The photoelectric effect

What is the photoelectric effect?

The photoelectric effect is the emission of electrons from a surface, usually a metal, when light or other electromagnetic radiation shines on it. The emitted electrons are called photoelectrons.

• Heinrich Hertz, 1887. While producing the radio waves that confirmed Maxwell's theory (7A), Hertz noticed sparks jumped his spark gap more easily when ultraviolet light shone on it.
• Philipp Lenard, 1902. Using a vacuum tube like the one below, Lenard measured the energy of the emitted electrons and found a result nobody expected: their maximum energy depended on the light's frequency, not its intensity.

The observations, and why waves fail

Einstein's photon model

In 1905, Einstein proposed that light itself is quantised. Light travels as a stream of particles of energy, later named photons, each with energy

$$E = hf = \frac{hc}{\lambda}$$

A photon interacts with a single electron and gives it all its energy, or none. Electrons are held in the metal, and the minimum energy needed to remove one from the surface is the work function, $\phi$. By conservation of energy, the electron leaves with kinetic energy of at most

$$K_{max} = hf - \phi$$

(Electrons deeper in the metal lose some energy on the way out, so they leave with less than $K_{max}$.)

Every observation now makes sense:

• Threshold. If $hf < \phi$, no single photon can free an electron, and adding more photons doesn't help because each acts alone. The threshold frequency is $f_0 = \phi/h$.
• Energy depends on frequency. $K_{max} = hf - \phi$ rises in a straight line with $f$.
• Intensity sets the number. Brighter light means more photons per second, so more electrons, but each photon still has the same $hf$.
• Instant emission. The energy arrives in one packet; there is no waiting to accumulate it.

Work functions are a few electronvolts. Typical values: caesium 2.1 eV, sodium 2.3 eV, calcium 2.9 eV, zinc 4.3 eV, platinum 5.6 eV. Recall that $1 \text{ eV} = 1.602 \times 10^{-19}$ J.

Stopping voltage

To measure $K_{max}$, make the collector negative relative to the emitter. The electric field now pushes electrons back. Only those with enough kinetic energy get across, so the current falls. At the stopping voltage $V_0$, even the fastest electrons are turned back and the current reaches zero. The work done by the field equals the maximum kinetic energy:

$$eV_0 = K_{max} = hf - \phi$$

With a positive collector, all emitted electrons are collected and the current levels off at a saturation current, which is proportional to intensity.

Measuring Planck's constant

Dividing $eV_0 = hf - \phi$ by $e$ gives $V_0 = \dfrac{h}{e}f - \dfrac{\phi}{e}$. A graph of stopping voltage (or $K_{max}$) against frequency is a straight line:

• gradient $= h/e$ (or $h$ for a $K_{max}$ graph in joules)
• x-intercept $=$ threshold frequency $f_0$
• y-intercept $= -\phi/e$ (or $-\phi$)

Every metal gives a line with the same gradient, shifted sideways by its different work function. Robert Millikan, who expected to disprove Einstein, spent ten years making these measurements with clean metal surfaces in vacuum. In 1916 he confirmed the equation exactly and measured $h$ to within 0.5%. Einstein received the 1921 Nobel Prize "especially for his discovery of the law of the photoelectric effect".

Photons carry momentum

Photons have no mass, but they do carry momentum:

$$p = \frac{h}{\lambda}$$

The momentum of one photon is tiny, but huge numbers of photons together exert a measurable radiation pressure. Comet tails point away from the Sun partly because sunlight pushes on the dust, and the Japanese spacecraft IKAROS (2010) sailed between planets on a 14 m wide solar sail. In 1923, Arthur Compton showed that X-ray photons bouncing off electrons lose momentum exactly as $p = h/\lambda$ predicts, which convinced most physicists that photons are real.

So what is light?

Interference, diffraction and polarisation (7E–7G) need waves. Black-body radiation and the photoelectric effect need quanta. Neither model alone explains everything. Light shows wave–particle duality: it travels and spreads like a wave, but exchanges energy and momentum in whole photons. Which behaviour you see depends on the experiment.