2.5.1 The First Failure: Black Body Radiation

We therefore begin this sub-section with Gustav Kirchhoff’s 1860 paper on The Relationship between the Emissivity and Absorptivity of Bodies for Heat and Light. It was known from experience that a body, which finds itself in an enclosure where the temperature is equal to its own, does not change its temperature through heat radiation. Consequently, Kirchhoff says, the body absorbs in a given time just as much radiation as it emits. The act of emission and absorption therefore locates heat radiation squarely in the realm of the inside of things and is proper concern of this essay. In his paper Kirchhoff sets out to examine the relationship between the emissivity and absorptivity of radiation of a given body. To define his terms, Kirchhoff imagines two screens placed one after another in front of a body $C$ , each of which has an opening of dimensions infinitely smaller than the distance between them. When the openings in them are aligned with each other, these screens trace out a beam of radiation emitted by, or fallen onto, the body. From that beam he considers only that portion which contains wavelengths between $λ$ and $λ + d λ$ and which are linearly polarized in a given direction. If the beam issues from the body, its intensity can be written as $E d λ$ where $E$ represents what Kirchhoff calls the emissivity of the body. If, on the other hand, the beam is allowed to fall upon the body, a part of it will be absorbed, another part transmitted, and the remainder reflected. The intensity of the radiation which has been absorbed, referred to the intensity of the incident radiation, represents the absorptivity $A$ of the body.

The quantities $E$ and $A$ depend in general on the nature and temperature of the body $C$ , on the position and shape of the openings defining the beam, on the wavelength $λ$ , and on the direction of polarization. Kirchhoff manages to demonstrate however that the ratio of $E$ to $A$ is given by:

\frac{E}{A} = J (T, λ) \frac{ω_{1} ω_{2}}{s^{2}}

where $ω_{1}$ and $ω_{2}$ are the surface areas of the two openings respectively, $s$ is the distance between them, and $J (T, λ)$ is a function which depends only upon the temperature of the body and the wavelength of the radiation it emits but is otherwise the same for all bodies. Since, by definition, a black body absorbs entirely the radiant energy which has fallen upon it, $A = 1$ , and $J (T, λ)$ thus represents the emissivity of such a body. This finding, that the emissivity of a black body is a universal function of its temperature and wavelength alone, was obtained on quite general grounds involving only thermal equilibrium and energy conservation considerations. It has therefore set up a long quest for the form of this very special function, a quest which ended forty years later with Planck’s formula.

The first result concerning the spectral distribution $J (T, λ)$ of a black body was obtained by Ludwig Boltzmann in March of 1884. In his paper Derivation of Stefan’s Law Regarding the Temperature Dependence of Heat Radiation from the Electromagnetic Theory of Light, Boltzmann considers a black body conceived as an absolutely empty space surrounded by walls that are impenetrable to heat radiation and which are kept at a fixed absolute temperature $T$ , and treats the radiation inside as an electromagnetic field. He points out that radiant heat, just as any propagating electromagnetic field, will therefore exert pressure upon the walls of a vessel containing it. Maxwell had proved that in the case of light, such a pressure is equal to one third of the energy density in the field. Borrowing this result as it applies to radiant heat and combining it with the second principle of thermodynamics, Boltzmann calculates that the energy density of radiant heat inside such a black body, energy which is proportional to the integral over all wavelengths of the spectral energy density $J (T, λ)$ defined above, is proportional to the fourth power of the temperature; this law was already experimentally advanced by Stefan and was found to be in good agreement with the data.

In 1893, Willy Wien succeeded in going beyond statements about radiant heat energy density to say something about the corresponding spectral density $J (T, λ)$ itself. In his paper The Superior Limit of Wavelengths which may Occur in the Thermal Radiation of Solids: A Conclusion from the Second Law of Thermodynamics, Wien demonstrates that the spectral density of a black body does not depend upon its temperature and the corresponding wavelength separately but rather upon their product only:

J (T, λ) = \frac{1}{λ^{5}} f (λ T)

These two results exhaust what can be obtained by thermodynamic arguments alone. To find the function $f (λ T)$ , some mental picture of the internal processes which produce and absorb radiant heat must be formed. In view of what has been said earlier in this chapter, that picture of the inside must be sought in the corpuscular theory of matter.

