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FROM THE HISTORY OF PHYSICS
ON THE HISTORY OF THE LAW OF BLACKBODY RADIATION
(On the Research of V. A. Mikhelson)
V. A. Sokolov
Last year marked 90 years since the birth of one of the outstanding Russian physicists—Vladimir Aleksandrovich Mikhelson, who, together with P. N. Lebedev, came from the school of A. G. Stoletov.
The sphere of V. A. Mikhelson’s scientific interests was quite broad and many-sided; within it, a special place must undoubtedly be assigned to his investigations into the theoretical explanation of the laws of thermal radiation of solid bodies, which played, historically, a very important role in the solution of this problem.
The first rudiments of the doctrine of thermal radiation arose as early as the end of the seventeenth and the beginning of the eighteenth centuries. Thus, at that time Newton, for example, was to some extent interested in questions of thermal radiation; in 1701 he formulated an approximate law stating that the quantity of radiant energy is proportional to the temperature difference between a body and its surroundings[^1]. Individual examples are known of still earlier manifestations of interest in this question, but the beginning of systematic and intensive research on thermal radiation falls at the end of the eighteenth century[^2], when, one after another, works appeared by Lambert, Scheele, Pictet, Leslie, Herschel, and many others. To this period also belong the introduction of the term “radiant heat” (Scheele), as well as the discovery of infrared rays (Herschel).
Without dwelling on the various numerous works on thermal radiation whose appearance is characteristic of the nineteenth century, we shall note only certain important stages in the development of this doctrine that are necessary for a fuller understanding of the question touched upon here.
In 1809 Prévost formulated a qualitative rule which stated that if two bodies absorb different quantities of radiant
energy, emission too must be different.\(^2\) But only half a century later, in 1859–1861, did Kirchhoff\(^3\) succeed in giving this rule the form of a strict quantitative law, playing a fundamental role in all questions of thermal radiation. The decisive significance in the discovery of this law was the Carnot–Clausius principle, which by that time had taken shape as one of the fundamental laws of physics—the second law of thermodynamics. It was precisely by using the second law of thermodynamics that Kirchhoff rigorously proved that the ratio of the radiating and absorbing powers of any body whatever that emits heat rays is a quantity identical for all bodies, and can be only a universal function of temperature and wavelength:
\[ \frac{E_{\lambda T}}{A_{\lambda T}}=\varepsilon_{\lambda T}. \tag{1} \]
Placing at the center of attention the function \(\varepsilon_{\lambda T}=f(\lambda,T)\), which represents the emissive power of an absolutely black body, Kirchhoff’s law thereby posed a new, very important and difficult problem: determining the form of this function. All that Kirchhoff himself could say on this point amounted approximately to the fact that, at low temperatures, this function is equal to zero for visible rays, differs from zero for longer waves, and at higher temperatures has finite values for visible rays as well.\(^3\) In addition, it followed from Kirchhoff’s investigations that, as the temperature rises, the values of the function for each wavelength \(\lambda\) increase in such a way that the curves of the energy distribution in the spectrum of a black body nowhere intersect, since otherwise there would be a violation of the second law of thermodynamics.
The year 1879 marks the appearance of Stefan’s law\(^4\) on the proportionality of the total radiation of a black body to the fourth power of the absolute temperature. In 1884 Boltzmann\(^5\) proved this law theoretically. But the Stefan–Boltzmann law, being an integral law, does not give a solution to the problem of the form of Kirchhoff’s function \(\varepsilon_{\lambda T}\), which requires establishing the law of the distribution of the energy of black-body radiation by wavelengths.
In 1886 Langley\(^6\) undertook the first experimental investigations of the character of the variation of the function \(\varepsilon_{\lambda T}\) and, with the aid of a spectrobolometer, found the distribution of energy in the spectrum of soot radiation at several different temperatures. In connection with this, the problem of the analytic expression of Kirchhoff’s function became still more urgent. At the same time the need to solve this problem arose above all from the quite definite demands of practice that had developed by that time, since, in connection with the beginning use of thermal light sources in the form of electric incandescent lamps and the development of spectral analysis, more and more sharply ...
there arises the need to know the concrete form of the function \(\varepsilon_{\lambda T}\), since only in that case could Kirchhoff’s law be used as a practically important computational formula. Naturally, the solution of the problem of finding an unambiguous analytical expression for this function had to be supplied by theory.
