Kirchhoff's Blackbody Radiation

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    What is light? Until the last half of the Seventeenth Century, natural philosophers had achieved essentially no success in answering that question. Isaac Newton (1642 Dec 24 - 1727 Mar 20) took the first great step toward a comprehensive answer to it in the early 1660's, when he used a glass prism to spread a ray of sunlight out into a band of colors that he could not break up further. He had been trying to gain an understanding of the chromatic aberration that surrounded objects viewed through a telescope with colored halos, an understanding that he hoped would enable him to redesign the refracting telescope to eliminate the aberration. He failed in that effort and invented the reflecting telescope, one based on the magnifying ability of a paraboloidal mirror, instead. Nonetheless, his discovery of the spectrum remained to tempt others to learn more about light.

    More than a century elapsed before natural philosophers made the next substantive advances in our collective understanding of light. Then, neatly bracketed by the years 1800, when Friedrich William Herschel (1738 Nov 15 - 1822 Aug 25) discovered infrared radiation, and 1900, when Max Karl Ernst Ludwig Planck (1858 Apr 23 - 1947 Oct 04) produced the final, full mathematical description of blackbody radiation, the Nineteenth Century came to Humanity as the era in which physicists answered the question more or less completely.

    We know how physicists, especially James Clerk Maxwell and Heinrich Hertz, worked out the electromagnetic nature of light and came to the brink of discovering the theory of Relativity. Now we want to know how physicists worked out the thermodynamic nature of light.

    That light must have a thermodynamic nature we know from direct experience; after all, we can feel that nature manifest in the warmth that sunlight brings into the world. That fact inspired Herschel to look at light in a new way. As Newton had done, he used a glass prism to spread a beam of sunlight into its constituent colors. Then he took an extra step and put the bulb of a thermometer into the spectrum, hoping thereby to determine how much each color in the sun's light contributes to the overall heating effect. When he had finished taking his data he moved the thermometer, for scientific completeness, into the dark zone below the red end of the spectrum. He expected to see that the heating effect had gone to zero: he saw that it hadn't. Well into that dark zone the thermometer showed the effect of more heat coming from the sun. Taking the Latin word for "below", Herschel named that phenomenon infra-red light.

    In 1814 Joseph von Fraunhofer (1787 Mar 06 - 1826 Jun 07) discovered that he could eliminate almost all of the blurring inherent in Newton's and Herschel's versions of the experiment. He passed a collimated beam of light through a narrow slit before sending it through a prism whose axis of symmetry ran parallel to the slit. When he brought sunlight into that apparatus he discovered thin bright and dark lines crossing the rainbow-colored spectrum. Fraunhofer got a clearer picture, but of what?

    Gustav Robert Kirchhoff (1824 Mar 12 - 1887 Oct 17) and Robert Wilhelm Bunsen (1811 Mar 31 - 1899 Aug 16) found out. In 1859 they built an improved version of Fraunhofer's apparatus, one with two prisms, that enabled them to locate more accurately Fraunhofer's lines in the spectrum. Relying on Bunsen's expertise as a chemist, the two men observed the spectra of light emanating from incandescent samples of the chemical elements known at the time and then discovered that they could match the bright lines in some of those spectra with some of the dark lines in the spectrum obtained from sunlight. From those matches they inferred the existence of the corresponding elements in the sun's atmosphere and even of two new elements, cesium and rubidium.

    In order to make that inference they had to assume into their premises a law of radiation that Kirchhoff had deduced in 1858 (and that the Scottish physicist, Balfour Stewart (1828 Nov 01 - 1887 Dec 19), had discovered independently the same year) and proven in 1861. That law includes a statement to the effect that a chemical element emits radiation preferentially at any wavelength at which it absorbs radiation preferentially. That statement comes from the principle of detailed balance as applied to radiation and we can deduce it much as Kirchhoff did.

    Rudolf Julius Emanuel Clausius (1822 Jan 02 - 1888 Aug 24) provided the key to theoretical radiation thermodynamics in 1850. In that year Clausius stated the second law of thermodynamics in the form "Heat cannot of itself pass from a colder to a hotter body". Only after fifteen years of further study did Clausius advance our knowledge of the second law far enough that he felt the need to define a new thermodynamic entity, which entity he named entropy. But his original statement gives us what we need here.

    We begin with Prévost's law, which states that any body with a temperature above absolute zero radiates heat, however faintly. That law, in turn, necessitates that any given body radiate heat at a rate that we can describe with a suitable monotonically increasing function of the body's absolute temperature and of nothing outside the body. That all comes from postulating the existence of heat radiation that bodies emit and absorb or reflect.

    In 1791 Pierre Prévost (1751 Mar 03 - 1839 Apr 08), a Swiss philosopher and physicist, presented his law of exchange, which states that all bodies radiate heat regardless of their temperature. He came to that law by way of postulating that cold is not a thing-in-itself but merely reflects a paucity of heat. In thermodynamics, according to Prévost, only heat has a real existence. Nonetheless, he assumed these propositions; he did not deduce them. We would like to deduce them.

    Let us assume, as Prévost did, the existence of heat radiation. We thus take an empirically determined fact as if it were an axiom. Now we assume that some bodies emit heat radiation and that some bodies absorb heat radiation: at this stage we don't know whether bodies emit or absorb heat radiation, but we certainly do not want to assume that they do both.

    Take a cold absorbing body and surround it with a hot emitting body that we keep at a temperature T. The absorbing body soaks up heat until it reaches the temperature T, then it must stop absorbing heat in order to conform to Clausius' law (Note that Prévost did not know of Clausius' law, but he likely had an intuitive understanding of its content, as do all of us). How can that happen?

