Wave–particle duality (A-level only)

AQA·Physics·3.12.2.5

de Broglie wavelength

De Broglie proposed that a particle with momentum has an associated wavelength. Its de Broglie wavelength is

λ=hp\lambda = \frac{h}{p}

Here, $$\lambda$$ is the wavelength, $$p$$ is the particle’s momentum and $$h$$ is the Planck constant. A particle with greater momentum has a shorter de Broglie wavelength: doubling the momentum halves the wavelength.

de Broglie wavelength of an accelerated electron

When an electron starts from rest and accelerates through a potential difference $$V$$, it gains kinetic energy $$eV$$, where $$e$$ is the magnitude of the electron’s charge. For a non-relativistic electron of mass $$m$$, its kinetic energy is also $$p^2/(2m)$$. Equating the two expressions gives

p22m=eV\frac{p^2}{2m} = eV

Rearranging gives the electron’s momentum:

p=2meVp = \sqrt{2meV}

Substituting this momentum into the de Broglie relation gives

λ=h2meV\lambda = \frac{h}{\sqrt{2meV}}

A larger accelerating potential difference gives the electron more kinetic energy and greater momentum, so its wavelength decreases. Specifically, the wavelength is inversely proportional to $$\sqrt{V}$$: increasing $$V$$ by a factor of four halves the wavelength.

Electron diffraction

A beam of low-energy electrons can form a diffraction pattern when it passes through a crystal. The atoms in the crystal are arranged in a regular pattern, with spacings comparable to the electrons’ wavelengths. The crystal therefore diffracts the electron waves.

The diffracted waves overlap. In some directions they reinforce one another, producing higher electron intensity; in others they reinforce one another less, producing lower intensity. The resulting pattern provides experimental evidence that electrons exhibit wave behaviour.

Electron speed and the diffraction pattern

An electron’s de Broglie wavelength depends on its momentum:

λ=hp\lambda = \frac{h}{p}

Here, $$\lambda$$ is the wavelength, $$h$$ is the Planck constant and $$p$$ is the electron momentum. Increasing the electron’s speed increases its momentum, so its wavelength decreases.

For the same crystal, a shorter wavelength produces smaller diffraction angles. The intensity pattern therefore contracts towards the centre. Decreasing the electron speed has the opposite effect: the wavelength increases, the diffraction angles become larger and the pattern spreads out.