Low-Energy Electron Diffraction Revision Notes | A Level AQA Physics

1. The Puzzle: Why Would Electrons Diffract?

Picture a beam of electrons fired at a metal crystal. Classical physics says electrons are particles — tiny charged balls — so you would expect them to bounce off the surface in a predictable, particle-like way. What actually happens is far stranger: the electrons produce a diffraction pattern, the hallmark of wave behaviour.

This is the central mystery that low-energy electron diffraction (LEED) both demonstrates and explains. To make sense of it, you need to connect two ideas: the de Broglie hypothesis (that moving particles have an associated wavelength) and the condition for observable diffraction (that the wavelength must be comparable to the gap size).

2. The de Broglie Wavelength: Particles as Waves

In 1924, Louis de Broglie proposed that any moving particle has an associated wavelength. The faster the particle moves — and therefore the greater its momentum — the shorter its wavelength. The relationship is captured in one compact equation.

For an electron accelerated through a potential difference of around 50–150 V, the resulting de Broglie wavelength falls in the range of 0.1–0.3 nm. This is precisely the same order of magnitude as the spacing between atoms in a crystal lattice — which is why low-energy electrons are so effective at producing diffraction patterns from crystal surfaces.

Notice the key implication: wavelength and speed are inversely related. Increase the electron's speed and its momentum $p = mv$ increases, so $\lambda = h/p$ decreases. This inverse relationship is the engine behind everything that follows in LEED.

3. The LEED Experiment: What Happens and What You See

Picture the apparatus: a gun fires a narrow beam of electrons at a clean, flat crystal surface inside a vacuum chamber. The electrons have been accelerated through a carefully controlled potential difference, giving them a precise, known kinetic energy — and therefore a precise momentum and de Broglie wavelength.

When the electrons strike the crystal surface, the regularly spaced rows of atoms act like a diffraction grating. The electrons scatter from these atomic rows and, because their wavelength is comparable to the atomic spacing, they interfere constructively in certain directions and destructively in others. A fluorescent screen placed around the crystal then lights up wherever electrons arrive — revealing a pattern of bright spots.

Those bright spots are the diffraction pattern. Their positions are determined by the atomic spacing of the crystal and the wavelength of the electrons. The pattern is not random — it has the same symmetry as the crystal surface itself, which is why LEED is used in surface science to determine crystal structure.

4. The Effect of Changing Electron Speed on the Diffraction Pattern

This is the heart of the AQA specification requirement, and it is where the de Broglie equation does real explanatory work. When you increase the accelerating voltage, you give the electrons more kinetic energy, which means greater speed and greater momentum. From $\lambda = h/mv$, a larger $mv$ gives a smaller $\lambda$.

Recall from your earlier study of wave diffraction that the angle of diffraction depends on the ratio of wavelength to gap size. When wavelength decreases relative to the atomic spacing, the diffraction angles decrease — the bright spots on the screen move closer together, converging towards the centre. The pattern becomes more compact.

Conversely, if you reduce the accelerating voltage, the electrons slow down, their momentum decreases, and their de Broglie wavelength increases. The diffraction angles increase, and the bright spots spread further apart on the screen. The pattern expands outward.

The atomic spacing of the crystal does not change — it is fixed by the crystal structure. So the only variable controlling the diffraction pattern is the electron wavelength, which you control via the accelerating voltage. This clean, controllable relationship is what makes LEED such compelling evidence for the de Broglie hypothesis.

5. Worked Calculation: Finding the de Broglie Wavelength

Exam questions on LEED often ask you to calculate the de Broglie wavelength of an electron accelerated through a given potential difference. The route involves linking the accelerating voltage to kinetic energy, then to momentum, and finally to wavelength. Here is a full walkthrough.

