F. Revolutions in Modern Physics: Quantum Mechanics and Special Relativity · Quantum MechanicsEvidence for Matter Waves: Electron Diffraction
F3.2 F2.2 F2.4 F1.1About 70 minutes
Learning goals- I can describe the Davisson-Germer and G. P. Thomson experiments and explain why they support a wave model of matter.
- I can use d sin θ = nλ together with λ = h/p to analyse electron diffraction results.
- I can predict how an electron diffraction pattern changes when the accelerating voltage changes.
- I can describe the single-electron double-slit experiment and connect it to the probability interpretation.
Words to know
electron diffraction
The spreading and interference of electron waves when they pass through or reflect from a regular structure such as a crystal.
crystal lattice
A regular, repeating arrangement of atoms in a solid; it acts like a diffraction grating for electrons, neutrons and X-rays.
lattice spacing (d)
The distance between neighbouring rows of atoms in a crystal, typically 0.1 nm to 0.4 nm.
diffraction grating equation
d sin θ = nλ: bright maxima occur where waves from neighbouring rows differ in path by a whole number of wavelengths.
Davisson-Germer experiment
The 1927 experiment that showed electrons scattered from a nickel crystal form a diffraction peak, confirming de Broglie's hypothesis.
Putting de Broglie to the test
A hypothesis is only an idea until an experiment tests it. If electrons really are waves with λ = h/(mv), they should diffract and interfere, just as light does through a grating. But a grating for 0.1 nm waves would need lines only about 0.1 nm to 0.3 nm apart, far finer than anyone can rule.
Nature provides one: a crystal. Its atoms sit in neat rows a fraction of a nanometre apart. X-ray scientists had already used crystals as gratings for X-rays since 1912. In 1927, two independent experiments did the same with electrons.
Remember from The Wave Nature of Light
For a diffraction grating with line spacing d, bright maxima occur at angles where d sin θ = nλ (n = 1, 2, 3, ...). A smaller wavelength gives smaller angles; a smaller spacing gives larger angles.
Davisson and Germer (1927)
At Bell Telephone Laboratories in New York, Clinton Davisson and Lester Germer fired a beam of slow electrons at a nickel crystal and measured how many bounced off in each direction. (After a lab accident let air into their apparatus, they heated the nickel to clean it, which turned it into a few large crystals. The scattering pattern changed dramatically, which is how they noticed something new.)
With 54 V electrons, they found a strong peak of scattered electrons at 50° from the incoming beam: a diffraction maximum, the hallmark of waves. The rows of atoms on the nickel surface are d = 0.215 nm apart, acting like a reflection grating.
Worked exampleExample 1: Checking de Broglie's prediction
Use Davisson and Germer's data (54 V electrons, surface atom spacing d = 0.215 nm, first-order maximum at θ = 50°) to find the electron wavelength from the diffraction pattern. Compare it with de Broglie's prediction.
- From diffraction (n = 1): λ = d sin θ = (0.215 nm)(sin 50°) = (0.215)(0.766) = 0.165 nm.
- From de Broglie: λ = h/√(2meV) ≈ 1.226 ÷ √54 = 0.167 nm.
- Difference: (0.167 − 0.165) ÷ 0.167 × 100% ≈ 1%.
Diffraction gives 0.165 nm; de Broglie predicts 0.167 nm. Agreement within about 1% was powerful evidence that electrons are waves.
G. P. Thomson and diffraction rings (1927)
In Scotland, George Paget Thomson sent faster electrons (thousands of volts) through very thin metal foils. A foil contains countless tiny crystals pointing in all directions, so the diffracted electrons formed rings on a photographic plate, exactly like the rings X-rays make passing through powdered crystals. The ring sizes matched λ = h/p.
Davisson and Thomson shared the 1937 Nobel Prize in Physics for discovering electron diffraction.
Many school labs have an electron diffraction tube: an electron gun fires electrons through a thin film of graphite onto a fluorescent screen. Two bright rings appear (from two different spacings of carbon atoms). Turn up the voltage, and the electrons' momentum increases, the wavelength shrinks and the rings get smaller: wave behaviour you can watch directly.
Worked exampleExample 2: Predicting the change in a ring
In an electron diffraction tube, a ring appears at a small angle θ when the voltage is 3.0 kV. The voltage is raised to 6.0 kV. By what factor does the ring's angle (and so its radius) change?
