$\>\!$ hysics  

April 3, 2003

Thermal Radiation:
Blackbody Radiation; the Stefan-Boltzmann Law,
and the Planck formula for Spectral Radiancy

I.  Introductory Thoughts

Refer to HRW Section 19-11

All objects give off radiation, i.e., electromagnetic waves, because of their temperature; we refer to this as thermal radiation. As well they absorb such radiation from their surroundings. If a body is hotter than its surroundings it emits more radiation than it absorbs, and tends to cool; if a body is cooler than its surroundings it absorbs more radiation than it emits, and tends to warm. Usually it will eventually come to thermal equilibrium with its surroundings: a condition in which its rates of absorbtion and emission of radiation are equal.

The spectrum of the thermal radiation from a hot solid body is continuous; that is, it emits electromagnetic radiation in a continuous range of wavelengths (or frequencies), over a rather large band of allowed wavelengths. It is usual for the amount of radiation at any given frequency to be different than at other frequencies, depending of course on the body, and the atoms that constitute it. We refer to the set of all these frequencies as its spectrum and are often interested not only in the range of frequencies but also the amount of radiation in some small frequency interval. In particular it is interesting to look at the portion of the spectrum where ``a majority" of the radiation is emitted by the body, i.e., the ``predominant'' part of its radiation spectrum. It seems reasonable that this would have something to do with what the body is made of and also what is its temperature. Most bodies with which we normally come into contact have the predominant portion of their radiation spectrum in the infrared, or far-infrared, portion of the spectrum, so that it is not visible to us. However, as the temperature of a body is increased more radiation is emitted, and also the dominant portion of its spectrum shifts so that its radiation begins to be visible to humans. We usually could describe this by saying that, as some object is heated more and more, the predominant color of the hot object shifts from dull red through bright yellow-orange to bluish, ``white heat."

As already mentioned, the radiation depend not only on the temperature but also the composition of the object. For example, at 2000 K a polished, flat tungsten surface emits radiation at a rate of 23.5 Watts/cm²; for molybdenum, however, the corresponding rate is only 19.2 Watts/cm². In each of these cases the rate increases somewhat if the surface is roughened, instead of being polished. Other differences appear if we measure the distribution in wavelengths of the emitted radiation. This dependence on materials makes the problem quite complicated. Hopefully it reminds us of some of the difficulties one has with describing gases of atoms. Gases of different sorts of atoms have different properties; the more elementary parts of their behavior could, however, be understood rather well by introducing, instead, the idea of an ideal gas, which was fairly easy to study, and could then be used as a beginning to understand more of the details of more complicated gases. This ideal gas was really an idealization, or abstraction, of the basic ideas surrounding the behavior of gases, which was fairly straight-forward to study because of their simplicity. Likewise, in considering ``the radiation problem,'' it is useful to introduce the concept of ``an ideal radiator." For such an object we assume that the spectrum of the emitted thermal radiation depends only on the temperature of the radiator and not on the material or the nature of the surface.

In the real world, a gas which behaves not very differently than an ideal gas can be obtained by considering a gas of monatomic atoms, such as helium, neon, etc., at temperatures that are not too high. Likewise, in the real world, one may obtain a very good approximation to an ideal radiator by considering the radiation emitted through a small hole of a hollow body, heated to some temperature. The radiation is emitted by the interior walls of the cavity, and becomes quite mixed before it manages to find the small hole to escape; as a result of this it depends very little on the actual material of the body, or its surface. We therefore refer to this radiation as . We can also consider this situation in reverse, and consider the radiation that is absorbed--rather than emitted--by this hole. As it is simply a hole--with a dark hollow on the far side--it absorbs all radiation that strikes it, independent, again, of its material. Therefore, often such radiation is referred to, instead, as . We will now spend the rest of this material trying, therefore, to understand the details of the spectrum of black-body radiation, and its dependence upon temperature. Such studies were made near the beginning of the twentieth century, and shortly thereafter.