    We can see three options for answering that question:

1. The emitting body stops radiating heat,

2. The absorbing body stops absorbing heat, and

3. The absorbing body emits heat at the same rate at which it absorbs heat.

    Now we have to find out which of those options actually describes Reality. To make Option 1 or 2 true to Reality we must have that one or the other of the two bodies determines the difference between their temperatures and responds accordingly. That would necessitate the existence of what Einstein called "spooky actions at a distance". It would also necessitate a more complicated mechanism at work in the situation. In light of that fact Occam's Razor ("Plurality should not be posited without necessity." - William of Occam (Ockham), 1284 - 1347)) would have us select Option 3, but Occam's Razor constitutes neither a necessary nor a sufficient condition for a proper deductive proof; it's inductive at best.

    Thus we infer that the qualities that make a body absorb heat radiation necessarily make that body emit heat radiation. But that is not a proper deduction, so if we want to add this piece of the puzzle to the Map of Physics, we will have to revisit this topic.

    Nonetheless, we can go ahead and use Prévost's law as a postulate in our reasoning. Now we can deduce Kirchhoff's law.

    Imagine a large body that has a deep cavity dug into it. Imagine further that we keep that body at some absolute temperature T and that we have put a small body at a different temperature into the cavity. If the small body has the higher temperature, then it will radiate heat faster than it absorbs heat so that there will be a net flow of heat from the hotter body to the colder body. And if the smaller body starts off with the lower temperature, then it must absorb heat faster than it emits heat in order to obey Clausius' rule. Eventually the system will come to thermal equilibrium; that is, both bodies will have the same temperature and the small body will emit heat as fast as it absorbs heat.

    No passive means can alter that equilibrium. Let's assume that we have coated the small body with a thin layer of a substance that reflects all but a small amount of the radiation at all wavelengths. Of the radiation falling onto the body, a fraction α will be absorbed and a fraction ε will be emitted. When the temperature of the body equals that of the surrounding walls, the body must emit as much heat as it absorbs, so we must have α=ε at that temperature. But we selected the temperature arbitrarily, so we must have α=ε true to Reality at all temperatures at which the small body is in thermal equilibrium with its surroundings. Thus we have the preliminary form of Kirchhoff's law of radiation: the emissivity of a body is equal to its absorptance at the same absolute temperature.

    So now imagine that we have coated the small body with a thin layer of a substance that reflects all radiation except that which has wavelengths within a certain narrow band. The body will absorb from the radiation field in the cavity only radiation with those wavelengths and it will reflect away the rest. It will come up to the temperature of the surrounding walls and then, in accordance with Prévost's law and the second law of thermodynamics, the body must radiate heat as fast as it absorbs it.

    We know that when a body absorbs energy, it scrambles that energy to convert it into heat. We should expect the same fact to apply to the absorption of heat radiation; that is, that heat manifested in a narrow band of wavelengths should be converted into heat manifested in all wavelengths when the body re-radiates it. Thus we expect our small body to radiate more or less uniformly over all wavelengths.

    We put that expectation to the proof by noting that for the body to radiate the heat coming into it, it would have to absorb enough heat to raise its temperature high enough that the wavelengths for which it has low-emissivity will radiate at a greater rate than the body is absorbing them from the radiation field. But that situation necessitates that the smaller body reach equilibrium at a temperature higher than that of the surrounding walls, which means that we could, in theory, connect the two bodies through a Carnot engine and generate work continuously, in violation of the second law of thermodynamics. We cannot have that situation come true to Reality, so we must infer that the smaller body, at equilibrium, radiates energy at each wavelength at the same rate at which it absorbs energy at that wavelength. Thus, we have Kirchhoff's law in full: the emittance of a body at any given wavelength equals that body's absorptance at that same wavelength at the same temperature.

    From that statement Kirchhoff inferred three rules that are also known as Kirchhoff's laws:

1. An incandescent solid or a gas under high pressure will produce a continuous spectrum,

2. A low-density gas will radiate an emission-line spectrum with an underlying emission continuum, and

3. Continuous radiation viewed through a low-density gas will produce an absorption-line spectrum.

    So now we see that Kirchhoff and Bunsen found that Fraunhofer had discovered a relationship between matter and light. All that physicists needed to do was to work out the details of that relationship. And the obvious place to start that work was the simplest case of that relationship.

    The simplest case is that in which matter and radiation interact in a way that does not depend upon any property of the matter. In 1860 Kirchhoff stated that such a case would involve a perfectly black body, a body that would absorb one-hundred percent of the radiation falling onto it at all wavelengths. Such a body would emit a continuous spectrum of radiation and Kirchhoff deduced that the body would emit that radiation in such a way that the intensity of the emission at any given wavelength would depend only upon the wavelength and the absolute temperature of the blackbody (a term that Kirchhoff coined in 1862). Mathematically, physicists would describe the power density at a given wavelength with the function u = u(λ, T).

    Kirchhoff also noted that if an experimenter were to take a body containing a large cavity and drill a small hole into the cavity, that hole would emit blackbody radiation. He reasoned that the hole would absorb radiation perfectly, because the radiation bouncing around inside the cavity would be absorbed by the walls before it returned to the hole, so it would have to radiate perfectly.

    Thus Kirchhoff gave physicists a theoretical object (the blackbody) and an experimental subject (the cavity radiator) that they could use to study what they came to call the normal spectrum. He also noted that a complete mathematical description of the normal spectrum, which Max Planck discovered in 1900, would lead to a deeper understanding of the nature of matter, of energy, and of the relationship between them. As we shall see in subsequent essays, he was right about that.


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