11.2 Einstein's Photoelectric Equation and the Photon

In 1916 Robert Millikan published the most careful photoelectric measurements ever made. He had spent a decade trying to prove Einstein wrong — he regarded the idea of "particles of light" as reckless — and instead found that every one of his results fell exactly on the straight lines Einstein had predicted in 1905. From their slope he obtained the Planck constant to within half a percent. Einstein's Nobel Prize (1921) was awarded for this law, not for relativity.

Section 11.1 ended with a puzzle: the energy of the photoelectrons depends on the colour of the light but not its brightness, there is a sharp threshold, and there is no delay. In this section we see how a single bold assumption — that light energy is quantised — explains all of it, how a graph of stopping potential against frequency measures both $h$ and the work function, and what kind of "particle" the photon is: how much energy and momentum it carries, and how many of them stream out of a laser pointer or arrive from the Sun.

Einstein's Photoelectric Equation

Max Planck had found in 1900 that the radiation from a hot body could only be explained if the energy exchanged between matter and radiation of frequency $\nu$ came in whole multiples of a smallest amount $h\nu$, where $h$ is a new constant of nature, the Planck constant, $$h=6.63\times10^{-34}\ \text{J s}.$$ Einstein went further: radiation itself consists of such packets, or quanta. Light of frequency $\nu$ is a stream of quanta each of energy $h\nu$; brighter light means more quanta per second, not bigger ones. In the photoelectric effect a quantum is absorbed completely by a single electron — all or nothing. The electron uses part of the energy to escape (at least $\phi_0$) and keeps the rest as kinetic energy.

A useful shortcut. With energies in eV and wavelengths in nm, $hc=1240$ eV nm (to three figures; with our rounded constants $hc/e$ comes to $1243$ eV nm, so answers can differ in the third figure). Then $$E(\text{eV})=\frac{1240}{\lambda(\text{nm})},\qquad \lambda_0(\text{nm})=\frac{1240}{\phi_0(\text{eV})}.$$ For example, $620$ nm light delivers $2.00$ eV per quantum. We say which form we use in every solution.

How the equation explains Law 11.1.

(i) Threshold. If $h\nu\lt\phi_0$, no single quantum can free an electron. An electron cannot store energy from several quanta (it loses it to the lattice almost at once), so increasing the intensity — more quanta of the same small size — does not help.

(ii) Current ∝ intensity. Intensity is proportional to the number of quanta arriving per second; each quantum frees at most one electron, so the number of photoelectrons per second, and the saturation current, are proportional to intensity.

(iii) $K_{\max}$ depends on $\nu$ only. $K_{\max}=h\nu-\phi_0$ contains no intensity, and increases linearly with $\nu$.

(iv) No delay. The energy arrives concentrated in one quantum, not spread over the wavefront; an electron that absorbs it can leave at once.

The Photon

Einstein's quanta of light behave in the photoelectric effect like particles: each is absorbed whole, at one place, by one electron. In 1923 Arthur Compton showed that X-ray quanta bounce off electrons like billiard balls, carrying momentum as well as energy. The quantum of light is now called the photon.

Because $h$ is so small, everyday light contains enormous numbers of photons and looks perfectly smooth and continuous; the graininess shows up only when light is very faint or when it interacts with single atoms and electrons. The wave description (Chapters 8 and 10) and the photon description are both correct — each explains a different set of experiments. Section 11.3 shows that matter, too, has this double character.

Exercises