Thermodynamic deduction versus quantum revolution: The failure of Richardson's theory of the photoelectric effect

Summary

Between 1911 and 1914, Owen Richardson formulated a theory of photoelectricity based on thermodynamics and statistical reasoning. Although this theory succeeded in accounting for most of the relevant phenomena and despite the lack of competing causal or descriptive accounts of the phenomena, it failed to attract other physicists. This paper seeks the reasons for the neglect of this theory in contemporary cultures of photoelectric research. Four main causes of neglect are identified: the relatively high number and the nature of the theory's assumptions, the contradiction of one of these assumptions with contemporary views, the failure to suggest new predictions or to account for hitherto unexplained regularities, and the view shared by many scientists that the problem of electromagnetic radiation required a radical solution that a descriptive theory could not provide. The expectation for a radical solution defines the revolutionary character of a research field. In the case of photoelectricity, it originated in a web of evidence to which other fields contributed. The very possibility of Richardson's theory shows that, taken separately, this phenomenon could receive an account which, unlike Einstein's light quantum, did not require deep changes in the conception of nature.

I am grateful to Leo Corry and a referee for their valuable comments and especially to Olivier Darrigol for his very helpful editorial comments. Part of the research for this paper was done while I was a Lady Davis fellow at the Hebrew University of Jerusalem. I thank the Library of the Max Planck Institute for the history of science in Berlin for supplying me sources for this research.

Notes

¹For a short discussion of the term, see Shaul Katzir, ‘From Explanation to Description: Molecular and Phenomenological Theories of Piezoelectricity’, Historical Studies in the Physical and Biological Sciences, 34 (2003), 69–94 (70–71).

²Owen W. Richardson, ‘The theory of photoelectric action’, Philosophical Magazine, 24 (1912), (570–74).

³In the twentieth century, ‘phenomenological’ considerations even implied quarks.

⁴On this theory, see Ole Knudsen, ‘O. W. Richardson and the Electron Theory of Matter, 1901–1916’, in Histories of the Electron: The Birth of Microphysics, edited by Jed Z. Buchwald and Andrew Warwick (Cambridge, MA, 2001), 227–53. Most contemporary references to Richardson's theory of photoelectricity mention its use of thermodynamics and statistics.

⁵On the fate of the phenomenological theory of piezoelectricity, see Katzir (note 1)

⁶Roger H. Stuewer, ‘Non-Einsteinian interpretations of the photoelectric effect’,in Historical & Philosophical Perspectives of Science, edited by Stuewer (Minneapolis, MN, 1970), 246–63; The Compton Effect: Turning Point in Physics (New York, 1975), 60–63. See also Bruce R. Wheaton, The Tiger and the Shark: Empirical Roots of Wave–Particle Dualism (Cambridge, 1983), 181–82, 193–94. By describing it only shortly and by his evaluation of the Sommerfeld–Debye theory (see below), Wheaton implies that Richardson's theory was not a serious alternative to Einstein's.

This section is based mostly on contemporary histories: Egon R. von Schweidler, ‘Über die lichtelektrischen Erscheinungen’, Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften 107 IIa (1898), 881–909, H. Stanley Allen, Photo-Electricity: the Liberation of Electrons by Light. With Chapters on Fluorescence & Phosphorescence, and Photo-Chemical Actions & Photography (London, 1913).

⁸Roger H. Stuewer, ‘Hertz's discovery of the photoelectric effect’, Actes du XIIIe Congrès International d'histoire des sciences, section VI (Moscow, 1971), 35–43; Wilhelm Hallwachs, ‘Lichtelektrizität’, in Handbuch der Radiologie, edited by Erich Marx (Leipzig, 1916), 245–561. Schweidler (note 7), 883 and passim.

⁹On early experiments: ibid., 886–87, 892–94; on the identification of the carriers: Bruce R. Wheaton, ‘Philipp Lenard and the photoelectric effect, 1889–1911’, Historical Studies in the Physical Sciences, 9 (1978), 299–322.

