G. P. Thomson Experiment

G. P. Thomson’s electron-diffraction experiments sent cathode rays, now understood as electron beams, through thin films and recorded ring patterns on a photographic plate. The result complemented the Davisson–Germer Experiment: electrons behaved as material particles in production and detection, but their propagation through ordered matter showed wave-like diffraction governed by de Broglie’s relation

$$\lambda=\frac{h}{p}.$$

The experiment is historically memorable because J. J. Thomson had helped establish the electron as a particle, while his son G. P. Thomson helped establish electron diffraction. That family contrast is a useful story, but the scientific point is sharper: independent diffraction geometries agreed with the same wavelength-momentum law.

Transmission Through Thin Films

Thomson’s route used relatively fast electrons transmitted through very thin films. A typical schematic contains four parts:

• an electron source and accelerating voltage;
• collimation to make a narrow beam;
• a thin film or foil acting as the diffracting sample;
• a photographic plate or screen downstream of the film.

For an electron accelerated through a potential difference $V$, the nonrelativistic kinetic-energy estimate is

$$K=eV, \qquad p\simeq\sqrt{2m_e eV}, \qquad \lambda\simeq\frac{h}{\sqrt{2m_e eV}}.$$

Thomson’s transmission experiments used higher electron energies than the low-energy nickel-surface scattering in Davisson–Germer. At kilovolt-scale voltages, the wavelength is much smaller than an atomic spacing, so diffraction angles are small. For electrons initially nearly at rest, $K=eV$ remains valid relativistically. Using $E_{\mathrm{total}}^2=p^2c^2+m_e^2c^4$ gives

$$\lambda =\frac{h}{ \sqrt{ 2m_e eV \left(1+\frac{eV}{2m_e c^2}\right) } }.$$

For the historical conclusion, the crucial observable was not merely that electrons passed through or scattered from the film. It was that the recorded intensity had concentric rings whose radii scaled as a diffraction pattern should when the electron wavelength was changed.

In a thin-film electron-diffraction geometry, many microscopic crystal orientations produce cones of diffracted electrons. A downstream plate cuts those cones in concentric rings, whose radii encode the de Broglie wavelength and lattice spacings.

Diffraction Rings

The ring pattern is the central signature of Thomson’s setup. In a single well-oriented crystal, diffraction may appear as spots or angular maxima tied to particular reciprocal-lattice directions. In a polycrystalline film, many small crystallites have different orientations. Each set of lattice planes that satisfies a Bragg condition sends electrons onto a cone around the incident beam. The photographic plate intersects the cones as rings.

In a simplified Bragg picture,

$$2d\sin\theta_B = n\lambda, \qquad n=1,2,\ldots,$$

where $d$ is a lattice-plane spacing and $\theta_B$ is the Bragg angle. In this elastic Bragg geometry the diffracted beam is deflected by $2\theta_B$ from the incident direction. If the plate is a distance $L$ from the film, the ideal flat-screen ring radius is

$$R = L\tan(2\theta_B).$$

For small diffraction angles,

$$R \simeq 2L\theta_B \simeq \frac{nL\lambda}{d}.$$

This last expression explains the basic experimental test. Increasing the accelerating voltage increases the momentum and decreases $\lambda$, so ring radii should shrink. Changing the film changes the relevant $d$ values, so the ring pattern should change in a way connected to the sample structure.

Wave Interpretation

The wave interpretation is not that the electron is a little classical ripple of charge passing continuously through every atom in the foil. The modern statement is that the electron state has phase. A periodic arrangement of scatterers adds amplitudes coherently in selected directions.

For elastic diffraction, the reciprocal-lattice form of the condition is

$$\mathbf{k}'-\mathbf{k} = \mathbf{G}, \qquad \lvert\mathbf{k}'\rvert = \lvert\mathbf{k}\rvert,$$

where $\mathbf{k}$ and $\mathbf{k}'$ are incident and outgoing electron wave vectors and $\mathbf{G}$ is a reciprocal-lattice vector. In a randomly oriented polycrystalline film, the possible $\mathbf{G}$ directions are distributed around the beam axis, producing rings rather than a single set of spots.

The measured intensity is a probability pattern:

$$I(R) \propto \lvert A(R)\rvert^2,$$

where $I(R)$ is the intensity per detector area and $A$ represents a coherent scattering amplitude before averaging over crystallites and beam spread. Counts in an annulus of width $dR$ are proportional to $2\pi R I(R)dR$; intensity per area and counts per radial interval are different quantities. The plate records localized interactions, but the distribution of many electrons reveals the phase structure of the quantum state.

Relation to J. J. Thomson

J. J. Thomson’s cathode-ray work at the end of the nineteenth century helped identify the electron as a subatomic charged particle. G. P. Thomson’s thin-film diffraction experiments then showed that those same electrons could produce wave-diffraction patterns.

The historical contrast should not be read as a contradiction between two experiments. It shows why classical categories were too rigid. The electron is not sometimes an ordinary particle and sometimes an ordinary wave. It is a quantum object whose preparation, propagation, and detection require a formalism with states, amplitudes, and measurement probabilities.

