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The photoelectric effect (lesson 4) gave light a particle aspect — a photon of energy hf. The natural next question was the symmetric one: if a wave can behave like a particle, can a particle behave like a wave? Louis de Broglie, in his 1924 Paris PhD thesis, made the bold conjecture that every particle has an associated wavelength λ = h/p, where p is its momentum. Within three years the prediction was tested experimentally, accidentally, by Clinton Davisson and Lester Germer in New Jersey and (independently and almost simultaneously) deliberately by George Paget Thomson — son of J. J. Thomson — at Aberdeen. Both teams observed unambiguous diffraction patterns when low-energy electrons were scattered off crystalline targets. Particles had wave properties; the wave-particle duality was symmetric; quantum mechanics, which had been a problem-by-problem rescue of classical theory, became the unified description of microscopic matter. This lesson covers de Broglie's hypothesis, the Davisson-Germer experiment, G. P. Thomson's foil-diffraction experiment, the electron-wavelength formula λ = h/√(2meV) for electrons accelerated through V volts, and the practical consequence — electron microscopy.
Spec mapping: This lesson covers part of AQA 7408 section 3.12.2 — de Broglie's hypothesis (λ = h/p), the Davisson-Germer experiment, electron diffraction through thin metal foils (G. P. Thomson), wave-particle duality, and the calculation of de Broglie wavelength for an electron accelerated through a known potential difference. (Refer to the official AQA specification document for exact wording.)
Synoptic links:
- Section 3.2.2 (particles and quantum phenomena): electron diffraction is the canonical Year-12 evidence for the wave nature of particles. This lesson revisits it in its 1924-27 historical context and extends the de Broglie wavelength calculation.
- Section 3.12.2 (photoelectric effect, lesson 4): photons → particles; particles → waves. The duality is symmetric and complete only with both halves.
- Section 3.12.3 (special relativity, lesson 7): for high-velocity electrons (e.g. in modern electron microscopes operating at 200+ kV), the non-relativistic formula λ = h/√(2meV) under-estimates λ; the relativistic correction is significant above ~50 kV. This is the same regime where relativity matters in lessons 6-7.
Louis de Broglie was, by background, a historian rather than a physicist — his older brother Maurice was the experimentalist of the family. His PhD thesis, Recherches sur la théorie des quanta (1924), was extraordinarily short by modern standards (about 70 pages) and contained one central idea: the relationship between a photon's wavelength and momentum, p = h/λ, which had been established by Einstein and Compton for light, should hold for all matter.
Rearranged:
λ = h / p
where λ is the "matter wave" wavelength associated with a particle of momentum p. For a photon (massless, momentum p = E/c = hf/c = h/λ) the relation is just the photon relation. For an electron of mass m_e and speed v ≪ c, the momentum is p = m_e v, so
λ = h / (m_e v)
If the electron has been accelerated from rest through a potential difference V, the kinetic energy is eV and (non-relativistically) ½m_e v² = eV, so v = √(2eV/m_e) and p = m_e v = √(2m_e eV). The wavelength is therefore
λ = h / √(2 m_e e V)
— a formula that, for typical laboratory voltages, gives wavelengths of order 0.01-10 nm. The hypothesis was striking because nothing in classical mechanics or in the existing quantum patches gave any reason to expect electrons to have wave properties. De Broglie's thesis examiners — Paul Langevin in particular — were unsure how seriously to take it; Langevin sent a copy to Einstein, who replied that de Broglie had "lifted a corner of the great veil." The thesis was accepted in November 1924.
Clinton Davisson and Lester Germer were experimentalists at Bell Telephone Laboratories in New York, studying the scattering of low-energy electrons (40-400 eV) from a nickel target. The work had nothing to do with de Broglie — Davisson was studying the secondary-emission properties of the surface for the development of vacuum tubes.
In 1925 a vacuum-system accident — a glass-handling slip that admitted air to the tube while the nickel target was hot — left the nickel surface heavily oxidised. To clean it, Davisson and Germer baked the nickel at high temperature for an extended period. The thermal treatment, by accident, transformed what had been polycrystalline nickel (many small grains, randomly oriented) into a single large crystal. When they resumed their scattering experiments on this newly crystalline target, the angular distribution of scattered electrons was completely different — sharp peaks at specific scattering angles, not the smooth distribution they had seen previously.
