Davisson-Germer Experiment

A low-voltage electron gun in a vacuum chamber fires electrons at about one-tenth of one percent of the speed of light onto a chunk of polished metal. A detector sweeps an arc, counting how many electrons bounce back at each angle. If electrons are tiny particles, you would expect a broad, structureless blur — roughly isotropic at small angles, falling off smoothly as the angle grows. You would not expect sharp peaks. You would certainly not expect the peaks to line up exactly with the prediction of a formula for a wave that is supposed to describe the electron.

But when Davisson and Germer ran this experiment in 1927 — partly by accident, following a vacuum failure that accidentally annealed their nickel target into a neat crystalline surface — that is exactly what they found. A sharp peak. At an angle that matched, to within the precision of their measurements, the prediction of the de Broglie formula \lambda = h/p. The experiment was one of the first direct, quantitative confirmations of the wave nature of matter. The article on the de Broglie hypothesis shows where the prediction came from. This article shows where the experiment found it.

The setup — what was actually in the chamber

A simple kinematic setup, but everything about the geometry matters.

• An electron gun: a heated filament emits electrons thermally, and an anode held at potential V (a few tens of volts) accelerates them to kinetic energy K = eV. The beam emerges through a hole in the anode as a roughly collimated stream of monoenergetic electrons.
• A crystalline nickel target: a single crystal of nickel, cut and polished so that a specific crystal face — the (111) face — is exposed to the beam. The atoms on this face form a regular hexagonal array with spacing d = 2.15 Å.
• A moveable detector (Faraday cup) at a fixed distance from the target that can rotate through angles \phi measured from the incident beam direction, counting the number of electrons scattered into a small solid angle around each angle.
• A high-vacuum chamber to keep the electron mean free path much longer than the apparatus dimensions — any residual gas would scatter the beam before it reached the target.

The experiment measures the scattered intensity I(\phi) as a function of angle, with the accelerating voltage V held fixed. Then V is changed, and the curve I(\phi) is remeasured.

At V = 54 V the scattered intensity curve shows a clear peak at \phi = 50°. At lower or higher voltages the peak sits at different angles. At still other voltages — notably those where \lambda does not match an integer multiple of the surface spacing — the peak disappears entirely. This is diffraction behaviour.

De Broglie's prediction — a one-line recap

From the de Broglie hypothesis, every particle has an associated wavelength

For a non-relativistic electron accelerated through potential V, kinetic energy is K = eV and momentum is p = \sqrt{2 m_e K} = \sqrt{2 m_e e V}. Substituting:

Plug in h = 6.626\times 10^{-34} J·s, m_e = 9.109\times10^{-31} kg, e = 1.602\times10^{-19} C:

Converting to ångströms (1 Å = 10^{-10} m):

Why: this is the operational form of de Broglie's formula, adapted to the lab. Every electron diffraction experiment uses it, and every value comes out comparable to inter-atomic spacings in solids. That is the miracle — atoms are about 10^{-10} m apart, and accessible lab voltages give electron wavelengths of about 10^{-10} m. The two numbers were built to match by nature.

For V = 54 V: \lambda = 12.27/\sqrt{54} = 12.27/7.35 = 1.669 Å.

Why the nickel surface acts as a grating — the Bragg-like condition

A crystal is a regular three-dimensional lattice of atoms. When a wave hits it, the wave scatters off every atom, and the scattered wavelets interfere. For most directions the wavelets are out of phase and cancel; for special directions they add in phase, producing a peak in the scattered intensity. This is identical to the picture you built in young's double slit experiment and diffraction grating, except with atoms as the slits.

For the Davisson-Germer geometry, the relevant atoms are the ones on the nickel surface — slow electrons do not penetrate deep into the metal. The surface atoms form a periodic array with spacing d = 2.15 Å (the nickel (111) face). The incident electrons come in at normal incidence. The detector sits at an angle \phi from the normal.

The two rays leaving the surface from neighbouring atoms are parallel (they both head toward the distant detector). The rightmost ray has to travel an extra distance equal to d \sin\phi before reaching the detector, relative to the leftmost ray.

Derivation of the diffraction condition

Step 1. Both incident rays arrive with the same phase (they are wavefronts of the same de Broglie wave coming down normally).

Why: the electron source is a long way away, so the incoming wave has plane wavefronts parallel to the surface. Two points on the surface are hit at the same instant.

Step 2. After scattering, the rays head to the detector in parallel directions. The ray from the right atom reaches the detector after travelling an extra path length.

Why: drop a perpendicular from the left atom onto the outgoing ray from the right atom. The right-triangle you get has hypotenuse d and one side d\sin\phi — that is the path difference.

Step 3. Constructive interference occurs when the path difference is a whole number of wavelengths.

Why: two rays meeting in phase must differ by an integer multiple of \lambda. n = 0 would be the forward direction (trivial); n = 1 is the first-order peak; higher n gives peaks at larger angles.

Step 4. The observed Davisson-Germer peak at V = 54 V is first-order, so n = 1:

With d = 2.15 Å and \phi = 50°:

Why: this is the wavelength the experiment says the electrons have, read off from the diffraction peak — with no reference to de Broglie's formula. It is a direct measurement of \lambda.

Step 5. Compare with de Broglie's prediction.

The two numbers agree to 1.3%.

Why: this is the moment the experiment clinches it. Two independent computations — one from the diffraction angle, one from the acceleration voltage and the de Broglie formula — agree to better than 2 percent. No classical particle picture predicts a peak at any angle, let alone at the angle de Broglie's wave formula demands. Electrons are waves.

