Radiation in Equilibrium with Matter

We often are told that when bodies are heated they radiate or that "hot" bodies radiate. True enough, but it is just as true that when bodies are cooled they radiate and that "cold" bodies radiate. All matter - gaseous, liquid, or solid - at all temperatures emits radiation of all frequencies at all times, although in varying amounts, possibly so small at some frequencies, for some materials, and at some temperatures as to be undetectable with today's instruments (tomorrow's, who knows?). Note that there is no hedging here: all means all. No exceptions. Never. Even at absolute zero? Setting aside that absolute zero is unattainable (and much lower than temperatures in the depths of the Antarctic winter or in the coldest regions of the atmosphere), even at absolute zero radiation still would be associated with matter because of temperature fluctuations. Temperature is, after all, an average, and whenever there are averages there are fluctuations about them.

Radiation emitted spontaneously, as distinguished from scattered radiation (see Ch. 3), is not stimulated by an external source of radiation. Scattered radiation from the walls of the room in which you read these words may be stimulated by emitted radiation from an incandescent lamp. Turn off the lamp and the visible scattered radiation vanishes, but the walls continue to emit invisible radiation as well as visible radiation too feeble to be perceptible.

We are interested in the spectral distribution of radiation - how much in each wavelength interval - emitted by matter. Consider first the simpler example of an ideal gas in a sealed container held at absolute temperature T (Fig. 1.1). When the gas is in equilibrium its molecules are moving in all directions with equal probability, but all kinetic energies E are not equally probable. Even if all the molecules had the same energy when put into the container, they would in time have different energies because they exchange energy in collisions with each other and the container walls. A given molecule may experience a sequence of collisions in which it always gains kinetic energy, which would give it a much greater energy than average. But such a sequence is not likely, and so at any instant the fraction of molecules with kinetic energy much greater than the average is small. And similarly for the fraction of molecules with kinetic energy much less than the average. The distribution of kinetic energies is specified by

Figure 1.1: At equilibrium, ideal gas molecules in a closed container at absolute temperature T have a distribution of kinetic energies (Fig. 1.2) determined solely by this temperature.

a probability distribution function f (E) which, like all distribution functions, is defined by its integral properties, that is, fE 2

is the fraction of molecules having kinetic energies between any two energies E\ and E2. Note that f does not specify which molecules have energies in a given interval, only the fraction, or probability, of molecular energies lying in this interval. If f is continuous and bounded then from the mean value theorem of integral calculus f 2 f (E) dE = f (E)(E2 - Ei),

where E lies in the interval (Ei, E2). If we denote Ei by E and E2 by E + A.E we have

1 re+ae

Because of Eq. (1.7) f (E) is sometimes called a probability density. When the limits of the integral in Eq. (1.5) are the same (interval of zero width) the probability is zero. The probability that a continuous variable has exactly a particular value at any point over the interval on which it is defined is zero, as it must be, for if it were not the total probability would be infinite.

A distribution function such as f (E) is sometimes defined by saying that f (E) dE is the fraction (of whatever) lying in the range between E and E + dE. This is sloppy mathematics because although E represents a definite number dE does not. Moreover, this way of defining a distribution function obscures the fact that f is defined by its integral properties. As we shall see, failure to understand the nature of distribution functions can lead to confusion and e error. It would be better to say that f (E) AE is approximately the fraction of molecules lying between E and E + AE, where the approximation gets better the smaller the value of AE.

You also often encounter statements that f (E) is the fraction of molecules having energy E per unit energy interval. This can be confusing unless you recognize it as shorthand for saying that f (E) must be multiplied by AE (or, better yet, integrated over this interval) to obtain the fraction of molecules in this interval. This kind of jargon is used for all kinds of distribution functions. We speak of quantities per unit area, per unit time, per unit frequency, etc., which is shorthand and not to be interpreted as meaning that the interval is one unit wide.

Gases within a sealed container held at constant temperature evolve to an equilibrium state determined solely by this temperature. In this state the distribution function for molecular kinetic energies is the Maxwell-Boltzmann distribution

where &b, usually called Boltzmann's constant, is 1.38 x 10~23 J K_1, and f is normalized

The limits of integration are symbolic: molecules have neither infinite nor zero kinetic energies; by zero is meant C kBT and by infinite is meant > kBT. Because of Eq. (1.9) f (E) is a probability distribution function.

The most probable kinetic energy Em is that for which f is a maximum, the energy at which its derivative with respect to E is zero:

As the temperature of the gas increases so does the most probable kinetic energy of its molecules. Figure 1.2 shows f relative to its maximum as a function of E relative to Em, a universal curve independent of temperature.

What does all this have to do with radiation? Because matter continuously emits radiation, a container with walls so thick that no photons leak from it will fill with a gas of photons (Fig. 1.3). The container is held at a fixed temperature T. At equilibrium the photons in the container, like gas molecules, do not all have the same energy (equivalently, frequency) but are distributed about a most probable value. The distribution function for the energies of photons in equilibrium with matter goes under various names and there are several versions of this function differing by a constant factor. Imagine a plane surface within the container. At equilibrium, the radiation field is isotropic, so regardless of how the surface is oriented the same amount of radiant energy crosses unit area in unit time. We consider only that radiant energy (photons) propagating in a hemisphere of directions either above or below the surface. The energy distribution function (or spectral distribution) is given by the Planck distribution (or Planck function)

hw3 1

o to

Energy/Energy at Peak

Figure 1.2: Distribution of kinetic energies of an ideal gas at equilibrium shown as a universal function independent of temperature. The kinetic energy relative to that at the peak of the distribution function, however, does depend on temperature.

The integral of this function over any frequency interval is the total radiant energy in that interval crossing unit area in unit time, called the irradiance (discussed in more detail in Sec. 4.2).

The Planck function is worthy of respect, if not awe, in that it contains not one, not two, but three fundamental (or at least believed to be so) constants of nature: the speed of light in a vacuum c, Planck's constant h, and Boltzmann's constant kB. You can't get much more fundamental than that.

The most probable photon energy is obtained by setting the derivative of Pe with respect to w equal to zero; the result is the transcendental equation where x = hw/kBT, the solution to which (obtained quickly with a pocket calculator) is x = 2.819. Thus the most probable photon energy is

Note the similarity of Eq. (1.11) to Eq. (1.8) and Eq. (1.13) to Eq. (1.10), which is not surprising given that both are distribution functions for gases, although of a different kind. The most striking difference between a gas of molecules and a gas of photons is that the number of molecules in a sealed container is conserved (barring chemical reactions, of course) whereas the number of photons is not. As the temperature of the container, which is the source of the photons, increases, the number of photons within it increases. Photons are not subject to the same conservation laws as gas molecules, which are endowed with mass.

At frequencies for which hw C kBT Eq. (1.11) can be approximated by

Figure 1.3: An opaque container at absolute temperature T encloses a gas of photons emitted by its walls. At equilibrium, the distribution of photon energies (Fig. 1.4) is determined solely by this temperature.

Folks interested in radiation of sufficiently low frequency (e.g., microwaves) sometimes express radiant power as a temperature. When first encountered this can be jarring until you realize that the Planck function is proportional to absolute temperature at such frequencies.

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