University Physics V · Foundations of Quantum Mechanics · 2.6
Electron diffraction
This is where λ = h/p stops being a slogan and becomes a measurement. A crystal hands the beam its own Fourier transform: bright directions exist only where the momentum transfer lands on a reciprocal-lattice vector, so every ring on the photograph is a reading of h. The same test runs unchanged from 54 eV electrons to C₆₀ molecules.
Build the model
Connect the measurement to the mechanism.
Take the beam seriously as the momentum eigenstate e(ikz) and a crystal becomes the measuring instrument for ħk. In the first Born approximation the amplitude to scatter into |k′⟩ is the Fourier transform of the potential at the momentum transfer q = k′ − k, and a periodic potential has a Fourier transform that vanishes everywhere except on the reciprocal lattice. So elastic beams leave the crystal only where ħ(k′ − k) = ħG — the Laue condition, which is Bragg's law 2d sin θ = nλ in disguise — and the whole geometry of the pattern follows from λ = h/p with no adjustable constant.
Davisson and Germer's 54 eV peak at 50° fits λ = 0.167 nm on nickel's 0.215 nm surface rows to 1%; a graphite tube's rings shrink as V⁻¹⁄²; and the same h then serves neutrons at 0.15 nm, helium atoms, and C₆₀ molecules at 2.5 pm, because the relation belongs to p̂ = −iħ∇ and never asks what is doing the travelling. The price of the clean picture is the single-scattering assumption: the Born series in powers of V̂ converges badly for slow electrons, whose mean free path is shorter than a few unit cells, so kinematic intensities fail at LEED energies and only the beam positions — fixed by periodicity, not by convergence — survive into dynamical theory.
- Simple definition
- Electron diffraction is elastic scattering of a beam prepared near a momentum eigenstate |k⟩, with intensity emerging only along directions where the momentum transfer ħ(k′ − k) equals ħG for a reciprocal-lattice vector G of the target.
- Example
- At 54 V the beam carries λ = 1.226/√54 = 0.167 nm; nickel's 0.215 nm surface rows then place first order at sin φ = 0.167/0.215 = 0.777, so φ ≈ 51° — Davisson and Germer measured 50°.
The dial calibrates the ruler: 54 V → 0.167 nm, 4 kV → 0.0194 nm — atomic spacings exactly.
h = 6.626 × 10⁻³⁴ J s, mₑ = 9.109 × 10⁻³¹ kg, e = 1.602 × 10⁻¹⁹ C; V in volts, non-relativistic
First-order scattering reads out the lattice's Fourier spectrum: the allowed beams are its harmonics, their brightness its structure factor |ṼG|².
q = k′ − k the momentum transfer; G a reciprocal-lattice vector, |G| = 2πn/d for the nth harmonic of plane spacing d
Laue and Bragg are one statement: momentum transfer quantised to ħG on the elastic shell.
θ measured from the planes, so the beam deviates by 2θ; d = 2πn/|G| in the same unit as λ
A surface conserves only k∥, so beam directions read the top layer's periodicity with the vacuum λ — the inner potential cannot move them, only brighten or dim them.
normal incidence; φ the exit angle from the surface normal, G∥ a vector of the top layer's 2D reciprocal lattice, d the row spacing, λ the vacuum wavelength
One photograph tests h with no free parameter: quadruple V and every ring halves, whatever the lattice.
r ring radius, L camera length, both in mm; small angles, since r = L tan 2θ and sin θ = λ/2d
Mass enters only through p: a 720 u molecule at 220 m s⁻¹ obeys the same grating equation as a 54 eV electron, with h unchanged.
C₆₀: m = 720 u = 1.196 × 10⁻²⁴ kg at v = 220 m s⁻¹ gives λ = 2.5 pm
Prepare the beam: a δ-normalised momentum eigenstate
The gun is the state-preparation device. Acceleration through V volts hands the electron kinetic energy eV, so p = √(2mₑeV), and in the drift space the beam is the laboratory approximation to the momentum eigenstate e(ikz) with ħk = p — an eigenfunction of p̂ = −iħ d/dz that lives outside L² and is normalised to ⟨k|k′⟩ = δ(k − k′), so 'the' wavelength is an idealisation that collimation and a narrow energy spread sharpen but never reach. It is a good one: the thermionic energy spread is a few tenths of an eV in 54, smearing λ by parts in a thousand. The numbers explain the choice of target. 54 V gives λ = 0.167 nm and 4 kV gives 0.0194 nm — atomic spacings, ten thousand times finer than a ruled optical grating — which is why the first matter-wave interferometer had to be a crystal.
Scatter once: the crystal answers with its Fourier transform
To first order in the interaction — the first Born approximation — the amplitude for |k⟩ → |k′⟩ is ⟨k′|V̂|k⟩ ∝ ∫ V(r) e(−iq⋅r) d³r = Ṽ(q), the Fourier transform of the potential at the momentum transfer q = k′ − k. That one line is the whole theory of diffraction geometry. A crystal's V(r) is periodic, and the Fourier transform of a periodic function is supported on a discrete comb: Ṽ(q) vanishes except at reciprocal-lattice vectors G, with |G| = 2πn/d for planes of spacing d. Elastic scattering adds the constraint |k′| = |k|, and squaring k′ = k + G under it collapses to Bragg's law 2d sin θ = nλ. Each surviving beam carries intensity proportional to |ṼG|², the structure factor — the pattern's positions read the lattice's periodicities, its brightnesses read what sits inside the unit cell.
