Electron, neutron and molecular diffraction

Nothing in D sin φ = nλ came from quantum mechanics. Send a wave at normal incidence onto a row of scatterers spaced D apart: every scatterer is struck on the same wavefront, so the only path difference is accumulated on the way out, and it is D sin φ between neighbours. Reinforcement demands that this be a whole number of wavelengths. That is the optical reflection grating, unchanged. The single quantum input is the substitution λ = h/p, and it turns the apparatus into a measuring instrument. You take D from X-ray crystallography, read φ off a detector, and the grating condition hands you λ; then you compute h/√(2mₑeV) from the accelerating voltage and see whether the two agree. In 1927, 54 eV electrons on nickel with D = 2.15 Å peaked at φ = 50°: the geometry gives 1.647 Å, de Broglie gives 1.669 Å. Note the ceiling built into the condition — sin φ cannot exceed 1, so the highest order is n = ⌊D/λ⌋, and if λ exceeds D nothing diffracts at all.

The same maximum is often quoted as Bragg reflection instead. Rays leaving at φ from a beam that came in along the normal are reflected off planes tilted φ/2 to the surface, so the glancing angle is θ = 90° − φ/2 = 65°, and those planes cut the surface rows at spacing d = D sin(φ/2) = 2.15 sin 25° = 0.909 Å. Then 2d sin θ = 2(0.909)(0.906) = 1.647 Å — the same number to every digit. It has to be, because 2⋅D sin(φ/2)⋅cos(φ/2) = D sin φ is a trigonometric identity: the two statements are one external geometry written twice, not two independent confirmations of anything. Treating their agreement as evidence is a mistake worth naming now, because the two readings stop being equivalent the moment the beam actually enters the crystal.

A conduction electron sits in a potential well. Nickel's mean inner potential is V₀ ≈ 15 V, so an electron accelerated through V arrives inside with kinetic energy e(V + V₀). At V = 54 V that is 69 eV, and λ falls from 1.669 Å outside to 12.264/√69 = 1.476 Å inside — the crystal has a refractive index μ = λₒᵤₜ/λᵢₙ = √(1 + V₀/V) = 1.130 for these electrons, and the beam refracts at the surface. Which reading survives now depends on which component of k the condition constrains. The surface-grating condition fixes only the component parallel to the surface, and that component is conserved across the step, so the vacuum wavelength is the correct one there and no correction is needed. The Bragg condition also fixes the perpendicular component, which the step does change; Snell's law in the form cos θ = μ cos θᵢₙ converts it into nλ = 2d√(sin²θ + V₀/V). Every Bragg peak is displaced, and fitting that displacement across a range of voltages is how V₀ is measured.

λ = h/√(2mE) inverts to E = h²/(2mλ²), so at the atomic scale the energy a probe needs is inversely proportional to its mass. To reach λ = 1.80 Å: a neutron needs 0.0253 eV, which is simply k_BT at 293 K; an electron needs 46.4 V; an X-ray photon, obeying E = hc/λ instead, needs 6.89 keV. That one line explains why neutron scattering exists as a field. Thermal neutrons leaving a moderator are already at the right wavelength and travel at 2.20 km s⁻¹, slow enough for a mechanical chopper to select them. They are uncharged, so they penetrate centimetres of material rather than nanometres and report bulk structure. They scatter from nuclei rather than electron clouds, so hydrogen — nearly invisible to X-rays — shows up strongly and neighbouring elements are distinguishable. And they carry a magnetic moment, so magnetic order diffracts too. Electrons scatter some 10⁴ times more strongly than X-rays, which is the opposite bargain: superb sensitivity, a probing depth of a few atomic layers, and a vacuum requirement.

Nothing in λ = h/p stops at the electron. A C₆₀ molecule of mass 720 u moving at 200 m s⁻¹ has p = 2.39 × 10⁻²² kg m s⁻¹ and λ = 2.77 pm — about 250 times smaller than the molecule's own 0.7 nm diameter. Diffracting it needs a 100 nm grating, which gives a first-order angle of only λ/D = 2.8 × 10⁻⁵ rad, so the fringes are 35 μm apart a metre and a quarter downstream. Two conditions then decide whether they appear at all. Transversely, the beam's coherence width at the grating, roughly λL/s for a source of width s at distance L, must span at least one period — which is why the collimating slits throw away almost the entire beam. Longitudinally, the coherence length λ²/Δλ must exceed the path difference nλ, so the highest visible order is about λ/Δλ. A thermal C₆₀ beam has Δv/v ≈ 60%, a coherence length under two wavelengths: the first orders survive and the rest wash out. A 54 eV electron beam with a 0.5 V spread has Δλ/λ = ½(ΔV/V) = 0.46%, a coherence length of 36 nm, and room for a couple of hundred orders.

The diffraction limit does not care what the wave is made of: in Rayleigh's form the smallest resolvable separation is dₘᵢₙ ≈ 0.61λ/(μ sin α), with α the collection half-angle and μ the refractive index of the medium — 1 in the vacuum a microscope column runs in. Put an electron wavelength into it and the numbers turn absurd. A 200 kV electron has λ = 2.51 pm — and note that the non-relativistic 12.264/√V would return 2.74 pm, 9% too long, because at 200 keV against a rest energy of 511 keV the factor √(1 + eV/2mₑc²) is no longer 1. A wavelength of 2.51 pm promises picometre resolution, and no microscope delivers it. Magnetic lenses have severe spherical aberration, which forces α down to about 10 mrad, and 0.61 × 2.51 pm / 0.010 = 0.15 nm — roughly what an uncorrected instrument achieves. Aberration correctors that open α to 40 mrad reach about 40 pm. So h/p sets the floor and the lens decides how far above it you land, the reverse of light optics, where the wavelength is the binding constraint.