¹⁰Allen (note 7), 68–79, 122–24; Schweider (note 7) 884–88; Robert A. Millikan, ‘The distinction between intrinsic and spurious contact e.m.f.s and the question of the absorption of radiation by metals in quanta’, Physical Review, 18 (1921), 236–44.

¹¹The actual experiment was slightly more complex and more accurate, since a small reverse current appeared below the null current owing to reflected light.

¹²Wheaton, ‘Lenard’ (note 9), 313–17, quotation on 317.

¹³Wheaton, ‘Lenard’ (note 9), 313–20, quotations on 317.

¹⁴Albert Einstein, ‘On a heuristic point of view concerning the production and transformation of light’, The collected papers of Albert Einstein—English translations, translated by Anna Beck, II, 86–103 (98).

¹⁵Albert Einstein, ‘On a heuristic point of view concerning the production and transformation of light’, The collected papers of Albert Einstein—English translations, translated by Anna Beck, II, p. 100. On Einstein's work, see Martin J. Klein, ‘Einstein's First Paper on Quanta’, Natural Philosopher, 2 (1963), 59–86, Thomas S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912 (Chicago, 1978), 170–82, Wheaton, Tiger (note 6) 105–109.

¹⁶Lenard himself did not suggest any relation between light frequency and the velocity of the emitted electrons; P. Lenard, ‘Ueber die lichtelektrische Wirkung’, Annalen der Physik, 8 (1902), 149–98.

¹⁷Ladenburg's motivation does not square with Wheaton's claim that ‘the triggering hypothesis … effectively rendered close experimental study of the relation [between frequency and energy] unnecessary’. Wheaton, Tiger (note 6), 108.

¹⁸Erich Ladenburg, ‘Über Anfangsgeschwindigkeit und Menge der photoelektrischen Elektronen in ihrem Zusammenhange mit der Wellenlänge des auslösenden Lichtes’, Physikalische Zeitschrift, 8 (1907), 590–94. In one method, he measured the positive potential needed to stop the electrons; in another, he measured the energy of the electrons directly; Erich Ladenburg and Karl Markau, ‘Über die Anfangsgeschwindigkeiten lichtelektrischer Elektronen’, Physikalische Zeitschrift, 9 (1908), 821–28. They assumed that for a pure resonance effect, the electrons’ velocity should follow Maxwell's distribution; Abram F. Joffé, ‘Eine Bemerkung zu der Arbeit von E. Ladenburg: ‘Über Anfangsgeschwindigkeit und Menge der photoelektrischen Elektronen usw’., Annalen der Physik, 24 (1907), 939–40 (939).

¹⁹Such experiments were conducted by A. W. Hull, Jacob Kunz (twice with conflicting results), David W. Cornelius (who found that the frequency is proportional to the square root of the velocity), J. R. Wright, Millikan with collaborators, and O. v. Bayer with A. Gehrts: David W. Cornelius, ‘The Velocity of Electrons in the Photo-electric Effect, as a Function of the Wave Lengths of the Light’, Physical Review, 1 (1913), 16–34; Jacob Kunz, ‘On the Initial Velocity of Electrons as a Function of the Wave-Length in the Photoelectric Effect’, Physical Review, 31 (1910), 536–44; O. v. Bayer and A Gehrts, ‘Die Anfangsgeschwindigkeiten lichtelektrisch ausgelöster Elektronen’, Verhandlungen der Deutschen physikalischen Gesellschaft, 12 (1912), 870–78; Wheaton, Tiger (note 6), 234–38.

²⁰This was the opinion of Allen (note 7) 134, 140, and of the authors of the experiments.

