Generated ERQ

✓ passed A.2 Forces and momentum × E.3 Radioactive decay 12 marks HL 3 passes 130.95s $0.7188

## ERQ · 12 marks · Topics: A.2 Forces and momentum + E.3 Radioactive decay **Stem.** A magnetic spectrometer is used to identify alpha particles emitted from a thin radium-226 source. Radium-226 decays to radon-222 with a Q-value (total kinetic energy released) of Q = 4.871 MeV when the daughter nucleus is left in the ground state, but a small fraction of decays leave the radon-222 in an excited state, reducing the Q-value to 4.671 MeV (the excitation energy is later released as a 0.200 MeV gamma photon). Alpha particles emerge through a collimator into a region of uniform magnetic field of magnitude B = 0.520 T, directed perpendicular to their velocity, and travel in circular arcs before striking a position-sensitive detector. Two distinct impact lines are observed on the detector, at distances d₁ = 0.402 m and d₂ = 0.394 m from the collimator slit, corresponding to the diameters of the two circular paths. Take the alpha particle mass as m_α = 6.645 × 10⁻²⁷ kg and charge q = 3.20 × 10⁻¹⁹ C. The source and detector are in vacuum and the parent nuclei are essentially at rest before decay. ### Part (a) State [2 marks] · AO1 · Topic: A.2 State the direction of the magnetic force on a positive alpha particle moving in the field, and explain why the speed of the alpha particle remains constant along its circular path. ### Part (b) Calculate [3 marks] · AO2 · Topic: A.2 Using the impact distance d₁ = 0.402 m, calculate the kinetic energy of the alpha particles that follow the larger-radius path, expressing your answer in MeV. ### Part (c) Determine [4 marks] · AO2/AO3 · Topic: A.2 + E.3 The recoiling radon-222 nucleus carries away a share of the Q-value. By applying conservation of momentum and energy to the decay (with the parent at rest), determine which of the two observed lines (d₁ or d₂) corresponds to decays leaving radon-222 in the **ground** state. ### Part (d) Suggest [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The kinetic energy calculated in (b) is slightly lower than the value predicted from the Q-value of the ground-state branch. Suggest **three** distinct physical reasons — at least one related to the radioactive source itself and at least one related to the spectrometer model — for this discrepancy. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Force is perpendicular to the velocity / directed towards the centre of the circle (radially inward); accept "given by F = qv × B" with correct sense indicated. [ECF: no] - M2: Because the force is always perpendicular to velocity, it does no work on the particle, so kinetic energy and hence speed remain constant. [ECF: no] ### Part (b) [3 marks] - M1: Equates magnetic force to centripetal force to obtain p = qBr, with r = d₁/2 = 0.201 m (radius identified). [ECF: no] - M2: Momentum p = (3.20 × 10⁻¹⁹)(0.520)(0.201) = 3.345 × 10⁻²⁰ kg m s⁻¹; kinetic energy T = p²/(2m_α) = (3.345 × 10⁻²⁰)²/(2 × 6.645 × 10⁻²⁷) = 8.42 × 10⁻¹⁴ J. [ECF: yes] - M3: Convert: T = 8.42 × 10⁻¹⁴ / 1.60 × 10⁻¹³ ≈ 0.526 / … = **4.78 MeV** (accept 4.77–4.79 MeV; correct unit, 3 s.f.). [ECF: yes] ### Part (c) [4 marks] - M1: Apply conservation of momentum: p_α = p_Rn (parent at rest), so T_Rn = p²/(2M_Rn) and the alpha shares the energy according to T_α/Q = M_Rn/(M_Rn + m_α). [ECF: no] - M2: Derive T_α = Q · M_Rn / (M_Rn + m_α) = Q · (222/226) for the alpha kinetic energy. [ECF: yes] - M3: Compute predictions: T_α(ground) = 4.871 × (222/226) = **4.785 MeV**; T_α(excited) = 4.671 × (222/226) = **4.589 MeV**. [ECF: yes] - M4: Identify d₁ (larger diameter ⇒ higher momentum ⇒ higher T_α ≈ 4.78 MeV from (b)) as the **ground-state** branch, since it matches the 4.785 MeV prediction; d₂ corresponds to the excited-state branch. [ECF: yes] ### Part (d) [3 marks] Award 1 mark for each distinct reason, up to 3. Must include at least one source-related and at least one spectrometer-related reason for full marks. - Source-related (accept any one): - Energy loss as the alpha traverses the finite thickness of the radium source material before escaping (self-absorption). [ECF: no] - Alpha emitted from a nucleus within the source layer loses energy ionising atoms in the source/window. [ECF: no] - Doppler/thermal motion of parent nuclei not truly at rest (very small effect — accept with caution). [ECF: no] - Spectrometer-related (accept any one): - Slight non-uniformity or fringing of the magnetic field at the edges of the field region means qvB = mv²/r is not exact along the whole path. [ECF: no] - Finite collimator slit width / detector resolution means the measured diameter d₁ is an average and may be systematically low. [ECF: no] - Residual gas in the "vacuum" causes small energy loss / scattering along the trajectory. [ECF: no] - Other accept: relativistic correction neglected (T ≈ 4.8 MeV gives v/c ≈ 0.05, small but non-zero); recoil correction was applied but nuclear masses were approximated by mass numbers 222/226. ### Marker notes - Part (b) alternative method: compute v = qBr/m_α = 2.516 × 10⁶ m s⁻¹, then T = ½m_αv² = 2.10 × 10⁻¹⁴ J … (check arithmetic); award full ECF if method consistent. - Part (c): a candidate who simply states "d₁ corresponds to ground state because higher Q gives higher T_α" without the recoil derivation earns M1 + M4 only (max 2/4). - Part (d) discriminator: do **not** award two marks both belonging to the same category (e.g. two source-thickness answers). Award of "experimental uncertainty in B or in d₁" is acceptable as a spectrometer reason but only once.

