## ERQ · 12 marks · Topics: E.2 Quantum physics (HL) + C.3 Wave phenomena
**Stem.** In a low-energy electron diffraction (LEED) experiment, electrons are accelerated from rest through a potential difference of 54 V and directed at normal incidence onto the (111) surface of a nickel crystal. The surface atoms form a regular two-dimensional array with adjacent atomic rows separated by d = 2.15 × 10⁻¹⁰ m. A fluorescent detector screen is placed parallel to the crystal surface at a distance L = 0.140 m from the sample. The first-order diffraction maximum is observed as a bright spot at a perpendicular displacement x from the central (zero-order) spot. The experiment is performed in ultra-high vacuum at room temperature. (Take mₑ = 9.11 × 10⁻³¹ kg, e = 1.60 × 10⁻¹⁹ C, h = 6.63 × 10⁻³⁴ J s.)
### Part (a) Outline [2 marks] · AO1 · Topic: E.2
Outline how the observation of a diffraction pattern in this experiment supports the de Broglie hypothesis.
### Part (b)(i) Calculate [3 marks] · AO2 · Topic: E.2
Calculate the de Broglie wavelength of an electron accelerated through 54 V.
### Part (b)(ii) Determine [3 marks] · AO2/AO3 · Topic: E.2 + C.3
Using the diffraction condition d sin θ = nλ for the first-order maximum, determine the displacement x of the first-order bright spot from the central spot on the detector screen.
### Part (c) Explain [2 marks] · AO3 · Topic: C.3
The accelerating potential is now doubled to 108 V. Explain, without further calculation, how the displacement x of the first-order maximum changes.
### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR
The displacement x measured in the actual experiment is slightly larger than the value calculated in (b)(ii). Suggest one physical reason for this discrepancy.
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## Mark Scheme
### Part (a) [2 marks]
- Mark 1: States that electrons (matter particles) exhibit wave-like behaviour / diffraction is a wave phenomenon [ECF: no]
- Mark 2: Links the spacing/angle of the observed pattern to a wavelength given by λ = h/p, confirming de Broglie's prediction [ECF: no]
### Part (b)(i) [3 marks]
- Mark 1: Uses eV = ½mv² (or eV = p²/2m) to find electron momentum; λ = h/p [ECF: no]
- Mark 2: Correct substitution: p = √(2 × 9.11 × 10⁻³¹ × 1.60 × 10⁻¹⁹ × 54) ≈ 3.97 × 10⁻²⁴ kg m s⁻¹ [ECF: yes]
- Mark 3: λ ≈ 1.67 × 10⁻¹⁰ m (accept 1.6–1.7 × 10⁻¹⁰ m), 2–3 s.f. with units [ECF: yes]
### Part (b)(ii) [3 marks]
- Mark 1: sin θ = λ/d → θ = arcsin(1.67 × 10⁻¹⁰ / 2.15 × 10⁻¹⁰) [ECF: yes from (b)(i)]
- Mark 2: θ ≈ 50.9° (accept 50°–52°); use of x = L tan θ [ECF: yes]
- Mark 3: x = 0.140 × tan(50.9°) ≈ 0.172 m (accept 0.17 m), 2–3 s.f. with units [ECF: yes]
### Part (c) [2 marks]
- Mark 1: Recognises that V doubled → p increases by √2 → λ decreases by factor √2 [ECF: no]
- Mark 2: Smaller λ gives smaller sin θ, hence smaller θ and smaller displacement x (x decreases, by roughly factor √2 for small angles) [ECF: yes]
### Part (d) [2 marks]
- Mark 1: Identifies a specific physical effect not accounted for in the model [ECF: no]
- Mark 2: Coherent physical explanation of how that effect increases x [ECF: no]
### Marker notes
- Alternative for (b)(i): combined formula λ = h/√(2meV) directly; full marks if substitution and final answer both shown.
- Alternative for (b)(ii): student may compute x via x = L sin θ / cos θ or use small-angle approximation — the small-angle form gives x ≈ 0.108 m and loses Mark 3 (angle is not small here); accept only if the student justifies/uses tan θ correctly.
- (d) discriminator: accept any one of —
• Thermal vibrations of surface atoms (Debye–Waller) effectively reduce d or broaden the maximum, shifting the observed peak outward.
• Inner potential of the crystal: electrons gain kinetic energy inside the solid, but on emerging the relevant momentum for the outgoing wave differs from the bulk value used.
• Surface reconstruction / lattice contraction at the (111) termination gives a row spacing slightly different from the bulk value 2.15 × 10⁻¹⁰ m.
• Finite size of the electron beam / detector geometry not perfectly parallel to surface.
• Relativistic correction to momentum (small at 54 eV but non-zero).
- Do NOT accept vague answers such as "experimental error" or "the equipment is not perfect" without a specific mechanism.