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Quanta and wave-particle duality

Two results broke the wave theory of light just after it had won. The ultraviolet catastrophe, in which classical physics predicted infinite energy from a hot body, and the photoelectric effect, in which bright red light ejects no electrons at all, both required light to be quantised. De Broglie then showed matter has a wavelength too.

Year 13AQA 3.12.2.4, 3.12.2.5, 3.12.2.6CIE 22.1.1, 22.1.2, 22.3.1, 22.3.2, 22.3.3, 22.3.4OCR A 4.5.1(a), 4.5.1(b), 4.5.2(b), 4.5.2(e), 4.5.3(a), 4.5.3(c)

Builds on The photoelectric effect and The nature of light.

IN THIS TOPIC

  • Describe the ultraviolet catastrophe and Planck's resolution in terms of quanta.
  • State the three failures of classical wave theory over photoelectricity, and why Einstein's answer mattered.
  • Use de Broglie's λ = h/√(2meV), and predict what happens to a diffraction pattern when the pd changes.
  • Estimate the pd needed for atomic-scale electron wavelengths.
  • Outline the TEM and the STM, and give the TEM's practical limitations.

COMMON MISCONCEPTION

Waves are waves and particles are particles, and nothing is ever both.

The ultraviolet catastrophe

The trouble began in a furnace. Classical wave theory, applied to the radiation inside a hot cavity, made a clean prediction for the black-body curve, and the prediction was absurd. Waves were free to carry any amount of energy, so ever-shorter wavelengths should carry ever more of it, and the predicted intensity climbs without limit towards the ultraviolet. A glowing coal should blast out unlimited ultraviolet and beyond. It plainly does not, and the failure earned the name ultraviolet catastrophe.

The ultraviolet catastrophe: classical theory climbs without limit at short wavelengths; the measured curve, and Planck's quanta, turn overclassical theory blows upwhat is measuredwavelengthintensitythe ultraviolet catastrophe, and Planck's escape: E = hf
FIG. 1Classical theory against reality, both computed. The measured black-body curve turns over and falls; the classical prediction runs off the top of the chart towards short wavelengths.

In 1900 Max Planck found the escape, and disliked it. Suppose energy is exchanged only in quanta, indivisible packets of size E = hf. At high frequencies each packet is expensive, so the short-wavelength modes are starved of energy and the curve turns over, exactly as measured. Planck introduced h as a mathematical fix. What it meant, he left carefully alone.

Einstein and the photoelectric verdict

The meaning arrived in 1905, through the photoelectric effect met in the core course. Classical wave theory fails three ways there, and an exam answer is expected to list all three. It cannot explain the threshold frequency, since a dim high-frequency source ejects electrons while an intense low-frequency one never does. It cannot explain why emission is instant, with no time needed to soak up wave energy. And it cannot explain why brighter light changes the number of electrons while leaving their maximum kinetic energy untouched.

Einstein's resolution took Planck's quanta literally. Light itself travels as packets, photons of energy hf, and one photon deals with one electron, all or nothing. Every photoelectric observation follows at once. The spec names the significance in one line. Electromagnetic radiation, so recently and so thoroughly proved a wave, also behaves as particles. The nature of light had turned again, and this time it refused to settle on either answer.

Matter waves

In 1923 Louis de Broglie completed the symmetry with an audacious guess. If waves behave as particles, particles ought to behave as waves, obeying the same momentum relation

p=hλp = \frac{h}{\lambda}ON THE AQA DATA SHEET

For an electron accelerated from rest through a pd V, the momentum follows from eV = ½mv2, and substituting gives the working form for this option:

λ=h2meV\lambda = \frac{h}{\sqrt{2meV}}ON THE AQA DATA SHEET

WORKED EXAMPLE

The wavelength that matched the atoms

Find the de Broglie wavelength of electrons accelerated through 54 V, the setting of the Davisson-Germer experiment.

λ = h/√(2meV) = 6.63 × 10−34 / √(2 × 9.11 × 10−31 × 1.60 × 10−19 × 54).

λ = 1.7 × 10−10 m, the size of an atomic spacing.

That coincidence is the experiment. Fired at a nickel crystal, whose atomic planes form a natural diffraction grating on exactly this scale, the electrons produced diffraction patterns. Particles, diffracting.

Electron diffraction rings shrink when the electrons speed up: a larger accelerating pd means more momentum, a shorter de Broglie wavelength and tighter rings50 V: slow electrons, wide rings200 V: rings half the sizeelectron diffraction through a thin crystalwaves diffract; speeding the particles up tightens the pattern
FIG. 2Low-energy electron diffraction rings, and the qualitative test the spec asks for: speed the electrons up and the rings tighten, because more momentum means a shorter wavelength.

The qualitative check confirms the formula's shape. Raise the accelerating pd and the rings shrink, since faster electrons carry more momentum and therefore a shorter wavelength. Quadruple the pd and the wavelength halves. Waves of matter respond to a voltage dial exactly as de Broglie said they must.