²¹Robert Pohl and Peter Pringsheim, ‘On the Long-wave Limits of the Normal Photoelectric Effect’, Philosophical Magazine, 26 (1913), 1017–24 (quote on 1019). The idea is elaborated in their 1914 textbook, Die lichtelektrischen Erscheinungen (Braunschweig, 1914), 63–64. A reply is in Arthur Ll. Hughes, ‘On the Long-wave Limits of the Normal Photoelectric Effect’, Philosophical Magazine, 27 (1914), 473–75. Their argument for Einstein's relation indicates the dependence of the understanding of photoelectricity on developments in other areas, especially regarding electromagnetic radiation. Bruce Wheaton showed that between 1911 and 1913, the previously separated fields of research of X-rays and light were united. The 1912 observation of X-rays interference strengthened previous developments. Wheaton, Tiger (note 6), 189–92, 198–203. On Millikan's experiment, see Millikan, ‘A Direct Photoelectric Determination of Planck's “h”‘, Physical Review 7 (1916), 355–88; Stuewer, Compton Effect (note 6), 72–75; Hasok Chang, ‘Can Planck's Constant be Measured with Classical Mechanics?’, International Studies in the Philosophy of Science, 11 (1997), 223–43; Wheaton, Tiger (note 6), 241.

²²On Thomson's opinion, cf. John L. Heilbron, ‘Lectures on the History of Atomic Physics 1900–1922’, in History of Twentieth Century Physics, edited by Charles Weiner (New York, 1977), 40–108 (56–57). Wheaton, Tiger (note 6), 136–37, 140–42.

²³On Thomson's opinion, cf. John L. Heilbron, ‘Lectures on the History of Atomic Physics 1900–1922’, in History of Twentieth Century Physics, edited by Charles Weiner (New York, 1977), 176–78. Ramsauer is quoted on 178.

²⁴The balance of probabilities appears to be in favour of regarding the normal photo-electric effect also [like the selective effect] as being due to resonance’, Allen (note 7), 150, and 118–29 for the discussion of the selective effect.

²⁵The experimental doubts about the exact relation did not seem to influence the theoretical suggestions.

²⁶Stuewer, ‘Non-Einsteinian interpretations’ (note 6), 252–53, Thomson's quote on 253. Based on a triggering hypothesis, Thomson's explanation presumed that the energy of the electrons originated in the atoms, despite Lenard and Ramsauer's finding.

²⁷Wheaton, Tiger (note 6), 180–89.

²⁸Kuhn (note 15), ch. 10 (first) quotation on 244, Wheaton, Tiger (note 6), 178–80 (second) quotation on 179.

²⁹Knudsen (note 4), 234–35.

³²Richardson, ‘Theory of photoelectric action’ (note 2), 574. Richardson's sympathy for Planck's theory implied no commitment

³⁰Owen W. Richardson, ‘Some applications of the electron theory of matter’, Philosophical Magazine, 23 (1912), 594–627, summarized in ‘The application of statistical principles to photoelectric effects and some allied phenomena’, Physical Review 34 (1912), 146–49 (read in December 1911). Richardson had previously shown that these electrons followed Maxwell's distribution of velocities well. This was the first measurement that confirmed a Maxwell–Boltzmann distribution function for any gas. Walter Kaiser, ‘Electron gas theory of metals: Free electrons in bulk matter’, in Histories of the Electron (see Knudsen note 4), 255–303 (264).

³¹Owen W. Richardson, ‘Some applications of the electron theory of matter’, Philosophical Magazine, 23 (1912), 594–627, summarized in ‘The application of statistical principles to photoelectric effects and some allied phenomena’, Physical Review 34 (1912), 146–49 (read in December 1911). Richardson had previously shown that these electrons followed Maxwell's distribution of velocities well. This was the first measurement that confirmed a Maxwell–Boltzmann distribution function for any gas. Walter Kaiser, ‘Electron gas theory of metals: Free electrons in bulk matter’, in Histories of the Electron (see Knudsen note 4), 617.

³³According to Richardson, ‘the change in U, the energy of the system, produced by a motion of the piston is where V is the volume below the piston and ‘φ the difference in the energy of the system due to the escape of a single electron’. ‘The corresponding increment of entropy is:

[S is the entropy, p the pressure] Since [d]S is a perfect differential when V and T are the independent variables [in modern terms, one can use the Maxwell relation for this case:

³³This is Richardson's implicit definition of w, equivalent to (in this case, a variation of volume induces the change in energy). With the perfect gas law p = nRT, this leads to equation (2). For the sake of consistency and clarity, I use T for the temperature and F for the kinetic energy throughout this paper, although Richardson and others employed different symbols in different publications. Richardson , ‘Applications of the Electron Theory of matter’ (note 30), 618–20 (619).