By 1927, Davisson had identified the peaks as Bragg-style diffraction maxima: the electrons were scattering off the regular planes of nickel atoms (lattice spacing d ≈ 0.215 nm) as if they were waves with a specific wavelength. For electrons accelerated through 54 V, the strongest peak occurred at a scattering angle of 50° from the surface normal — and the de Broglie formula λ = h/√(2m_e eV) gives, for V = 54 V, λ ≈ 0.167 nm, which matches the Bragg condition for nickel's lattice spacing within a few percent. De Broglie's prediction was experimentally confirmed.
graph TD
A["Electron gun<br/>(V = 54 V)"] --> B["Single-crystal Ni target"]
B --> C["Detector arc<br/>(measures intensity vs angle)"]
B -.->|"Bragg<br/>diffraction"| D["Sharp peaks<br/>at specific θ"]
style B fill:#1d4ed8,color:#fff
style D fill:#27ae60,color:#fff
Working independently at the University of Aberdeen, George Paget Thomson (son of J. J. Thomson, who had discovered the electron 30 years earlier) developed a different experimental approach: he passed a beam of higher-energy electrons (10-60 keV) through a very thin polycrystalline foil — gold, aluminium, tin, celluloid — and recorded the scattered electrons on a photographic plate beyond the foil.
The pattern recorded was a series of concentric rings — the signature of diffraction from a polycrystalline target, in which crystallites of all orientations contribute to a Debye-Scherrer-style pattern. The same pattern is produced by X-ray diffraction from the same foil, and the wavelengths can be compared. For 30 keV electrons, the de Broglie wavelength is approximately λ = 7 × 10⁻¹² m = 7 pm — comparable to the X-ray wavelengths used in crystallography of the period.
J. J. Thomson (the father, 1897) had discovered the electron as a particle; G. P. Thomson (the son, 1927) discovered it as a wave. Davisson and G. P. Thomson shared the 1937 Nobel Prize in Physics for the experimental discovery of electron diffraction. De Broglie himself had already received the 1929 Nobel Prize for the theoretical prediction.
An electron is accelerated from rest through a potential difference V = 100 V. Calculate (a) its kinetic energy in J and in eV, (b) its momentum, (c) its de Broglie wavelength.
Use h = 6.63 × 10⁻³⁴ J s, m_e = 9.11 × 10⁻³¹ kg, e = 1.60 × 10⁻¹⁹ C.
Solution.
(a) KE = eV = (1.60 × 10⁻¹⁹)(100) = 1.60 × 10⁻¹⁷ J = 100 eV.
(b) Non-relativistically, KE = p²/(2m_e), so p = √(2m_e × KE) = √[2 × (9.11 × 10⁻³¹) × (1.60 × 10⁻¹⁷)] = √(2.92 × 10⁻⁴⁷) = 5.40 × 10⁻²⁴ kg m s⁻¹.
(c) λ = h/p = (6.63 × 10⁻³⁴)/(5.40 × 10⁻²⁴) = 1.23 × 10⁻¹⁰ m = 0.123 nm.
For comparison, a typical visible-light wavelength is 500 nm — about 4,000 times larger. A 100 V electron has the same wavelength scale as the spacing between atoms in a crystal, which is why electrons diffract off crystal planes (this is the basis of every electron-microscope and electron-diffraction technique used in materials science today).
A transmission electron microscope (TEM) operates at an accelerating voltage of V = 80 kV. Calculate the de Broglie wavelength of the electrons (use the non-relativistic formula, then comment on the relativistic correction).
Solution.
