The Photoelectric Effect and Wave-Particle Duality

The Photoelectric Effect and Wave-Particle Duality

The photoelectric effect provides direct evidence that electromagnetic radiation has a particle nature (photons). Combined with diffraction and interference evidence for waves, this leads to wave-particle duality — one of the foundations of quantum physics.

The Photoelectric Effect

When electromagnetic radiation of sufficiently high frequency shines on a metal surface, electrons are emitted. These are called photoelectrons.

Key Observations

1. Below a certain frequency (threshold frequency f₀), no electrons are emitted — regardless of the intensity of the light

2. Above the threshold, electrons are emitted immediately (no time delay, even at very low intensity)

3. Increasing intensity increases the number of photoelectrons but not their maximum kinetic energy

4. Increasing frequency increases the maximum kinetic energy of the photoelectrons

Why the Wave Model Fails

Classical wave theory predicts:

• Energy should accumulate gradually — there should be a time delay before emission (not observed)
• Any frequency should work if the intensity is high enough (not observed)
• Higher intensity should increase the energy of individual electrons (not observed)

Einstein's Photon Explanation (1905)

Light consists of discrete packets of energy called photons. Each photon has energy:

E = hf = hc/λ

where:

• h = Planck's constant = 6.63 × 10⁻³⁴ J s
• f = frequency (Hz)
• c = speed of light (m s⁻¹)
• λ = wavelength (m)

Einstein's Photoelectric Equation

hf = φ + E_k(max)

or equivalently: E_k(max) = hf − φ

where:

• hf = energy of the incident photon
• φ = work function of the metal — the minimum energy needed to free an electron from the surface
• E_k(max) = maximum kinetic energy of the emitted photoelectron

The threshold frequency f₀ is when E_k(max) = 0:

φ = hf₀ → f₀ = φ/h

Worked Example

The work function of sodium is 2.28 eV. Find the threshold frequency and the maximum kinetic energy of photoelectrons when illuminated with UV light of wavelength 250 nm.

φ = 2.28 eV = 2.28 × 1.6 × 10⁻¹⁹ = 3.648 × 10⁻¹⁹ J

f₀ = φ/h = 3.648 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ = 5.50 × 10¹⁴ Hz

For λ = 250 nm: E_photon = hc/λ = (6.63 × 10⁻³⁴ × 3.0 × 10⁸) / (250 × 10⁻⁹) = 7.956 × 10⁻¹⁹ J

E_k(max) = E_photon − φ = 7.956 × 10⁻¹⁹ − 3.648 × 10⁻¹⁹ = 4.31 × 10⁻¹⁹ J = 2.69 eV

The Stopping Potential

The maximum kinetic energy can be measured using a stopping potential V_s:

E_k(max) = eV_s → eV_s = hf − φ

Plotting V_s against f gives a straight line with gradient h/e and y-intercept −φ/e.

The Electronvolt

1 eV = the energy gained by an electron accelerated through a p.d. of 1 V:

1 eV = 1.6 × 10⁻¹⁹ J

This is a convenient unit for atomic and subatomic energies.

Wave-Particle Duality

All matter and radiation exhibits both wave and particle properties:

Light: Shows wave behaviour (diffraction, interference, polarisation) and particle behaviour (photoelectric effect, Compton scattering)

Electrons: Show particle behaviour (tracks in cloud chambers, discrete charge) and wave behaviour (electron diffraction)

de Broglie Wavelength

In 1924, de Broglie proposed that all moving particles have an associated wavelength:

λ = h/p = h/(mv)

where:

• λ = de Broglie wavelength (m)
• h = Planck's constant
• p = momentum (kg m s⁻¹)
• m = mass, v = velocity

Worked Example — Electron Wavelength

Find the de Broglie wavelength of an electron accelerated through 150 V.

E_k = eV = 1.6 × 10⁻¹⁹ × 150 = 2.4 × 10⁻¹⁷ J

½mv² = E_k → v = √(2E_k/m) = √(2 × 2.4 × 10⁻¹⁷ / 9.11 × 10⁻³¹) = √(5.27 × 10¹³) = 7.26 × 10⁶ m s⁻¹

λ = h/(mv) = 6.63 × 10⁻³⁴ / (9.11 × 10⁻³¹ × 7.26 × 10⁶) = 1.00 × 10⁻¹⁰ m = 0.100 nm

This is comparable to atomic spacing, so electron diffraction from crystals is observable.

Electron Diffraction

When a beam of electrons passes through a thin polycrystalline graphite film, a diffraction pattern of concentric rings is produced on a fluorescent screen — identical in form to X-ray diffraction patterns.

This confirms that electrons behave as waves. The ring pattern matches the prediction from the de Broglie wavelength and the atomic spacing of the graphite.

Key evidence: Increasing the accelerating voltage → faster electrons → shorter λ → diffraction rings get smaller (closer to the centre), exactly as predicted by λ = h/(mv).

Significance

• The photoelectric effect proves the particle nature of light
• Electron diffraction proves the wave nature of matter
• Together they establish wave-particle duality: everything exhibits both wave and particle properties, but which is observed depends on the experiment
• This duality is central to quantum mechanics and cannot be explained by classical physics

Energy Levels and Photon Emission

Atoms have discrete energy levels. When an electron transitions between levels:

ΔE = hf → f = (E₂ − E₁)/h

This produces photons of specific frequencies → line spectra. Each element has a unique set of energy levels and therefore a unique emission spectrum — used for chemical identification.