There had been a number of proposals made before the turn of the 19^th Century concerning these internal processes. Willy Wien himself suggested one, H.A. Lorentz suggested another, and J.H. Jeans yet another. Max Planck, following a more detailed path than Wien had done, obtained the same result as Wien. The results obtained by Lorentz and Jeans appeared to fit the available data for relatively large values of the product between wavelength and absolute temperature; those obtained by Wien and Planck, corresponded to the measured spectral density when that product was very small. Since it was Planck’s work which eventually catapulted the corpuscular theory of matter from its classical form into the quantum one, we shall focus on the latter two contributions.

In his On the Energy Distribution in the Emission Spectrum of a Black Body published in 1896, Willy Wien took the provisional position that the heat radiation inside a black body issues from a perfect gas whose molecules obey Maxwell’s law of distribution of velocities; that each molecule emits radiation whose wavelength is a function of its velocity; and that the intensity of the radiation whose wavelength lies between $λ$ and $λ + d λ$ is proportional to the number of molecules that emit radiation of that wavelength. Then, he found that

f (λ T) = F (λ) e^{\frac{- g (λ)}{T}}

where $F (λ)$ and $g (λ)$ are two unknown functions. Referring to Boltzmann’s law establishing the temperature dependence of the energy density in a black body and his own results of 1893, Wien proved that:

f (λ T) = C e^{- \frac{c}{λ T}}

a result in good agreement with the then available data obtained by Friedrich Paschen.

The succession of results described above marks out the slow but sure increase in the explanatory power of the corpuscular concept as it applied to the thermal equilibrium established by the heat radiated between bodies. Throughout, it was taken for granted that radiant heat was light, but that identity passed essentially unexploited. Max Planck took it upon himself to remedy the lack in a series of papers he started in 1895. If radiant heat was light, and light was an electromagnetic wave, then one may be able to determine the spectral distribution for the former by studying the interaction between the constituents of bodies which emit and absorb electromagnetic waves and the surrounding electromagnetic field into which they are embedded. Since according to Kirchhoff, the spectral distribution of black body radiation was independent of the nature of the bodies involved, Planck focused his attention upon the simplest realization of a molecule able to interact with such a wave, an electric charge oscillating around a fixed point; he called such a system a resonator. According to Maxwell’s theory of electromagnetic phenomena, a resonator could both emit energy in the form of an electromagnetic wave as well as absorb it from a surrounding electromagnetic field, processes which resembled the exchanges of heat by radiation:

I have begun these inquires with the intention of obtaining a more detailed insight into the processes of emission and absorption of heat radiation, and therefore of the temperature equalization they cause, from the stand point of the electromagnetic theory of light.
Max Planck, Concerning Irreversible
Radiation Processes (1897)

In the first three of his papers communicated to the Royal Prussian Academy of Sciences, Planck carried his detailed analysis far enough to convince himself that he was on the right path towards explaining the structure of black body radiation. And then, in 1897 Boltzmann objected:

Mr. Planck has developed a series of formulas which have certainly proven themselves to be useful for calculations of experiments with electric resonators and are also appropriate for the theory of the dispersion of light. However, I cannot agree with the consequences which he draws from them for the explanation, or schematic representation, of irreversible processes.
Ludwig Boltzmann, Concerning Irreversible
Radiation Processes (1897)

The point which Boltzmann makes is that Maxwell’s equations are reversible while radiant heat processes, which always facilitate the attainment of thermal equilibrium in a closed system of bodies, are manifestly not:

When an arbitrary electric resonator finds itself in a field, the statement remains valid that the Maxwell equations are nowhere violated if at a given moment all electric forces and polarizations in the field and in all resonators remain unchanged but the direction of time and all magnetic forces and polarizations are simply reversed.

