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Wave-particle duality

The photoelectric effect caught light behaving as particles. Electron diffraction catches matter behaving as waves. Neither label survives on its own, and one small equation, lambda equals h over mv, connects the two worlds.

Year 12AQA 3.2.2.4CIE 22.3.1, 22.3.2, 22.3.3, 22.3.4OCR A 4.5.3(a), 4.5.3(b), 4.5.3(c)EDEXCEL 7, 86, 87, 90

Builds on The photoelectric effect.

IN THIS TOPIC

  • State the two-way evidence, electron diffraction for the wave nature of particles and the photoelectric effect for the particle nature of light.
  • Use the de Broglie wavelength λ = h/mv, finding v from a kinetic energy where needed.
  • Explain how and why the diffraction changes when a particle's momentum changes.
  • Say why wave behaviour never shows up in everyday objects.

COMMON MISCONCEPTION

Light is a wave, and electrons are particles.

The evidence runs both ways

By the photoelectric effect, light, the textbook wave, delivers its energy in particle-like photons. The reverse result came from electrons. Fire a beam of them through a thin sheet of graphite and they do not land on the screen as a single spot. They arrive in concentric rings, a diffraction pattern, the signature behaviour of waves.

Electrons fired through a thin crystal land in diffraction rings: particles behaving as wavesthin graphiteelectronsrings: an interference pattern
FIG. 1Electrons through thin graphite produce rings on the screen: diffraction, from things with mass and charge.

So electron diffraction shows that particles possess wave properties, while the photoelectric effect shows that electromagnetic waves have a particulate nature. Learn that two-way sentence in both directions. The details of any particular diffraction apparatus are not examined.

The de Broglie wavelength

de Broglie's proposal gave that wave a wavelength, set by the particle's momentum.

λ=hmv\lambda = \frac{h}{mv}ON THE AQA DATA SHEET

where mv is the momentum. The smallness of the Planck constant is what keeps the world looking normal. Everyday objects carry so much momentum that their wavelengths are unmeasurably tiny. An electron's momentum is small enough to give a wavelength around the spacing of atoms, which is the scale needed to diffract off a crystal lattice.

The de Broglie wavelength shrinks as momentum grows, so faster particles diffract lesssmall momentum: long wavelengthlarge momentum: short wavelengthλ = h / mv: more momentum, less diffraction
FIG. 2Two de Broglie waves: small momentum, long wavelength; large momentum, short wavelength. λ = h/mv.

The equation also predicts what happens when you change the beam. Accelerate the electrons harder and their momentum rises, so λ shrinks. Diffraction is strong only while the wavelength is comparable to the spacing it meets, so the rings tighten towards the centre. More momentum, less diffraction. You can test that by turning a voltage knob, and it holds.

WORKED EXAMPLE

Why crystals, of all things

How fast must an electron travel for its de Broglie wavelength to match a crystal's atomic spacing of 1.0 × 10−10 m?

Diffraction needs the wavelength comparable to the gap, so crystals are the natural grating for electrons.

Rearrange λ = h/mv for v: v = h/mλ = 6.63 × 10−34/(9.11 × 10−31 × 1.0 × 10−10) = 7.3 × 106 m s−1.

About a hundred and fifty volts of accelerating pd reaches that speed. Electron diffraction therefore fits on a school bench, with no national laboratory required.

GUIDED PRACTICE

Why you have never diffracted

Find the de Broglie wavelength of a 60 kg person walking at 1.5 m s−1, and use the answer to explain why people do not diffract through doorways.

Show the working

λ = h/mv = 6.63 × 10−34/(60 × 1.5) = 7.4 × 10−36 m.

Twenty-five powers of ten smaller than an atom, let alone a doorway. Diffraction needs a gap near the wavelength, and nothing in the universe has a gap that small. The wave behaviour has not gone away. At large momentum it simply becomes too slight to measure.

INDEPENDENT PRACTICE

Electron against proton

An electron and a proton travel at the same speed. Which has the longer de Broglie wavelength, and by what factor? (The proton is about 1800 times more massive.)

Show the working

In λ = h/mv, at equal speed the wavelength is inversely proportional to mass.

The electron's wavelength is about 1800 times longer. Lighter particles are the wavier ones, and that is one reason the electron microscope became the practical instrument.

Ideas on probation

Duality is also the course's case study in how physics changes its mind. The wave picture of light stood alone for a century before the photoelectric effect showed light exchanging energy in quanta, something no pure wave account explains, while interference and diffraction still demand the wave description. The particle picture of the electron lasted barely thirty years. Claims of that size earn acceptance only through peer review and independent validation by the scientific community, and duality is the reminder that even a settled classification stays open to evidence.

ASSESSMENT FOCUS

  • The evidence pairing must point the right way. Electron diffraction shows particles behaving as waves, and the photoelectric effect shows light behaving as particles. Swap them and you score zero.
  • In λ = h/mv the denominator is the momentum. Given a kinetic energy instead, find v from ½mv2 first and then multiply by m.
  • “Explain how the pattern changes if the accelerating pd increases” wants four links in order. Momentum rises, λ = h/mv falls, diffraction decreases, the rings move inwards.
  • Diffraction matters only when λ is comparable to the gap or the spacing, and for a thrown ball the momentum is so large that any real gap outsizes λ by thirty powers of ten and more. That comparison sentence is usually a mark on its own.

CHECK YOURSELF

An electron (m = 9.11 × 10−31 kg) moves at 3.3 × 106 m s−1. Find its de Broglie wavelength, and state what happens to the diffraction pattern if the electrons are accelerated to a higher speed.

Show a hint

Momentum first. Then remember what diffraction needs from a wavelength.

Show the answer

Momentum first. mv = 9.11 × 10−31 × 3.3 × 106 = 3.0 × 10−24 kg m s−1.

Then λ=hmv\lambda = \frac{h}{mv} = 6.63 × 10−34 / (3.0 × 10−24) = 2.2 × 10−10 m, about one atomic spacing, and a crystal lattice therefore diffracts the beam well.

At higher speed the momentum grows, λ shrinks, and the electrons diffract less: the rings contract towards the centre.

Everything carries both behaviours. Momentum sets the wavelength; the apparatus decides whether the wave side shows.

WORKBOOK

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Question workbook

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Question workbook

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Answer book

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  • State the two-way evidence, electron diffraction for the wave nature of particles and the photoelectric effect for the particle nature of light.
  • Use the de Broglie wavelength λ = h/mv, finding v from a kinetic energy where needed.
  • Explain how and why the diffraction changes when a particle's momentum changes.
  • Say why wave behaviour never shows up in everyday objects.

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