The first in the history of physics who attempted to determine theoretically the form of the function \(\varepsilon_{\lambda T}\) was V. A. Michelson, and in this, as O. D. Khvolson correctly notes,\(^1\) lies his immortal merit, since “he was the first to give an impetus to the development of one of the important problems of physics.”
Indeed, V. A. Michelson’s work, which appeared in 1887 and was published simultaneously in Russia and abroad, “An Attempt at a Theoretical Explanation of the Distribution of Energy in the Spectrum of a Solid Body,”\(^7,8\) was not only the first and fairly successful attempt at an approximate solution of the problem of the form of the function \(\varepsilon_{\lambda T}\), but it also indicated the path that led Wien to a more exact approximation and Planck to such a solution of this question as is still considered final. This path consists in applying the methods of statistical physics to that mass ensemble of elementary radiators which a heated solid body represents.
In his investigations Michelson set himself the task which is best formulated in his own words: “The present article has the aim, first, of indicating in general the possibility of applying the theory of probabilities to molecular optics and, second, of revealing that, on the basis of this theory, even with the simplest and most general assumptions about the motion of atoms, we can obtain results that characterize the phenomenon rather fully from its qualitative side.”\(^7,8\)
In accordance with this formulation of the question, Michelson proceeded from the following simplified conceptions. He considered such a solid body all the atoms of which are in approximately identical conditions, regarding the molecules of the body as so close to one another that the interactions between atoms of neighboring molecules are just as strong and continuous as the interactions between atoms of one and the same molecule. The atoms of such a body cannot have oscillations of a definite period, and such bodies must have the same continuous spectrum. Under the described conditions each atom can move within a certain infinitely small sphere of radius \(\rho\). Assuming that within its sphere the atom can move freely and, upon reaching its surface, rebounds inward like an elastic ball, and considering that atoms, by means of frequent disturbances, can rapidly exchange velocities, one may suppose that under such conditions there is established a distribution of velocities expressed by Maxwell’s law.
Before applying this law, Michelson, considering the possible motions of an atom inside an elastic spherical shell, arrives at the following relation between the velocity \(v\) and the period of oscillation of the atom \(\tau\), or the frequency \(\nu\):
\[ \nu = \frac{4\rho}{\tau} = av, \tag{2} \]
i.e., he assumes the frequency of oscillations to be proportional to the velocity of the atoms’ displacements.
Proceeding further from Maxwell’s law for the distribution of velocities and using substitution (2) for \(v\), he obtains the number of atoms \(n_\tau\) in a unit volume whose period of oscillation lies between \(\tau\) and \(\tau + d\tau\). In order to pass from the thermal oscillations of atoms to their radiation, or, as Michelson puts it, “to the oscillations of the ether,” he assumes that the intensity of radiation of atoms in a given spectral interval is proportional, first, to the number \(n_\tau\) of atoms in a unit volume of the body which produce waves of the given period, and, second, to some function of the kinetic energy of these atoms
\[ \varphi\left(\frac{mv^2}{2}\right) = \varphi\left(\frac{8m\rho^2}{\tau^2}\right) = \psi\left(\frac{1}{\tau^2}\right). \tag{3} \]
Expanding this function in a series and retaining only the first term, he obtains:
\[ \psi\left(\frac{1}{\tau^2}\right) = A\tau^{-2p}. \tag{4} \]
Taking this into account, one may write for the intensity:
\[ I_\tau = A n_\tau \tau^{-2p}. \tag{5} \]
Substituting the value \(n_\tau\) calculated from Maxwell’s law and carrying out the corresponding transformations, Michelson obtains a certain general expression for the intensity as a function of the temperature \(T\) and the wavelength \(\lambda\).