Microscopes from matter waves

Duality turned out to be useful as well as strange. A microscope cannot resolve detail much smaller than the wavelength it uses, which condemns light microscopes at around 10−7 m. Electron waves shrug that limit off: a modest bench pd already brings λ down to atomic scale.

GUIDED PRACTICE

The TEM's working wavelength

Estimate the de Broglie wavelength in a transmission electron microscope running at 100 kV.

Show the working

λ = h/√(2meV) = 6.63 × 10−34 / √(2 × 9.11 × 10−31 × 1.60 × 10−19 × 105) = 3.9 × 10−12 m.

Thousands of times shorter than visible light, and at this pd the electrons are fast enough that the classical formula is itself starting to err, a warning the relativity lessons will make precise.

Two microscopes built on matter waves: the transmission electron microscope, and the scanning tunnelling microscope feeling a surface atom by atomelectron gunmagnetic lensthin specimenmagnetic lensviewing screensharp tipatoms of the surfaceTEMSTMTEM: electrons pass through a sliver of specimenSTM: a tunnelling current maps the surface atom by atom
FIG. 3The two instruments. The TEM fires electrons through a sliver of specimen, focused by magnetic lenses onto a screen; the STM holds a sharp tip a nanometre above a surface and reads the tunnelling current.

The transmission electron microscope is a light microscope rebuilt for electron waves. An electron gun supplies the source, and magnetic coils act as lenses, focusing the beam through a very thin specimen and projecting the transmitted pattern onto a fluorescent screen, all in vacuum. Denser regions scatter more electrons and show darker.

Those two words, thin and vacuum, are the TEM's limitations, and questions ask for them. The specimen must be thin enough for electrons to pass through, so it needs slicing and mounting. It must survive a vacuum, so nothing living can be imaged alive. And the electron beam itself heats and damages delicate samples.

The scanning tunnelling microscope abandons lenses entirely. A metal tip sharpened to almost a single atom is held about a nanometre above a conducting surface, close enough for electrons to tunnel across the gap. The tunnelling current falls off so steeply with distance that atom-height bumps change it measurably, and scanning the tip across the surface maps it atom by atom.

INDEPENDENT PRACTICE

The voltage for an atom

Estimate the anode voltage needed to give electrons a de Broglie wavelength of 1.0 × 10−10 m, the order of the size of an atom.

Show the working

Rearranging: V = h2/(2meλ2) = (6.63 × 10−34)2 / (2 × 9.11 × 10−31 × 1.60 × 10−19 × (1.0 × 10−10)2).

V ≈ 150 V. Atomic-scale resolution costs less voltage than a cathode-ray television once did, and that is why electron diffraction turned up by accident within four years of de Broglie's guess.

ASSESSMENT FOCUS

  • The ultraviolet catastrophe fits in one sentence. Classical wave theory lets every wavelength carry any energy, so it predicts intensity rising without limit at short wavelengths, contrary to the measured curve.
  • Planck's move is about exchange. Energy is emitted and absorbed in quanta of E = hf, which starves the high-frequency modes. Use the word "quanta", and say what h sets the size of.
  • The photoelectric failures come in threes. Threshold frequency, instant emission, and intensity affecting the number of electrons but not their maximum kinetic energy. Einstein's photon answers each one, and the significance is light's particle nature.
  • λ = h/√(2meV) assumes acceleration from rest and non-relativistic speeds. Quote answers to two significant figures, and expect the wording "estimate" once the pd runs into tens of kilovolts.
  • TEM and STM answers want principles, not engineering. Magnetic lenses focusing electrons through a thin specimen for the TEM, plus its thin-and-vacuum limitations. A tunnelling current across a nanometre gap, mapped by scanning, for the STM.

CHECK YOURSELF

In a low-energy electron diffraction experiment the accelerating pd is doubled. Explain what happens to the ring pattern, and why an electron microscope can resolve detail no light microscope can reach.

Show a hint

Follow the chain pd, momentum, wavelength. Then remember resolution follows wavelength.

Show the answer

Doubling V multiplies the momentum by √2, so λ = h/p falls by a factor of √2 and the rings shrink by that factor. Faster electrons, shorter waves, tighter pattern.

Resolution is limited to roughly the wavelength used. Electrons at even 150 V carry λ ≈ 10−10 m, thousands of times shorter than visible light's 5 × 10−7 m.

An electron microscope therefore resolves atomic-scale structure that no arrangement of glass lenses could ever reach. Wave-particle duality, turned into an instrument.

Planck quantised energy at E = hf, and Einstein made the packets real as photons.

de Broglie reversed the idea with λ = h/p, and matter waves became microscopes.

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  • Describe the ultraviolet catastrophe and Planck's resolution in terms of quanta.
  • State the three failures of classical wave theory over photoelectricity, and why Einstein's answer mattered.
  • Use de Broglie's λ = h/√(2meV), and predict what happens to a diffraction pattern when the pd changes.
  • Estimate the pd needed for atomic-scale electron wavelengths.
  • Outline the TEM and the STM, and give the TEM's practical limitations.

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