Were there serious attempts to model the photoelectric effect classically?

Question

Today we see the photoelectric effect as one of the simplest pieces of empirical evidence that leads to quantum physics. The historical development of the subject, however, seems to have involved much more complicated and indirect evidence such as the blackbody radiation curve.

In textbooks and laboratory manuals, a fact about the photoelectric effect that is often presented prominently is that the stopping voltage is independent of the intensity of the light. This fact is easy to explain with the photon model. One also often sees in textbooks statements that according to "the classical theory," the stopping voltage should have increased with the intensity of the light.

However, it is not at all obvious to me that any such classical theory can be constructed. Specifically, I doubt that one can construct a classical model that makes the following predictions: (P1) the photoelectric effect occurs with solid metals, (P2) there is a finite stopping voltage, and (P3) this stopping voltage increases with the intensity of the light. Note that if the model doesn't predict P1 and P2, then we can't even meaningfully ask whether it predicts P3.

What was the actual historical situation? I can imagine at least five scenarios.

(a) The photoelectric effect in a solid metal cathode was a total Kuhnian anomaly. Physicists trained within the classical paradigm would not have seen any promising lines of attack, and would have been unable to construct even a partially successful model. Therefore they ignored it and worked on other problems.

(b) There was some actual classical model that was published or publicly discussed, which predicted P1-P3.

(c) People built and discussed models that gave predictions similar to P1-P3 in the case of a free electron, or perhaps a gaseous cathode. The models just couldn't be extended to a solid metal cathode.

(d) Textbook authors who refer to "the classical model" are using sloppy language or have not studied the actual history; there is actually some model that predicts P1-P3, but it's not a purely classical model. It's some kind of hybrid such as the Bohr-Kramers-Slater theory.

(f) Looking through the history section of the WP article on the photoelectric effect, it seems that textbook presentations may be giving a misleading impression of the historical sequence. It seems that the evidence prior to 1905 was mostly very crude and qualitative. Because of the paucity of quantitative data, there may have been little motivation to embark on serious model-building.

Can anyone clarify what actually happened?

Einstein's 1905 paper has the following (translation by D. Ter Haar?) on this topic:

As far as I can see, our ideas are not in contradiction to the properties of the photoelectric action observed by Mr. Lenard. [Under the quantum hypothesis] the velocity distribution of the electrons ... will be independent of the intensity of the incident light ...

He does not follow up by claiming that such an observation would contradict the classical theory. Einstein references Lenard's 1902 paper, which showed that the stopping potential increased with the frequency of the light but was independent of intensity.

Briefly, here is the reason why it's not obvious to me that any classical model can produce P1-P3. I'm not claiming that I'm right -- I'm just trying to explain why I have doubts about the existence of any such model. In a classical model of a solid metal cathode, the conduction electrons are a classical gas with a Maxwellian velocity distribution. Visible light has a wavelength much longer than the mean separation between electrons, and therefore all it really does is heat the cathode. Regardless of temperature, the Maxwellian distribution has a high-velocity tail whose energy is greater than the work function, so electrons are emitted from the surface at some rate. There is no sharp cutoff to this tail, and therefore there is no well-defined stopping voltage.

Answer 1

"Serious" in the OP sense is probably too high a bar. In 1900-s the situation was very much in flux as to what classical physics could and could not explain. Even Planck's and Einstein's ideas, that we now associate with quantum mechanics, were incorporated into seemingly classical approaches at the time. But what dominated the scene were qualitative hypotheses, rather than carefully constructed theories with spelled out predictions.

Between 1902, when Lenard established the independence of photoeffect on the intensity of light, and 1911, the prevailing explanation was Lenard's own "triggering hypothesis". According to it, "the light only has an initiating action, rather like that of the fuse in firing a loaded gun", as Lenard himself put it in his 1906 Nobel lecture. It went with, his also, model of the atom that anticipated Bohr's in some aspects, but both were qualitative, not quantitative.

Although Lenard's experiments inspired Einstein's 1905 "photoeffect" paper, his light quanta did not get much traction until 1911, when Lenard himself disproved the triggering hypothesis by demonstrating that ionization in gases by ultraviolet light did not produce strong ionization unaccompanied by absorption of the light. More precise experiments with gas were only performed in 1908 by Langevin and Bloch, and with metals in 1914 by Millikan. A detailed historical account is Wheaton's Philipp Lenard and the Photoelectric Effect, 1889-1911:

"Lenard's largely empty atom, which emits characteristic spectral lines when free electrons return to equilibrium in the dynamic atomic structure, is suggestive of the atom proposed five years later by Niels Bohr. But there are fundamental differences. Lenard's was never a quantitative atom theory. It was primarily because of his remarkable experimental skill that his ideas about atoms drew support in Germany. His triggering hypothesis for the photoelectric effect required that each electron within the atom has a velocity which is related in a consistent way to its mechanical frequency. The specific relation of frequency to velocity for each cathode material could only be determined by experiment. The photoelectric effect did not predict what the relationship would be, it only made it accessible to observation. The triggering hypothesis formed the bridge which connected the photoelectric effect to the issues of atomic structure.

In 1909 Lenard's triggering hypothesis was included by a knowledgeable reviewer among "the generally accepted truths of physics." Albert Einstein, who characterized Lenard's experimental work on the photoeffect as "pathfinding," proposed an alternative interpretation in 1905. But Einstein's light quantum, which explained Lenard's results at the cost of rejecting classical wave radiation, had little support early in the century... Einstein had predicted that a continuous linear relation between frequency and stopping potential would be found. But the empirical data was in doubt until 1915. With an appropriate atom model, like those proposed by Thomson in 1910 and 1913, even the Einstein law could be reconciled with the triggering mechanism."

Even Einstein's light quantum ideas seemed to be reconcilable with classical physics at the time. Einstein and Lorentz corresponded about light quanta in 1909. Lorentz just worked out the classical theory of blackbody radiation in painstaking detail, and sharply criticized "point-like quantities of energy or at least energy quantities concentrated in very small spaces", that Einstein seemed to postulate. Classical explanations for discretization of energy in the photoeffect seemed achievable. Even Einstein distanced himself from "point-like light quanta" and offered a description of them as "singularities that are surrounded by a vector field whose strength decreases with increasing distance. The field energy is then related somehow to the number of these singularities." See Kox's Hendrik Antoon Lorentz’s struggle with quantum theory.

Similar dynamic happened with Planck's blackbody radiation formula. Statistical mechanics in general, and the equipartition theorem in particular, were under suspicion. Planck, and even Jeans and Lorentz, had doubts about the universality of the equipartition theorem. It did not become clear that classical theory leads to the "ultraviolet catastrophe" until Lorentz's 1908 Rome lecture. According to Kuhn's source study, Planck too was under the impression that he was doing classical physics until 1906. In 1908, when Lorentz admitted that "only with the help of the hypothesis of energy elements one can arrive at the correct radiation-law", the focus shifted from purely classical approach to identifying the source of "discontinuity": was it in the matter, in its interaction with electromagnetic field, or in the field itself? And later to reconciling Maxwell's waves with Planck's quanta, or even getting rid of waves altogether, as Duane suggested in connection with the X-ray diffraction.