De Broglie Waves and Electron Diffraction

By 1924 radiation had a settled double life. Interference and diffraction made light a wave; the photoelectric effect and the Compton effect made it a stream of photons carrying energy $E = h f$ and momentum $p = h / λ$ . Louis de Broglie, in his doctoral thesis, proposed that the symmetry runs both ways: if a wave can behave like a particle, a particle should behave like a wave. Matter, he argued, carries a wave whose wavelength is fixed by the particle's momentum through the same relation that holds for light.¹

The de Broglie relations

De Broglie assigned to a particle of total energy $E$ and momentum $p$ a wave of frequency and wavelength

f = \frac{E}{h}, λ = \frac{h}{p} .

For a photon the two relations are not new. A photon has zero rest energy, so its energy and momentum satisfy $E = p c$ , and Einstein's quantization $E = h f$ gives

E = p c = h f = \frac{h c}{λ} ⟹ λ = \frac{h}{p},

which is the second de Broglie relation. The content of the hypothesis is that the identical relation holds for an electron, a proton, a baseball — anything with momentum. De Broglie confirmed by a relativistic argument that $λ = h / p$ transfers unchanged from massless to massive particles.

The frequency relation $f = E / h$ resists direct experimental test, because a free particle's total energy $E$ includes its rest energy and the zero of energy is a matter of convention. Wavelength, tied to momentum, is the measurable quantity, and every experiment below measures $λ$ .

Matter waves and the Bohr condition

De Broglie's first success was to make the Bohr model's quantization rule look inevitable. Bohr had postulated, without deeper reason, that the electron's orbital angular momentum is quantized in units of $ℏ$ :

m v r = n ℏ = \frac{nh}{2 π}, n = 1, 2, 3, \dots

Rewrite this with $p = m v$ and $ℏ = h /2 π$ :

n \frac{h}{2 π} = p r ⟹ n \frac{h}{p} = 2 π r ⟹ nλ = 2 π r .

The quantization condition says that an integer number of de Broglie wavelengths fits around the circumference of the orbit. A stable orbit is a standing wave: the electron wave joins onto itself in phase after one loop. Any non-integer number of wavelengths interferes destructively with itself on successive loops and cancels. Bohr's arbitrary rule becomes the resonance condition for a wave confined to a ring.¹

A stable Bohr orbit holds an integer number of de Broglie wavelengths (here n = 3) around its circumference; the wave closes on itself in phase.

The scale of matter waves

The wave properties of light went unnoticed until slits comparable to the wavelength were available; the same holds for matter. Diffraction spreads a wave of wavelength $λ$ through a slit of width $a$ into angles $θ$ with $sin θ \sim λ / a$ . When $λ ≪ a$ the spreading is unobservable and the wave travels in straight rays. Because $h$ is so small, $λ = h / p$ is minuscule for any macroscopic object, so its diffraction is undetectable.

An electron is different. Give it a modest kinetic energy and its wavelength lands near atomic dimensions, where crystals supply a natural diffraction grating.

Wavelength of an accelerated electron

Accelerate an electron from rest through a potential difference $V_{0}$ . It gains kinetic energy $E_{k} = e V_{0}$ , and for $e V_{0} ≪ m c^{2}$ the motion is nonrelativistic, so

E_{k} = \frac{p ^{2}}{2 m} = e V_{0} ⟹ p = 2 m e V_{0} .

Substituting into $λ = h / p$ and writing it through the electron's rest energy $m c^{2} = 0.511 \times 1 0^{6} eV$ and $h c = 1.24 \times 1 0^{3} eV\cdotnm$ ,

λ = \frac{h}{p} = \frac{h c}{2 ( m c ^{2} ) e V _{0}} = \frac{1.226}{V _{0}} nm (V_{0} in volts, e V_{0} ≪ m c^{2}) .

The coincidence of the electron wavelength with the crystal-plane spacing $d \approx 0.1$ – $0.2 nm$ is the whole reason the hypothesis could be tested in the 1920s. A crystal is a three-dimensional diffraction grating already used to diffract X-rays of the same wavelength.

The Davisson-Germer experiment

C. J. Davisson and L. H. Germer confirmed de Broglie's relation directly in 1927. They fired low-energy electrons from an electron gun at a single crystal of nickel and measured the scattered intensity as a function of the angle $ϕ$ between the incident beam and the detector.

The Davisson-Germer apparatus. Electrons from a gun strike a nickel single crystal; a movable detector records scattered intensity versus angle.

For $54 eV$ electrons the intensity peaked sharply at a scattering angle $ϕ = 5 0^{\circ}$ . To explain the peak, treat the regularly spaced atomic planes as Bragg reflectors. Constructive interference between waves reflected from successive planes of spacing $d$ requires the Bragg condition, with $θ$ the grazing angle to the planes:

nλ = 2 d sin θ .

Bragg reflection. Waves scattered from successive atomic planes of spacing d reinforce when their path difference is a whole number of wavelengths.