Then, imposing the condition that the isothermal curves do not intersect and taking into account Stefan’s law (to determine \(p\) and to give the radiation the character of black radiation), he finds a possible value of the function \(\varepsilon_{\lambda T}\) in the following form:
\[ \varepsilon_{\lambda T} = c_1 T^{\frac{3}{2}} \lambda^{-6} e^{-\frac{c_2}{\lambda T}}, \tag{6} \]
where \(c_1\) and \(c_2\) are constants. The curves calculated from this equation reproduce, in general features, the course of the experimental curves; the only difference is that Michelson’s theoretical curves have a somewhat steeper decline than the experimental ones.
furthermore, that there was no need for complete quantitative agreement with experiment.
The expression for \(\lambda_{\max}\) is obtained from Michelson’s law (6), by taking the derivative \(\dfrac{d\varepsilon_{\lambda T}}{d\lambda}\) and setting it equal to zero, which gives:
\[ \lambda_{\max}^{2} T = \mathrm{const}. \tag{7} \]
Thus, Michelson came close to the displacement law \(\lambda_{\max} T = \mathrm{const}\), although he did not obtain it.
In 1896 W. Wien\({}^{9}\), with whom Michelson was personally and rather closely acquainted, modified Michelson’s derivation and obtained an expression for the function \(\varepsilon_{\lambda T}\) which described the experiment with greater accuracy than Michelson’s formula.
At the same time, Wien’s work is only a certain alteration and extension of Michelson’s ideas, in accordance with somewhat more rigorous arguments.
In order to form a clearer idea of the connection that exists between the investigations of Michelson and Wien, it is best to turn to Michelson’s later review papers, where Michelson emphasizes what was new that Wien had done in the direction of further developing his original considerations.
Thus, in one of these articles, speaking of Kirchhoff’s function, Michelson writes as follows:
“The first attempt to ascertain the form of this function on the basis of the properties of irregular oscillations was made by me in 1887. Starting from Maxwell’s law on the most probable distribution of velocities among particles and using the substitution \(v = \dfrac{c}{\tau}\), where \(\tau\) is the period of oscillation, I obtained a certain general relation between the periods and the corresponding energy, which made it possible, under certain special assumptions and taking Stefan’s law into account, to determine a function satisfying all the requirements that we could then impose on Kirchhoff’s function \(\varepsilon_{\lambda T}\).
Subsequently, in 1896, W. Wien somewhat modified my derivation, replacing the substitution \(v = \dfrac{c}{\tau}\) by the following one: \(v^{2} = \dfrac{c}{\tau}\). In other words: whereas I assumed the number of oscillations per second for each radiation \(\left(\dfrac{1}{\tau}\right)\) to be proportional to the velocity of motion of the atoms emitting it, W. Wien assumes the number of oscillations to be proportional to the square of the velocity”\({}^{10,8}\).
Michelson adds that the replacement of the substitution was made by Wien not arbitrarily, but “in order to bring the law of energy distribution in the spectrum of irregular radiation into agreement with those consequences of thermodynamics which, with the aid of Doppler’s principle, had been obtained by him as early as 1893”\({}^{10,8}\).
“... in order finally to ascertain the form of this function,” Michelson also writes, “Wien, following my example, makes use of Maxwell’s law, which determines the most probable distribution of velocities among gas molecules”¹¹, ⁸.
Considering Michelson’s application of Maxwell’s law to the atoms of a solid to be only inexact, Wien imagines, as the emitting body, a certain peculiar “ideal gas”⁹, which in general does not alter Michelson’s fundamental premise, since in fact the formula is nevertheless derived for its application to a solid body; what is new in Wien’s reasoning is chiefly only the substitution \(v^3 = \dfrac{c}{\tau}\). Moreover, in order to make the radiation of such a “gas” completely black, Wien encloses it in a closed, perfectly reflecting shell.
