Black-Body Radiation
39
With the designs by Lummer, Kurlbaum and Pringsheim (1898/1903) the black
body attained its more or less final shape and has been used for radiation research
in the following decades, remaining occasionally in use to this day.
Primary Literature
1. W. Wien, O. Lummer: Methode zur Prüfung des Strahlungsgesetzes absolut schwarzer Körper.
Annalen der Physik, 3rd ser., 56 (1895) 451–456; reprint in: idem: Das Wiensche Verschiebungsgesetz. Die Verwirklichung des schwarzen Körpers. Ostwalds Klassiker vol. 228,
Frankfurt/Main 1997.
2. O. Lummer, E. Pringsheim: Die Strahlung eines “schwarzen Körpers” zwischen 100◦ und
1,300 ◦ C. Annalen der Physik, 3rd ser., 63 (1897) 395–410.
3. O. Lummer, E. Pringsheim: Der electrisch geglühte “absolut schwarze” Körper und seine Temperaturmessung. Annalen der Physik, 4th ser., 17 (1898) 106–111.
4. O. Lummer, E. Pringsheim: Die strahlungstheoretische Temperaturskala und ihre Verwirklichung bis 2,300 ◦ C. Annalen der Physik, 4th ser., 1 (1903) 3–13.
Secondary Literature
5. D. Hoffmann: On the experimental context of Planck’s foundation of quantum theory. Centaurus
43 (2001) 240–259.
6. H. Kangro: Early History of Planck’s Radiation Law. Taylor and Francis, London 1976.
Black-Body Radiation
Clayton Gearhart
Hot objects give off light and heat in the form of electromagnetic radiation whose
character changes with temperature. Black-body radiation is such electromagnetic radiation in equilibrium with its material surroundings. By the late 1800s,
it was a lively research topic for both theoretical and experimental physicists.
Samuel Pierpont Langley (1834–1906) in the United States, and a group of experimental physicists in Germany centered around the Physikalisch-Technische
Reichsanstalt (PTR) in Charlottenburg, had developed sophisticated techniques for
studying this radiation. Part of their motivation was practical – establishing better
absolute temperature scales, and measuring light intensities, at high temperatures
( Black Body).
In December 1900 and January 1901, the German physicist Max Planck (1858–
1947) published three short papers in which he derived a new equation to describe
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Black-Body Radiation
black-body radiation—one that ever since has given excellent agreement with observation. This derivation was the culmination of research Planck had begun in the
mid-1890s. In a series of lengthy papers, Planck had combined thermodynamics,
in which he was an acknowledged authority, with the new electromagnetic theory
of James Clerk Maxwell (1831–1879). He considered the electromagnetic field in
equilibrium with what he called “resonators” – electric dipoles oscillating in simple harmonic motion – which represented the material cavity containing the field.
By late 1899, he had found a new and more rigorous derivation of Wien’s law,
an equation describing black-body radiation discovered in 1896 by his friend and
colleague Wilhelm Wien (1864–1928), and seemingly in good agreement with experiment.
By mid-1900, however, physicists at the PTR had found systematic deviations
between Wien’s law and their latest experiments. Planck went back to work, and
by the end of the year, had produced his new radiation law, which takes the familiar form
hν
8πν 2
,
uν = 3 hν/ kT
−1
c e
where c is the speed of light, and uν is the energy density of the electromagnetic
field as a function of the frequency ν and the absolute temperature T . This equation also contains two new fundamental constants of nature, h and k – today we
call them Planck’s constant and Boltzmann’s constant – to which Planck attached the greatest importance. They played a central role in his system of natural
units for length, mass, time, and temperature, which as he said in 1899, “necessarily retain their significance for all times and for all cultures, even alien and
non-human ones.”
However, Planck’s derivation was decidedly mysterious. It relied on a 1877 paper by the Austrian physicist Ludwig Boltzmann (1844–1906), relating entropy and
probability, now famous but little known in 1900. Today it is summarized in the
equation S = k log W , inscribed on Boltzmann’s tombstone in Vienna. Boltzmann
had begun with a physically unrealistic picture, in which he divided the energy of a
gas into finite “energy elements” (as Planck later called them), which he distributed
among the molecules of an ideal gas. This step allowed him to use combinatorials
to calculate the probabilities of microscopic states and relate them to the entropy of
a gas. Planck applied a similar scheme to his resonators, though he persisted in his
absolute interpretation of entropy and the second law of thermodynamics, in sharp
contrast to Maxwell’s and Boltzmann’s probabilistic viewpoint.
In 1877, Boltzmann had replaced his artificial scheme with the more realistic one
of partitioning molecules among arbitrarily small cells in phase space, thereby recovering the standard description of an ideal gas. Planck, by contrast, could make his
derivation work only by retaining these finite “energy elements” and assigning them
the specific size hν. In 1900, he said nothing about the striking differences between
the two derivations, though he certainly understood what Boltzmann had done.
Today we call these energy elements “quanta,” and over the last century, physicists have developed the strange new theory called quantum mechanics to describe
nature at the atomic level. But in 1900, all this was yet to come. The “energy
Black-Body Radiation
41
elements,” whatever they might be, had no obvious interpretation in the physics
of the day. Planck in 1900 said virtually nothing about how to interpret them physically. Both his contemporaries and later historians found it difficult to grasp his
meaning.
