Matter waves and electron diffraction

Planck and Einstein gave light E = hf, and Compton's recoil confirmed p = h/λ. De Broglie read the pair as a statement about every particle: a free particle of energy E and momentum p carries a wave with ω = E/ħ and k = p/ħ, so λ = 2π/k = h/p. The step is more than analogy. A plane wave's phase kx − ωt counts the crests passing a point, and a count cannot depend on the observer, so (ω/c, k) must transform between frames exactly as (E/c, p) does, and one constant has to serve light and matter alike. The wavelength therefore belongs to the motion, not to the object: an electron at 1.00 × 10⁶ m s⁻¹ has λ = h/mₑv = 0.727 nm, several atomic diameters, while a 0.145 kg ball at 40 m s⁻¹ has 1.1 × 10⁻³⁴ m, far below any structure that could diffract it.

The experimenter sets a voltage, not a momentum. An electron released from rest and accelerated through V gains K = eV, and while K ≪ mₑc² its momentum is p = √(2mₑK), so λ = h/√(2mₑeV). Check the dimensions before trusting it: mₑeV has units kg J = kg² m² s⁻², its root is a momentum in kg m s⁻¹, and J s divided by kg m s⁻¹ is m. Inserting the constants gives the working form λ = 1.226 nm/√V with V in volts: 0.388 nm at 10 V, 0.167 nm at 54 V, 0.0388 nm at 1 kV. Remember the inverse square root. Four times the voltage halves λ, so bracketing the 0.1 to 0.4 nm spacings of real crystals takes only about 10 to 150 V, which is why slow electrons became the natural probe of a crystal surface.

Davisson and Germer fired 54 eV electrons normally onto a nickel crystal and swept a detector through the backscattered directions; the count rate peaked 50° from the incident beam. Electrons this slow penetrate only a few atomic layers, so the scatterers act as a reflection grating made of the surface rows, whose spacing, d = 0.215 nm, was already known from X-ray diffraction. Waves leaving neighbouring rows toward angle φ differ in path by d sin φ and reinforce when d sin φ = nλ. The peak therefore implies λ = 0.215 nm × sin 50° = 0.165 nm, and de Broglie predicts 0.167 nm: agreement to 1.3%, inside the width of their broad peak. The peak also rose and fell as the voltage passed through 54 V, as a wavelength sweeping past a fixed grating condition requires. One crystal, probed with photons and then with electrons, gave the same spacing.

The grating condition carries its own window. A first-order beam needs sin φ = λ/d ≤ 1, so a wave longer than the spacing produces no diffracted beam at all, only the specular one; on nickel's rows that happens below 32.5 V. At the other end, when λ ≪ d, the orders sit at φ ≈ nλ/d, pressed against the direct beam. Every real beam has an angular spread from its collimation and energy spread, and once nλ/d falls inside it the orders cannot be separated and the particle behaves as a ray. That is the classical limit, the same step that takes wave optics to geometric optics as λ → 0. The ball's 10⁻³⁴ m through a 1 m gap is hopeless. To push the window to heavier particles, slow them: C₆₀ molecules at 200 m s⁻¹ have λ = 2.8 pm, and a 100 nm grating turns them through 28 μrad, small but measurable. What decides is λ/d, never size alone.

A neutron has no charge, so it passes through centimetres of metal and scatters from nuclei rather than from electron clouds; that makes it the probe for bulk structure, for light atoms such as hydrogen, and, through its magnetic moment, for magnetic order. It samples the whole three-dimensional lattice, so the condition is Bragg's, 2d sin θ = nλ, with d the spacing of a family of planes and θ the glancing angle; the beam turns through 2θ. The wavelength comes from temperature. A moderator at T gives the neutrons a Maxwell distribution whose most probable speed carries kinetic energy kT, so λ = h/√(2mₙkT): at 300 K, kT = 25.9 meV and λ = 0.178 nm, matched to interatomic spacings by thermal equilibrium alone. Since λ = h/mₙv = ht/(mₙL), timing the flight over a path L measures λ directly: a 0.178 nm neutron needs 0.450 ms per metre.

An electron microscope accelerates through 100 to 300 kV, where K is no longer small beside mₑc² = 511 keV. λ is still h/p, but p comes from the invariant (pc)² = K² + 2Kmₑc², so λ = hc/√(K² + 2Kmₑc²). The slow-limit result exceeds this by the factor √(1 + K/2mₑc²): 4.8% at 100 kV (3.88 pm against 3.70 pm) and 13.7% at 300 kV. Both limits check, K ≪ mₑc² returning h/√(2mₑK) and K ≫ mₑc² returning hc/K, the photon's value. Wavelength alone would allow δ ≈ 0.61λ/α = 0.23 nm with α = 10 mrad, but magnetic lenses have spherical aberration, a blur Cₛα³ that grows fast with aperture. With Cₛ = 1.0 mm the two blurs are equal at α = 6.9 mrad, each 0.33 nm, and the best resolution scales as λ³⁄⁴ Cₛ¹⁄⁴: about 90 wavelengths, set by the lens.