Davisson–Germer: Electron Diffraction

Swing the detector at 54 V. Start from the 1927 setup and drag the detector angle from 10° out to 90°. The reading falls away from the beam, then climbs to a bump near 50° and drops again. Watch the amber dots: near 50° a steady trickle of electrons is heading into the aperture; at 35° or 65° almost none are.

Park the detector at 50° and sweep the voltage. Come down to 40 V and the bump has slid outward, past the detector; go up to 65 V and it has slid inward. It sits on the detector at about 54 V. Keep raising the voltage: near 220 V a second, weaker bump arrives at the same angle — the second order, at four times the voltage.

Drop to 30 V. The bump is gone from every angle. Not weaker — absent. Below 32 V the electron’s wavelength is longer than the 0.216 nm between rows of nickel atoms, and a grating cannot diffract a wave longer than its own spacing anywhere but straight back.

A crystal is a ruler with marks a fraction of a nanometre apart. Anything that diffracts from it has a wavelength, and the angle of the peak reads that wavelength off the ruler: D sin θ = λ. At 54 V the peak at 50° reads 0.165 nm.

In 1924 de Broglie had proposed that a particle of momentum p carries a wave of length h/p. An electron that has fallen through 54 volts has a definite momentum, and the formula gives 0.167 nm — with no adjustable constant in it, only h, the electron’s mass and charge, and the voltage on the gun. The crystal measured the wavelength of an electron, and it came out where de Broglie said it would to about one percent.

That is the whole result. The electrons were not asked to do anything wave-like; the crystal simply treated them the way it treats X-rays, and the number matched. The same year G. P. Thomson fired faster electrons through thin metal foils and photographed the rings X-rays would make. Thomson’s father had won the Nobel prize for showing the electron is a particle; the son shared one with Davisson for showing it is also a wave.

λ = h / √(2 m e V). Kinetic energy eV, momentum √(2m·eV), wavelength h over that. In nanometres and volts it collapses to λ ≈ 1.226/√V, which is worth memorising: 100 V is 0.123 nm, about the size of an atom, and that is why electron microscopes see atoms.

D sin θ = nλ. The rows of atoms on the (111) surface, D = 0.216 nm apart, act as a reflection grating for electrons that only get a few layers deep. Textbooks often quote the same peak as Bragg’s law, 2d sin φ = λ with d = 0.091 nm and φ = 65°. Those are not a different nickel: d = D sin(θ/2) is the spacing of the planes that bisect the incoming and outgoing beams, and φ = 90° − θ/2 is the glancing angle on them. The “Bragg form” readout above converts one into the other, and the two are the same equation.

The bottom panel is what X-rays of the electron’s wavelength do to nickel. It looks nothing like the top one: a comb of sharp lines instead of one broad bump. X-rays pass through microns of metal, so they diffract from the full three-dimensional lattice, every family of planes (111), (200), (220)… adding its own line at 2d sin θ = λ. A 54 eV electron is stopped within a couple of atomic layers and sees only the surface rows — a one-dimensional grating a handful of rows wide, hence one peak, and a wide one.

The comparison is the point of the panel, not a claim that the two pictures should match. The electrons and the X-rays agree about the size of the ruler; they disagree about how much of it they can see.

Clinton Davisson and Lester Germer were not looking for electron waves. At Bell Labs they were mapping how electrons bounce off nickel, and in 1925 a liquid-air bottle burst in the lab and oxidised their target. They cleaned it by baking it at high temperature, which turned the fine-grained polycrystalline nickel into a handful of large crystals — and the smooth scattering curves they had been recording sprouted bumps.

It took Davisson a trip to the 1926 Oxford meeting of the British Association, where Born was talking about de Broglie’s waves and Schrödinger’s equation, to realise what the bumps were. He and Germer went back, cut a single crystal to expose the (111) face, fired electrons straight at it and swung a Faraday cup around the beam — the arrangement drawn above — and found the 54 V / 50° peak in early 1927.

The geometry, the wavelength formula, the row spacing and the grating condition are the real experiment’s, and the numbers are checked inlib/viz/electronDiffraction.test.tsagainst de Broglie’s 0.167 nm, Davisson and Germer’s 50°, 54 V and 2.15 Å, and the published Cu Kα powder card for nickel.

The shape of the intensity curve is a model: a finite grating of a few rows under a falling single-atom scattering amplitude, plus a smooth background. It puts the peak in the right place with the right width; it does not reproduce the finer structure of the real polar plots, which need multiple scattering and the crystal’s inner potential. That inner potential is also why the model peak sits at 50.7° while the measured one was at 50°: the electron is refracted as it enters the metal, and the 1927 paper spends a section on exactly that.

The X-ray panel is a powder pattern — many small crystals at every orientation — not the single crystal above, because that is the form in which nickel’s X-ray lines are tabulated and checkable. Line positions are exact; the relative heights leave out thermal motion and absorption and are only approximate.