Over the next decade, scientists slowly came to terms with these new ideas
( Quantum theory, early period). If Planck’s energy elements do become arbitrarily
small, for example, Planck’s law goes over to the Rayleigh-Jeans law,
uν = 8πν 2 /c3 kT , in which the radiation density increases without limit at short
wavelengths—an effect Paul Ehrenfest (1880–1933) later dubbed the “ultraviolet
catastrophe.” Physicists developed an increasingly sophisticated understanding of
this theme and its relation to equipartition in the first decade of quantum theory.
Planck contributed to these efforts in his 1906 book, Lectures on the Theory of
Heat Radiation, in which he presented h as the “elementary quantum of action,”
since its units were those of action, the product of energy and time. He also showed
that h is the size of a finite “elementary domain” in phase space, a step that made his
combinatorial assignments of probability more plausible. Hendrik Antoon Lorentz
(1853–1928), Paul Ehrenfest, Henri Poincaré (1854–1912) and others also explored
the foundations of black-body radiation, and showed that it necessarily involved a
sharp and inescapable break with earlier physical theory.
For many years, Planck pointed out the need for a physical interpretation of his
theory, but was reluctant to advance one himself. Only in 1909 did he state publicly that the energies of his resonators were restricted to integer multiples of hν.
But in that same year, Lorentz showed that under some circumstances, it would take
an implausibly long time to absorb one quantum of radiation from a Maxwellian
electromagnetic field. Neither Lorentz, Planck, nor most other physicists were prepared to accept the alternative of “light quanta” that Albert Einstein (1879–1955)
had proposed in 1905 ( Light quanta; Quantum theory, early period).
In 1911, therefore, Planck proposed what became known as his “second quantum
theory,” in which resonators absorbed energy continuously, but emitted energy in
quanta only when they reached the boundaries of finite cells in phase space, where
their energies became integral multiples of hν. This theory also led Planck to his
new radiation law. But in this version, resonators possessed a “zero-point” energy,
the smallest average energy that a resonator could take on. Thus, for the first time,
physicists contemplated systems whose energy did not go to zero at the absolute zero
of temperature. This zero-point energy soon took on a life of its own, appearing in
the early 1920s in the context of both Planck’s first and second theories, and after
1925, finally finding a secure home in modern quantum mechanics.
Albert Einstein took perhaps the most radical view of black-body theory, beginning with his famous paper of 1905, in which he suggested that light consists of
“a finite number of energy quanta that are localized in points of space, move without dividing, and can be absorbed or created only as a whole.” ( light quanta;
Quantum theory, early period) In succeeding years, black-body radiation and its
connection to light quanta remained at the center of Einstein’s thoughts. In 1909,
for example, it was at the heart of his analysis of fluctuations – random variations
in energy and momentum – in which he argued that light sometimes behaved like
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Black-Body Radiation
a wave and sometimes like a particle, and that the dual wave and particle nature
of light was inescapable – he spoke of “a kind of fusing of the wave and emission
theories of light.”
In 1916, he found a new derivation of Planck’s radiation law, his famous and
influential “A and B coefficients” argument that involved assumptions on the “stimulated emission” of light and set down the underlying principles of the laser, not
invented until decades later. And in 1924, he understood immediately the significance of a paper sent to him by the then-unknown Indian physicist Satyendra
Nath Bose (1894–1974), who had found yet another derivation of Planck’s radiation law – one that implicitly suggested that Einstein’s light quanta were not
independent particles. Einstein translated Bose’s paper into German and arranged
for its publication. He also saw its implications for the seemingly unrelated topic
of quantum ideal gases, and published the papers describing what is now known as
Bose-Einstein condensation, experimentally confirmed only recently ( Quantum
statistics, Bose-Einstein-statistics).
In short, although black-body theory was not the whole of early quantum theory,
it remained a continuing source of inspiration and new discoveries. Please see also
the Reference Specific heats.
Primary Literature
1. M. Planck (annotated by Hans Kangro; transl. D. ter Haar and Stephen G. Brush): Planck’s
Original Papers in Quantum Physics (Taylor & Francis, London, 1972; German and English
Edition)
2. M. Planck (ed., with an introduction, by Alan A. Needell): The Theory of Heat Radiation
(American Institute of Physics, College Park, MD, 1988) This work contains both the 1906
edition, never translated into English, and the very different 1913 edition, based on Planck’s
second theory.
Secondary Literature
3. O. Darrigol: From c-Numbers to q-Numbers (University of California Press, Berkeley, 1992);
Statistics and combinatorics in early quantum theory. Historical Studies in the Physical and Biological Sciences 19, 17–80 (1988); The Historians’ Disagreement over the Meaning of Planck’s
Quantum. Centaurus 43, 219–239 (2001)
4. C. A. Gearhart: Planck, the Quantum, and the Historians. Physics in Perspective 4, 170–215
(2002)
5. M. J. Klein: Paul Ehrenfest: Vol. 1, The Making of a Theoretical Physicist (North Holland,
Amsterdam, 1970)
6. T. S. Kuhn: Black-Body Theory and the Quantum Discontinuity, 1894–1912 (Oxford University
Press, New York, 1978)
7. J. J. Prentis: Poincaré’s Proof of the Quantum Discontinuity of Nature. American Journal of
Physics 63, 339–350 (1995)
8. R. Staley: On the Co-Creation of Classical and Modern Physics. Isis 96, 530–558 (2005)
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