The historic experiment of the photoelectric effect

Before reinterpreting the photoelectric effect in terms of energy bands, it is useful to retrace the original experiment: discovered in 1887 by Heinrich Hertz, studied systematically by Philipp Lenard (Nobel 1905) and finally explained by Albert Einstein (1905, Nobel 1921 precisely for this explanation, not for relativity).

The apparatus

A vacuum tube contains two metal electrodes: a photosensitive plate (the cathode, typically sodium, potassium or zinc) and a collector (the anode). A transparent window allows the plate to be illuminated with monochromatic light of adjustable frequency $f$ and intensity $I$. A variable generator applies a voltage $V$ of arbitrary sign between cathode and anode: if $V>0$ the electric field accelerates towards the collector the electrons emitted from the plate; if $V<0$ (braking voltage) it repels them. An ammeter measures the current $i$ in the circuit.

Apparatus of the photoelectric effect. Light strikes the plate (cathode K); the electrons extracted cross the vacuum to reach the collector, generating a current measured by the ammeter. The variable voltage $V$ can accelerate or brake the photoelectrons.

Lenard’s observations (1902)

1. For each metal there exists a threshold frequency $f_0$ below which no current is observed at all, however much the light intensity is increased.
2. For $f>f_0$ the current appears instantaneously (in times $<10^{-9}$ s), even at very weak intensities.
3. Increasing the intensity $I$ (at fixed $f$) the current grows linearly, but the maximum kinetic energy of the photoelectrons does not change.
4. Increasing the frequency $f$ (at fixed $I$) the maximum kinetic energy of the photoelectrons grows linearly with $f$, regardless of $I$.
5. The maximum kinetic energy is measured via the stopping voltage $V_s$, the minimum braking voltage that cancels the current: then $E_\text{cin,max}=eV_s$.

Why these are baffling for classical physics

In the classical view light is an electromagnetic wave, and its energy is proportional to the intensity $I$, not to the frequency. An intense wave should therefore always be able to extract electrons, provided one waits long enough for the electron oscillator to accumulate enough energy. Instead: a sharp threshold in frequency, no appreciable delay, kinetic energy independent of intensity. None of this fits the wave picture.

Einstein’s hypothesis (1905)

Light is made of quanta — photons — of energy $E=h\,f$. A photon that strikes an electron in the metal gives it all of its energy, or none at all: partial transfers do not exist. If the photon’s energy exceeds the extraction energy $W$ (the metal’s work function), the electron leaves with maximum kinetic energy:

$\ev{E_\text{cin,max} = h\,f - W}$

If instead $hf<W$, the electron stays trapped, and sending more photons at the same frequency is useless: each photon acts individually. Here is the explanation of the threshold:

$f_0 = \frac{W}{h}$

Here is why the maximum kinetic energy does not depend on the intensity: $I$ only determines the number of photons per second — that is, the current — not the energy of each one. And here is why there is no delay: the first photon with $hf>W$ extracts an electron immediately.

Maximum kinetic energy of the photoelectrons as a function of frequency. The line has slope $h$ (Planck’s constant, the same for all metals) and intercept $-W$ on the vertical axis (dependent on the metal). Below $f_0$ there is no emission.

The line representing $E_\text{cin,max}=hf-W$ thus has slope equal to Planck’s constant $h$ — the same for every metal! — and intercept $-W$ on the vertical axis, which instead depends on the metal. Measuring the stopping voltage as a function of frequency for different metals yields parallel lines: a very clean test of Einstein’s formula, carried out with great precision by Robert Millikan in 1916.

A concrete numerical case — the stopping voltage of sodium illuminated in the ultraviolet — is worked through in full among the exercises of this section.

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Topics: Fisica quantistica Concepts: Effetto fotoelettrico · Fotone

Related exercises: Perché i raggi X non sono solo luce intensa · Ricavare h e W dal grafico fotoelettrico · Fotoelettroni dal sodio a 345 nm