II.  The Stefan-Boltzmann Law

The total radiated power per unit area of a cavity aperture, summed over all wavelengths, is called its . It depends on the temperature T, has units of Watt/m², and is given by a simple formula, given below, which involves $\sigma$, the Stefan-Boltzmann constant, which is a universal constant, independent of material, surface, or temperature. We note, however, that ordinary hot objects always radiate less efficiently than do cavity radiators; therefore, we associate with them an additional, positive quantity, $\epsilon$, referred to as the emissivity, which is always less than 1, and depends on the material, the surface, and (usually) the temperature as well. Although it is called the emissivity, it also is used to describe the absorption of radiation by that material as well:

$$\hbox{for a blackbody, or cavity aperture:~~} I(T) = \sigma T^4\;,\quad \sigma = 5.670 \times 10^{-8} W/m^2-K^4 \;,\eqno(2.1)$$

$$\hbox{for a real material:~~} I(T) = \epsilon\sigma T^4\quad \;.\eqno(2.2)$$

III.  The Spectral Radiancy

While the total radiant intensity, or intensity per unit area is a very interesting quantity, much more interesting is the various proportions of that intensity emitted (or absorbed) in a particular band of wavelengths. Therefore we introduce the spectral radiancy, $R(\lambda)$, which tells us how the intensity of the cavity radiation varies with wavelength, for a given temperature.

$R(\lambda)\,\Delta\lambda$   gives the radiated power per unit area that lies within a band of wavelengths that extends from $\lambda$ to $\lambda + d\lambda$.

Therefore we must certainly have the normalization requirement on the spectral radiancy that

$$\sigma\,T^4 = I(T) = \int_0^\infty R(\lambda) \,d\lambda \;.\eqno(3.1)$$

The figure below shows the measured spectral radiancy for cavity radiation at three selected, rather hot temperatures, and then one below that just at ordinary room temperature. Although they look quite similar, notice the difference in scale in the horizontal and vertical axes. (The horizontal axis is wavelength measured in microns = 1000 nanometers, while the vertical axis, labelled $P$, is proportional to the spectral radiancey, but with an arbitrary scale to make things fit nicely, but the same in all of the graphs.) The total area under each curve is the appropriate radiant intensity; therefore, it increases as the temperature increases, as we can see.

We also notice that the curves begin at zero intensity at zero wavelength, near gamma rays and X-rays, then rise to a single maximum value, and then fall, rather more slowly, to zero as the wavelength increases toward very large values, such as radio waves.

These curves were measured in the nineteenth century, and many people endeavored to understand them. An early effort in this direction was made by Wilhelm Wien (Germany, 1864-1928) who was able to show that the wavelength at which the maximum of the curve occurred was simply inversely proportional to the temperature. He was awarded the 1911 Nobel prize for his research in this area; the Wien displacement law says, very simply, that

$$\lambda_{_{max}} = 2897.8/T \;,\hbox{~~measured in microns (or micro-meters).}\; \eqno(3.2)$$

After the verification of Wien's Law, there was considerable effort to derive a simple formula that would describe the complete shape of the curves for the spectral radiancy. In Sept., 1900 there were two different suggested formulas for the shape of the curve, both derived from what were considered very basic principles. Unfortunately the curves definitely did NOT fit the experimental curves.

The first one was originally derived by Lord Rayleigh, with later work by Einstein and James Jeans. It was based quite carefully on the well-understood (classical) physics of that time. It fit the curves quite well in the limit of very long wavelengths; however, it had what was referred to as an ``ultraviolet catastrophe,'' since it became infinite as the wavelength become smaller and smaller, leading to the very silly prediction that cavity radiators produced incredibly large amounts of energy at very low wavelengths.

The second proposed relationship was due to Wien. His formula was very much better, since it fit the curves well at short wavelengths (which included visible light), passed the curves through a maximum at the correct place, but then deviated substantially from the experimental values when the wavelengths became very large. However, a worse problem with Wien's formula was that it was not actually based on a deep understanding of the physics, but simply a ``guess" based upon a belief that there should be an analogy between the behavior of the radiation of electromagnetic radiation and the behavior of the speeds of molecules in an ideal gas.

IV.  Planck's Law for the Spectral Radiancy

At this point, Max Planck (German, 1858-1947) made an inspired interpolation between the two formulas that turned out to fit the data at all wavelengths. In October of 1900 he published the following guess, and then managed, two months later, to create a derivation of his equation, which included the speed of light, c, and a brand-new constant, h, which we now refer to as :

$$R(\lambda) = {2\pi c^2 h\overwithdelims()\lambda^5}{1\over e^... ...T} - 1}\;,\quad h = 6.626 \times 10^{-34} \hbox{J-s.}\eqno(4.1)$$

Here the constant k is a previously known one, the Boltzmann constant, which we studied in our discussions of thermodynamics. It has the following value, where I also give the value of Planck's constant in some other units:

$$k = 1.381 \times 10^{-23} \hbox{~J/K}\,,\qquad h = 4.14 \times 10^{-15} \hbox{~eV-s.}\eqno(4.2)$$

It can be checked that Planck's formula does indeed satisfy the constraint imposed by equation (3.1), namely the area under its curve is indeed the entire radiant intensity at that temperature.