The formula obtained by Wien has the following form:
\[ \varepsilon_{\lambda T}=c_1\lambda^{-5}e^{-\frac{c_2}{\lambda T}}. \tag{8} \]
This formula is still widely used in those cases when one has to find the distribution of energy in the visible and, in general, short-wave part of the spectrum at not very high temperatures, since under these conditions it gives good agreement with experiment. But it is not justified at high temperatures for large wavelengths, i.e. for large values of the product \(\lambda T\).
As a result of this, Planck¹¹ at one time complicates Wien’s formula (8) by attaching to it the factor
\[ \frac{1}{1-e^{-\frac{c_2}{\lambda T}}} \tag{9} \]
and obtains, instead of (8), the equation
\[ \varepsilon_{\lambda T}=c_1\lambda^{-5}\frac{1}{e^{\frac{c_2}{\lambda T}}-1}, \tag{10} \]
which, with a rather high degree of accuracy, agrees with the experimental data over the entire region of the spectrum.
With Planck’s formula there was completed that important investigation which had been begun by our Russian physicist V. A. Michelson, and the most significant consequence of all these investigations was the appearance of the theory of quanta.
Planck, as is well known, did not confine himself to proposing his formula as an empirical one, but then tried to give also its theoretical derivation, extending for this purpose Michelson’s indicated idea of applying the methods of statistics and linking it with the electromagnetic theory of light.
It is worth noting here that Michelson, in his work “An Attempt at a Theoretical Explanation of the Distribution of Energy in the Spectrum of a Solid Body,” quite clearly also pointed to the possibility of a parallel use of electrodynamics in the derivations. Thus, speaking of the proportionality of the radiation intensity to function (3), he notes in a special footnote that “a theoretical investigation of this function on the principles of the electromagnetic theory of light is conceivable” \(^{7,8}\).
As is known, instead of the oscillating atoms considered by Michelson, Planck \(^{12}\) proceeds from the idea of radiating electric resonators. On the basis of electrodynamical considerations, the volume density of radiation can be represented in the form
\[ u=\frac{8\pi\nu^2}{c^3}\,\overline{u}, \tag{11} \]
where \(\overline{u}\) is the mean energy of a resonator oscillating in a stationary manner.
Assuming that the total energy of all resonators consists of a very large number of further indivisible energy elements \(\varepsilon\), Planck, by applying statistics, and also using Wien’s general displacement law, arrives at the following expression for the mean energy of a resonator:
\[ \overline{u}=\frac{\varepsilon}{e^{\frac{\varepsilon}{kT}}-1}. \tag{12} \]
It then turns out that the energy element must be directly proportional to the number of oscillations:
\[ \varepsilon=h\nu. \tag{13} \]
In this case, for a corresponding value of the constant \(h\), called “Planck’s constant,” the Planck formula (10), obtained on the basis of (11) and (12), agrees well for any wavelength \(\lambda\) and any temperature \(T\).
This was the first, “theoretical,” discovery of light quanta. As is known, the subsequent fruitful development of the doctrine of quanta found irrefutable experimental proofs for itself.
Although Michelson did not explicitly express the idea of quanta in his work, this hypothesis was nevertheless the logical completion of precisely that chain of important investigations which had been begun by V. A. Michelson, and in this sense it is difficult to overestimate the latter as an outstanding predecessor of Wien and Planck.
That Wien’s works are only a further extension of Michelson’s original ideas, as was shown above, cannot raise any doubts. Planck’s works, however, in their
in turn constitute a direct continuation and further development of Wien’s work, and consequently also of Michelson’s. If one looks closely at Planck’s investigations, it cannot be denied that the hypothesis of light quanta is the offspring of the statistical method in molecular optics, the general idea of applying which to this field had first been indicated by Michelson. Planck’s merit, therefore, consists in the further elaboration and realization of this idea.
In his Essays on Spectral Analysis8,11 Michelson says: “For the theoretical determination of the form of the functions \(\varepsilon_{\lambda T}\)—owing to the complete irregularity of black radiation—it is necessary to turn to the theory of probability, as I definitely indicated as early as 1887.” And further: “The universal role and the very necessity of the existence of such a function follow from those general laws of the theory of probability which characterize the completely irregular distribution of any elements whatever in a system subject to the law of large numbers.”