Davisson–Germer, read as a surface grating
At 54 eV an electron penetrates only a few atomic layers, so the working model of the nickel target is its top (111) plane: rows of atoms d = 0.215 nm apart acting as a two-dimensional grating, with constructive exit at d sin φ = nλ, the angle φ measured from the surface normal down which the beam arrives. First order at λ = 0.167 nm predicts φ = 51°; they measured 50° at 54 V, and sin φ tracked V⁻¹⁄² as the voltage moved. Do not blame the 1° residual on refraction. A surface has two-dimensional translational symmetry, so the wavevector component parallel to it is conserved across the interface: every exit direction obeys k sin φ = |G∥| with the vacuum k, whatever the electron does inside the metal. The residual sits inside the width of the broad peak and the collector's angular resolution, and a single volt of uncertainty in the true accelerating voltage — contact potentials are of that order — moves λ by 0.9% on its own. What the mean inner potential of order +10 V does is different: it shortens λ inside the crystal, so the interlayer condition that makes a beam bright is met at a lower voltage than the vacuum Bragg formula predicts, which is the refractive index Davisson and Germer had to introduce for their intensities. Even the famous accident teaches the model: the peaks appeared only after an oxidation mishap forced a re-anneal that grew large crystallites. The periodicity, not the nickel, does the diffracting.
The ring test: r ∝ V⁻¹⁄² on polycrystalline graphite
A polycrystal fires every orientation at once, so for each allowed spacing d some crystallite sits at the Bragg angle and the diffracted beams fill a cone of half-angle 2θ: rings on the screen, radius r = L tan 2θ ≈ Lλ/d. Graphite's two strong spacings, d₁₀ = 0.213 nm and d₁₁ = 0.123 nm, give two rings whose radii keep the fixed ratio 0.213/0.123 = 1.73 at every voltage — a lattice fingerprint the dial cannot touch. What the dial does control is λ: at 4.00 kV with camera length L = 135 mm the rings sit near 12.3 mm and 21.3 mm, and a run from 1.5 to 5 kV plotted as r against V⁻¹⁄² is a straight line through the origin with slope Lh/(d√(2mₑe)). Fit that slope and you have measured h with a teaching tube — or, taking h as known, measured graphite's lattice.
Neutrons, atoms, C₆₀: the relation is about p̂
Nothing in q = G names the electron. Neutrons from a room-temperature moderator carry λ = h/√(3mₙkBT) ≈ 0.15 nm at 293 K, matched to crystal planes by the same accident of scale, and neutron diffraction is now routine crystallography. Helium atoms at thermal speeds diffract from surface rows. The 1999 Vienna experiment pushed the mass four orders further: C₆₀ at 220 m s⁻¹ has λ = h/(mv) = 2.5 pm — four hundred times smaller than the molecule's own 1 nm diameter — and still built fringes 31 µm apart behind a 100 nm grating. The wavelength belongs to the centre-of-mass momentum operator, so the object's size is irrelevant to whether its amplitude interferes. What does matter is coherence: the C₆₀ left the oven near 900 K and radiating, but its thermal photons are micron-wavelength — far too long to resolve which slit, so no which-path record is written and the fringes survive.
Where single scattering fails
The Born amplitude is the first term of a series in powers of V̂, and for electrons that series is on probation. At 50–100 eV the mean free path between scatterings is about a nanometre — the electron scatters again before it leaves — so the kinematic intensities |ṼG|² are simply wrong at low energy, and LEED analysis solves the multiple-scattering (dynamical) problem numerically, inner potential and absorption included. X-rays couple some four to six orders of magnitude more weakly per atom, which is why their intensities stay kinematic and trustworthy, and why electrons, in exchange, see surfaces that X-rays sail through. The saving grace for this topic: multiple scattering off the same lattice still transfers only sums of ħG, so ring and spot positions survive dynamical theory untouched. Trust the geometry to test λ = h/p; trust the intensities only after the hard calculation.
Change one variable at a time
Make the relationship visible.
Sweep V from 5 down to 1.5 kV: every ring grows as V⁻¹⁄², the λ = h/p signature. Then tune the trial spacing until the arrowed circle lands on the inner solid ring — only d = 0.213 nm works, and at every voltage, which is how a diffraction photograph reads a lattice constant off the film.
WAVELENGTH λ = h/p0.0245 nm
RING r, d = 0.213 nm15.5 mm
RING r, d = 0.123 nm26.9 mm
TRIAL RING r20.7 mm
Live interpretationWAVELENGTH λ = h/p: 0.0245 nm. RING r, d = 0.213 nm: 15.5 mm. RING r, d = 0.123 nm: 26.9 mm. TRIAL RING r: 20.7 mm
Catch the common trap
Explain before calculating.