λ = h / √(2 m_e e V) Denominator: √[2 × (9.11 × 10⁻³¹) × (1.60 × 10⁻¹⁹) × (80 × 10³)] = √[2.33 × 10⁻⁴⁴] = 1.527 × 10⁻²² λ = (6.63 × 10⁻³⁴)/(1.527 × 10⁻²²) = 4.34 × 10⁻¹² m = 4.34 pm
The relativistic correction at 80 kV is significant — the electron's kinetic energy is 80 keV, which is about 16% of its rest energy m_e c² = 511 keV. The relativistic formula
p = (1/c)√[(KE + m_e c²)² − (m_e c²)²]
gives a momentum about 8% higher than the non-relativistic value, so the actual de Broglie wavelength is about 8% shorter than 4.34 pm — closer to 4.0 pm.
At this wavelength, a TEM can resolve features of order λ — about 1,000× smaller than the diffraction-limited resolution of a visible-light microscope (which is set by ~250 nm = λ_vis/2). This is why electron microscopes can image atomic columns in crystalline materials, while optical microscopes cannot.
By 1927 — just three years after de Broglie's thesis — wave-particle duality had moved from speculative philosophy to experimental fact. The summary picture, still essentially correct today:
| Entity | Particle aspect | Wave aspect |
|---|---|---|
| Photon | Photoelectric effect, Compton scattering, photon counting | Interference (Young), diffraction, polarisation |
| Electron | Cathode rays (Thomson 1897), cloud-chamber tracks, point detection | Davisson-Germer (1927), G. P. Thomson (1927), electron interference |
| Neutron | β-decay products, point detection | Neutron diffraction (Mitchell, 1936) — used in crystallography |
| Atom | Mass spectrometry, isolated trapping | Atom interference (1990s) — even fullerenes (C₆₀) have been shown to diffract |
The unifying principle: every quantum object has a wavelength λ = h/p and a frequency f = E/h. For macroscopic objects, the wavelength is so small as to be unobservable: a 1-kg ball moving at 1 m s⁻¹ has λ = h/(1 × 1) = 6.6 × 10⁻³⁴ m, which is roughly 10⁻²⁵ times smaller than an atomic nucleus and far below any conceivable measurement. The wave aspect of matter is utterly negligible at everyday scales — but it dominates the behaviour of electrons in atoms, semiconductors, and any structure on the sub-nanometre scale.
An electron is accelerated from rest through a potential difference V = 200 V in a school electron-diffraction tube. Take e = 1.60 × 10⁻¹⁹ C, m_e = 9.11 × 10⁻³¹ kg, h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹.
(a) Kinetic energy of the electron.
The work done by the accelerating field equals the kinetic energy gained:
KE = eV = 1.60 × 10⁻¹⁹ × 200 = 3.20 × 10⁻¹⁷ J
(b) Momentum of the electron.
Using KE = p² / (2m), so p = √(2 m KE):
p = √(2 × 9.11 × 10⁻³¹ × 3.20 × 10⁻¹⁷) p = √(5.830 × 10⁻⁴⁷) p = 7.64 × 10⁻²⁴ kg m s⁻¹
(c) de Broglie wavelength.
λ = h / p = (6.63 × 10⁻³⁴) / (7.64 × 10⁻²⁴) = 8.68 × 10⁻¹¹ m ≈ 87 pm
(d) Comparison with X-ray wavelengths.
Typical laboratory X-rays (e.g. Cu Kα at 0.154 nm = 1.54 × 10⁻¹⁰ m) have wavelengths in the range 10⁻¹⁰ m — the same order of magnitude as the 87 pm de Broglie wavelength of the 200-V electron. This is exactly the wavelength range needed to resolve atomic-lattice spacings, which are typically 0.1–0.5 nm in crystalline solids. The fact that low-kV electrons have wavelengths comparable to X-rays is why electron diffraction off polycrystalline graphite (as in the school demonstration tube) produces concentric-ring diffraction patterns analogous to X-ray powder diffraction. The Davisson-Germer 1927 experiment used 54-V electrons (giving λ ≈ 167 pm) and a nickel single crystal, observing the predicted Bragg-condition reflection peaks; G. P. Thomson's parallel 1927 experiment used 17-kV electrons and a thin gold foil, observing ring patterns. Both were awarded share of the 1937 Nobel Prize.
(e) Transmission electron microscopy at 200 kV.
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