Kirchhoff’s Radiation Law

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    In 1859 Robert Gustav Kirchhoff (1824 Mar 12 – 1887 Oct 17) set the science of radiation thermodynamics, a subset of the science of physics, onto a proper mathematical foundation and thereby initiated the process that culminated in Max Planck’s derivation of a complete and correct description of the spectrum of blackbody radiation forty-one years later. Kirchhoff got the radiation thermodynamics ball rolling by presenting a simple rule: in a system at thermal equilibrium, a body that absorbs radiation with a certain efficiency, emits radiation with that same efficiency. But Kirchhoff understood that a certain subtlety underlay that simplicity.

    Imagine that the vapor of a chemical element fills a certain volume. A narrow beam of light containing all wavelengths passes through that volume, then passes through a prism and spreads out into a spectrum on a screen calibrated to show the wavelengths of the light falling on it. On that spectrum dark lines represent the wavelengths of light that the vapor has absorbed due to its particular nature. Then we turn off the light and heat the vapor to incandescence. If a narrow beam of light emitted by the vapor follows the path of the first beam and goes through the prism, then bright lines will occupy the positions where the dark lines appeared. A more complicated version of that experiment, conducted with sodium vapor, inspired Kirchhoff to devise his radiation law.

    To begin to deduce the result of that experiment imagine a tube whose inner wall neither emits nor absorbs radiation. Two cylindrical slugs occupy opposite ends of that tube and contact with appropriate heat reservoirs maintains them both at the same temperature. Slug-1 emits radiation with a presumed intensity of E1 watts per square centimeter and Slug-2 absorbs a fraction A2 of that emission. Likewise, Slug-2 emits radiation with an intensity of E2 watts per square centimeter and Slug-1 absorbs a fraction A1 of it, reflecting the rest. Because, by assumption, the system exists in thermal equilibrium with itself (its parts all have the same temperature), the second law of thermodynamics (in Rudolf Clausius’s version: net heat will not, of itself, flow from a cold body to a warmer body) necessitates that the amount of radiation coming from one slug must equal the radiation coming from the other: if that did not happen, then one slug would send more radiation to the other slug than the other slug sends back to it, become cooler as a result, and thereby destroy the equilibrium. The amount of radiation coming off Slug-1 equals the sum of E1 and the amount of E2 reflected from the slug, both multiplied by the area of the slug’s face. The analogous sum describes the radiation coming off the face of Slug-2. Equating those two sums and, because both slugs’ faces have the same area, dividing out the area yields

E1+E2(1-A1)=E2+E1(1-A2).

(Eq’n 1)

Subtracting E1+E2 from both sides of that equation and dividing the result by minus one yields

E2A1=E1A2.

(Eq’n 2)

Finally, dividing that equation by the product A1A2 yields

(E1/A1)=(E2/A2).

(Eq’n 3)

That equation expresses the fact that the emissivity of a given body stands in direct proportion to that body’s absorptivity. To the extent that a body’s emissivity depends upon certain properties of the body, to the same extent and in precisely the same way the absorptivity will depend upon those properties. Put more simply, a good absorber of radiation is also a good emitter of radiation.

    A simple experiment demonstrates how the absorptivity of a body controls the body’s emission of radiation. Imagine a thermally insulated pipe filled with hot liquid and put into a cold environment, the colder the better. The flat caps closing the ends of the pipe consist of a thermally conductive material with the exterior faces painted, one white and the other black. Thermometers positioned the same short distance from the end caps on the line passing through the center of the pipe receive radiation from the caps and heat up as a consequence. People who have actually conducted this experiment have found that the thermometer near the black cap displays a higher temperature than does the thermometer near the white cap, indicating that the black cap, which has an absorptivity close to unity, emits heat radiation at a rate faster than does the white cap, which has an absorptivity close to zero.

    But we can take the analysis further, much as Kirchhoff did. The following analysis yields the principle of detailed balance as a rather obvious extension of Kirchhoff’s radiation law.

    Imagine putting a body at some temperature inside an enclosure, an oven, kept at the same temperature. In that circumstance the second law necessitates that the body and the oven remain in thermal equilibrium with each other by way of the radiation field inside the oven; specifically, it requires that the body emit thermal radiation at the same power at which it absorbs thermal radiation, lest the body’s temperature rise or fall and thereby set up a violation of the second law.

    Now imagine that body enclosed within a thin shell made of a material that reflects all radiation perfectly except for radiation in a narrow band of frequencies, which special radiation it transmits perfectly. Reflecting and transmitting perfectly means that the shell does not absorb any of the radiation striking it. The presence of the shell cannot affect the intrinsic properties of the body; specifically, it cannot alter the body’s emissivity or absorptivity. Neither can the presence of the shell affect the nature of the radiation inside the oven, lest it create the possibility of violating the second law.

    Those facts necessitate that at any given frequency the body emit radiation with the same power with which it absorbs radiation at that frequency. Assume that the body absorbs radiation with nearly one hundred percent efficiency in the band of frequencies that pass through the shell. The radiation emanating from the walls of the oven strikes the shell and the radiation in the band of special frequencies passes through the shell and gets absorbed by the body. We expect, in consequence of that absorption, the body to heat up and, thus, to radiate more energetically at all frequencies, most of which reflect off the inside of the shell and go back into the body, thereby making the body heat up further in blatant violation of the second law. That expectation would come true to Reality if the body did not radiate energy at each frequency exactly as efficiently as it absorbs heat radiation at that frequency. But in a system, as in our example, at thermal equilibrium within itself each part must, over any band of frequencies, either broad or narrow, both reflect and radiate exactly as much power as it receives in that band. Thus we come to understand what physicists call the principle of detailed balance, which explains why sodium vapor, for example, absorbs radiation of certain frequencies strongly, creating dark lines in its absorption spectrum, and also radiates strongly at those same frequencies when heated to incandescence, creating bright lines in its emission spectrum.

    Finally, assume, as Kirchhoff did, the existence of a body that absorbs radiation of all frequencies with perfect, one-hundred-percent efficiency. That body will absorb absolutely all of the radiation falling on it, reflecting absolutely no radiation, so it will appear perfectly black. Given Kirchhoff’s law, we expect that such a body, when heated, will emit perfectly white radiation, though that light would not necessarily appear white to creatures whose eyes can only detect light over a narrow range of frequencies. In 1862 Kirchhoff proposed calling that emission blackbody radiation and physicists refer to its distribution of radiated power as the normal spectrum.

    Determining the algebraic description of the normal spectrum became one of the great projects of Nineteenth Century physics. As the purest form of emitted radiation, completely independent of the nature of the material emitting it, it would stand, physicists assumed, as the gateway to a full understanding of the ultimate nature of what we now call electromagnetic radiation. Max Planck validated that assumption at the end of 1900 when he presented his formula describing the blackbody spectrum and the weird hypothesis that he had to make in order to derive the formula. That weird hypothesis led to the development of the quantum theory, but that’s a different essay.

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