A polycrystalline graphite target throws its inner diffraction ring at radius 14 mm when the accelerating voltage is 1.0 kV. The voltage is raised to 4.0 kV with the geometry untouched. Where does the inner ring now sit?
Choose an answer to test the model.
Practice & worked examples
Reason from the model, then test the result.
EasyElectrons are accelerated through 54 V onto a nickel crystal whose surface atom rows are spaced d = 0.215 nm. Find the de Broglie wavelength, use the surface-grating condition d sin φ = nλ to predict the first-order angle, and compare with the peak Davisson and Germer observed at φ = 50°.
- Kinetic energy E = eV = 54 × 1.602 × 10⁻¹⁹ J = 8.65 × 10⁻¹⁸ J, far below mₑc² = 511 keV, so the non-relativistic p = √(2mₑE) is safe.
- p = √(2 × 9.109 × 10⁻³¹ × 8.65 × 10⁻¹⁸) = 3.97 × 10⁻²⁴ kg m s⁻¹.
- λ = h/p = 6.626 × 10⁻³⁴ / 3.97 × 10⁻²⁴ = 1.67 × 10⁻¹⁰ m = 0.167 nm — the shortcut 1.226/√54 nm returns the same number.
- First order: sin φ = λ/d = 0.167/0.215 = 0.777, so φ = 51°.
- The 1° gap from the measured 50° is a 1.2% mismatch in λ, within the peak's width and within a 1 V uncertainty in the true accelerating voltage, which alone shifts λ by 0.5/54 = 0.9%. It is not refraction: the surface conserves k∥, so the inner potential leaves this exit angle untouched.
Answerλ = 0.167 nm and predicted φ ≈ 51° against the observed 50° — agreement at the 1% level the surface-grating model can claim.
MediumA teaching diffraction tube accelerates electrons through 4.00 kV onto polycrystalline graphite; the screen sits L = 135 mm downstream. Predict the ring radii for the d₁₀ = 0.213 nm and d₁₁ = 0.123 nm spacings, and decide whether the relativistic correction to λ matters here.
- λ = 1.226 nm/√4000 = 0.01939 nm — eleven times smaller than d₁₀, so every Bragg angle is small.
- Inner ring: sin θ = λ/(2d₁₀) = 0.01939/0.426 = 0.0455, θ = 2.61°, and the beam deviates by 2θ = 5.22°.
- r = L tan 2θ = 135 × 0.0913 = 12.3 mm; the small-angle shortcut r ≈ Lλ/d = 135 × 0.01939/0.213 gives 12.3 mm as well.
- Outer ring: r ≈ Lλ/d₁₁ = 135 × 0.01939/0.123 = 21.3 mm; the flat-screen tan 2θ pushes it to 21.5 mm, a 1% correction the curved bulb of a real tube partly undoes.
- Relativistic bracket from the previous topic, [1 + eV/(2mₑc²)]⁻¹⁄²: eV/(2mₑc²) = 4.00 keV/1022 keV = 0.39%, so λ shrinks by 0.20% — about 0.02 mm on the inner ring, invisible against a ~0.5 mm ring width.
Answerr(0.213 nm) ≈ 12.3 mm and r(0.123 nm) ≈ 21.3 mm; the relativistic correction (−0.2% in λ) sits far below what the photograph can resolve.
HardIn the 1999 Vienna experiment, C₆₀ molecules (m = 720 u) at the most probable speed v = 220 m s⁻¹ passed a nanofabricated grating of period a = 100 nm, with the detector plane L = 1.25 m downstream. Find λ, the first-order diffraction angle, and the fringe spacing at the detector, and compare λ with the molecule's own 1 nm diameter.
- m = 720 × 1.6605 × 10⁻²⁷ = 1.196 × 10⁻²⁴ kg, so p = mv = 1.196 × 10⁻²⁴ × 220 = 2.63 × 10⁻²² kg m s⁻¹.
- λ = h/p = 6.626 × 10⁻³⁴ / 2.63 × 10⁻²² = 2.52 × 10⁻¹² m = 2.5 pm.
- Grating equation a sin θ = nλ, first order: θ₁ ≈ λ/a = 2.52 × 10⁻¹² / 1.00 × 10⁻⁷ = 2.5 × 10⁻⁵ rad = 25 µrad — which is why the beam must first be collimated to about 10 µrad.
- Fringe spacing on the detector: x = Lλ/a = 1.25 × 2.52 × 10⁻⁵ = 3.1 × 10⁻⁵ m = 31 µm, resolved by scanning an ionising laser across the far field.
- λ/diameter = 2.5 pm / 1 nm ≈ 1/400: the fringes come from the centre-of-mass amplitude, and nothing requires the wavelength to exceed the object that carries it.
Answerλ ≈ 2.5 pm, θ₁ ≈ 25 µrad, fringes every ≈ 31 µm at L = 1.25 m — clean interference from a molecule 400 times larger than its own de Broglie wavelength.