting and absorbing electromagnetic radiation that originates in the random thermal motion of the electrically charged particles that constitute matter. This remarkable phenomenon is a clear proof of the atomic constitution of matter and its electrical nature. In the present discussion, we shall not have to appeal to electromagnetic theory, which is surprising, but mainly to thermodynamics and statistical mechanics.

When the matter is heated to about 500°C the radiation begins to become visible to the eye as a dull red glow, and may be sensed as warmth by the skin. Further heating rapidly increases the amount of radiation, and its perceived colour becomes orange, yellow and finally white. This radiation is called heat radiation, a literal translation of the German Wärmestrahlung, or thermal radiation, or black body radiation.

Bodies may exchange thermal energy by heat radiation, approaching equilibrium when they are at the same temperature. In most practical cases, however, when the bodies are surrounded by air, convection is a much more important means of energy exchange. If the bodies are in contact, energy can move by conduction as the thermal motions directly interact. Bodies not in contact in a vacuum have only the radiation mechanism for heat energy transfer. An obvious example is the Earth in space, receiving heat radiation from the Sun which is partly absorbed and partly reflected, and emitting heat radiation to space. Equality of these energy fluxes determines the equilibrium temperature of the Earth.

Heat radiation is observed to have a continuous spectrum of energy per unit frequency, rising rapidly to a maximum (as f2 at small f), then decreasing exponentially for large f. This spectrum is a function of T alone, and at any frequency the radiation always increases with T. The explanation of this spectrum on the basis of thermodynamics alone was sought, but was never found. Max Planck found a simple derivation in 1900 by introducing the arbitrary assumption that the oscillators in equilibrium with the radiation took only discrete energies nhf, where n = 0, 1, 2, ... and h = 6.626 x 10-34 J-s is Planck's constant. This was considerably before the discovery of quantum mechanics, so it was theoretically inexplicable. It is celebrated as one of the origins of quantum theory. The interpretation is now different, as we shall soon see when we consider the radiation itself as a system of quantized oscillators.

The system generally considered for studying heat radiation is a material enclosure or "cavity" of volume V of arbitrary nature that is at a uniform temperature T. The volume V contains normal modes of electromagnetic radiation that behave as harmonic oscillators. An oscillator of frequency f has energy levels E = (n + 1/2)hf. The energy for n = 0, hf/2, is called the zero point energy, which has observable effects. Because it will not influence any of our present considerations, it is normally neglected, and we avoid having to consider its infinite value when we have an infinite number of oscillators.

A cavity source makes a practical realization of an ideal black body to a very good approximation, which can be compared with other sources. A shiny metallic surface, on the other hand, approximates an ideal reflector, which radiates very poorly. The approximation is not quite as good, however. A Dewar flask uses a vacuum between inner and outer containers to eliminate convection and conduction, and the surfaces are silvered to prevent radiative transfer of energy.

The number of modes between f and f + df in a system of volume V is (8πV/c3)f2df, called the density of states. This is normally a very large number, roughly equal to the number of cubes with sides equal to wavelength that will fit into V. It is an approximate result, of course, but where it is not accurate it has very little effect on our conclusions. For a derivation, see the References. Most earlier work on thermal radiation used the wavelength as the independent variable, but now it seems that frequency is preferred. Of course, λf = c is the relation between the two variables.

Now we must relate the state of the mode oscillators to the thermodynamic temperature T. Boltzmann showed us how to do this, by means of the quantity exp(-E/kT), which is proportional to the probability of a state of energy E in equilibrium at the temperature T. We have E = nhf, so the average value of n at temperature T will be given by <v> = Σnexp(nhf/kT) / Σexp(nhf/kT), where the sums are from 0 to infinity. The sums are easily evaluated as the sum of a geometric series.

This is done at the right. Note that the numerator sum is the derivative of the denominator sum with respect to x = hf/kT. The spectrum is the product of the density of states and the average energy of each mode. Note the limits for small f and for large f, which have the observed behaviour. The total energy density is obtained by integrating over frequency from 0 to infinity. The result that this is proportional to T4 is Stefan's Law, which can be derived from thermodynamics alone.

At the left, it is illustrated that the product of the wavelength at the maximum of the spectral energy density per frequency interval u(f) and the temperature T is a constant. This result is Wien's Law, and can be derived from thermodynamics (see Planck's book). However, the present derivation is very much simpler and clearer. If we find the spectral energy density per unit wavelength interval, u(λ), we have a different function, though it is of the same general shape. The product of the wavelength at the maximum of this curve and the temperature is again a constant, but a different one. Here we have used the angular frequency ω = 2πf for variety, as is done by Kittel. There is often some confusion of what is meant by λmax, so it is best to be careful.

