Wave nature of electrons and de Broglie wavelengthEdexcel International A Level Physics: Revision notes
Section 1
Electrons behave as waves
Light shows both wave and particle behaviour. In 1924 de Broglie proposed that particles such as electrons also have a wave nature. Experiments in the 1920s confirmed this.
A wave property can only be shown by a wave effect: diffraction and interference. If electrons give these effects, they are behaving as waves.
Section 2
Electron diffraction experiments
In an electron diffraction tube, electrons from a heated filament are accelerated by a high potential difference and pass through a thin film of polycrystalline graphite in a vacuum. On a fluorescent screen a pattern of concentric rings appears.
- The layers of carbon atoms act as a diffraction grating, because their spacing (about 10⁻¹⁰ m) is similar to the electron wavelength.
- Diffracted electrons interfere to give rings of high and low intensity.
- Particles would simply form a central spot, so the rings are evidence for the wave nature of electrons.
The tube is evacuated so that electrons are not scattered by air molecules.
Do not say the electrons are 'waves' instead of particles. Say that the diffraction pattern shows they can behave as waves, and name diffraction and interference.
Section 3
The de Broglie equation
The wavelength associated with a moving particle is its de Broglie wavelength:
where J s is the Planck constant, is the momentum and the mass.
- A larger momentum means a shorter wavelength.
- Doubling the speed halves the wavelength.
- Everyday objects have a huge momentum, so λ is far too small to detect: wave effects are only seen for tiny particles such as electrons.
Wavelength is inversely proportional to momentum: if p doubles, λ halves.
Section 4
Worked example
Find the de Broglie wavelength of an electron moving at 4.0 × 10⁶ m s⁻¹ (mass 9.11 × 10⁻³¹ kg).
λ = h / mv = 6.63 × 10⁻³⁴ / (9.11 × 10⁻³¹ × 4.0 × 10⁶) = 1.8 × 10⁻¹⁰ m
This is similar to atomic spacings, so electrons of this speed are diffracted by crystals.
If you are given kinetic energy instead of speed, first use ½mv² = to find v, then use λ = h/mv.
Section 5
Using the wave nature of electrons
Detail smaller than the wavelength cannot be resolved, so a shorter wavelength gives better resolution. Fast electrons have wavelengths far shorter than visible light (about 5 × 10⁻⁷ m), so an electron microscope can see much finer detail than a light microscope.
- Faster electrons: more momentum, smaller λ, rings closer together in a diffraction tube.
- The wavelength must be similar to the spacing for noticeable diffraction.
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Wave nature of electrons and de Broglie wavelength
- A physics technician is demonstrating an electron diffraction tube. Electrons from a heated filament are accelerated and directed at a very thin polycrystalline graphite film inside an evacuated glass tube. A fluorescent screen beyond the film shows a pattern of concentric bright rings.Explain why the ring pattern is evidence for the wave nature of electrons.2 marks
- A teacher compares the behaviour of electrons with that of larger moving objects such as a tennis ball. She uses the de Broglie equation, with Planck constant h = 6.63 × 10⁻³⁴ J s and electron mass 9.11 × 10⁻³¹ kg, and considers an electron moving at 4.0 × 10⁶ m s⁻¹ in a vacuum.Calculate the de Broglie wavelength of the electron moving at 4.0 × 10⁶ m s⁻¹.2 marks
- Electrons in a beam have a kinetic energy of 4.0 × 10⁻¹⁷ J. The beam is directed at a crystal in which the spacing between neighbouring layers of atoms is 0.21 nm. A student wants to know whether the electrons will be noticeably diffracted by the crystal.Calculate the speed of an electron with kinetic energy 4.0 × 10⁻¹⁷ J.3 marks
Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).