It must be regretted that V. A. Michelson, having at the time indicated the idea—later to prove so fruitful—of applying statistics to the solution of the problem of black radiation, confined himself only to the first approximation on the path toward its realization and did not have the opportunity to carry it through to the end. Intensive and strenuous work on his dissertation led to a serious impairment of his health, and the pulmonary tuberculosis that subsequently developed deprived Michelson of working capacity for almost an entire decade, until prolonged and persistent treatment brought about a relative recovery of his organism. The appearance of the works of Wien, and then of Planck, removed the problem of black radiation from the agenda.
Thus, the investigation begun by V. A. Michelson was completed by Western European scientists. But, as the history of this question briefly reviewed here shows, it would be entirely wrong to detach the achievements of Wien and Planck from the work of our compatriot. This was well understood at the time by many representatives of Western European physics as well. Thus, Lummer, with regard to Michelson’s work under consideration, wrote in 1900 that it “opened the way for a whole series of other, extremely important investigations”13; by this he had in mind the works of Wien and Planck. How great the interest was with which this work of Michelson’s was received in Europe may be judged from one of K. Ångström’s letters to him, in which he wrote: “...I shall never forget our remarkable meeting in Unter den Linden, when, not having heard your name, I began to ask about Vladimir Michelson, whose work on the distribution of spectral energy had delighted me.”13
The appearance of the “Attempt at a Theoretical Explanation of the Distribution of Energy in the Spectrum of a Solid Body” was in its time due
thus also noted in Russia, where, for these studies, the author was awarded the V. P. Moshin Prize by the Society of Lovers of Natural Science.
A general assessment of the significance of Mikhelson’s investigations briefly discussed here can, it seems to us, be given quite exhaustively in the words of A. S. Predvoditelev, who writes that “if the quantitative relations obtained by Mikhelson in analyzing the laws of the distribution of energy over the spectrum of black radiation are erroneous, then the qualitative aspect of his work is so profound that it cannot be excluded from the history of the development of the theory of black radiation of solid bodies.
After Mikhelson, very little remained to be done in order to obtain quantitative relations correctly describing experiment”[^14].
Subsequently Mikhelson returned more than once to questions of thermal radiation, providing excellent historical-critical analyses and surveys of the general development of this problem. Among such works should be included above all the already cited Essays on Spectral Analysis[^8],[^11] and “A Survey of the Latest Investigations in the Thermodynamics of Radiant Energy”[^8],[^10].
CITED LITERATURE
- O. D. Khvolson, Course of Physics, vol. II, GIZ, 1923.
- P. S. Kudryavtsev, History of Physics, vol. I, Uchpedgiz, 1948.
- G. Kirchhoff, Pogg. Ann. 109, 299 (1860).
- I. Stefan, Wiener Akad. Ber. 2, 391 (1879).
- L. Boltzmann, Wied. Ann. 22, 291 (1884).
- S. Langley, Phil. Mag. 22, 149 (1886).
- V. A. Mikhelson, ZhRFKhO 19, 79 (1887); J. de Physique (2), 6, 462 (1887); Phil. Mag. (5), 25, 425 (1888).
- V. A. Mikhelson, Collected Works, vol. 1, “Novyi agronom” Press, Moscow, 1930.
- W. Wien, Wied. Ann. 58, 662 (1896).
- V. A. Mikhelson, A Survey of the Latest Investigations in the Thermodynamics of Radiant Energy, Publ. of the Russian Physico-Chemical Society, St. Petersburg, 1902.
- V. A. Mikhelson, Fizicheskoe obozrenie, 2, 165, 231, 273 (1901).
- M. Planck, Drude’s Ann. 4, 553 (1901).
- I. A. Zdanovsky, Biographical Sketch of V. A. Mikhelson (published in [^8]).
- Essays on the History of Physics in Russia, ed. A. K. Timiryazev, Uchpedgiz, 1949.