Critic verdicts

Iteration 0 — failed

ao_ladder fail

The ERQ violates the cognitive ladder. Part (a) is correctly AO1 (State). However, part (b)(i) and (b)(ii) are both AO2/AO3 calculation/derivation tasks, leaving no distinct AO2 middle tier—and critically, part (d) jumps directly to AO3 (Suggest) without any intermediate AO2 task. The ladder should progress: AO1 → AO2 → AO3, but here it goes AO1 → AO2/AO3 → AO2/AO3 → AO3, missing a clear structural separation and an explicit AO2-only or mixed AO2-dominant middle section.

FIX: Restructure as follows: (a) State [2 marks, AO1]; (b) new AO2 part [2 marks]: Calculate/determine a preliminary quantity (e.g., the radius of curvature from the chord, or the charge-to-mass ratio needed). Then (c) Show that/Derive [4 marks, AO2/AO3]: the full momentum and KE with comparison to decay prediction. Finally (d) Suggest [4 marks, AO3]: reasons for the spread and energy deficit. Alternatively, retain the current parts but insert a short (b)(i)-revised as a pure AO2 Calculate task (e.g., 'Calculate the radius of the circular path') before the Show that, creating a clearer three-tier ladder.
cross_theme fail

Topic E.3 (Radioactive decay) is invoked only in Part (b)(ii) M3 as a calculation assumption (conservation of momentum in the decay process) and in Part (d) as a discriminator. However, neither application genuinely blends the two themes: Part (b)(ii) uses decay kinematics merely to generate a comparison value, not to solve the spectrometer problem itself; Part (d) mentions excited nuclear states but asks candidates to 'suggest reasons' for observations rather than requiring decay physics to be integrated into the momentum-spectrometer analysis. The core physics work (Parts a, b(i), d-spread) depends entirely on A.2 (magnetic force, circular motion, momentum). E.3 appears as filler or a final check rather than as a co-equal driver of the solution.

FIX: Redesign (b)(ii) and (d) to require active integration of both topics. For example: (b)(ii) could ask students to explain why the measured KE differs from the two-body decay prediction by invoking both momentum conservation in the decay AND energy loss mechanisms in the spectrometer; (d) could ask students to predict the energy spread expected from a mixture of ground-state and first-excited-state decays, then calculate how the resulting momentum spread translates into a chord spread using the Lorentz force formula. This would force simultaneous use of A.2 (force, momentum, kinematics) and E.3 (decay energy, excited states, branching ratios) rather than treating them sequentially or as independent checks.
discriminator pass

Part (d) explicitly asks students to 'Suggest reasons' for experimental observations (energy spread and sub-predicted energies) by critiquing both the decay process and the spectrometer model. This requires students to identify and discuss limitations of the idealized treatment (e.g., nuclear excitation, energy loss, collimator assumptions), which is genuine model-critique work rather than further calculation.
rule_of_one fail