³⁴According to Richardson, ‘the change in U, the energy of the system, produced by a motion of the piston is where V is the volume below the piston and ‘φ the difference in the energy of the system due to the escape of a single electron’. ‘The corresponding increment of entropy is:

[S is the entropy, p the pressure] Since [d]S is a perfect differential when V and T are the independent variables [in modern terms, one can use the Maxwell relation for this case:

³⁴This is Richardson's implicit definition of w, equivalent to (in this case, a variation of volume induces the change in energy). With the perfect gas law p = nRT, this leads to equation (2). For the sake of consistency and clarity, I use T for the temperature and F for the kinetic energy throughout this paper, although Richardson and others employed different symbols in different publications. Richardson , ‘Applications of the Electron Theory of matter’ (note 30), 619–22, 625, quotation on 622.

³⁶The expression is

where c is the speed of light, α the proportion of the electrons absorbed in the metal from those that reach it, β a constant calculable from the kinetic theory of gases, and ω the potential energy of an electron (notice that this is not w), 624.

³⁵The expression is

where c is the speed of light, α the proportion of the electrons absorbed in the metal from those that reach it, β a constant calculable from the kinetic theory of gases, and ω the potential energy of an electron (notice that this is not w). Ibid., 623.

³⁷On the experiment: Owen W. Richardson and Karl T. Compton, ‘The photoelectric effect’, Philosophical Magazine, 24 (1912), 575–94 (first quote there on 590). The results are also mentioned in Richardson, ‘Theory of Photoelectric Action’ (note 2), 574 (second quote there on 571). In 1918, Richardson recalled that

when these experiments [with Compton] were started I thought it improbable that equation would turn out to be correct, on account of the very grave objections to the form of quantum theory on which it had up to that time been based by Einstein. After making the experiments I felt that there was no reasonable doubt about its entire validity

cf. ‘The photo-electric action of X-rays’, Proceedings of the Royal Society of London, 94A (1918), 269–80, on 269–70. Since Compton and Richardson started to work on their experiment in 1911, and the latter referred to ‘new experimental results’ in his first theoretical paper, it is possible that his discussion of the plausibility of Einstein's equation there was influenced by preliminary experimental results.

³⁸Ibid. Richardson derived a similar expression without performing the integration in his first paper on the photoelectron theory: ‘Applications of the Electron Theory’ (note 30), 621.

³⁹Despite Knudsen's claim (note 243–44), this is not a triggering hypothesis theory.

⁴⁰Note that the left-hand side of (5) has the form of a Laplace transform only if Wien's approximation is used for the blackbody law. Apparently, Richardson could not have derived the threshold frequency without this approximation. Richardson justified the use of Wien's law: ‘We know’, he wrote, ‘that [the number of released electrons] is zero for small values of v, and it may for most substances in all probability be put equal to zero over the part of the spectrum for which [Wien's law] is not a sufficiently accurate representation of the facts’. Richardson, ‘Applications of the electron theory of matter’ (note 30), 622.

⁴⁰In the same year, Henri Poincaré used the properties of the inverse Laplace transform to show that Planck's and similar laws must lead to discontinuity. However, Planck's distribution function is part of the Laplace transform instead of the original function (in the integrand) in Richardson's reasoning. A few months earlier, Paul Ehrenfest's proof of the necessity of the quantum hypothesis used the Laplace transform in a somewhat different manner (he did not solve equations with them). Henri Poincaré, ‘Sur la théorie des quanta’, Comptes rendus 153 (1911), 1103–108 (presented on 4 December 1911); Paul Ehrenfest, ‘Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?’ Annalen der Physik, 36 (1911), 91–118, especially 106–107, 115–17.