- λ ∝ 1/√V, so λ(new) = λ(old) × √(3.0/6.0) = 0.71 λ(old).
- For small angles, sin θ ≈ θ and d sin θ = λ gives θ ∝ λ.
- The radius on the screen is roughly proportional to θ.
The ring angle and radius shrink to about 0.71 (1/√2) of their original size.
Electrons one at a time
The most dramatic version is the double-slit experiment with electrons. Claus Jönsson first showed electron double-slit interference in 1961. In 1989, Akira Tonomura's team at Hitachi sent electrons through an electron "biprism" (which acts like a double slit) one at a time and recorded where each landed. Just like single photons in lesson 7, each electron arrived at one spot, and thousands of spots built up interference fringes.
So electrons are not little marbles, and they are not spread-out waves either. Like photons, they travel as waves and are detected as particles, and the wave gives the probability of where each one lands. Wave-particle duality applies to matter too.
The effect is not limited to electrons. Neutrons, helium atoms and even molecules as large as carbon-60 "buckyballs" (60 carbon atoms, shown in Vienna in 1999) have produced interference patterns.
Worked exampleExample 3: Fringe spacing for electrons
Suppose 100 V electrons pass through two slits 500 nm apart and hit a screen 1.0 m away. Estimate the spacing of the bright fringes.
- λ ≈ 1.226 ÷ √100 = 0.123 nm = 1.23 × 10⁻¹⁰ m.
- Double-slit fringe spacing (small angles): Δy = λL/d.
- Δy = (1.23 × 10⁻¹⁰ m)(1.0 m) ÷ (5.0 × 10⁻⁷ m) = 2.5 × 10⁻⁴ m.
About 0.25 mm. Tiny but measurable: real experiments magnify the pattern with electron lenses.
Just for funJ. J. Thomson won the 1906 Nobel Prize for work showing the electron behaves as a particle. His son, G. P. Thomson, shared the 1937 Nobel Prize for showing it behaves as a wave. Both were right, which must have made for interesting family dinners.
Hands-on activityA grating you already own
You need: an old CD or DVD (the shiny side), a flashlight or phone light, and a white wall. Optional, with a grown-up: a red laser pointer. Safety: never shine a laser into anyone's eyes, and don't look at its reflection from the disc.
- Hold the CD under a ceiling light and tilt it. Notice the rainbow bands: the tightly packed track spirals act as a reflection grating, just as rows of atoms do for electrons.
- With a grown-up: in a dim room, shine the laser pointer straight at the CD's shiny side from about 10 cm away, so the reflected spots land on a wall or sheet of paper about 30 cm away. You should see a bright central spot and a spot on each side.
- Measure the distance from the CD to the wall (L) and from the central spot to one side spot (y). Find the angle with tan θ = y/L.
- Use d = λ/sin θ with λ = 650 nm (red laser) to estimate the track spacing.
What to look for: CDs have track spacing of about 1.6 μm, so a red laser gives first-order spots near 24°. DVDs (about 0.74 μm) give much larger angles. Davisson and Germer did the same thing in reverse: they knew the spacing of the atoms and measured the angle to find the electrons' wavelength.
Your turn
1What did Davisson and Germer observe that showed electrons behave as waves?
A strong diffraction maximum (peak) of scattered electrons at a particular angle (50° for 54 V electrons), which only waves interfering from regular rows of atoms can produce.
2Electrons with λ = 0.123 nm strike a crystal surface with row spacing 0.215 nm. At what angle is the first-order maximum?
sin θ = λ/d = 0.123 ÷ 0.215 = 0.572; θ ≈ 35°.
3Electrons accelerated through 100 V give a first-order maximum at 30° from a crystal. Find the row spacing.
λ = 1.226 ÷ √100 = 0.1226 nm. d = λ/sin θ = 0.1226 ÷ 0.500 = 0.245 nm.
4In an electron diffraction tube, what happens to the rings when the voltage is decreased? Explain.
The rings get larger. Lower voltage → lower momentum → longer wavelength → larger diffraction angles.
5Why did Davisson and Germer need a crystal rather than a ruled grating?
Electron wavelengths (~0.1 nm) are about the size of atoms. Only the rows of atoms in a crystal are spaced closely enough to act as a grating.
6In Tonomura's single-electron experiment, what is the evidence for (a) particle behaviour and (b) wave behaviour?
(a) Each electron is detected at one spot. (b) The spots build up an interference pattern.
