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Planck’s Theory of Radiation

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The Bumpy Road

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Abstract

In this chapter I explore Planck’s radiation theory from his preliminary studies (1896) through his more mature Pentalogy (1897–1899). Planck viewed the problem of the black-body radiation very differently from Wien and the majority of his contemporaries. In particular, Planck was not primarily interested in deriving a radiation law. Instead, he considered heat radiation as an ideal case to support his strict view of thermal irreversibility. He wanted to prove that electromagnetic radiation in a cavity, when suitably stimulated, reaches irreversibly a form of stable thermal equilibrium. Initially, Planck thought that this statement could be demonstrated as a consequence of the electromagnetic features of the problem. Boltzmann jumped in and showed that this could not possibly be the case. In the second part of the Pentalogy, Planck changed strategy. He modified the morphology of his theory to accommodate new resources and gave a more central role to some symbolic practices, notably Fourier series. The central move of the reorganization of his theory was the introduction of the hypothesis of natural radiation as a way to draw a divide between the macroscopic and microscopic state. Planck obtained his argument for irreversibility, but he had to pay a prize for it: his entire program depended essentially on the validity of Wien’s law.

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Notes

  1. 1.

    Important theoretical research on the propagation of spherical waves had been already carried out independently by Rowland (1884). For this reason, it is not uncommon to find in the literature the term ‘Rowland-Hertz oscillator’.

  2. 2.

    On this point see (Buchwald 1998).

  3. 3.

    This function, which is actually a potential, is currently known as the Hertz vector. From this vector it is easy to derive the usual scalar and vector potentials (Bremmer 1958; Essex 1977).

  4. 4.

    Planck assumes that no Joule effect takes place, and hence the energy emitted is due to the oscillation and not to transformation into other kinds of energy.

  5. 5.

    In particular, he shows that if the resonator is small with regard to the wavelength of the interacting radiation, it is possible to simplify the expression for the dipole moment and its derivative (Darrigol 1992). Ultimately, Planck’s theory of resonator relies on a clear-cut distinction between two lengths of different orders of magnitude: the characteristic wavelength involved in the resonator-field interaction and the geometric lengths ascribed to the material system (size of the cavity, linear dimension of the resonator). The latter must be much larger than the former to apply the usual laws of refraction and reflection without the intervention of diffraction phenomena.

  6. 6.

    Planck arrives at the second order from the usual third order equation by considering only real solutions.

  7. 7.

    Here one can see at work an example of the difference between ‘physics of problems’ and ‘physics of principle’ recently outlined by Seth (2010).

  8. 8.

    These kinds of assumptions, similarly to the approximations on the linear size of the resonator, are common in Planck’s theory and, more generally, in theories concerned with drawing macroscopic consequences from a microscopic model.

  9. 9.

    Note that this consequence also follows from Wien’s thermodynamic argument that only higher frequencies can produce thermal effects (Wien 1893). Though, Planck does not quote Wien: he wants to make a general argument on the behavior of an oscillating structure separate from the nature of heat radiation.

  10. 10.

    The complete definition of the SZA is more complex; see (Ehrenfest and Ehrenfest 1911, pp. 4–13).

  11. 11.

    Boltzmann (1877, 1909, Vol. II, pp. 164–223). According to common wisdom, Boltzmann did not realize the probabilistic implications of his theory until Loschmidt’s reversibility objection (Klein 1973; Kuhn 1978; Brown et al. 2009). However, this claim does not stand before a careful examination of Boltzmann’s papers during the period of 1868–1877. I have argued elsewhere that Boltzmann was aware since the late 1860 s that irreversibility was a matter of probability (Badino 2011).

  12. 12.

    Indeed, Planck was right: Maxwell’s distribution is the only one that fulfills the SZA applied to reversed collisions (Ehrenfest and Ehrenfest 1911, pp. 11–13). In that case, however, the SZA ceases to be a probabilistic assumption.

  13. 13.

    Boltzmann to Eilhard Wiedemann, 20 March 1896, (Höflechner 1994, Doc. 427).

  14. 14.

    That such a time exists, follows from the condition above that only the harmonics with very high frequency and short period play a role in heat radiation.

  15. 15.

    The amplitude \(C_n\) is related to the amplitude of the Hertz vector, while \(\theta _n\) are its phase constants. The index \(a = 1, 2, 3, \dots \) comes from an ingenious rearrangement of the harmonics that Planck operates to simplify the calculation of the Poynting vector. The cross products can be ordered in groups of harmonics separated by an increasing distance a.

