Generated ERQ

✓ passed E.1 Structure of the atom × C.2 Wave model 10 marks SL 3 passes 111.77s $0.7649

``` ## ERQ · 10 marks · Topics: E.1 Structure of the atom + C.2 Wave model · Archetype: theory_application **Integration:** primary=E.1 Structure of the atom, secondary=C.2 Wave model (strength: supporting) **Stem.** A student reproduces a Rutherford scattering experiment in which a collimated beam of alpha particles is directed at a thin gold foil (Z = 79) in an evacuated chamber. A movable detector records the count rate of scattered alpha particles as a function of scattering angle θ. The alpha particles each have kinetic energy 5.0 MeV, charge +2e, and mass 6.64 × 10⁻²⁷ kg. The radius of a gold nucleus is approximately 7.0 × 10⁻¹⁵ m. The Coulomb constant is k = 8.99 × 10⁹ N m² C⁻², and Planck's constant is h = 6.63 × 10⁻³⁴ J s. ### Part (a) State and Outline [2 marks] · AO1 · Topic: E.1 State the composition of an alpha particle, and outline one observation from Rutherford scattering that provides evidence for a small, dense, positively charged nucleus. ### Part (b) Calculate [3 marks] · AO2 · Topic: E.1 A 5.0 MeV alpha particle is directed head-on toward a gold nucleus. Assuming all of the initial kinetic energy is converted to electric potential energy at the point of closest approach, calculate the distance of closest approach. ### Part (c)(i) Calculate [2 marks] · AO2 · Topic: E.1 + C.2 Calculate the de Broglie wavelength of a 5.0 MeV alpha particle. ### Part (c)(ii) State [1 mark] · AO2 · Topic: C.2 By comparing your answer in (c)(i) with the radius of the gold nucleus, state whether diffraction of the alpha particle by the nucleus is expected to be significant. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR · Topic: E.1 + C.2 A second experiment uses alpha particles of much higher kinetic energy (e.g. 100 MeV). Suggest why, at sufficiently high projectile energies, the classical Rutherford scattering model is expected to break down. --- ## Mark Scheme ### Part (a) [2 marks] — State + Outline - M1: Alpha particle consists of 2 protons and 2 neutrons (equivalently: a helium-4 nucleus) [no ECF] - M2: A small fraction of alpha particles were scattered through large angles (>90°)/back-scattered, indicating the positive charge and most of the mass are concentrated in a very small, dense nucleus [no ECF] ### Part (b) [3 marks] — Calculate - M1: Equates initial KE to Coulomb PE: E_k = k(2e)(79e)/d, with E_k = 5.0 × 10⁶ × 1.60 × 10⁻¹⁹ = 8.0 × 10⁻¹³ J [no ECF] - M2: Rearranges and substitutes: d = (8.99 × 10⁹)(2)(79)(1.60 × 10⁻¹⁹)² / (8.0 × 10⁻¹³) [no ECF] - M3: d = 4.5 × 10⁻¹⁴ m (accept 4.5 × 10⁻¹⁴ to 4.6 × 10⁻¹⁴ m) [ECF from M1/M2] ### Part (c)(i) [2 marks] — Calculate - M1: Momentum from p = √(2mE_k) = √(2 × 6.64 × 10⁻²⁷ × 8.0 × 10⁻¹³) = 1.03 × 10⁻¹⁹ kg m s⁻¹ [ECF from (b) M1 for E_k in J] - M2: λ = h/p = 6.63 × 10⁻³⁴ / 1.03 × 10⁻¹⁹ = 6.4 × 10⁻¹⁵ m (accept 6.4–6.5 × 10⁻¹⁵ m) [ECF from M1] ### Part (c)(ii) [1 mark] — State - M3: λ (~6.4 × 10⁻¹⁵ m) is comparable to the nuclear radius (~7.0 × 10⁻¹⁵ m), so diffraction effects are expected to be significant / not negligible [ECF from (c)(i)] ### Part (d) [2 marks] — Suggest (proposal + warrant per §4.4.1) - M1 (Proposal): At higher kinetic energy the distance of closest approach decreases (d ∝ 1/E_k), so the alpha particle penetrates inside the gold nucleus - M2 (Warrant): Inside the nucleus the inverse-square Coulomb law no longer describes the interaction (the strong nuclear force acts and the nucleus can no longer be treated as a point charge), so the classical hyperbolic trajectory used by Rutherford fails to predict the observed angular distribution ### Marker notes - Alternative acceptable answer for (d): M1 (Proposal): At higher E_k the de Broglie wavelength decreases (λ = h/√(2mE_k)) but remains comparable to nuclear dimensions; M2 (Warrant): wave/diffraction effects produce an angular distribution (diffraction minima) inconsistent with the classical point-particle Coulomb prediction. - (a) M2: accept "most alpha particles passed straight through, showing the atom is mostly empty space" only if paired with a statement that large-angle scattering implies a concentrated nucleus. - (b): accept use of d = 2k(2)(79)e²/(mv²) route with equivalent substitution. - (c)(i): accept non-relativistic treatment only (v/c ≈ 0.05, so γ ≈ 1). - ECF: any consistent numerical error in E_k (in J) propagated through (b), (c)(i), (c)(ii) earns full method marks. ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly implements the cognitive ladder: Part (a) uses State/Outline (AO1), parts (b)–(c) use Calculate/Show that (AO2), and part (d) uses Suggest with evaluation of assumptions (AO3). The progression is appropriate for 10 marks at SL, with increasing demand and proper integration of both topics.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 'Suggest' command-term structure. The scheme combines M1 (proposal) and M2 (justification with estimate) into two separate marks, but M1 as written is a pure descriptive proposal without explicit physics-based justification linking the de Broglie wavelength concept to the validity assumption. The justification logic is deferred entirely to M2, which means M1 alone does not satisfy the 'proposal + brief physics-based justification' requirement for a single Suggest marking point. Additionally, M2's ECF annotation '[ECF from (b)]' appears on a reasoning/estimation mark rather than a method-mark step that depends on a preceding calculation, creating ambiguity about whether ECF is correctly applied per §4.6.

FIX: Restructure Part (d) into two complete Suggest marking points, each containing proposal + justification: M1: 'The de Broglie wavelength λ = h/p must be compared with nuclear radius; if λ ≤ nuclear size, the alpha particle exhibits wave behaviour and cannot be treated as a classical point projectile, invalidating the hyperbolic-trajectory assumption.' M2: 'Calculate λ = h/√(2mE_k) ≈ 6–7 × 10⁻¹⁵ m for 5.0 MeV [ECF from (b) energy]; since λ is comparable to the nuclear radius (~7 × 10⁻¹⁵ m), diffraction effects are significant and the classical model breaks down.' Alternatively, award M1 for proposal + single-sentence justification, and M2 for numerical estimate with comparison to nuclear size, clearly separating conceptual justification from quantitative validation.
cross_theme fail

Part (d) treats C.2 Wave model only as an assumption-checking tool, not as an integral physics concept used in solving the main problem. The de Broglie wavelength is invoked in the final part to validate or critique the E.1 classical model, but wave mechanics is not actively used to derive or explain the scattering process itself. The question fails the genuine blending requirement because C.2 appears only as a validation criterion at the end, not as a co-equal method or framework for understanding the physics.

FIX: Restructure the ERQ so that wave-particle duality and/or the wave model is woven into the core solution pathway. For example: (i) have students calculate both the classical distance of closest approach (E.1) AND the de Broglie wavelength (C.2) in the main body; (ii) ask them to use both to estimate a quantum correction factor or tunneling probability; or (iii) ask them to use the wave model to explain why the Rutherford cross-section formula breaks down at certain angles/energies. Alternatively, redesign part (d) to require derivation or application of wave concepts (e.g., uncertainty principle, wave packet width) to determine when diffraction becomes important, rather than merely citing λ as a validity check.
discriminator pass

Part (d) explicitly asks students to critique the classical Rutherford model by evaluating an assumption (treating the alpha particle as a point projectile). The part requires justification via de Broglie wavelength comparison with nuclear size, directly addressing whether diffraction effects invalidate the model. This is authentic model critique.
rule_of_one pass

All four parts show coherent mark-to-award-criterion alignment. Part (a) has 2 marks with 2 criteria (composition + nuclear model support); part (b) has 3 marks with 3 criteria (energy conversion, substitution, numerical answer); part (c) has 3 marks with 3 criteria (energy conservation, PE formula, algebraic rearrangement); part (d) has 2 marks with 2 criteria (de Broglie proposal + numerical justification). No divergences exceed 1.
topic pass

Part (d) genuinely requires both topics: E.1 (classical Coulomb scattering model and nuclear structure) provides the physical scenario, while C.2 (de Broglie wavelength and wave behaviour) is essential to evaluate the validity of the point-particle assumption. The calculation of λ = h/p directly invokes quantum wave properties and compares it meaningfully to nuclear dimensions, showing both topics are substantively integrated rather than superficially named.

