Photons and the Photoelectric Effect

The photoelectric effect is the emission of electrons from a metal surface when light of sufficiently high frequency falls on it; Einstein explained it in 1905 by treating light as photons of energy E = hf, so that the maximum kinetic energy of the emitted electrons is KE_max = hf − φ, where φ is the work function of the metal. In brief: there is a threshold frequency f₀ = φ ÷ h below which no electrons are emitted, however intense the light. Above it, KE_max increases linearly with frequency and is independent of intensity; the number of electrons (the current) is proportional to intensity. Emission is instantaneous. These observations cannot be explained by the classical wave theory of light.

Solar cells, light sensors and digital camera detectors are based on photoelectric effects. The experiments of 1887 to 1902 produced results that contradicted classical physics, and Einstein’s explanation, for which he received the 1921 Nobel Prize in Physics, established the photon concept. This guide describes the experiment and its observations, explains the failure of the wave theory and Einstein’s equation, compares the predictions in a table, works through numerical examples and corrects common mistakes.

What are the observations?

In the experiment, light shines on a metal plate (the emitter) in an evacuated tube, and the electrons emitted (photoelectrons) are collected by a second electrode. A variable voltage between them can be used to measure their kinetic energy: the stopping potential V_s is the reverse voltage that just stops even the most energetic electrons, so KE_max = eV_s. Results:

• Electrons are emitted only if the frequency exceeds a threshold frequency f₀ that depends on the metal; below it, there is no emission even with very intense light.
• Above the threshold, the maximum kinetic energy of the electrons increases with the frequency of the light and does not depend on its intensity.
• The photocurrent (the number of electrons per second) is proportional to the intensity of the light.
• The emission starts immediately (within about 10⁻⁹ s), even for very dim light.

Why does the wave theory fail?

According to the classical wave theory, the energy of light depends on its intensity (amplitude squared), not on its frequency. Therefore, intense light of any frequency should eject electrons, and the kinetic energy of the electrons should increase with the intensity. Dim light should need a measurable time to deliver enough energy to an electron (energy is spread over the whole wavefront, so an electron would have to accumulate energy over minutes). The experiments contradict each of these predictions: the threshold frequency, the independence of KE from intensity, and the instant emission were all unexplained.

What is Einstein’s explanation?

Einstein proposed in 1905 that light consists of localized quanta, later called photons, each of energy E = hf. A photon is absorbed entirely by a single electron, in an all-or-nothing interaction. The electron needs a minimum energy, the work function φ, to escape from the surface (typically 2 to 5 eV). The excess energy becomes kinetic energy:

hf = φ + KE_max, or KE_max = hf − φ = eV_s.

• If hf < φ, no electrons are emitted: the threshold frequency is f₀ = φ ÷ h (threshold wavelength λ₀ = hc ÷ φ).
• A more intense beam has more photons per second, so more electrons per second, but each electron receives the energy from a single photon, hence KE_max is unchanged.
• Emission is immediate because the energy is delivered in a single event.

A graph of KE_max (or V_s) against frequency is a straight line, with gradient h (or h ÷ e), intercept on the frequency axis f₀ and intercept on the energy axis −φ. Robert Millikan measured it carefully in 1914 to 1916, obtaining h to within 1 percent, in spite of his initial skepticism about the photon idea.

Wave and photon predictions compared

Worked example: sodium and a stopping potential

Sodium has a work function of about 2.3 eV. Use hc = 1,240 eV·nm.

1. Threshold wavelength: λ₀ = hc ÷ φ = 1,240 ÷ 2.3 = 539 nm. Red light (700 nm) or yellow (590 nm) does not eject electrons, however bright. The threshold frequency is f₀ = c ÷ λ₀ = 5.6 × 10¹⁴ Hz.
2. Violet light of 400 nm: photon energy E = 1,240 ÷ 400 = 3.10 eV. KE_max = 3.10 − 2.3 = 0.80 eV = 1.28 × 10⁻¹⁹ J.
3. Stopping potential: V_s = KE_max ÷ e = 0.80 V.
4. Maximum speed: v = √(2KE ÷ m) = √(2 × 1.28 × 10⁻¹⁹ ÷ 9.11 × 10⁻³¹) = 5.3 × 10⁵ m/s.
5. Shorter wavelength. With ultraviolet light of 250 nm: E = 4.96 eV, KE_max = 2.66 eV, V_s = 2.66 V. The energy of the electrons rises with frequency, as predicted.
6. Doubling the intensity of the 400 nm light: the number of photons per second doubles, so the photocurrent doubles, while V_s remains 0.80 V.
7. Finding h. A graph of V_s against f has a gradient of 4.1 × 10⁻¹⁵ V·s: h = e × gradient = 1.602 × 10⁻¹⁹ × 4.1 × 10⁻¹⁵ = 6.6 × 10⁻³⁴ J·s.

How was it discovered?

Heinrich Hertz noticed in 1887 that ultraviolet light helps sparks to jump between electrodes. Wilhelm Hallwachs and Philipp Lenard studied the effect in 1888 to 1902; Lenard found that the energy of the electrons does not depend on the intensity of the light. Einstein explained it in 1905, in the paper in which he proposed light quanta. The photon (a name coined in 1926 by Gilbert Lewis) became fully accepted after Arthur Compton’s experiment on X-ray scattering in 1923. The photoelectric effect is the basis of photomultipliers, photodiodes, CCD and CMOS image sensors, and solar cells (in which the electrons are not ejected from the material, but excited into the conduction band: the internal photoelectric effect).

Which mistakes are common?

Mistake 1: Thinking that brighter light gives faster electrons

Intensity changes the number of electrons per second. The maximum energy depends on the frequency.

Mistake 2: Saying that low-frequency light would work if left on long enough

Below the threshold frequency, no photon has enough energy, and waiting does not help.

Mistake 3: Mixing the work function and the kinetic energy

φ is the minimum energy needed to escape. The kinetic energy is what remains of the photon energy.

Mistake 4: Forgetting that KE_max is a maximum

Electrons from deeper inside the metal lose some energy, so they emerge with less kinetic energy.