However, from our point of view, the important part of this radiation law is the changes that Planck had to make in the physics of the times in order to obtain this equation, and to see why the Rayleigh-Jeans law did not properly agree with the experimental data, at short wavelengths! Rayleigh, Jeans, and also Planck analyzed the interplay between the radiation in the cavity volume and the atoms that make up the cavity walls. He assumed that these atoms behave like tiny oscillators, each with a characteristic frequency of oscillation. These oscillators radiate energy into the cavity, and absorb energy from it. Therefore, one should be able to deduce the characteristics of the cavity radiation from the characteristics of the oscillators that generate it.

The next step, taken by Rayleigh, and Jeans, was to suppose that the energy of these tiny oscillators is a smoothly continuous variable; i.e., one may have any (positive) value whatsoever for their energy. When this did not work, Planck decided, instead, to make a very unusual assumption:

atomic oscillators may not emit or absorb any energy E, but only energies from a discrete set, made of integer multiples of a smallest possible value:

$$E = n h f\;,\quad n = 1, 2, 3, 4, \ \ldots\quad . \eqno(4.3)$$

In this equation, f is the basic oscillator frequency. Today, we describe this assumption by saying that

the energy of an atomic oscillator is quantized.

Today we would also refer to the integer n in the formula as a quantum number. Because the equation involves an integer, and is linear in the frequency, we see that the allowed values for the energy are evenly spaced, i.e., the difference between any two adjacent values is just hf, the same no matter which pair of adjacent values we choose. (This is the same idea as the system of coinage in the United States, where every amount of money we are able to hand over to another person is required to be some (integer) number of pennies.)

Planck himself was never able to create an argument he would believe for ``why" this was true; however, it fit the data exactly, so that it must in fact be true. It is this beginning that created what today we call ``quantum physics." It is worth noting that the very, very small size of Planck's constant h causes this difference between two adjacent values to usually be very, very small. In particular, since $hf = hc/\lambda$, it is this very small size that allowed the Rayleigh-Jeans theory to fit the data very well for large values of $\lambda$, where hf is indeed very small relative to the energies involved.

Example:

To acquire some feeling for how very small these quantities can be, note that an ordinary spring or an ordinary pendulum is an example of an oscillator, albeit not the sort that you expect in a cavity radiator. Therefore, this new assumption must apply to it as well. We consider the following problem:

A 300 gram body, connected to a spring with force constant 3.0 N/m, is oscillating with an amplitude of 10 cm. Treat this system as a quantum oscillator and find the energy interval between adjacent energy levels, and the quantum number that describes its oscillations.

We proceed as follows:

(a) 

The frequency of the spring is just

$$f = {\sqrt{k/m}\over 2\pi} = {\sqrt{(3.0 N/m)/(0.3 kg)}\over 2(3.1416)} = 0.50 \hbox{~~cycles/sec .}$$

Then the total mechanical energy of the oscillating system is

$$E = {1\over 2}kA^2 = (0.5)(3.0 N/m)(0.10 m)^2 = 0.015 J .$$

As friction acts on the system the amplitude of the oscillations should die away. Quantum theory predicts that the energy must decrease in ``jumps" rather than discretely. The predicted size of the ``jumps" is given by

$$\Delta E = hf = (6.63 \times 10^{-34})(0.50) = 3.3 \times 10^{-34} J .$$

We see that this number is incredibly small relative to the energy of the system, so that we would be totally unable to discern that the system is actually acting in a quantized manner.

(b) 

We may also calculate the quantum number. In this case it is just the inverse of the ratio above, of ``jump size'' to total energy:

$$n = E/hf = 4.6 \times 10^{31} .$$

This is of course an enormous number, so that a change of just 1 would be quite un-noticeable.

Therefore, we see that the quantization of energy simply does not show up for large-scale oscillators. The smallness of Planck's constant makes the graininess in the energy much too fine to detect in those experiments. This is quite similar to the statement that we do not ordinarily observe the fact that the air in the room is actually made up of many, many individual molecules. Nonetheless, we can indeed perform experiments in which this graininess is noticeable, and even important. Obviously the behavior of the spectral radiancy at very short wavelengths is one such case. The phenomena involved with the photoelectric effect, and the Compton effect, are others.

This text adapted (slightly) from Halliday, Resnick, and Krane (Ch. 49).