Light arrives in lumps, and brightness only changes how many

A clean metal plate in a vacuum, connected so that any electron leaving it can be counted. Light falls on it. The question is what decides whether electrons come off, and the answer is not the one a wave picture gives.

Four features of that plot were measured before there was a theory for any of them, and all four are wrong on the wave picture.

There is a threshold. Below a certain frequency, nothing comes out. Not a little, not slowly — nothing.

Above it, emission is instantaneous. Within the resolution of the apparatus, which by 1916 was nanoseconds, electrons appear as soon as the light does.

Brightness changes the number and not the energy. Doubling the intensity doubles the current and leaves the maximum electron energy exactly where it was.

The maximum energy rises linearly with frequency, with a slope that is the same for every metal.

Why a wave cannot do this #

The wave picture is not vague about what it predicts, which is what makes the disagreement useful.

A light wave carries energy spread over its wavefront. An electron in the metal is a small target — an atom is a tenth of a nanometre across — so it intercepts a tiny fraction of that flux and must accumulate energy over time before it has enough to escape. The accumulation time can be estimated: a dim source of one microwatt per square metre delivers, to an atomic cross-section, something like 10⁻²⁴ watts. Freeing an electron costs a few electronvolts, or about 10⁻¹⁹ joules. The wait is around 10⁵ seconds, or a day.

What the wave picture has to offer is energy spread continuously along and across a front, so an electron sitting somewhere in the metal is exposed to whatever small fraction of that front happens to cross it. Nothing in such a picture can deliver two electronvolts to one place in a nanosecond at low intensity — the energy has to be collected over an area and a time, and the arithmetic gives minutes where the experiment gives no measurable delay at all. Nothing in it distinguishes one frequency as capable and another as not, either, because a wave of any frequency can be made as intense as one likes.

Nothing of the kind is seen. Lenard’s measurements in 1902 put an upper bound in the microseconds, and later work pushed it to nanoseconds. That alone is awkward. The rest is worse.

On the wave picture, a brighter source delivers energy faster, so the accumulation is quicker and each electron ends up with more of it — the energy should rise with intensity. And there should be no threshold at all: a low-frequency wave delivers energy more slowly per cycle but still delivers it, so a long enough wait at any frequency should produce electrons.

The lump picture, and what it explains at once #

Einstein’s 1905 proposal was that light of frequency f is absorbed in indivisible amounts of hf, using the constant Planck had introduced five years earlier for a completely different problem. One lump goes to one electron. The electron spends some of it escaping the metal — the work function φ, a property of the surface — and keeps the rest:

$$E_{\max} = hf - \phi.$$

Every one of the four features follows immediately and without adjustment. The threshold is where hf first exceeds φ. Emission is instantaneous because there is nothing to accumulate: an electron either meets a quantum large enough or it does not. Intensity is the number of quanta per second, so it sets the current and cannot set the energy. And the graph is a straight line whose slope is h and whose intercept is −φ.

The formula also explains why the lines for different metals are parallel. The slope belongs to the light; the intercept belongs to the surface. A metal with a loosely bound electron sea has a low work function and a low threshold — caesium’s is 2.14 eV, which corresponds to 580 nanometres, so caesium emits under yellow light and is why photocathodes are made of it. Zinc’s 4.33 eV puts its threshold in the ultraviolet, which is why the classic lecture demonstration needs a mercury lamp and fails through a sheet of glass.

Reading the surface off the intercept #

The negative part of the vertical axis on the opening figure needs a word, because no electron ever has negative kinetic energy.

The measured points all lie in the upper right quadrant, above the threshold. The line is then extrapolated down to zero frequency, and where it crosses is −φ. That crossing is not an observation; it is a construction, and its value comes entirely from the slope and the position of the measured points. What makes it trustworthy is that the same φ appears in a second place — the threshold frequency, φ/h, which is directly observed — and the two agree.

This is how a work function is measured to this day, and it is a good example of the site’s recurring move: a quantity that cannot be observed directly is obtained as the intercept of something that can. The technique has a sharp practical edge, because φ is exquisitely sensitive to what is on the surface. A monolayer of oxide moves it by an electronvolt. Millikan spent ten years on the experiment partly because keeping a clean metal surface in the vacuum technology of 1910 meant shaving the specimen in situ with a magnetically operated knife.

The arithmetic that makes it usable #

Working with the equation in joules and hertz is possible and nobody does it. The convenient form comes from expressing the quantum’s energy through its wavelength,

$$E = \frac{hc}{\lambda} = \frac{1239.8\ \text{eV nm}}{\lambda},$$

which turns every threshold into a wavelength and every wavelength into a threshold by one division. Caesium’s 2.14 eV becomes 579 nanometres, which is yellow. Zinc’s 4.33 eV becomes 286 nanometres, which is deep ultraviolet. The energies of visible light run from about 1.65 eV at the red end to 3.26 eV at the violet, and that narrow band — a factor of two — is the whole of what an eye responds to and very nearly the whole of what ordinary chemistry costs.

The same number explains a practical fact about sources.