⁴¹Note that the left-hand side of (5) has the form of a Laplace transform only if Wien's approximation is used for the blackbody law. Apparently, Richardson could not have derived the threshold frequency without this approximation. Richardson justified the use of Wien's law: ‘We know’, he wrote, ‘that [the number of released electrons] is zero for small values of v, and it may for most substances in all probability be put equal to zero over the part of the spectrum for which [Wien's law] is not a sufficiently accurate representation of the facts’. Richardson, ‘Applications of the electron theory of matter’ (note 30), 622.

In the same year, Henri Poincaré used the properties of the inverse Laplace transform to show that Planck's and similar laws must lead to discontinuity. However, Planck's distribution function is part of the Laplace transform instead of the original function (in the integrand) in Richardson's reasoning. A few months earlier, Paul Ehrenfest's proof of the necessity of the quantum hypothesis used the Laplace transform in a somewhat different manner (he did not solve equations with them). Henri Poincaré, ‘Sur la théorie des quanta’, Comptes rendus 153 (1911), 1103–108 (presented on 4 December 1911); Paul Ehrenfest, ‘Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?’ Annalen der Physik, 36 (1911), 572–74, quote on 574.

⁴²Richardson, ‘Applications of the Electron Theory’ (note 30), 617.

⁴³Richardson, ‘Theory of Photoelectric Action’ (note 2), 573. The same assumption appears in Einstein's theory, but there it follows from the light-quantum hypothesis. Hughes, Die Lichtelektrizität, translated by Max Iklé (Leipzig, 1915), 64–65.

⁴⁴Richardson, ‘The Theory of Photoelectric and Photochemical Action’, Philosophical Magazine, 27 (1914), 476–88.

⁴⁵Discussing their determination of Planck's constant by photoelectric measurements, in the same issue of the Philosophical magazine, Richardson and Compton considered the possibility of different values: ‘It may be, however, that w ₀ is slightly smaller for the electrons emitted photoelectrically by the complete radiation, than it is for the electrons which are emitted thermionically’ (Richardson and Compton (note 37), 592). On Richardson's view of the assumption, cf. Richardson, ‘Photoelectric and Photochemical Action’ (note 44), 487–88.

⁴⁶On the value of the exponent: Richardson, ‘Theory of Photoelectric Action’ (note 2), 572–3. See the derivation of the term on 571. At the end of the paper, Richardson claimed that his equation for the energy of the emitted electrons (7) was ‘independent of the particular value of the index 2 of T in equation (4)’. Ibid., 575. Since equations (5) and (7) are Laplace transforms of the same kind, F (the ratio of the transformed function) should be of the kind: Φ = x/2(hν-w ₀)where x is the exponent of the temperature in equation (4). The quotation from 1914 in Richardson, ‘Photoelectric and Photochemical Action’ (note 44), 480–81.

⁴⁷This does not include assumptions, laws etc. that were commonly accepted at the time, such as the existence of electrons.

⁴⁸See, for example, the case of piezoelectricity, Katzir (note 1).

⁴⁹Owen W. Richardson and Karl T. Compton, ‘The photoelectric effect II’, Philosophical Magazine, 26 (1913), 549–67, on 566. An experiment carried out in Uppsala led to similar results: Simon Werner, Über lichtelektrische Elektronenemission bei Kathodenzerstäubungsschichten (Uppsala, 1913), on 61.

⁵⁰Richardson, ‘The theory of photoelectric and photochemical action’, Physical Review, 3 (1914), 64–65 (presented on 29 November 1913) summary of ‘Photoelectric and Photochemical Action’ (note 44). Instead of admitting that he had changed parts of the first theory, Richardson claimed that he only ‘amplified the discussion of some points’ (ibid. p. 476). On the term atom, 476 and passim. The inclusion of X- and γ-rays in the new theory indicates the growing identification of them with regular light. In 1912, Richardson still included the proviso ‘if they differ from light only in degree and not in kind’. Richardson, ‘Theory of Photoelectric Action’ (note 2), 574. He no longer did so in the second theory.

⁵¹Richardson, ‘Photoelectric and Photochemical Action’ (note 44), 478. The high-frequency approximation enters the step from his equation (10) to (11) on 479.