Part (b)(i) is marked as 3 marks but the mark scheme lists 3 award bullets that describe a single logical chain (recognize geometry → apply force equation → evaluate numerically). However, the third bullet (M3) is ambiguous and conflates two separate pathways (correct calculation vs. incorrect radius interpretation), making it unclear whether this represents one or two distinct operations. More critically, Part (b)(ii) is marked as 4 marks but only 4 bullets are listed, yet M3 and M4 together constitute comparison logic that should be one operation, not two separate awards. The mark-to-bullet ratio breaks down under strict dimensional analysis: (b)(ii) should either be explicitly split into (a) KE calculation from (b)(i) [2 marks] + (b) momentum-conservation prediction [1 mark] + (c) comparison [1 mark], or consolidated to 3 marks. Additionally, Part (d) is marked 3 marks with exactly 3 bullets, but the marker notes state 'accept any two distinct reasons'—this introduces a pass/fail threshold misalignment (candidates answering only 2 complete reasons would score only 2/3, not 3/3).

FIX: Restructure as follows: (b)(i) keep at 3 marks; rewrite M3 to unambiguously award for 'correct numerical answer with units within acceptable range' (one operation). (b)(ii) split into [2 marks]: M1–M2 for KE calculation from momentum; [2 marks]: M3–M4 for momentum-conservation prediction and comparison (or restructure to explicit sub-parts). Part (d): either (i) reduce to 2 marks with M1 and M2 as the two required reasons, or (ii) clarify that M1, M2, M3 are all independent and any two suffice for full credit—use conditional phrasing 'Award 3 marks if all three reasons present; 2 marks if any two distinct reasons are stated.'
topic fail

Topic E.3 (Radioactive decay) is invoked only nominally in parts (b)(ii) and (d). Part (b)(ii) asks students to compare measured kinetic energy with a prediction from 'conservation of momentum in the decay'—a single-line calculation using mass ratio and Q-value. Part (d) mentions excited nuclear states (decay physics), but the main scenario tests magnetic deflection (A.2 Forces and momentum) throughout. The decay process itself is never analysed: no decay equation, no energy distribution, no recoil dynamics are examined. E.3 concepts are superficially touched rather than genuinely integrated into the problem's physics.

FIX: Restructure to require genuine E.3 content: (1) ask students to identify which decay branch (via Q-value and recoil kinematics) produced the observed alpha energy; (2) explore why the energy spectrum of alphas from a thin Ra-226 source shows two discrete lines (corresponding to ground and excited states of Rn-222), and use the spectrometer to measure the energy difference; or (3) use the momentum measurement to determine the Q-value of the decay and compare with tabulated values, explicitly working through the recoil correction. This ensures A.2 (magnetic momentum spectrometry) and E.3 (decay energetics and branching) are both substantially tested.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly: (a) is AO1 (State), (b) is AO2 (Calculate), (c) is AO2/AO3 (Show that/Determine), and (d) is AO3 (Suggest). Command terms align with cognitive demand at each tier, progressing from recall through procedural calculation to synthesis and evaluation of experimental limitations.
cross_theme pass

The ERQ genuinely blends both A.2 (Forces and momentum) and E.3 (Radioactive decay). Part (c) is the critical integration point: it uses momentum conservation from the decay (E.3) to derive the kinetic energy formula, then applies the magnetic spectrometer model (A.2) to identify which decay branch produces which line. Part (d) reinforces the blend by requiring discrimination of causes between decay physics and spectrometer mechanics. Both topics are essential to solving the question, not merely name-checked.
discriminator pass

Part (d) explicitly asks students to critique the model by identifying and explaining reasons for deviations between theory and experiment. It discriminates between decay physics (E.3) and spectrometer assumptions (A.2), requiring students to evaluate which model assumptions break down and why—a classic model-critique flavor.
rule_of_one fail

Part (c) is marked [4 marks] but the mark scheme lists 4 award criteria (M1–M4), which appears aligned. However, M2 conflates two distinct operations: (i) setting up the energy equation Q = T_α + T_Rn and (ii) algebraic manipulation to derive the final formula. M3 and M4 are also problematic: M3 computes both ground and excited predictions (two calculations), yet M4 requires computing the measured energy from (b) and comparing both predictions to identify the branch. This makes M3 and M4 overlap and ambiguous in their boundaries. The scheme should cleanly separate: formula derivation (1 mark), ground-state calculation (1 mark), excited-state calculation (1 mark), identification via comparison (1 mark). As written, the mark boundaries are unclear and do not cleanly isolate 4 distinct operations.