The 1937 Nobel Prize in Physics recognized C. J. Davisson and G. P. Thomson for the experimental discovery of electron diffraction by crystals. The pairing matters: two independent experimental routes made the case much stronger than either route alone.

Evidence for Matter Waves

Thomson’s experiment strengthened the matter-wave hypothesis in several ways.

First, it used transmission through thin films rather than reflection-like scattering from a nickel surface. This reduced the possibility that the Davisson–Germer result was an idiosyncrasy of one apparatus, one target, or one angular convention.

Second, the ring geometry was visually close to X-ray powder diffraction. Electrons, which had been treated as charged material particles, were producing patterns naturally interpreted with wavelengths and lattice spacings.

Third, the wavelength scale agreed with the de Broglie relation. The electron momentum came from the accelerating voltage, while the diffraction ring radii were read from the plate. Agreement between those independent quantities tested the wavelength law.

The combined lesson of Electron Diffraction, Davisson–Germer scattering, and Thomson thin-film rings is that wave-like propagation is not confined to light. Matter beams can interfere and diffract, while still being detected as discrete localized events.

Common Mistakes

• Saying J. J. Thomson proved the electron was a particle and G. P. Thomson proved it was a wave. The better statement is that quantum electrons show particle-like detection and wave-like propagation.
• Treating rings as mysterious circular paths. The rings are the intersection of diffraction cones with a detector plane.
• Forgetting the role of many crystallite orientations in a thin film.
• Applying the nonrelativistic wavelength formula blindly at high accelerating voltages.
• Confusing diffraction from a polycrystalline film with a two-slit interference experiment. Both involve phase, but the geometries and evidence are different.
• Reading the Bragg formula as a complete dynamical theory of electron scattering.

Exercises

1. A nonrelativistic electron is accelerated through $20\,\mathrm{kV}$. Estimate its de Broglie wavelength using $\lambda(\mathrm{nm})\simeq1.226/\sqrt{V}$ with $V$ in volts.

Solution

Substitute $V=20000$:

$$\lambda \simeq \frac{1.226\,\mathrm{nm}}{\sqrt{20000}} \simeq 0.0087\,\mathrm{nm}.$$

This is much smaller than a typical lattice spacing, so the diffraction angles are small. At $20\,\mathrm{kV}$ the correction factor is $[1+20/(2\times511)]^{-1/2}\simeq0.9904$, so the relativistic wavelength is about $0.00859\,\mathrm{nm}$, nearly $1\%$ smaller. Whether that correction matters depends on the measurement precision.

2. In a thin-film diffraction setup, take $L=10\,\mathrm{cm}$, $d=0.20\,\mathrm{nm}$, and $\lambda=0.0087\,\mathrm{nm}$. Estimate the first-order small-angle ring radius.

Solution

For small angles,

$$R \simeq \frac{nL\lambda}{d}.$$

With $n=1$,

$$R \simeq \frac{(0.10\,\mathrm{m})(0.0087\,\mathrm{nm})} {0.20\,\mathrm{nm}} \simeq 4.4\times10^{-3}\,\mathrm{m}.$$

So the first ring radius is about $4.4\,\mathrm{mm}$.

3. Why does a polycrystalline thin film tend to produce rings rather than a small number of isolated diffraction spots?

Solution

A polycrystalline film contains many microscopic crystallites with different orientations. For a fixed plane spacing and wavelength, each suitable orientation can diffract into a direction satisfying the Bragg condition. Rotating those directions around the incident beam forms a cone; the detector plane cuts the cone as a ring.

4. Explain why Thomson’s experiment and the Davisson–Germer experiment were complementary evidence for matter waves.

Solution

Davisson and Germer used low-energy electrons scattered from a nickel crystal, while Thomson used transmitted electrons through thin films. The apparatus, sample geometry, and recorded pattern were different, but both connected electron momentum to diffraction through $\lambda=h/p$. Agreement in independent geometries made the matter-wave interpretation harder to dismiss as an apparatus-specific effect.

References

• C. Davisson and L. H. Germer, “Diffraction of Electrons by a Crystal of Nickel,” Physical Review 30, 705-740, 1927, DOI: 10.1103/PhysRev.30.705.
• L. de Broglie, “Recherches sur la théorie des quanta,” Annales de Physique, 10th series, 3, 22–128, 1925, DOI: 10.1051/anphys/192510030022.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• Nobel Prize Outreach, The Nobel Prize in Physics 1937.
• G. P. Thomson and A. Reid, “Diffraction of Cathode Rays by a Thin Film,” Nature 119, 890, 1927, DOI: 10.1038/119890a0.
• G. P. Thomson, “Experiments on the Diffraction of Cathode Rays,” Proceedings of the Royal Society A 117, 600-609, 1928, DOI: 10.1098/rspa.1928.0022.
• G. P. Thomson, Electronic Waves, Nobel Lecture, June 7, 1938.