The arithmetic that makes this usable is also what makes the demonstration awkward. A tungsten filament at 3000 K puts out overwhelmingly infrared — quanta well under one electronvolt, far too small to free an electron from any metal — so a caesium photocell in front of a lamp responds only to the thin sliver of the spectrum left of 579 nanometres. That is why the classic demonstration needs a mercury arc, and why a hot filament is a poor ultraviolet source at any brightness whatever.

There is a wider point in that comparison. Raising a source’s temperature raises the number of quanta everywhere and the energy of the typical one, and the two are not separately adjustable — a thermal source has one knob. An electrical discharge, a light-emitting diode and a laser all break that constraint by not being thermal, which is why they can put substantial power at one energy without being hot.

The discovery inside the experiment that proved the opposite #

The effect was found in 1887 by Hertz, in the course of the experiments that established light is an electromagnetic wave.

He was generating sparks across one gap and detecting the radiation with a second spark gap some distance away, and he noticed that the receiving spark jumped more readily when it was in view of the transmitting one. Enclosing the receiver in a dark box suppressed it; a quartz window restored it; a glass window did not. Quartz passes ultraviolet and glass absorbs it, so what was helping the second spark along was the first spark’s ultraviolet light.

Hertz reported it carefully and moved on — his subject was the waves, and this was an oddity in the apparatus. The oddity was the first sighting of the photoelectric effect, and it was found inside the definitive demonstration that light is a wave, by an experimenter who had every reason to treat the wave picture as settled.

It is worth pausing on the shape of that. The evidence against the sufficiency of a model very often shows up as a nuisance in the experiment establishing the model, because that is the experiment being run with enough care to notice. Hertz’s glass window is one of the cleaner examples in physics: a control that failed, for a reason that took eighteen years and a different physicist to explain.

What it cost, and who paid it #

Millikan’s own account is the interesting part of the history, because he set out to disprove the equation and succeeded in confirming it to one per cent.

He regarded the light-quantum hypothesis as untenable — it appeared to discard the wave theory of light, which had a century of interference and diffraction evidence behind it and which nothing in Einstein’s paper explained. His 1916 measurements produced the straight line, the correct slope and a value of h agreeing with Planck’s. He published them while writing that the underlying theory “cannot be regarded as resting on any sort of a satisfactory theoretical foundation”.

He was not being obtuse. The objection was serious and it is the thing this essay owes an answer to.

Where the picture stops #

The lump picture, taken as “light is a stream of small bullets”, is wrong, and it is worth saying exactly how.

A stream of independent particles does not produce interference fringes. It does not diffract around an obstacle, and it gives no account of why a beam split in two and recombined can cancel. Those effects are not marginal; they are how the wave theory was established, and any picture that discards them is worse than the one it replaces.

What the photoelectric effect actually establishes is narrower and stranger: light is absorbed in units of hf. It says nothing directly about how light travels. The modern reading is that the field propagates as a wave and interacts in quanta, and that the two facts are not in tension because the wave is not a wave of energy density — it is a wave whose square gives the probability of a quantum arriving.

The reconciliation is an experiment rather than an argument, and it is worth stating even though it belongs elsewhere. Light arriving one quantum at a time still builds an interference pattern: each arrival is a single dot in one place, and the fringes are what the accumulation looks like. The wave decides where the dots are likely and the lump decides that there are dots at all. What actually settled the matter for the physicists who accepted Millikan’s straight line and still declined to accept its explanation was momentum — a quantum carrying $hf$ also carries $h/\lambda$, and the collision that shows it is the next rung.

Two further limits are worth stating.

The equation gives the maximum energy, not the typical one. Most emitted electrons come from below the top of the metal’s occupied band and lose more than φ on the way out, so the actual spectrum runs from zero up to hf − φ. The straight line is a bound, and the figure’s dashed ramp between the stopping voltage and zero is a model of that spread rather than a measurement of it.

And single-quantum absorption is not a law of nature but a statement about intensity. At the field strengths a focused pulsed laser reaches, an electron can absorb two or more quanta within one atomic response time, and emission below the classical threshold is routine. Multiphoton photoemission was demonstrated in 1964 and the threshold rule stops applying — not because the quantum picture fails, but because the assumption that the quanta arrive one at a time does.

The argument that does not need photons #

Millikan’s objection was serious, and this essay has so far answered it by saying that the two pictures are reconciled at a higher level. There is a sharper and more uncomfortable thing to say, which is that the photoelectric effect does not actually prove what it is universally said to prove.

The wave picture demolished above is a classical atom in a classical field: an electron treated as a small target soaking up energy until it has enough. That picture is indeed hopeless, and every one of the four measurements refutes it.

Now quantise the atom and leave the field classical. The electron is not a target accumulating energy; it is a quantum system with bound states and a continuum, driven by an oscillating classical field. First-order perturbation theory then gives a transition rate that is constant in time, proportional to the square of the field amplitude, and nonzero only when the driving frequency satisfies $\hbar\omega$ greater than the binding energy.

Read off what that predicts. There is a threshold, because below it no final state is available. Emission is instantaneous, because a transition rate is constant from the moment the field is switched on and nothing is being accumulated. Intensity sets the rate and not the energy, because the rate carries the field amplitude squared and the energy is fixed by the frequency. And the maximum kinetic energy is $\hbar\omega$ minus the binding energy, which is the straight line.

