The Photoelectric Effect

Chem 3240 · Lecture 1.3

Davit Potoyan

The Puzzle

• Shine radiation on a metal: electrons fly off
• Only above a threshold frequency \nu_0
• Below \nu_0: no electrons, no matter how bright
• Frequency increases left to right

Why Classical Physics Fails

• Classically the energy of a wave scales with its amplitude: E \propto A^2, with no role for frequency
• Experiment says no, twice: KE ignores brightness, and electrons leave instantly even in dim light
• Intensity is the wrong knob. Frequency is the gatekeeper

Enter the Photon

• Planck quantized the oscillators (seen then as a math trick); Einstein quantized light itself: a stream of discrete packets, photons
• One packet is absorbed whole by one electron, instantly: no soaking-up time, and brightness only sets how many packets arrive

Energy of a Photon

E_{photon} = h\nu = \frac{hc}{\lambda}

• E_{photon}: energy of one photon
• \nu: frequency of that photon
• Both matter and radiation are quantized, by the same relation: energy = (Planck constant) times frequency

Kinetic Energy: Frequency vs Intensity

• Need \nu > \nu_0 to eject at all; then KE grows linearly with frequency, with slope h
• KE does not depend on intensity: below \nu_0, brighter light still ejects nothing

Electric Current: Frequency vs Intensity

• Above threshold, frequency does not change the current
• Intensity sets the current: more photons, more electrons, rising linearly

How Photons Explain It All

• Intensity = number of photons (sets the current)
• Frequency = energy per photon (sets the KE)
• n photons per second carry total energy nh\nu
• One photon ejects one electron, if it carries enough energy

The Photoelectric Equation

One photon hands all its energy to one electron: part pays the exit fee, the rest is speed.

E_{photon} = W_0 + KE \qquad\Longrightarrow\qquad h\nu = h\nu_0 + \frac{mv_e^2}{2}

• Work function W_0 = h\nu_0: minimum energy to free an electron (material dependent)
• \nu_0: threshold frequency; below it, no energy transfers
• Excess energy becomes electron kinetic energy KE = mv_e^2/2

Live: pick a metal, dial the light

The line is KE_{max} = h\nu - W_0: its slope is always h, only the threshold moves with the metal. The dot is your light: on the line means electrons out, on the axis means nothing, however bright.

viewof W0 = Inputs.select(new Map([["cesium (W0 = 2.14 eV)", 2.14], ["sodium (2.28 eV)", 2.28], ["potassium (2.30 eV)", 2.30], ["zinc (4.33 eV)", 4.33], ["copper (4.70 eV)", 4.70], ["platinum (5.65 eV)", 5.65]]), {label: "metal"})

viewof lam = Inputs.range([150, 800], {step: 5, value: 400, label: "light wavelength (nm)"})

{
  const hEV = 4.1357e-15, c = 2.998e8;
  const nu0 = W0 / hEV / 1e14;
  const nuL = c / (lam * 1e-9) / 1e14;
  const KE = hEV * nuL * 1e14 - W0;
  const line = d3.range(0, 25, 0.1).map(n => ({x: n, y: n > nu0 ? hEV * n * 1e14 - W0 : 0}));
  return Plot.plot({
    width: 950, height: 320,
    x: {label: "frequency (10^14 Hz)", domain: [0, 25]},
    y: {label: "KE_max (eV)", domain: [-0.3, 6.5]},
    marks: [
      Plot.rect([{x1: c / 750e-9 / 1e14, x2: c / 380e-9 / 1e14, y1: -0.3, y2: 6.5}], {x1: "x1", x2: "x2", y1: "y1", y2: "y2", fill: "gold", fillOpacity: 0.15}),
      Plot.ruleY([0], {stroke: "#333"}),
      Plot.ruleX([nu0], {stroke: "#999", strokeDasharray: "2,3"}),
      Plot.line(line, {x: "x", y: "y", stroke: "#C8102E", strokeWidth: 2.5}),
      Plot.dot([{x: nuL, y: Math.max(KE, 0)}], {x: "x", y: "y", r: 8, fill: KE > 0 ? "#107895" : "#999"})
    ]
  });
}

md`Photon energy **${(4.1357e-15 * 2.998e8 / (lam * 1e-9)).toFixed(2)} eV** at ${lam} nm vs work function **${W0.toFixed(2)} eV**: ${(4.1357e-15 * 2.998e8 / (lam * 1e-9)) > W0 ? `electrons fly off with KE_max = **${(4.1357e-15 * 2.998e8 / (lam * 1e-9) - W0).toFixed(2)} eV**` : "**no electrons**, no matter how bright the light"}`

Why It Matters

• Cornerstone of quantum theory
• Solar cells and photovoltaics
• Photoelectron spectroscopy
• Night vision and detectors

Takeaway

Light is quantized into photons: frequency sets each photon’s energy (and the ejected electron’s kinetic energy), while intensity sets only how many electrons fly off.