Therefore, according to Boltzmann, Planck’s efforts to explain radiant heat on the basis of electromagnetic theory were bound to fail.

This shot over the bow initiated an inconclusive exchange of papers between the two men, at the end of which Planck continued his attempt to explain the irreversible progress towards thermal equilibrium with his system of resonators residing in an enclosure. He soon found, however, that to get where he was going he had to make a crucial assumption concerning the ontology of a radiation field. His calculations indicated that the energy of a resonator does not depend directly upon the intensity of the activating wave but involves the amplitudes and the phases of all the spectral components of the latter. Thus he found that as long as one does not know the particulars amplitudes and the particular phases of the individual partial oscillations of the activating wave, one cannot find a specified, generally valid, connection between the energy of the resonator and the intensity of the activating wave. But, the phenomena of absorption and emission of heat radiation speak on behalf of such a connection truly existing in nature. He therefore averaged the resonator energy over the unknown amplitudes and phases assuming that each of them was statistically independent of all the others. This amounted to replacing the radiation field interacting with the resonator by a new entity which he called ‘disordered radiation’. The assumption is most clearly described by Albert Einstein:

We expand the $z$ -component of the electric force $(Z)$ at an arbitrary point of the space under consideration between time limits $t = 0$ and $t = T$ (where $T$ stands for a time very much larger than all oscillation periods considered) into a Fourier series:
$Z = \sum_{ν = 1}^{ν = \infty} A_{ν} s i n (2 π ν \frac{t}{T} + α_{ν})$
where $A_{ν} \geq 0$ and $0 \leq α_{ν} \leq 2 π$ . Imagining that at the selfsame point in space such an expansion has been accomplished arbitrarily often with a stochastically chosen initial point in time, one would obtain various values for the system of quantities $A_{ν}$ and $α_{ν}$ . There exist then for the frequency of occurrence of the various combinations of values for the quantities $A_{ν}$ and $α_{ν}$ a statistical probability $d W$ of the form:
$d W = f (A_{1}, A_{2}, \dots, α_{1}, α_{2}, \dots,) d A_{1} d A_{2} \dots d α_{1} d α_{2} \dots$
The radiation is then considered disordered when:
$f (A_{1}, A_{2}, \dots, α_{1}, α_{2}, \dots) = F_{1} (A_{1}) F_{2} (A_{2}) \dots f_{1} (α_{1}) f_{2} (α_{2}) \dots$
that is, when the probability of a given value of one quantity $A$ , for instance $a$ , is independent of the value possessed by another quantity $A$ , for instance $x$ .
Albert Einstein, On a Heuristic Point of View Concerning the Production and Transformation of Light, (1905)

This averaging-out of the unknown amplitudes and phases of the activating wave recalls the averaging over the unknown mechanical configurations of the molecules of an ideal gas which was required to establish a direct relationship between pressure and volume. There, as here, the relationship was demanded by experience and the theory was therefore required to involve an average over the locations and velocities of all molecules assuming they were statistically independent of each other.

Be that as it may, introducing disordered radiation allowed Planck to obtain the irreversibility he sought and he eventually managed to derive Wien’s law for the spectral distribution of black body radiation. The date was March of 1900. Unfortunately, by October 25^th of that same year, measurements performed by H. Rubens and F. Kurlbaum with long wavelengths at different temperatures substantially modified the known dependence of the energy spectrum of black body radiation upon the product between wavelength and temperature. Planck’s five-year quest employing the corpuscular theory of the structure of matter to explain black body radiation had thus failed. This failure did not come alone however. Two other failures were exercising the scientific community at the time: the failure to explain the spectrum of the light emitted by an atom and the failure to explain the photoelectric effect. The great crisis of the turn of the 19^th Century thus announced itself.