All four, with no quantisation of the light anywhere. The result was obtained by Wentzel and by Beck in 1926 and pressed hard by Lamb and Scully in a 1969 paper titled, pointedly, The photoelectric effect without photons.

So the correct statement is narrower than the textbooks make it. The photoelectric effect is decisive evidence that matter has discrete energy levels and that transitions between them are driven at resonance. It is entirely consistent with a field that is not quantised at all, and Einstein’s inference — that light itself comes in lumps — was right for reasons the experiment does not supply.

What does supply them is a different measurement, and it is worth naming because it is the one that closes the argument. Send light from a single emitter onto a beam splitter with a detector on each output, and count how often both fire at once. Any classical field, however feeble, sometimes delivers enough to both detectors simultaneously, and there is a bound on how anticorrelated the two outputs can be. Light from a single atom violates that bound: the two detectors essentially never fire together, because there is one indivisible excitation and it goes one way or the other.

That experiment was done — for resonance fluorescence in 1977 and in a clean single-photon version in 1986 — and no classical field of any description reproduces the result. Photon antibunching, the Lamb shift and spontaneous emission itself all require the field to be quantised. The photoelectric effect does not, and saying so is not a diminishment of it: it remains the measurement that established $h$ outside thermodynamics and the one that showed transitions happen at a frequency rather than after a wait.

The prize, and the word #

Two small pieces of history round the subject off, and both are more than trivia.

Einstein’s Nobel Prize, awarded in 1922 for the year 1921, cites “his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect”. Relativity is not mentioned. The committee had deliberated for years and had declined to award for relativity, which it regarded as insufficiently established and which was the subject of noisy public opposition; the photoelectric law had Millikan’s measurements behind it and was safe. Einstein was travelling in Asia and did not attend, and when he gave his Nobel lecture the following July he spoke about relativity anyway.

The word photon is younger than the idea by twenty-one years, and it was coined for something else. Gilbert Lewis proposed it in a letter to Nature in 1926 for a hypothetical “atom of light” that was conserved — neither created nor destroyed, merely transferred between atoms, in the way a chemist’s atoms are conserved through a reaction. That concept is wrong: photons are created and annihilated constantly, which is the first thing a quantised field does. The name was adopted within months and the theory it belonged to was never adopted at all.

Einstein’s own term was Lichtquant, a light quantum, which says less and claims less. The drift from “quantum of light” to “photon” is a small piece of evidence for the argument in the section above: a name that sounds like a particle encourages a picture of a small bullet, and the picture is the one the interference figure on this page exists to refuse.

Made into an instrument #

The effect is not now studied for its own sake; it is used.

Photomultipliers convert one quantum into a measurable pulse by letting the freed electron strike a series of biased plates, each releasing several more. Gains of 10⁶ are ordinary, and the device counts individual photons — which is only a coherent thing to say because of the argument above.

Photoelectron spectroscopy inverts the equation. Fixing f with a monochromatic source and measuring the electrons’ energy spectrum gives the distribution of binding energies inside the material, because every electron’s kinetic energy is hf minus whatever it was bound by. X-ray sources reach core levels and identify elements; ultraviolet sources reach the valence band and map the band structure directly. The straight line on the opening figure, read backwards, is one of the standard tools of surface science.

Solar cells are the same energetics with a different destination for the electron, and the same threshold sets their ceiling: light below the semiconductor’s band gap passes through unabsorbed, and light far above it wastes the excess as heat. That single trade-off is why a silicon cell’s theoretical limit is 33 per cent and not 100.

What the picture cannot show #

The graph shows a maximum energy against a frequency and hides the two things that matter most about the process.

It hides the electron’s origin. Nothing on the figure indicates that the electrons come from a range of depths in a solid with a range of binding energies, and that φ is a property of the surface’s last atomic layer rather than of the bulk. Two samples of the same metal with different crystal faces exposed have work functions differing by several tenths of an electronvolt, and the figure has one line per metal.

It also hides the quantum’s fate. The drawing has a frequency going in and an energy coming out; the absorption itself — a field mode losing exactly one excitation while a bound state becomes a free one — has no representation at all. The picture is a bookkeeping statement, and a correct one, about a process it does not depict.

Where the ladder goes next #

The rungs from here: the momentum a quantum carries, which is the second and independent piece of evidence and the one that convinced the holdouts; the photon’s lack of rest mass, and why E = hf and p = h/λ are consistent with the relativistic energy–momentum relation only for something moving at c; spontaneous and stimulated emission, and the laser; photon statistics, where the difference between a laser beam and a thermal source shows up in how the arrivals are spaced rather than in their spectrum; and the threshold’s mirror image in absorption, where an atom refuses every frequency except a few.

The claim to carry forward is the division of labour, which is the same shape as the one a wave on a string obeys and means something entirely different. The source’s brightness fixes how many. The source’s frequency fixes how much each one carries. Nothing at all fixes the two together, and every attempt to explain the effect by a quantity that mixes them fails on one of the four measurements above.