  16. 16.

    The integers \(k_n\) are related to the natural period of the resonator. Essentially, they select the acceptable solutions in terms of the resonance interaction. The amplitude \(D_n\) is related to the field amplitude \(C_n\) via special phase parameters.

  17. 17.

    A clarification of this argument would come in the fourth paper. There, Planck shows that the Fourier series of the radiation intensity is time-dependent in two different ways: in the main series, the time appears in periodic terms, while in the Fourier coefficients, time is the argument of functions that are in general aperiodic. The equilibration process acts precisely on these aperiodic terms (Planck 1898b, 1958, Vol. I, p. 541).

  18. 18.

    In fact, the energetic difference between the primary and the secondary field is precisely the contribution of the resonator.

  19. 19.

    The parameter \(\gamma \) confines the series to terms very close to the characteristic frequency of the resonator.

  20. 20.

    If the damping process lasts too long, the resonator will still be in vibration when a new train of waves impinges on it, and the energy given off will be the superposition of the radiations that arrived at different times instead of the component of a single wave.

  21. 21.

    The parameter \(\delta \) plays the same role as the parameter \(\gamma :\) it constraints the series in order that only frequencies very close to the characteristic frequency carry a non-negligible energy.

  22. 22.

    On the technical aspects of this debate see (Dias 1994). Culverwell had already objected in the same vein in a previous paper (Culverwell 1890), which had apparently gone unnoticed.

  23. 23.

    A similar argument was proposed by (Bryan 1894a) in support of (Watson 1894).

  24. 24.

    (Boltzmann 1895b), see Sect. 3.2.3.

  25. 25.

    Thomas Kuhn has suggested that the molar/molecular distinction allowed Boltzmann to combine statistical and dynamical phraseology because it corresponds to the micro/macro distinction (Kuhn 1978, pp. 54–60). This terminology, however, was not new. J.J. Thomson, for instance, used a similar distinction to stress the uncontrollability of the molecular behavior (Thomson 1887). The molar/molecular distinction meant therefore something slightly different from the micro/macro divide, because the emphasis was preferably on the possibility of acting on microscopic states. It is precisely this point that Boltzmann wants to make with the concept of molecular chaos.

  26. 26.

    Ultimately, Boltzmann’s point amounts to distinguishing between equilibrium as a state described by Maxwell’s distribution and as the end point of a mechanical trajectory. In other words, it concerns the epistemic surplus of two sets of symbolic practices: probabilistic tools and mechanical methods.

  27. 27.

    In effect, Kuhn regards the introduction of the HNR as the entering of statistics into radiation theory. Darrigol is more cautious and points out that Planck did not need to install Boltzmann’s full package in his theory (Darrigol 1992, pp. 51–54).

  28. 28.

    On this point see also (Seth 2010, pp. 119–126).

  29. 29.

    I elaborate on the probabilistic consequences of this hypothesis in (Badino 2009); see also (Needell 1980).

  30. 30.

    Subsequently, Planck published a long paper in the Annalen that essentially reports the result of the fifth part of the Pentalogy with some small technical improvements (Planck 1900, 1958, Vol. I, pp. 614–667).

  31. 31.

    Letter from Planck to Runge, 14 October 1898; see also (Planck 1958, Vol. I, p. 623).

  32. 32.

    One formal difficulty that Planck does not succeed in overcoming is the dependence of the spectral component on the damping constant of the analyzing resonator. At the end, he is forced to drop it on the grounds that an acceptable spectral component cannot possibly depend on the features of the ideal apparatus used to measure it.

  33. 33.

    It is interesting to note that Planck does not write explicitly the relation between energy density and average energy of the resonator:

    figure a

    which is usually associated with his electromagnetic radiation theory. To be sure, this relation can be easily derived from Planck’s equations, but it is worth stressing that, at this stage, energy density and resonator energy are not the two macroscopic quantities he wants to relate. It is much more important for his irreversibility argument to work with the polarized intensity.

  34. 34.

    For a discussion of this aspect see (Badino and Robotti 2001).

  35. 35.

    Criticisms against this procedure were raised in (Burbury 1902; Ehrenfest 1905); see below Sect. 4.7.1.

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    Massimiliano Badino

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Badino, M. (2015). Planck’s Theory of Radiation. In: The Bumpy Road. SpringerBriefs in History of Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-20031-6_3

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