7Name one particle other than the electron that has been diffracted.
For example neutrons, helium atoms, or C₆₀ molecules (buckyballs).
More practice: warm-up
1In what year did Davisson and Germer, and G. P. Thomson, confirm electron diffraction?
2What acts as the diffraction grating in electron diffraction experiments?
A crystal (rows of atoms)
3Write the grating equation.
4What pattern does a beam of electrons make after passing through a thin film of many tiny crystals?
5What is the wavelength of an electron accelerated through 54 V?
More practice: on your own
1Through what voltage must electrons be accelerated to produce a first-order maximum at 45° from a crystal with d = 0.215 nm?
λ = d sin θ = 0.215 × 0.707 = 0.152 nm. √V = 1.226 ÷ 0.152 = 8.07, so V ≈ 65 V.
2A ring in an electron diffraction tube has radius 2.0 cm at 4.0 kV. Estimate its radius at 9.0 kV (small angles).
Radius ∝ λ ∝ 1/√V. r = 2.0 × √(4.0/9.0) = 2.0 × 0.667 = 1.3 cm.
3Neutrons with λ = 0.18 nm hit a crystal with d = 0.30 nm. Find the first- and second-order angles.
n = 1: sin θ = 0.18/0.30 = 0.60, θ ≈ 37°. n = 2: sin θ = 0.36/0.30 = 1.2 > 1, so no second-order maximum exists.
4Why is the 1% agreement in the Davisson-Germer analysis so convincing?
The wavelength found purely from the diffraction geometry (d and θ) matched the wavelength predicted from the electron's momentum (h/p), two completely independent routes. A coincidence at that precision is very unlikely, and repeating at other voltages gave the same agreement.
5Explain why the electron double-slit experiment done one electron at a time rules out the idea that interference comes from electrons pushing on each other.
With only one electron in the apparatus at a time, there is nothing to interact with. The pattern still builds up, so each electron's own wave must pass through both slits and interfere with itself.
Challenge
1In a Davisson-Germer-type experiment with d = 0.215 nm, the first-order peak is at 50° for 54 V. At what voltage would the first-order peak appear at 30°?
λ = 0.215 × sin 30° = 0.1075 nm. √V = 1.226 ÷ 0.1075 = 11.4, so V ≈ 130 V.
2An electron diffraction tube uses graphite with spacing 0.213 nm. The screen is 0.14 m from the graphite. At 5.0 kV, find the wavelength and the radius of the first ring. (The electron is deflected through 2θ, where 2d sin θ = λ. Use small angles.)
λ = 1.226 ÷ √5000 = 0.0173 nm. sin θ = λ/(2d) = 0.0173 ÷ 0.426 = 0.0407, θ = 2.33°, 2θ = 4.66°. r = L tan 2θ = 0.14 × 0.0815 = 0.011 m (about 1.1 cm).
3Evaluate this statement: "Electron diffraction proves electrons are waves, not particles."
Too strong. Electron diffraction proves electrons have wave properties (they diffract and interfere with λ = h/p). But electrons are also detected as single particles with definite charge and mass, landing at single points. The correct conclusion is wave-particle duality: electrons show both behaviours, depending on the experiment.
Quiz
1. The Davisson-Germer experiment showed that
2. Why are crystals used as gratings for electron waves?
3. The accelerating voltage in an electron diffraction tube is increased. The rings
4. 54 V electrons give a first-order peak at 50° from a crystal with d = 0.215 nm. The wavelength found from the diffraction is about
5. When electrons are sent through a double slit one at a time, what is observed?
6. Which pair of physicists shared the 1937 Nobel Prize for electron diffraction?
Big idea (F3.2): electron diffraction is the experimental evidence for de Broglie's matter waves. Crystals act as gratings; the measured wavelengths match λ = h/p. Electrons, like photons, show wave-particle duality.
How to help: do the CD activity together (a grown-up should handle the laser). Ask your teen to explain how this is "the same idea" as Davisson and Germer, and what is different (light vs electrons, tracks vs atoms).
Common difficulties:
- Using the voltage in the grating equation directly. First find λ from V (λ ≈ 1.226/√V nm), then use d sin θ = nλ.
- Forgetting that a higher voltage gives a shorter wavelength and smaller rings.
- Calculator in radian mode when finding sin θ.
- Claiming electrons are "really" waves. They show both wave and particle behaviour.