Iteration 1 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 (Outline) requiring knowledge of the nuclear model; parts (b) and (c) are AO2 (Calculate) involving energy conservation and de Broglie relations; part (d) is AO3 (Suggest) requiring synthesis of wave and classical models to predict breakdown of the Rutherford formula. The progression from recall → calculation → evaluation is strict and properly sequenced.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 'Suggest' structure. The command requires 'a proposal + a brief physics-based justification', but M1 and M2 as written are two separate complete suggestions (each with proposal + justification), not a single structured pair. Additionally, M1 conflates two distinct physical mechanisms (wavelength decrease and distance decrease) into one bullet without clear separation of proposal from warrant.

FIX: Restructure Part (d) to award marks for a single coherent suggestion with explicit proposal-and-warrant structure. Example: M1: 'Proposal: At higher kinetic energy, the de Broglie wavelength decreases (λ = h/√(2mE_k) decreases as E_k increases).' M2: 'Warrant: When λ becomes comparable to or smaller than the nuclear radius (~7 × 10⁻¹⁵ m), the alpha particle's wave nature (diffraction, wave packet spread) cannot be neglected, and the classical point-particle Coulomb hyperbolic trajectory fails to predict the observed angular distribution (diffraction fringes appear; Coulomb law breaks down inside the nucleus).' Alternatively, accept M1 as the primary suggestion and M2 as a secondary alternative, but clearly label them as such. Ensure both M1 and M2 explicitly state the physical mechanism (wave diffraction, wavelength-to-size comparison, or Coulomb breakdown) rather than leaving it implicit.
cross_theme pass

The ERQ genuinely integrates both E.1 (Structure of the atom) and C.2 (Wave model). E.1 dominates parts (a)–(b) and (c)(ii) via Rutherford scattering and Coulomb distance-of-closest-approach; C.2 is substantively employed in (c)(i) via de Broglie wavelength calculation and critically in (d) where wave-model concepts (diffraction, wavelength comparison to nuclear size, breakdown of point-particle assumption) are the explicit physics driving the answer. Part (d) specifically requires students to use C.2 reasoning to explain why the classical (E.1) model fails—the two topics are genuinely blended rather than sequential or decoupled.
discriminator pass

Part (d) explicitly asks the student to suggest a reason why the classical Rutherford formula breaks down at higher energies using the wave model. The mark scheme accepts diffraction of the alpha matter wave, breakdown of the point-particle model due to wave packet size becoming comparable to nuclear dimensions, or penetration invalidating the Coulomb potential — all of which are clear critiques of the model's assumptions. This satisfies the dimension's requirement for model critique.
rule_of_one fail

Part (c) claims 3 marks but contains 4 distinct award bullets: M1 and M2 apply to sub-part (c)(i) [2 marks], while M3 applies to sub-part (c)(ii) [1 mark]. The mark scheme conflates the two sub-parts into a single list, obscuring the allocation. Additionally, M1 and M2 for (c)(i) describe two separate operations (momentum calculation and wavelength calculation), which correctly fills the 2-mark sub-part, but the overall structure is ambiguous.

FIX: Restructure Part (c) to separate the mark scheme by sub-part: create a sub-heading for (c)(i) [2 marks] with M1 (momentum) and M2 (wavelength), then a separate sub-heading for (c)(ii) [1 mark] with M3 (comparison statement). Alternatively, if the scheme intends all three bullets to apply to the full 3-mark part, rewrite to show clearly that M1 covers both momentum and wavelength calculations as a single operation, then M2 covers the numerical answer, and M3 covers the statement in (c)(ii)—but this would not match the current structure.
topic pass

The ERQ genuinely integrates both topics. E.1 (Structure of the atom) is tested through the Rutherford scattering scenario, Coulomb interactions, and nuclear model interpretation (parts a, b, c(ii)). C.2 (Wave model) is tested through de Broglie wavelength calculation and explicit comparison of wavelength with nuclear dimensions (part c(i)), and critically in part (d) where the breakdown of classical scattering due to wave effects at smaller wavelengths is the conceptual linchpin. Neither topic is superficial: both drive the physical reasoning and cannot be omitted without breaking the question's logical structure.

Iteration 2 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly: Part (a) requires AO1 (State) + AO1 (Outline); Parts (b)–(c)(ii) require AO2 (Calculate, State); Part (d) requires AO3 (Suggest with assumptions discrimination). Each tier is sequenced correctly from simple recall through calculation to evaluation-level reasoning.
command_structure fail

Part (d) fails the AO3 'Suggest' command-term structure. Per §4.4.1, 'Suggest' requires a proposal + a brief physics-based justification. The mark scheme provides M1 (proposal) and M2 (warrant), but M2 is a long compound statement mixing two distinct ideas (strong force + failure of point-charge model) without clear causal linkage to the proposal. The warrant should explicitly connect 'why the classical model breaks down' to the physical mechanism triggered by the proposal. Additionally, the alternative answer conflates two different physical mechanisms (wavelength decrease + diffraction effects) without clear separation of proposal from warrant structure.

