How this instrument works
Louis de Broglie argued in his 1924 Sorbonne thesis that if light could act as particles, particles ought to act as waves, and he gave every moving body a wavelength equal to Planck's constant divided by its momentum. Einstein, sent that thesis for an opinion, replied that de Broglie had lifted a corner of the great veil. Three years later Clinton Davisson and Lester Germer, rebuilding a nickel target at Bell Labs after a coolant flask shattered and oxidised it, accidentally baked their sample into large crystal grains and watched a scattered electron beam split into diffraction peaks. George Thomson saw the same rings through thin metal foils at Aberdeen. De Broglie took a Nobel Prize in 1929; both experimentalists shared one in 1937.
Matter wavelength is not physical size. It is phase scale: how far a particle travels before its quantum phase advances one full cycle, and therefore how finely spaced an obstacle must be to bend it. Light, slow particles get long wavelengths; heavy or fast ones get short ones. An electron at 1000 km/s measures about 0.73 nm, roughly three atomic diameters, which is exactly why crystals work as diffraction gratings for electron beams. Room-temperature neutrons sit near 0.18 nm and serve the same role on hydrogen-rich samples that X-rays handle badly. A 160 gram cricket ball at 40 m/s lands near 10⁻³⁴ m, and no aperture that narrow exists anywhere.
Two limits are worth respecting. First, m·v is ordinary momentum, so this form drifts once speed approaches c: inside a 100 kV transmission electron microscope an electron already moves at 0.55c, and the simple expression overstates its wavelength by about five per cent. Swap m·v for γm·v and accuracy returns. Second, the formula says nothing about photons, which carry no rest mass and instead take their momentum from E/c. One happy detail: since the 2019 revision of SI base units, h is no longer measured but defined as exactly 6.62607015 × 10⁻³⁴ J·s, and the kilogram is now pinned to it rather than the reverse.
- Enter Particle mass in kilograms, or switch the unit to grams. The electron rest mass, 9.1093837015 × 10⁻³¹ kg, is already loaded as a default.
- Enter Particle speed in m/s or km/h. Hold it well below 3 × 10⁸ m/s, or relativistic momentum takes over and this non-relativistic form drifts.
- Read De Broglie wavelength in metres, then switch its unit to nanometres or micrometres for a figure you can compare against atomic spacing.
- Sanity-check what comes back: a wavelength dwarfing your apparatus means no diffraction, while anything near 0.2 nm will spread across a crystal lattice.
Worked example — an electron at 1000 km/s
Take one electron, Particle mass 9.1093837015 × 10⁻³¹ kg, at a Particle speed of 1.0 × 10⁶ m/s. That is a thousand kilometres per second, roughly what an electron picks up falling through 2.84 volts — modest, by laboratory standards. Its momentum is p = 9.1093837015 × 10⁻³¹ × 1.0 × 10⁶ = 9.1094 × 10⁻²⁵ kg·m/s.
Divide Planck's constant by that momentum: λ = 6.62607015 × 10⁻³⁴ ⁄ 9.1094 × 10⁻²⁵ = 7.27389510325 × 10⁻¹⁰ m. Set De Broglie wavelength to nanometres and it reads 0.7274 nm.
Which is the entire reason anyone cares. Around 0.73 nm spans several atomic diameters, close to plane spacings inside a nickel crystal. Send that electron at such a crystal and it does not merely bounce off; it fans out into bright and dark rings, behaving as X-rays of similar wavelength do. Push speed to 2.0 × 10⁶ m/s and wavelength halves to 0.3637 nm — strictly inverse, no exceptions.
Questions
Does a moving car have a de Broglie wavelength?
Yes, and it is hopelessly beyond measurement. A 1500 kg car at 20 m/s carries 30 000 kg·m/s of momentum, giving roughly 2 × 10⁻³⁸ m — around twenty-three orders of magnitude below a proton's radius. Wave behaviour only shows itself when wavelength is comparable to whatever obstacle stands in front of it, and nothing built or found is that fine. Interference has been demonstrated for molecules of some 2000 atoms, which is about where laboratory technique currently stalls.
Should I use rest mass or relativistic mass?
Use momentum. λ = h/p is the exact statement; λ = h/(m·v) holds only while speed stays small beside light. Past about 0.1c the gap opens beyond one per cent, and at 100 kV — standard for transmission electron microscopy — true wavelength is 3.70 pm against the naive 3.88 pm. Multiply m·v by the Lorentz factor γ = 1/√(1 − v²/c²) and agreement is restored at any speed.
Why does this formula fail for photons?
A photon carries no rest mass, so m·v is meaningless for it. Photons still obey λ = h/p perfectly well — their momentum is p = E/c, which folds the relation into λ = hc/E. Matter waves and light waves come from one rule; only the route to momentum differs between them. Use this instrument for electrons, neutrons, atoms and molecules, and the energy form for anything travelling at c.
What speed gives an electron a wavelength of 0.1 nm?
About 7.27 × 10⁶ m/s. Rearranged, v = h/(m·λ), so shrinking wavelength tenfold from 0.727 nm demands ten times as much speed. In practice you reach it with voltage rather than a dial: accelerate an electron through roughly 150 volts and it arrives almost exactly at 0.1 nm, which is why low-energy electron diffraction instruments used for surface studies operate in that voltage band.
How does wavelength relate to kinetic energy?
Inversely, through a square root. Because E = p²/(2m) for a slow particle, λ = h/√(2mE) — quadruple energy and wavelength halves. Electron microscopes chase ever-higher accelerating voltages for this reason, since resolution improves as wavelength shrinks. Returns diminish, though: going from 100 kV to 400 kV buys less than a factor of two.
What mistake do people most often make here?
Typing in a molar mass. Kilograms per mole is not kilograms — divide by the Avogadro constant, 6.02214076 × 10²³, before anything enters Particle mass. Carbon-12 is 0.012 kg/mol but 1.99 × 10⁻²⁶ kg per atom, a gap of twenty-four orders of magnitude. Running a close second: supplying kinetic energy where speed belongs. Convert first with v = √(2E/m).