de Broglie matter waves
& electron diffraction

From the photon's momentum to the electron's wavelength

Compton scattering fixed p = h/λ for light. de Broglie's step was to assert the same relation for matter with nothing adjusted: any particle of momentum p has λ = h/p. For an electron accelerated from rest through V, energy conservation gives eV = p²/2mₑ, so p = √(2mₑeV) and λ = h/√(2mₑeV). Check the dimensions before trusting it: mₑeV is kg × J = kg² m² s⁻², whose root is kg m s⁻¹, a momentum, and J s divided by kg m s⁻¹ is m. With the constants in, λ = 1.226 nm/√(V/1 V), so 54 V gives 0.167 nm and 10 kV gives 12.3 pm. The same formula explains why nobody had noticed: a 0.145 kg ball at 40 m s⁻¹ has λ = 1.1 × 10⁻³⁴ m, diffraction angles of order λ/d vanish, and the straight-line trajectories of classical mechanics are the λ → 0 limit, just as ray optics is the short-wavelength limit of wave optics.

A crystal is a grating with a known period

A ruled grating has lines about a micrometre apart, ten thousand times the 0.1 nm wavelengths here; the planes of atoms in a crystal sit 0.1 to 0.4 nm apart, exactly the right scale. Treat each plane as a weak partial mirror. Waves reflected from neighbouring planes a distance d apart leave in step when the extra path, 2d sin θ, is a whole number of wavelengths: 2d sin θ = nλ, with θ the glancing angle between the beam and the planes. Three features of this condition do the work. The beam is deflected through 2θ, so that is the angle a detector records. A reflection needs sin θ ≤ 1, so it exists only for λ ≤ 2d/n. And d is measured independently, by X-ray diffraction from the same crystal, so an electron experiment has no free parameter: elastic diffraction at that same wavelength and from the same periodic structure has the same nominal geometrical conditions. Different interactions can make different beams visible and give different intensities.

Davisson and Germer: one strong beam at 50°

Davisson and Germer sent 54 eV electrons normally onto nickel and observed a strong outgoing beam about 50° from the outward surface normal. For surface rows spaced D = 0.215 nm, the normal-incidence grating condition D sin φ = λ gives 0.165 nm, compared with the de Broglie prediction 0.167 nm. D is a surface-row spacing, not the interplanar d of bulk Bragg diffraction. Parallel wavevector conservation fixes the ideal surface beam’s exit direction using the vacuum wavelength; an inner potential is not an explanation of this small residual. Internal refraction and multiple scattering matter when interpreting the complete voltage-dependent pattern and intensities.

Thomson's rings, and a test only the voltage can run

G. P. Thomson sent fast electrons through thin polycrystalline films. Crystallites at many orientations produce diffraction cones, one for each accessible plane spacing, and a flat screen cuts the cones into rings with R = L tan(2θ). At small angles, R ≈ Lλ/d, so ring-radius ratios approach inverse plane-spacing ratios. For non-relativistic momentum, λ ∝ V^(−1/2), making R√V approximately constant—not exactly constant with the full screen geometry. At the tens-of-kilovolts energies of Thomson’s experiments, use the relativistic wavelength when the correction is significant compared with measurement precision. At 50 kV the non-relativistic estimate is 2.42% too large. Magnetic deflection of the pattern identifies the charged electron beam rather than neutral X-rays. The test is the wavelength-dependent diffraction geometry; peak intensities need a scattering model as well.

Same relation, heavier particles

Nothing in λ = h/p mentions charge or the electron, so the prediction extends to anything with momentum. At fixed kinetic energy λ = h/√(2mK) falls as 1/√m: a neutron, 1839 times heavier, carries 1/43 of an electron's wavelength. To reach 0.18 nm it needs only K ≈ 25 meV, the thermal energy of a room-temperature moderator, at about 2200 m s⁻¹. Neutrons scatter from nuclei rather than from electron clouds, so they locate the hydrogen atoms X-rays barely see, their own magnetic moment probes magnetic order, and being uncharged they can penetrate deeply into many materials: the same Bragg geometry, a different probe. For C₆₀, 720 u moving at 220 m s⁻¹, λ = h/mv = 2.5 pm, roughly 400 times smaller than the molecule's 1 nm diameter; a grating of 100 nm period diffracts it into a first order only 25 μrad from the beam, and that fringe pattern was measured in 1999. Mass makes the angle small, not zero: the classical limit is λ/d → 0, not a change in the physics.

Where the simple model stops

The wavelength relation λ = h/p survives; the approximations used to calculate p and the diffraction pattern have limits. At 54 V the non-relativistic wavelength is only 0.0026% high, but at 80 kV it is 3.84% high relative to the relativistic value. A kinematic crystal model adds amplitudes from single scattering and predicts nominal Bragg directions; electron intensities often require dynamical multiple-scattering theory. Finite crystal size, disorder and beam spread broaden features, and multiple scattering can change which beams become visible. Do not infer exact peak heights or all selection rules from the simple geometry alone. A crystal’s inner potential changes internal momentum and bulk diffraction conditions. For a periodic surface, the ideal exit directions remain constrained by parallel wavevector conservation and the vacuum wavelength.

de Broglie relation

λ in m; p in kg m s⁻¹; h = 6.626 × 10⁻³⁴ J s

At equal momentum, particles have equal wavelength. For elastic scattering from the same periodic structure, this fixes the same nominal geometrical diffraction conditions, not identical intensities or visibility of every beam.

Electron accelerated from rest through V

mₑ = 9.109 × 10⁻³¹ kg; e = 1.602 × 10⁻¹⁹ C; V in volts; from eV = p²/2mₑ

Numerically λ = 1.226 nm/√(V/1 V): 54 V gives 0.167 nm and 3.0 kV gives 22.4 pm. Halving λ costs four times the voltage.

Bragg condition

d = plane spacing in m; θ = glancing angle between beam and planes; n = 1, 2, 3, …

θ is measured to the planes; the deflection is 2θ. For this selected plane family and order, a reflection requires nλ ≤ 2d.

Ring radius on a screen at distance L

R and L in m; θ from the Bragg condition; one cone per set of planes in a polycrystalline film

At small angles, R ≈ Lλ/d for first order. With non-relativistic momentum, R√V is approximately constant at fixed d; exact screen geometry and relativistic momentum both modify that scaling.

Relativistic correction

mₑc² = 511 keV; the bracket is 1.00005 at 54 V and 1.049 at 50 kV

Relative to the exact wavelength, the non-relativistic estimate is 0.29% high at 6 kV, 2.42% at 50 kV and 4.78% at 100 kV. Compare this correction with the required precision; no universal voltage makes it safe to ignore.

Any particle of kinetic energy K

m in kg; K in J; a neutron with K = 25 meV has λ = 0.18 nm

At equal K, λ ∝ 1/√m: a neutron carries 1/43 of an electron's wavelength, which is why neutron diffraction runs at thermal energies, not kilovolts.