⁵²Richardson, ‘Photoelectric and Photochemical Action’ (note 44), 478. The high-frequency approximation enters the step from his equation (10) to (11) on 479, 478–80, quote on 477.

⁵³Richardson, ‘Photoelectric and Photochemical Action’ (note 44), 478. The high-frequency approximation enters the step from his equation (10) to (11) on 479, 487–88. Another reason for this conclusion was that ‘the results which we have been given are not easily harmonized with the values of the specific heats of bodies at low temperatures’ (ibid.). Circa 1911, it was clear that the specific heat of solids approaches zero at low temperatures, against the classical Dulong–Petit law, and in conformity with Einstein's and Debye's quantum theories. Cf. Kuhn (note 15), 210–20.

⁵⁴Richardson, ‘Photoelectric and Photochemical Action’ (note 44), 481.

⁵⁵Richardson mentioned Einstein's first publication on the subject (which did not include the assumption to which he objected) in a note added in proof to his second 1912 paper (‘Theory of Photoelectric Action’ (note 2), 574). Since Richardson derived the relation between energy and frequency in that publication, the two theories were most probably independent.

⁵⁶Albert Einstein, ‘Thermodynamische Begründung des photochemischen Äquivalentgesetzes’, Annalen der Physik, 37 (1912), 832–38, Albert Einstein, ‘Nachtrag zu meiner Arbeit: “Thermodynamische Begründung des photo-chemischen Äquivalentgesetzes”‘, Annalen der Physik, 38 (1912), 881–84, Albert Einstein, ‘Déduction thermodynamique de la loi de l’équivalence photochemique’, Journal de physique, 3 (1913), 277–82, quote on 277.

⁵⁷Einstein used the term ‘classical thermodynamics’ despite his employment of statistical averages: ‘Déduction thermodynamique’ (note 56), 277. It is unclear, however, whether he thereby meant to distinguish his reasoning from fully fledged statistical mechanics or from quantum theory. ‘Classical’ began to designate non-quantum a year earlier, but in 1909 Einstein still applied ‘classical’ to macroscopic thermodynamics. On the use of the term ‘classical’, see Richard Staley, ‘On the co-creation of classical and modern physics’, Isis 96 (2005), 530–58.

⁵⁸Like Richardson's first theory, Einstein's derivation was based on Wien's law. Like the former, the latter later showed that the derivation could be done with Planck's law. Martin J. Klein, A. J. Kox, Jürgen Renn and Robert Schulmann, ‘Einstein on Photochemical Equivalence’, in The Collected Papers of Albert Einstein (Princeton, NJ, 1995) vol. 4, 109–13. On the effect of light of a particular frequency: Einstein, ‘Déduction thermodynamique’ (note 56), 287–88; On the specific heat, Einstein, ‘Thermodynamische Begründung’ (note 56), 836.

⁵⁹Richardson, ‘Photo-electric action of X-rays’ (note 56), 269.

⁶⁰Wheaton, Tiger (note 6), 109, 169–72; Olivier Darrigol, ‘Quantum theory and atomic structure, 1900–1927’, in The Modern Physical and Mathematical Sciences, edited by Mary J. Nye (Cambridge, 2003), 331–49 (335).

⁶¹For example, Allen (note 7), 153–57; Hallwachs (note 8), 466–67; Hughes (note 43), ‘Report on Photo-Electricity: Including Ionizing and Radiating Potentials and Related Effects’, Bulletin of the National Research Council 2 (1921), 83–167; Schweidler, ‘Photoelektrizität’, in Handbuch der Elektrizität und des Magnetismus, edited by Leo Graetz (Leipzig, 1923), 131–92 (189). Pohl and Pringsheim, Die lichtelektrischen Erscheinungen (note 21) does not refer to Richardson's theory.

⁶²The temporal order determined the attribution of the relation to Einstein.

