It deals with radiation and the energy properties of light and is very revolutionary.

— Albert Einstein, letter to Conrad Habicht (May 1905)

In May 1905 Einstein wrote to his friend Conrad Habicht announcing four papers. Of the four, he called only one revolutionary, and it was not relativity: it was the paper on light, sent to the Annalen der Physik in March. It was for this paper that Einstein won the Nobel Prize.

A mystery with light and metals

At the end of the nineteenth century physicists knew that light is an electromagnetic wave. They also knew that when light strikes certain metals, it knocks electrons out of the surface: this is the photoelectric effect, studied in detail by Philipp Lenard in 1902. But the effect had three features that wave theory could not explain:

  • Colour matters, brightness does not. Ultraviolet light knocks electrons out of zinc; red light knocks out none, however intense it is.
  • The energy of each electron depends only on the colour. Stronger light knocks out more electrons, but each comes out with the same maximum energy.
  • There is no waiting. Even with very faint light, the first electrons come out immediately.

In a wave, energy spreads smoothly over the wavefront. A strong wave ought to shake the electrons harder, whatever the colour, and a weak wave ought to take time to build up enough energy. None of this happened.

Packets of energy

In 1900 Max Planck had found the correct formula for the light emitted by a hot body with a mathematical trick he himself distrusted: he supposed that energy was exchanged in portions, quanta, each with energy

$$E=hf,$$

where $f$ is the frequency of the light and $h\approx6.626\times10^{-34}$ J·s is Planck’s constant. For Planck it was a calculating device.

Einstein took the idea seriously: light itself would be made of packets of energy $hf$, which would later be called photons. With this, the three mysteries are solved at once. An electron is knocked out when it absorbs one photon. To leave the metal it spends a minimum energy $\phi$, called the work function, characteristic of the metal. What is left becomes energy of motion:

$$\boxed{K_{\max}=hf-\phi.}$$
  • If $hf<\phi$, no single photon has enough energy to knock out an electron, and sending more photons does not help: red light, of low frequency, does not work.
  • The energy of each electron depends on $f$, the colour; the intensity only changes the number of photons, and so the number of electrons.
  • A single photon is enough, so there is no waiting time.

The calculation in electronvolts

For atoms the joule is a huge unit. One uses the electronvolt (eV), the energy an electron gains in crossing 1 volt: $1\ \text{eV}\approx1.602\times10^{-19}$ J. With the frequency written as $f=c/\lambda$, where $\lambda$ is the wavelength, there is a handy rule:

$$E=\frac{hc}{\lambda}\approx\frac{1240\ \text{eV·nm}}{\lambda}.$$

In a metal with $\phi=2.0$ eV, violet light of $\lambda=400$ nm has photons of $\frac{1240}{400}=3.1$ eV, and the electrons come out with up to $3.1-2.0=1.1$ eV. Red light of $700$ nm has photons of $\frac{1240}{700}\approx1.77$ eV, below 2.0 eV: no electrons come out, not even with a floodlight.

Millikan tries to prove it wrong

The idea of photons seemed absurd to almost all physicists, Planck included, because it contradicted a hundred years of experiments showing that light is a wave. The American Robert Millikan spent ten years measuring the photoelectric effect with great care, convinced he would find a flaw in Einstein’s equation. In 1916 he published the result: the equation was right in every detail, and his measurements gave a value of $h$ that agreed with Planck’s. Millikan wrote that he confirmed it “in spite of its unreasonableness”.

The 1921 Nobel Prize in Physics was given to Einstein “for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect”. Millikan won the 1923 prize, partly for the same measurements.

Exercise: what is the longest wavelength that still knocks electrons out of a metal with a work function of 4.3 eV, like zinc?

The limit is when $hf=\phi$, that is, $\frac{1240}{\lambda}=4.3$, and $\lambda=\frac{1240}{4.3}\approx288$ nm. That is ultraviolet light: visible light runs from about 400 to 700 nm. That is why ordinary light does not knock electrons out of zinc, as Hertz and Lenard had observed.

Exercise: a 1-watt green lamp, with λ = 530 nm, emits how many photons per second?

Each photon has $\frac{1240}{530}\approx2.34$ eV, or $2.34\cdot1.602\times10^{-19}\approx3.75\times10^{-19}$ J. One watt is one joule per second, so there are

$$\frac{1}{3.75\times10^{-19}}\approx2.7\times10^{18}\ \text{photons per second}.$$

There are so many that light looks continuous, like water from a tap, even though it is made of drops.

Sources: Albert Einstein, “On a heuristic point of view concerning the production and transformation of light”, Annalen der Physik 17 (1905), translated in Einstein’s Miraculous Year, ed. John Stachel (Princeton, 1998) · 1921 Nobel Prize in Physics, Nobel Foundation · Robert A. Millikan, “A direct photoelectric determination of Planck’s h”, Physical Review 7 (1916).

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