Worked example: Calculating the de Broglie Wavelength of an Accelerated Electron

An electron is accelerated from rest through a potential difference of 100 V. Calculate the de Broglie wavelength of the electron. (Mass of electron $m_e = 9.11 \times 10^{-31}$ kg, charge of electron $e = 1.60 \times 10^{-19}$ C, Planck constant $h = 6.63 \times 10^{-34}$ J s)

1. $E_k = eV = 1.60 \times 10^{-19} \times 100 = 1.60 \times 10^{-17}$ J — When a charge $e$ moves through potential difference $V$, it gains kinetic energy $E_k = eV$. [M1] for using this relationship correctly.
2. $E_k = \frac{p^2}{2m}$, so $p = \sqrt{2m E_k}$ — This rearrangement avoids having to find speed separately. It links energy directly to momentum, which is what the de Broglie equation needs. [M1] for correct rearrangement.
3. $p = \sqrt{2 \times 9.11 \times 10^{-31} \times 1.60 \times 10^{-17}}$ $= \sqrt{2.915 \times 10^{-47}}$ $= 5.40 \times 10^{-24}$ kg m s$^{-1}$ — Check: $2 \times 9.11 \times 10^{-31} = 1.822 \times 10^{-30}$. Then $1.822 \times 10^{-30} \times 1.60 \times 10^{-17} = 2.915 \times 10^{-47}$. Square root gives $5.40 \times 10^{-24}$ kg m s$^{-1}$. [A1] for correct momentum.
4. $\lambda = \frac{h}{p} = \frac{6.63 \times 10^{-34}}{5.40 \times 10^{-24}}$ — Now substitute directly into $\lambda = h/p$. [M1] for correct formula.
5. $\lambda = 1.23 \times 10^{-10}$ m $= 0.123$ nm — $6.63 \div 5.40 = 1.228$, and $10^{-34} \div 10^{-24} = 10^{-10}$, giving $1.23 \times 10^{-10}$ m. [A1] for correct answer with units. This is 0.123 nm — well within the range of atomic spacings in crystals, confirming why diffraction is observable.

Answer: $\lambda = 1.23 \times 10^{-10}$ m (0.123 nm)

Notice that 0.123 nm sits comfortably in the range of typical atomic spacings (0.1–0.5 nm). This is not a coincidence — it is precisely why low-energy electrons (tens to hundreds of eV) are chosen for LEED experiments. X-rays used in X-ray crystallography have wavelengths in the same range, which is why both techniques can probe crystal structure.

6. LEED as Evidence for Wave-Particle Duality

Earlier in this topic you encountered the photoelectric effect, which showed that electromagnetic radiation — classically a wave — can behave as a stream of particles (photons). LEED is the mirror image of that argument: electrons, classically particles, behave as waves. Together, these two results establish wave-particle duality as a universal feature of quantum objects.

The critical point for your exam is that the diffraction pattern produced in LEED cannot be explained by treating electrons as classical particles. Classical particles bouncing off a surface would produce a uniform scatter, not a regular pattern of bright spots. The spots arise from constructive interference — a wave phenomenon — and their positions are quantitatively predicted by the de Broglie wavelength.

Historically, the Davisson-Germer experiment (1927) provided the first direct experimental confirmation of de Broglie's hypothesis. Davisson and Germer observed that electrons scattered from a nickel crystal produced intensity maxima at angles consistent with diffraction, with the wavelength matching the de Broglie prediction. LEED is the modern, refined version of this foundational experiment.

7. Exam Technique and Common Mistakes

AQA questions on LEED tend to fall into two categories: qualitative explanation questions (describe or explain what happens to the pattern when speed changes) and calculation questions (find the de Broglie wavelength given a voltage). Both reward precise, linked reasoning — not vague statements about waves and particles.

For qualitative questions, the mark scheme expects a clear chain: voltage change → speed change → momentum change → wavelength change → diffraction angle change → pattern change. Missing any link in that chain typically costs a mark. Write it out step by step, even if it feels over-explained.

Low-energy electron diffraction sits at the heart of the wave-particle duality topic because it provides direct, controllable, quantitative evidence that particles with momentum $p$ behave as waves with wavelength $\lambda = h/p$ — and that you can tune the wavelength, and therefore the diffraction pattern, by adjusting the electron speed through the accelerating voltage.