The geometry of the crystal ties the plane spacing $d$ to the surface atomic spacing $D$ by $d = D sin α$ , where $α$ is the angle the planes make with the surface. Working the trigonometry through, the Bragg condition becomes a relation in the measured scattering angle $ϕ = 2 α$ :

nλ = 2 d sin θ = 2 D sin α cos α = D sin 2 α = D sin ϕ .

For nickel, X-ray diffraction gives $D = 0.215 nm$ . The $n = 1$ peak at $ϕ = 5 0^{\circ}$ then implies

λ = D sin ϕ = (0.215 nm) sin 5 0^{\circ} = 0.165 nm .

The de Broglie prediction for $54 eV$ electrons is $λ = 1.226/ 54 nm = 0.167 nm$ . The two agree to about one percent, the small deficit coming from refraction of the electron wave as it enters the metal, which shortens $λ$ slightly inside the crystal.

Scattered intensity versus detector angle for 54 eV electrons. The peak near 50 degrees matches the Bragg condition for the de Broglie wavelength.

Davisson and Germer then held the detector fixed and varied the accelerating voltage. Because $λ \propto V_{0}^{- 1/2}$ , plotting intensity against $1/ λ \propto V_{0}^{1/2}$ produced a series of equally spaced peaks at successive integers $n$ , and the measured wavelengths tracked $λ = 1.226/ V_{0} nm$ across the whole range up to $400 eV$ .

Measured electron wavelength versus one over the square root of the accelerating voltage. The data fall on the de Broglie line of slope 1.226 nm.

Thomson diffraction and other particles

In the same year G. P. Thomson (son of J. J. Thomson, who had shown the electron to be a particle) demonstrated the wave nature of $10$ – $40 keV$ electrons in transmission. Passing the beam through a thin polycrystalline metal foil produced concentric diffraction rings, exactly like the Laue rings from X-rays through the same foil. The many randomly oriented crystallites each satisfy the Bragg condition at some grazing angle $θ$ , and each scatters into a cone of half-angle $2 θ$ about the beam, so the pattern is a set of rings.

Electron transmission through a polycrystalline foil gives concentric rings, each ring a Bragg cone from crystallites at one orientation.

The hypothesis is not confined to charged particles. Because neutral atoms and molecules cannot be accelerated electrostatically, their wavelengths were harder to reach, but Stern and Estermann diffracted thermal beams of helium atoms and hydrogen molecules from a lithium fluoride crystal in 1930, finding $λ \approx 0.1 nm$ as predicted from their thermal energy of about $0.03 eV$ . Diffraction of protons and neutrons followed. In every case the measured wavelength matched $λ = h / p$ .

Particle	Typical energy	de Broglie $λ$	Diffractor
Electron (Davisson-Germer)	$54 eV$	$0.167 nm$	Ni single crystal
Electron (Thomson)	$10$ – $40 keV$	$\sim 0.05 nm$	metal foil
He atom (Stern-Estermann)	$0.03 eV$	$\sim 0.10 nm$	LiF surface
Thermal neutron	$0.06 eV$	$0.12 nm$	polycrystalline Cu

Relativistic wavelengths

At high energy the nonrelativistic $λ = 1.226/ V_{0} nm$ fails, and one must start from the exact energy-momentum relation of special relativity,

E^{2} = (p c)^{2} + (m c^{2})^{2} .

Writing $E = E_{0} + E_{k}$ with rest energy $E_{0} = m c^{2}$ and solving for $p$ ,

p = \frac{1}{c} 2 E_{0} E_{k} + E_{k}^{2} ⟹ λ = \frac{h}{p} = \frac{h c}{2 E _{0} E _{k} + E _{k}^{2}} .

Dividing through by $E_{0}$ expresses the result in units of the particle's Compton wavelength $λ_{c} = h / m c$ , giving a single curve valid at any energy:

\frac{λ}{λ _{c}} = \frac{1}{2 ( E _{k} / E _{0} ) + ( E _{k} / E _{0} ) ^{2}} .

For $E_{k} ≪ E_{0}$ this reduces to the nonrelativistic $λ = h / 2 m E_{k}$ ; for $E_{k} ≫ E_{0}$ it approaches $λ \approx h c / E_{k}$ , the photon-like limit. A $150 GeV$ cosmic-ray proton, with $E_{0} = 0.938 GeV$ and $λ_{c} = 1.32 fm$ , has $E_{k} / E_{0} = 160$ , giving $λ / λ_{c} \approx 6 \times 1 0^{- 3}$ and $λ \approx 8 \times 1 0^{- 3} fm$ — short enough to probe the interior of a nucleus.

The de Broglie hypothesis leaves one question open. A single wave of definite wavelength extends over all space, yet a particle is found at one place. The next lesson resolves this by superposing waves into a localized packet and asking what, physically, the wave amplitude represents.

Tipler & Llewellyn, §5-1 — the de Broglie relations $f = E / h$ , $λ = h / p$ , their derivation from the photon relations, and the reading of Bohr's angular-momentum quantization as the standing-wave condition $nλ = 2 π r$ . ↩ ↩²