The big idea: An electron microscope can image a virus far too small for any light microscope — because its speeding electrons behave like waves with a tiny wavelength.
Momentum sets the wavelength: Every moving particle has this de Broglie wavelength, set by its momentum p: the bigger the momentum, the shorter the wavelength (λ = h/p).
- de Broglie wavelength (m)
- Planck constant, 6.63×10⁻³⁴ J s
- momentum of the particle (kg m s⁻¹)
An electron (mass 9.11 × 10⁻³¹ kg) moves at 2.0 × 10⁶ m s⁻¹. Find its de Broglie wavelength.
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Electrons make diffraction patterns: Fire a beam of electrons at a thin crystal and you get a diffraction pattern of rings. That is exactly what waves do at gaps about the size of their wavelength.
Why it proves the wave nature
- The atomic spacing in the crystal (~10⁻¹⁰ m) matches an electron's de Broglie wavelength, so the effect shows up.
- This is direct proof that particles have a wave nature.
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Big, everyday objects have huge momentum, so their de Broglie wavelength is far too small to ever notice. Wave behaviour only shows up for tiny particles like electrons.
A 0.16 kg cricket ball is bowled at 40 m s⁻¹. Find its de Broglie wavelength, and comment.
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Reading a diffraction pattern: Electrons fired through a thin film of atoms spread into rings, just as light spreads through a narrow gap. Where the intensity first drops to a minimum is fixed by the electrons' de Broglie wavelength.
- accelerating potential difference (V)
- electron mass, 9.11 × 10⁻³¹ kg
The first minimum: For diffraction by a gap or spacing b, the first minimum is at sin θ ≈ λ/b. A shorter wavelength (faster electrons) gives a smaller angle: the rings close in.
Worked example — electrons through a crystal film
Electrons are accelerated from rest through 150 V and pass through a film whose atoms are 2.1 × 10⁻¹⁰ m apart. Find the angle of the first minimum.
Solution
- de Broglie wavelength:
- First minimum:
Final answer
θ ≈ 29°.
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How this is tested — matter waves are HL only (E.2) and split cleanly by paper:
Paper 1A
- A quick λ = h/p.
- Or 'why do electrons diffract but cricket balls don't?'
Paper 2
- Find a de Broglie wavelength (often after the momentum first).
- Or the angle of the first minimum, sin θ ≈ λ/b, for diffracted electrons.
The classic trap: Find the momentum p = mv first, then λ = h/p. A bigger momentum gives a shorter wavelength — the relation is inverse.
Three easy marks: (1) Find the momentum p = mv first, then λ = h/p. (2) A bigger momentum → a shorter wavelength. (3) Diffraction needs a gap about the size of the wavelength.
A proton has a momentum of 3.0 × 10⁻²¹ kg m s⁻¹. Determine its de Broglie wavelength.
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See the answer broken into clear points, with guidance on how each point earns marks.