FIX: Restructure Part (d) mark scheme as follows: **Primary path (Coulomb breakdown):** - M1 (Proposal): At higher E_k, the distance of closest approach decreases (d ∝ 1/E_k), allowing the alpha particle to penetrate inside the nucleus. - M2 (Warrant): The Coulomb inverse-square law applies only outside the nucleus; inside, the strong nuclear force dominates and the nucleus cannot be treated as a point charge. Therefore, the classical hyperbolic trajectory predicts an angular distribution that no longer matches observations. **Alternative path (wave effects):** - M1 (Proposal): At higher E_k, the de Broglie wavelength decreases (λ = h/√(2mE_k)) toward nuclear dimensions. - M2 (Warrant): When wavelength becomes comparable to nuclear size, diffraction effects produce additional scattering peaks and minima inconsistent with the classical point-particle Coulomb prediction. Each alternative must have explicitly separate proposal and warrant bullets, with the warrant directly explaining why the classical model fails.
cross_theme pass

Both E.1 (Structure of the atom) and C.2 (Wave model) are genuinely integrated throughout the question. E.1 dominates parts (a)–(b) and (d), while C.2 appears substantively in (c)(i)–(c)(ii) where de Broglie wavelength is calculated and compared to nuclear radius to assess diffraction significance. Part (d) offers an alternative warrant that explicitly blends both: quantum wave effects (diffraction minima from C.2) break down the classical Rutherford model when λ becomes comparable to nuclear size. The second topic is not mere filler; it is woven into the physical explanation of model failure.
discriminator pass

Part (d) clearly asks students to suggest why the classical Rutherford model breaks down at high energies, requiring critique of the model's assumptions (inverse-square law, point nucleus, classical mechanics). The mark scheme explicitly requires both a proposal and warrant, demonstrating genuine model limitation analysis rather than calculation.
rule_of_one fail

Part (c)(ii) is marked as [1 mark] but contains only 1 award criterion (M3: the comparison statement). However, a 1-mark State question should have exactly 1 criterion, which it does. The real problem is part (d): it is marked [2 marks] but the mark scheme lists 2 criteria (M1 Proposal, M2 Warrant) that describe a single integrated answer, not 2 distinct independent operations. For a 2-mark Suggest question, the scheme should explicitly separate 2 independently awardable steps (e.g., identification of a physical effect + explanation of breakdown, or calculation of effect + application to Rutherford model). The current scheme conflates proposal and warrant into one reasoning chain.

FIX: Part (d): Either (i) restructure as a 3-mark question with 3 bullet criteria (e.g., M1: states d decreases with E_k, M2: states distance becomes comparable to/enters nucleus, M3: states Coulomb law fails and/or strong force dominates), or (ii) rewrite the mark scheme to make two fully independent 2-mark criteria explicit—for example: M1: Identifies that at high E_k the alpha particle penetrates the nucleus / distance of closest approach becomes very small [AO3]; M2: Explains that inside the nucleus Coulomb interaction is no longer inverse-square / strong nuclear force acts / wave effects become significant, invalidating classical Rutherford prediction [AO3].
topic pass

The ERQ genuinely integrates both topics across multiple parts. E.1 (Structure of the atom) is primary and tested throughout (composition, nuclear charge, Rutherford evidence, closest approach). C.2 (Wave model) is substantively invoked in (c)(i)–(c)(ii) via de Broglie wavelength and diffraction, and in part (d) where wavelength scaling and diffraction effects explain the breakdown of the classical model at high energy. Neither topic is merely nominal; both are essential to the physical scenario.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A student performs a Rutherford scattering experiment using alpha particles directed at a thin gold foil. The apparatus detects scattered alpha particles at various angles. The student must explain the atomic structure revealed by the experiment, calculate the distance of closest approach for an alpha particle, and then critically evaluate whether the wave nature of the alpha particle affects the validity of the classical scattering model.
Cross-pollination part: d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the composition of an alpha particle and outline the evidence from Rutherford scattering that supports a nuclear model of the atom
b	3	AO2	An alpha particle (charge +2e, mass 6.64 × 10⁻²⁷ kg) is directed head-on toward a gold nucleus (charge +79e). Calculate the distance of closest approach if the alpha particle has initial kinetic energy 5.0 MeV
c	3	AO2	Show that the impact parameter b for 90° scattering is given by b = kZz e² / (2E_k), where Z and z are nuclear and projectile charges respectively
d	2	AO3	Suggest why the de Broglie wavelength of the alpha particle must be considered when evaluating the limitations of the classical Rutherford scattering model, and explain whether diffraction effects would be significant at the gold nucleus

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