⁶³In October 1922, Arthur H. Compton wrote that the equation for the energy of the photo-electrons ‘was first proposed by Einstein … but was shown by Richardson to be a direct consequence of Planck's radiation formula’, quoted in Stuewer, Compton Effect (note 6), 217. Otto Stuhlmann Jr and Karl T. Compton, ‘The Photoelectric Properties and Contact Resistances of Thin Cathode Films’, Physical Review, 2 (1913), 199–219; Karl T. Compton, ‘Note on the Velocity of Electrons Liberated by Photoelectric Action’, Physical Review, 1 (1913), 382–92; Otto Stuhlmann Jr, ‘On the Asymmetric Emission of Photo-Electrons From Thin Films of Platinum. I’, Physical Review 4 (1914), 195–207. Stuhlmann explained that the asymmetry in the velocities of emitted electrons (in and against the direction of the incident light) observed in thin films could not be used to dismiss Richardson's theory. In 1922, Arthur Compton judged that Richardson's theory explained the asymmetry qualitatively: cf. Stuewer, Compton Effect (note 6), 197. A year earlier, however, Hughes had concluded that the asymmetry was spurious (‘Report’ (note 61), 120–22).

⁶⁵Hughes, ‘Report’ (note 61), 110–15, quotations on 110 and 114–15.

⁶⁴William Wilson referred to Richardson's expression for the photoelectric and thermionic current (right side of equation (5)) in his study of the photoelectric current due to black-body radiation: ‘The complete photo-electric emission from the alloy of sodium and potassium’, Proceedings of the Royal Society of London, 93 (1917), 359–72. Hallwachs ‘Lichtelektrizität’ (note 43), 467 and a later discussion of experimental results in an appendix from 1914 on 532. Following his interest, Werner briefly described and explained Richardson's theory. He described how the theory leads to the equation for the energy of the electron , while he mentioned Einstein only in a footnote, Werner (note 49).

⁶⁶Hughes, Lichtelektrizität (note 37), 64–65, Richardson, ‘Photo-Electric Action of X-Rays’ (note 37).

⁶⁸Robert A. Millikan, The Electron: Its Isolation and Measurement and the Determination of Some of Its Properties (Chicago, 1963), 237–38. Millikan viewed Einstein's theory as a radicalization of Thomson's ‘ether-string theory’ (221–23).

⁶⁷Millikan, ‘A Direct Photoelectric Determination of “h”’ (note 21), 385–87, quotations on 383, 387.

⁶⁹This was the opinion of most experts, at least after the rejection of the triggering hypothesis.

⁷⁰Thomson's models included non-classical assumptions such as a force ‘confined to a limited number of radial tubes in the atom’, Stuewer, ‘Non-Einsteinian interpretations’ (note 6), 253.

⁷¹In 1913, Voigt pointed out a peculiar change in the value of the pyroelectric coefficient at low temperature, which was beyond the scope of the phenomenological theory. His suggestion that this is because a quantum effect did not have any follow-up. Cf. Jagdish Mehra, The Solvay Conferences on Physics: Aspects of the Development of Physics Since 1911 (Dordrecht, 1975), 88. S. Boguslawski suggested a quantum theory of pyroelectricity in 1914: cf. Walter G. Cady, Piezoelectricity: An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals (New York, 1964), 780.

⁷²Jochen Büttner, Jürgen Renn and Matthias Schemmel, ‘Exploring the limits of classical physics: Planck, Einstein, and the structure of a scientific revolution’, Studies in History and Philosophy of Modern Physics, 34 (2003), 37–59; Olivier Darrigol, ‘Statistics and combinations in early quantum theory’, Historical Studies in the Physical and Biological Sciences, 19 (1988), 17–80.

⁷³Some physicists, including Richardson (quotation on p. 18), viewed ionization and the chemical action of these rays as part of the photoelectric theory broadly understood.

⁷⁴Millikan, The Electron (note 68), 220–37.

⁷⁵Einstein's light quantum originated in the study of blackbody radiation, not in photoelectricity. The success of the light-quantum hypothesis to explain photoelectricity was insufficient to convince most physicists to accept it. The hypothesis was generally accepted only following the accumulation of other evidence such as the Compton Effect. Cf. Stuewer, Compton Effect (note 6), Wheaton, Tiger (note 6).