Watch the peak move — an interactive tour of \lambda \propto 1/\sqrt{V}

The cleanest way to internalise Davisson-Germer is to watch the peak angle change as you dial the voltage. Higher V means higher momentum means shorter \lambda. Shorter \lambda means \sin\phi = \lambda/d is smaller. So the peak sits at a smaller angle.

Every point on that curve is a prediction. Davisson and Germer verified several of them — enough to rule out any interpretation except the wave one. Their paper reported peaks at multiple voltages, all sitting on this curve to within experimental error.

Comparison with X-ray diffraction

Long before matter waves were a thing, X-ray crystallography had been doing the same geometric trick with X-rays. Von Laue in 1912 and the Braggs in 1913 showed that shining X-rays on a crystal produced sharp diffraction peaks obeying

where \theta is the angle between the incoming ray and the crystal plane (not the normal), and d is the interplanar spacing. The difference in geometry — planes vs surfaces, angle from plane vs from normal — changes the factor of 2 and swaps the sine of an angle with the sine of the complementary angle, but the underlying physics is the same: waves scattering off a regular lattice produce sharp angular peaks.

The significance of Davisson-Germer is that the same crystal gives the same kind of peak whether you aim it with an X-ray of wavelength \sim 1 Å or an electron of wavelength \sim 1 Å. The target does not care what kind of wave you send at it; it only cares about the wavelength. That is the cleanest possible demonstration that the electron, when it is propagating, is a wave in every sense that X-rays are.

This equivalence is what makes electron microscopes possible. In an electron microscope the de Broglie wavelength of a 100 keV electron is about 0.04 Å — a hundred times smaller than visible light's wavelength and small enough to resolve individual atoms. The Transmission Electron Microscopes (TEMs) at AIIMS and at major IIT materials-science labs work by shining a monochromatic electron wave through a sample and collecting its diffracted image. The physics under the hood is direct descendant of Davisson-Germer.

Worked examples

Common confusions

• "The electron really did travel through one specific path and bounced off one specific atom." That is the classical particle picture and it cannot produce the sharp peaks that are observed. A lone particle would give a featureless angular distribution. The sharp peak at \phi = 50° is a property of the wave associated with the electron, which interacts with the entire array of surface atoms simultaneously.
• "Diffraction means each electron splits into pieces." No. Like in the double-slit experiment, each electron arrives at the detector as a single discrete event; but the probability of arriving at each angle is governed by the wave interference of all possible paths. Over many electrons, the probability distribution — which is the observed count rate — shows the diffraction peaks.
• "The Bragg condition 2d\sin\theta = n\lambda should apply." That form applies to X-rays scattering off parallel planes inside a crystal, with \theta measured from the plane. For the Davisson-Germer surface geometry, the relevant scatterers are atoms in a surface row, and the angle \phi is measured from the surface normal. Different geometry, different formula — but same physics of wave interference from a regular array.
• "The formula should include relativistic corrections." At V = 54 V the electron's kinetic energy is 54 eV, which is tiny compared with its rest energy of 511 keV. Non-relativistic mechanics is accurate to better than one part in 10^4. Relativistic corrections matter only at voltages above a few kV — and there is a modified formula, \lambda = h/\sqrt{2 m_e e V (1 + eV/2 m_e c^2)}, for those cases.
• "The experiment only works for electrons." Not at all. Neutrons, atoms, and even molecules (buckminsterfullerene, C_{60}) have been diffracted off crystals by the same basic method. Every matter wave obeys \lambda = h/p. The experimental difficulty rises for heavier particles only because their wavelengths shrink — a C_{60} molecule at thermal velocities has \lambda \sim 5 pm, which demands diffraction gratings finer than atomic spacing, and specific techniques (nano-fabricated gratings, inertia-reduced beams) were developed to make it work in the 1990s.

Why this mattered for quantum mechanics

The de Broglie hypothesis was, at the start, a bold guess in a PhD thesis. It had beautiful symmetry with the photon case — if light of wavelength \lambda has momentum h/\lambda, why shouldn't matter of momentum p have wavelength h/p? — but symmetry is not evidence. Several physicists at the time, including de Broglie's thesis committee, were unsure what to make of it.

Davisson-Germer made matter waves experimentally undeniable. The very same electron that had been treated, for twenty-five years since J.J. Thomson's experiments, as a classical charged particle — a tiny billiard ball with mass, charge, and velocity — turned out to obey a wave equation when you asked it to scatter off a crystal. Three months after Davisson-Germer, George Thomson (J.J.'s son!) published a complementary experiment: he fired higher-energy electrons through a thin gold foil and saw a full set of Debye-Scherrer rings — the same ring pattern X-rays give when they pass through a polycrystalline sample. Thomson the elder had proved the electron was a particle (1897); Thomson the younger proved it was a wave (1927). Both father and son received Nobel prizes. The physics was right both times.

This was the experimental foundation on which Schrödinger's wave equation (1926) could be trusted. Before Davisson-Germer, Schrödinger's equation was a brilliant mathematical guess whose physical meaning was disputed. After Davisson-Germer, the wave function of an electron was an object the experiments measured directly.

Going deeper

Where this leads next

• de Broglie hypothesis — the theoretical prediction Davisson-Germer confirmed: every particle has wavelength h/p.
• Photoelectric effect — the wave-particle duality of light; electrons as particles absorbing discrete photons.
• Photons and wave-particle duality of light — the companion experimental story for light: photons carry momentum and interfere with themselves.
• Bohr's model of the hydrogen atom — the first quantum atomic model, whose quantisation of angular momentum L = n\hbar finds a natural interpretation as standing matter waves in a circular orbit.
• Young's double slit experiment — the template double-slit interference argument, now applicable to electrons, neutrons, and atoms.
• Diffraction grating — the wave-optics analog of the crystal grating Davisson and Germer used, with the same fundamental formula d\sin\phi = n\lambda.