Photoelectric Effect

The experiment

UV light shines on a clean metal surface. A circuit measures current — electrons emitted from the metal travel to a collector electrode.

Observations (puzzling classically):

1. Below a threshold frequency f₀, NO electrons are emitted, regardless of light intensity.
2. Above threshold, electrons emitted essentially instantly (≪ 1 second).
3. Maximum KE of emitted electrons depends on f, not intensity.
4. Number of emitted electrons (current) depends on intensity, not f.

Classical wave theory predicts the opposite — should be able to accumulate energy from waves over time, regardless of frequency.

Einstein's photon picture

Light consists of discrete energy packets — photons — with energy:

E_photon = h · f

where h = 6.626 × 10⁻³⁴ J·s (Planck's constant). Each photon interacts with ONE electron.

If hf > φ (work function): electron escapes with kinetic energy:

KE_max = hf − φ

If hf < φ: no escape, regardless of intensity. Threshold frequency:

f_0 = φ / h

Work functions of common metals

Metal	φ (eV)	Threshold λ (nm)
Cesium	1.95	636 (visible red)
Potassium	2.30	539 (green)
Sodium	2.36	525 (green-yellow)
Calcium	2.87	432 (violet)
Aluminum	4.08	304 (UV)
Copper	4.70	264 (UV)
Gold	5.10	243 (UV)
Platinum	5.65	219 (UV)

Cesium responds to visible light; most metals require UV.

JavaScript — photoelectric calculations

const h = 6.626e-34;  // J·s
const c = 3e8;
const eV = 1.602e-19;

// Photon energy (J)
function photonEnergyJ(frequency) { return h * frequency; }
function photonEnergyEv(frequency) { return h * frequency / eV; }
function photonWavelength(energy_eV) { return c / (energy_eV * eV / h); }

// Visible light photon energies
console.log(`Red 700 nm: ${photonEnergyEv(c/700e-9).toFixed(2)} eV`);    // 1.77
console.log(`Green 550 nm: ${photonEnergyEv(c/550e-9).toFixed(2)} eV`);  // 2.25
console.log(`UV 200 nm: ${photonEnergyEv(c/200e-9).toFixed(2)} eV`);      // 6.20

// Photoelectric: max KE
function maxKineticEnergy(photon_eV, work_function_eV) {
  // Returns KE in eV; 0 if photon below threshold
  const KE = photon_eV - work_function_eV;
  return Math.max(0, KE);
}

// Sodium (φ = 2.36 eV) hit by green (2.25 eV)
console.log(`Na + green: ${maxKineticEnergy(2.25, 2.36)} eV`);  // 0 — below threshold!

// Sodium hit by UV (4 eV)
console.log(`Na + 4 eV UV: ${maxKineticEnergy(4, 2.36).toFixed(2)} eV`);  // 1.64

// Stopping voltage (voltage to stop fastest electron)
function stoppingVoltage(photon_eV, work_function_eV) {
  return Math.max(0, photon_eV - work_function_eV);  // V
}

// Threshold wavelength for given work function
function thresholdWavelength(work_function_eV) {
  // λ = hc/φ ; convert eV to J
  const phi_J = work_function_eV * eV;
  return h * c / phi_J;
}

console.log(`Cs threshold: ${(thresholdWavelength(1.95) * 1e9).toFixed(0)} nm`);  // 636 (red)
console.log(`Cu threshold: ${(thresholdWavelength(4.70) * 1e9).toFixed(0)} nm`);  // 264 (UV)

// Number of photoelectrons per second (rough)
function photoelectronRate(intensity_W_per_m2, area_m2, photon_eV, efficiency = 0.1) {
  // Rate = intensity × area × efficiency / photon_energy
  const photon_E_J = photon_eV * eV;
  return intensity_W_per_m2 * area_m2 * efficiency / photon_E_J;
}

// 1 W/m² UV (4 eV) on 1 cm² with 10% quantum efficiency
console.log(`Photoelectron rate: ${photoelectronRate(1, 1e-4, 4).toExponential(2)} per sec`);

// Solar cell efficiency limit (Shockley-Queisser style — simplified)
function maxEfficiency(bandgap_eV, sun_temp_K = 5800) {
  // Rough: efficiency = bandgap / (k_B T_sun × N) accounting for spectrum
  // Real Shockley-Queisser limit ~33% for Si bandgap 1.1 eV
  return Math.min(1.0, 0.4);  // heuristic; real calculation more complex
}

Where photoelectric matters

• Solar cells. Direct conversion of light to electricity. Modern silicon ~20-25% efficient; multi-junction ~46%.
• Photodetectors. Cameras, optical fiber receivers, infrared sensors, X-ray detectors.
• Photomultiplier tubes. Single-photon detection (cascade amplification of photoelectrons). Used in particle physics, low-light astronomy.
• Image sensors. CCDs and CMOS sensors capture light by photoelectric effect at each pixel.
• Photoemission spectroscopy. Measure work functions, band structure of materials by analyzing emitted electrons.
• Solar wind/space. UV from Sun causes photoelectric effects on spacecraft surfaces, influencing electrostatics.
• Education. Foundational quantum experiment; introduces photons, work function, quantum vs classical.

Common mistakes

• Believing intensity should affect electron energy. Classical view; wrong. Intensity affects current (number); frequency affects max energy.
• Forgetting work function depends on material. Different metals have different φ; same light might eject from cesium but not copper.
• Using wrong units. Common: photon energy in eV, work function in eV, voltage in V. Don't mix with J.
• Treating one photon as ejecting multiple electrons. Photoelectric is one-photon-one-electron at typical intensities. Multiphoton effects exist at very high intensities (lasers).
• Confusing photoelectric with Compton. Photoelectric — photon absorbed, electron ejected. Compton — photon scatters off electron, both continue (with shifted frequencies).
• Forgetting the threshold. Below f₀, NO emission no matter intensity. Crucial point distinguishing quantum from classical.