FIX: Restructure Part (c) M2–M4 as follows: M2: 'Derive T_α = Q·M_Rn/(M_Rn + m_α) from momentum and energy conservation' (formula shown); M3: 'Calculate T_α (ground) = 4.871 × 222/226 = 4.785 MeV and T_α (excited) = 4.671 × 222/226 = 4.602 MeV'; M4: 'Calculate measured T_α from d₂ (approximately 4.60 MeV) and identify inner line as excited-state branch by comparison to predicted values.' Alternatively, if retaining the current structure, explicitly state in M3 that it is two separate sub-calculations (one per branch) and clarify in M4 that it requires both the momentum calculation from (b) and the predictions from M3.
topic pass

The ERQ genuinely integrates both A.2 (Forces and momentum) and E.3 (Radioactive decay). Parts (a)–(b) test the magnetic force and circular motion (A.2 core). Part (c) requires conservation of momentum from the decay process (E.3) applied to derive the alpha kinetic energy formula, then uses this with the spectrometer data (A.2) to identify which decay branch produced each line. Part (d) explicitly discriminates between decay physics (E.3: self-absorption, excited-state energy loss) and spectrometer model assumptions (A.2: field uniformity, slit width), forcing candidates to recognize that both topics are essential to explaining the observations.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: Part (a) uses State [AO1], Part (b) uses Calculate [AO2], Part (c) uses Determine [AO2/AO3 with derivation], and Part (d) uses Suggest [AO3 requiring evaluation of physical assumptions]. Command terms and cognitive demands escalate appropriately across all four parts.
cross_theme pass

The ERQ successfully blends A.2 (Forces and momentum) and E.3 (Radioactive decay). Part (c) is the critical integration point: it applies conservation of momentum and energy to the radioactive decay process (E.3) to predict two distinct kinetic energies for alpha particles, then uses the magnetic deflection formula from A.2 to identify which observed line corresponds to ground-state decay. Both physics concepts are genuinely required to solve the question, not merely name-checked.
discriminator pass

Part (d) explicitly asks students to 'Suggest three distinct physical reasons' for a discrepancy between calculated and predicted kinetic energy, requiring critique of the model assumptions (source thickness, field uniformity, collimator resolution, residual gas, etc.). This is a genuine model-critique question that goes beyond mere calculation.
rule_of_one fail

Part (d) has 3 marks but the mark scheme lists 4 distinct award criteria (3 categorical reason bullets plus a fourth 'Other accept' bullet). The structure violates the contract: a 3-mark part must map to exactly 3 award bullets, not 4. Additionally, the scheme's conditional logic ('up to 3... must include at least one source and one spectrometer') makes the actual award criteria ambiguous and non-linear.

FIX: Restructure Part (d) to have exactly 3 clear, independent award criteria. Option 1: Rewrite as M1 (any valid source-related reason), M2 (any valid spectrometer-related reason), M3 (a third distinct reason from any category); remove the 'Other accept' as a separate bullet and fold it into the conditional logic. Option 2: Change part to [4 marks] and explicitly list 4 bullet-point award criteria (e.g. M1–M4 for source effect, spectrometer effect, third reason, and correct reasoning). The current hybrid format with conditional categorization and an open 'Other' clause does not yield a deterministic 1–1 mapping between marks and operations.
topic pass

The ERQ authentically integrates both A.2 (Forces and momentum) and E.3 (Radioactive decay). Parts (a)–(b) require circular motion under magnetic force and momentum-energy relations (A.2). Part (c) critically applies conservation of momentum and energy to the two-body decay process (E.3) to determine which observed line corresponds to which decay branch—the alpha particle kinematics depend directly on the recoil of the daughter nucleus. Part (d) extends the physics by requiring explanation of discrepancies using source and spectrometer physics. Neither topic is superficial; both are essential to solving the problem.

Planner output (stage 1)

Archetype: real-world device
Scenario: A particle detector uses a magnetic field to deflect charged particles produced by radioactive decay. Alpha particles (helium nuclei) emitted from a radium-226 sample enter a uniform magnetic field region perpendicular to their initial velocity. Students must analyze the motion of these particles using momentum and circular motion principles, then evaluate how the decay process and particle energy distribution affect the detector's ability to distinguish particle types.
Cross-pollination part: d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State
b	3	AO2	Calculate
c	4	AO2\|AO3	Derive
d	3	AO3	Evaluate

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