describe and interpret qualitatively the evidence provided by electron diffraction for the wave nature of particles

Wave‑Particle Duality

At the turn of the 20th century it became clear that light and matter cannot be described solely as particles or solely as waves. The concept of wave‑particle duality states that every quantum entity exhibits both wave‑like and particle‑like properties, the dominant behaviour depending on the experimental arrangement.

De Broglie Hypothesis

Louis de Broglie proposed that any particle with momentum \$p\$ has an associated wavelength \$\lambda\$ given by

\$\lambda = \frac{h}{p}\$

where \$h\$ is Planck’s constant. For an electron of kinetic energy \$K\$ (non‑relativistic), \$p = \sqrt{2m_e K}\$, so

\$\lambda = \frac{h}{\sqrt{2m_e K}}\$

This wavelength is typically of the order of a few picometres for electrons accelerated through a few hundred volts – comparable to inter‑atomic spacings in crystals, making diffraction observable.

Electron Diffraction Experiments

Davisson–Germer Experiment (1927)

Electrons were accelerated through a potential \$V\$, giving them kinetic energy \$eV\$ and wavelength \$\lambda = h/\sqrt{2m_e eV}\$. The beam struck a nickel crystal and the intensity of scattered electrons was measured as a function of angle \$\theta\$.

Key observations:

• A pronounced maximum in intensity appeared at angles satisfying the Bragg condition for X‑rays:

  \$2d\sin\theta = n\lambda\$

  where \$d\$ is the spacing between crystal planes and \$n\$ is an integer.

• The measured angles matched the wavelengths calculated from de Broglie’s formula, confirming that electrons behave as waves.

Low‑Energy Electron Diffraction (LEED)

In modern surface‑science labs, electrons of 20–200 e \cdot are directed onto a crystalline surface. The resulting diffraction pattern is recorded on a phosphor screen.

The pattern consists of a set of bright spots whose positions are related to the reciprocal lattice of the surface, exactly as predicted for wave diffraction.

Qualitative Interpretation of the Evidence

When electrons are treated as particles, we would expect a smooth angular distribution determined only by classical scattering (Rutherford‑type). The appearance of discrete, angle‑dependent intensity maxima is inexplicable in a purely particle picture.

In the wave picture:

1. The crystal acts as a three‑dimensional diffraction grating.
2. Each electron’s wavefunction is coherently scattered by the periodic array of atoms.
3. Constructive interference occurs when the path difference between waves reflected from successive planes equals an integer multiple of the wavelength, leading to the Bragg condition.

Thus the experiment directly demonstrates that electrons possess a wavelength and interfere, satisfying the wave description.

Comparison of Particle vs. Wave Predictions

Key Take‑aways for A‑Level Students

• Electron diffraction provides direct, quantitative evidence for the wave nature of matter.
• The de Broglie wavelength calculated from the electron’s kinetic energy matches the wavelength inferred from diffraction angles.
• The Bragg condition, originally derived for X‑ray diffraction, applies equally to electrons, reinforcing the universality of wave behaviour.
• Understanding diffraction patterns allows us to probe crystal structures and to confirm fundamental quantum principles.