Physics · Atomic and quantum physics · Grade 11

De Broglie waves

An electron is a particle, so why does it pass through a crystal and draw wave-like rings on the screen?

The situation

A biologist cannot see viruses in an ordinary microscope: a virus is about 100 nm across, and a microscope cannot resolve details smaller than about half the wavelength of light (≈ 200 nm). In an electron microscope, though, the virus is seen clearly — because an accelerated electron is also a wave, with a wavelength smaller than an atom. But an electron is a particle with mass and charge! If you fire electrons one at a time, random dots appear on the screen, yet once thousands of dots have built up, the interference fringes typical of waves emerge. How can a particle be a wave, and what does its wavelength depend on?

Topic

Wave properties of particles. Electron microscope and a double slit with electrons passing one at a time

Key ideas

λ = h/p
Electron diffraction
Uncertainty principle

Formulas

λ = \frac{h}{p} = \frac{h}{\sqrt{2 m e U}} \approx \frac{1.226}{\sqrt{U [V]}} nm

De Broglie wavelength: for an electron accelerated through a voltage $U$ (at tens of kV a relativistic correction is added).

2 d \sin θ = n λ, D \approx \frac{2 L λ}{d}

The Wulff–Bragg condition and the ring diameter on the electron diffraction pattern ( $L$ — distance from the graphite to the screen, $d$ — spacing between planes).

a \sin φ = n λ

Davisson–Germer: the rows of atoms on the crystal surface ( $a = 0.215$ nm) act like a diffraction grating.

Δ x_{scr} = \frac{λ L}{d}

Double slit: the spacing of the interference fringes on the screen ( $d$ — slit separation, $L$ — distance to the screen).

Δ x Δ p_{x} \geq \frac{ℏ}{2}, v_{gr} = \frac{d ω}{d k} = v, v_{f} = \frac{ω}{k} = \frac{v}{2}

The Heisenberg uncertainty relation; the group and phase velocities of a free-electron wave packet.

Theory

Duality and the de Broglie hypothesis

In interference and diffraction light behaves as a wave, but in the photoelectric and Compton effects it behaves as a particle (a photon) — this is wave–particle duality. In 1924 Louis de Broglie proposed the opposite idea: every particle also has wave properties, $λ = h / p$ . For an electron accelerated through 100 V, $λ = 0.123$ nm — of the same order as the spacing of atoms in a crystal, so a crystal acts as a diffraction grating for electrons. For a 60 g tennis ball moving at 1 m/s, however, $λ \approx 10^{- 32}$ m — no instrument could detect that: for large objects quantum physics passes over into classical physics (the correspondence principle).

Thomson and Tartakovsky: rings

In an electron diffraction tube, electrons pass through a thin layer of graphite. Graphite consists of tiny crystallites pointing in all directions. In any crystallite whose planes are at an angle satisfying the Wulff–Bragg condition $2 d \sin θ = n λ$ , the electrons are deflected through $2 θ$ ; all the orientations together give a cone, and on the screen a ring ( $d_{1} = 0.213$ nm, $d_{2} = 0.123$ nm). If $U$ increases, $λ$ decreases and the ring diameters shrink. In 1927–1928 G. P. Thomson and, independently of him, P. S. Tartakovsky passed electrons through thin polycrystalline metal films and obtained exactly the same rings. At the same $λ$ , X-rays (the Debye–Scherrer method) and neutrons give exactly the same rings. In a three-dimensional crystal all three Laue conditions must be satisfied at once; waves reflected from some planes cancel because of how the atoms are arranged (structure factor $F = 0$ ), so not every $d$ gives a ring.

The Davisson–Germer experiment

C. Davisson and L. Germer directed electrons at a nickel crystal at normal incidence and measured the current of reflected electrons as a function of angle. At 54 eV a sharp peak appeared in the direction $φ \approx 50 °$ . The spacing of the atomic rows on the surface is $a = 0.215$ nm, so $λ = a \sin 50 ° \approx 0.165$ nm, while the de Broglie formula gives $h / p = 0.167$ nm. The peak is strong only within a certain range of energies: waves reflected from deeper layers must also add in phase (the second Laue condition). Model: 6 surface rows and 8 layers stacked on top of each other ( $c = 0.203$ nm, the amplitude falls by a factor of 0.55 in each layer); the acceleration of the electron inside the crystal (the inner potential) is ignored, the background is approximate, and the current is in relative units.

Statistical interpretation: electrons one at a time

In an electron microscope the beam can be weakened so much that only one electron is in the column at a time. Each electron leaves a single dot on the screen — like a particle — and where it lands cannot be predicted in advance. Yet thousands of dots draw out the interference pattern. Born's interpretation: $| ψ (x) |^{2} d x$ is the probability of finding the electron at that place; the electron passes through both slits as a wave. If we measure which slit each electron passes through, the interference disappears: the pattern becomes the sum of two single-slit patterns — measurement changes the state. This is Bohr's complementarity principle: wave and particle properties complement each other, but both cannot be seen together in one experiment.

The uncertainty relation and the wave packet

An electron at a definite place is not a single sine wave but a superposition of waves with different $k$ , that is, a wave packet. The narrower the packet, the wider the range of $k$ needed: $Δ x Δ k \geq 1 / 2$ , that is, $Δ x Δ p \geq ℏ / 2$ (Heisenberg). The same happens at a single slit: the slit fixes the electron's position to within $Δ x \approx a$ , and at once its transverse momentum is spread over $Δ p_{x} \approx h / a$ — narrow the slit and the spot on the screen widens. The centre of the packet moves at the group velocity $v_{g r} = d ω / d k = p / m$ , that is, at the particle's own speed, while the wave crests shift more slowly at the phase velocity $v_{f} = ω / k = v / 2$ . A free packet spreads with time: $σ (t) = σ_{0} \sqrt{1 + {(ℏ t / 2 m σ_{0}^{2})}^{2}}$ , while the momentum distribution does not change.

About the electron microscope and the model

The electron microscope was built in 1931 by E. Ruska and M. Knoll: magnetic lenses with copper coils focus electrons just as glass lenses focus light. At 100–300 kV, $λ$ is a few picometres, and the resolution, because of lens aberrations, is about 0.05–0.2 nm. Model: the electron diffraction tube has $L = 13.5$ cm and a spherical screen; for the double slit, the slit-to-screen distance is taken as $L = 1$ m and a ±60 µm section of the screen is shown magnified (in real experiments the pattern is magnified by electron lenses). The flight of the electrons in the column is slowed down enormously; on the wave-packet track, 1 s = 10 fs. The brightness of the rings and fringes is relative.

Did you know?

Joseph John Thomson received the Nobel Prize in 1906 for his research on the conduction of electricity through gases — work that showed the electron to be a particle. His son George Paget Thomson was awarded the Nobel Prize in 1937, together with C. Davisson, for discovering the diffraction of electrons — that is, the wave nature of the electron.
Louis de Broglie set out the idea that particles have wave properties in his doctoral thesis in 1924; when experiments confirmed it, he received the Nobel Prize in Physics in 1929.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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