Work Function and Threshold Frequency: Photoelectric Guide
In the photoelectric effect, light falling on a clean metal surface can eject electrons. Each metal needs light of at least a certain frequency before any electrons come out, no matter how bright the light is. This minimum frequency is called the threshold frequency.
The work function explains this threshold. It is the minimum energy needed to pull an electron out of the metal. This concept matters because it proved light delivers energy in packets (photons) and it underlies solar cells and light detectors.
Key Concepts
Work function (φ₀ or W₀): Minimum energy to remove an electron from the surface with zero kinetic energy, measured in joules or electronvolts (1 eV = 1.602 × 10⁻¹⁹ J).
Threshold frequency (ν₀): Lowest frequency that can cause emission for a given metal. Below it, no emission occurs.
Threshold wavelength (λ₀): Longest wavelength that can cause emission, λ₀ = c / ν₀.
Photon energy: E = h ν, where h = 6.626 × 10⁻³⁴ J·s is Planck's constant.
Formulas
Work function: φ₀ = h ν₀ = h c / λ₀. Units: h in J·s, ν in Hz (s⁻¹), c in m/s, λ in metres, giving φ₀ in joules. Assumes emission from the surface with no extra kinetic energy at threshold.
Einstein photoelectric equation: h ν = φ₀ + K_max, or h c / λ = φ₀ + (1/2) m v_max². K_max is the maximum kinetic energy of emitted electrons. Valid for a single photon absorbed by a single electron, neglecting thermal energies.
Worked Example
The work function of a metal is 2.0 eV. Find its threshold wavelength. (h c ≈ 1240 eV·nm).
Step 1: Use λ₀ = h c / φ₀.
Step 2: λ₀ = 1240 eV·nm / 2.0 eV = 620 nm.
Step 3: Threshold frequency ν₀ = c / λ₀ = 3.00 × 10⁸ / 620 × 10⁻⁹ ≈ 4.84 × 10¹⁴ Hz.
Light with wavelength longer than 620 nm (redder) cannot eject electrons from this metal, however intense it is.
Common Misconceptions
- Misconception: Very bright red light can eject electrons if left long enough. Correct: Below threshold frequency, no electrons are ejected even with high intensity or long exposure.
- Misconception: Work function is the same for all metals. Correct: Each metal has its own work function (e.g. caesium is low, platinum is high).
- Misconception: Emitted electrons all have the same energy. Correct: K_max is the maximum; many electrons lose energy inside and emerge slower.
Real-World Applications
Low work function materials coat cathodes in vacuum tubes and electron microscopes. Solar cells and photodiodes are chosen with suitable energy thresholds for the light they must detect, and UV detectors use metals that respond only to ultraviolet photons.
Recap: the work function is the exit price for an electron. Light must bring at least that much energy per photon, setting a sharp threshold frequency.