The Sun's radiation has T = 6000K (5778K more accurately), so λmax = 850 nm. The Earth radiates at about T = 300K (288K more accurately), so λmax = 17μm. These are the maxima of the energy density per unit frequency interval. For the maxima of the energy density per unit wavelength interval, these figures are 491 nm and 10μm. An incandescent lamp radiating at 3000K has the maximum of u(f) at 1.7 μm or maximum of u(λ) at 1.0 μm, in the near infrared. It should be remembered that the spectra are very broad and these wavelengths give only locate them roughly. The extreme limits of the eye's sensitivity are 380nm and 750nm, so visual appearance is often deceiving with respect to ultraviolet or infrared properties.

The standard illuminants for colorimetry are: illuminant A, gas-filled tungsten lamp at 2848K; illuminant B, noon sunlight at 4800K; and illuminant C, average daylight at 6500K. All of these are approximately black-body spectra. Carbon particles or a Welsbach mantle assume the temperature of the otherwise nonradiating blue flame of the fuel.

So far we have considered only the energy density u. All this energy is moving at speed c = 2.99792 x 108 m/s, and since it is isotropic, in solid angle 4π sr. The flux in W m-2 sr-1 is then (c/4π)u. The total energy flux through an area dA is found by integrating over a solid angle 2π, remembering that the effective area at an angle θ is cos θ dA. The geometric integral has the value π, so the total flux, or exitance, is just (c/4)u. This situation can be considered as a hole of area dA in the wall of the volume containing the radiation.

An actual surface will not absorb all the radiation falling on it. Some will be reflected, diffusely or specularly, and some may be transmitted. A fraction α, usually a function of frequency will, however be absorbed. It is quite reasonable to consider that this gives the efficiency of the interaction with the atomic charges whose thermal motion is affected by the radiation. This is made quantitative by the important Kirchhoff's Law, which states that a surface of a given α will radiate with that same fraction of the radiation we have just calculated exiting from a hole in a thermal enclosure. That is, the exitance we have calculated is that of a body with α = 1 at all frequencies, called a black body. The walls of our cavity can have any absorptivity α--as long as it is not exactly zero, the radiation will come into equilibrium. The fraction of the black body radiation that a certain body will radiate is called the emissivity ε, and Kirchhoff's Law is ε = α at all frequencies.

The emissivity is usually a function of frequency. For example, a white paint may have a large emissivity at infrared frequencies (0.93 is quoted) while it appears white to the eye, so that it is effectively black as far as thermal radiation is concerned.

Now let us find the pressure of the radiation on the confining surface. This can be found from the free energy F of the radiation, according to the prescriptions of thermodynamics. At the left, it is shown how to find the free energy and from it the pressure, using the statistical mechanics of the radiation system. A sum we have seen before, the sum of the probabilities of all states, is called the partition function, and from it the free energy can be found. The partition function for the complete system is the product of the partition functions for each mode. Since we take the logarithm to find the free energy, this becomes a sum which we can easily evaluate using the density of states. An integration by parts then gives the energy density divided by 3. The pressure is the derivative of F with respect to V, holding T constant, which is very easy to do here. The final result is that the pressure is the energy density divided by 3. This is often called the Maxwell radiation pressure, and can also be calculated from electrodynamics, which is much more difficult.

It is easy to show that the photon picture of the radiation gives the result almost at once. The amount of energy directed in one cartesian direction is cu/6. The momentum of a photon of energy E is E/c, so the momentum flux is u/6. Since the momentum is reversed on reflection, the total momentum flux, which is the pressure, is 2 x u/6 = u/3. Although this is for a perfectly reflecting wall, it has the same value for a black wall, since a black wall will emit exactly the same energy flux as it absorbs. It is also easy to see that the pressure cannot depend on the emissivity of the wall by considering a cylindrical container with one end inside shiny, the other end black. If the pressure were different on the two ends, the cylindrical system would accelerate in one direction or the other without any external force, which is impossible.

With this result for the pressure, we can now easily prove Stefan's Law from thermodynamics alone. We assume only that u is a function of T only, and that the pressure is u/3. The proof is shown in the box on the right.

At stellar temperatures (greater than 107K) the radiation pressure is large enough to support the star against gravitational collapse. When the nuclear fuel is exhausted, the decrease in temperature allows the star to collapse to a white or black dwarf.

There has been some enthusiasm for painting roofs white to reduce heat absorption from sunlight. The paint should be selected with care, since as we have seen its infrared emissivity (and so its absorptivity) may be high--that is, the white paint will be black in the infrared, and may actually absorb more heat than the previous surface. A shiny metal roof may be a better choice.

### References

Max Planck, The Theory of Heat Radiation, 2nd Ed. (New York: Dover, 1959).

P. M. Morse, Thermal Physics (New York: Benjamin, 1961) Chapter 25.

C. Kittel, Thermal Physics (New York: John Wiley & Sons, 1969) Chapter 15.

Thermal Radiation Online course; good plots of u(λ).