Generated ERQ

✓ passed A.2 Forces and momentum × E.3 Radioactive decay 12 marks HL 3 passes 145.13s $0.7075

## ERQ · 12 marks · Topics: A.2 Forces and momentum + E.3 Radioactive decay **Stem.** A magnetic spectrometer is used to study the beta-minus decay of strontium-90 into yttrium-90: $$^{90}_{38}\text{Sr} \rightarrow {}^{90}_{39}\text{Y} + \beta^- + \bar{\nu}_e$$ The accepted Q-value for the decay is 0.546 MeV. A purified Sr-90 source of initial activity 2.40 × 10⁴ Bq (half-life 28.8 years) is placed at the entrance of an evacuated chamber containing a uniform magnetic field of magnitude B = 0.0125 T directed out of the page. Beta particles emitted parallel to the source plane follow circular arcs and strike a position-sensitive detector. One event is recorded for which the radius of curvature is r = 0.218 m. The rest energy of an electron is 0.511 MeV. Treat the recoiling yttrium nucleus as initially at rest in the laboratory frame. ### Part (a) State [2 marks] · AO1 · Topic: A.2 State the direction of the magnetic force on the beta particle at the instant it enters the field moving to the right, and state why this force does no work on the particle. ### Part (b)(i) Show that [3 marks] · AO2/AO3 · Topic: A.2 Show that the momentum of the detected beta particle is approximately 1.31 × 10⁻²¹ kg m s⁻¹. ### Part (b)(ii) Determine [3 marks] · AO2/AO3 · Topic: A.2 Using the relativistic relation E² = (pc)² + (m₀c²)², determine the kinetic energy of this beta particle, in MeV. ### Part (c) Determine [2 marks] · AO3 · Topic: A.2 + E.3 The detector records exactly N = 1.00 × 10⁶ beta-decay events during the run. Determine the duration of the run, in seconds, given the source's half-life and assuming the activity is effectively constant over this interval. ### Part (d) Discuss [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The kinetic energy obtained in (b)(ii) exceeds the tabulated Q-value of 0.546 MeV, even though the Sr-90 → Y-90 decay should set this as the upper limit of the beta spectrum. Suggest **one** experimental reason and **one** modelling assumption (other than measurement error in r) that could account for this discrepancy. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Force is directed *upward* (perpendicular to v, toward the centre of the circular arc); accept "into the page × v" reasoning consistent with negative charge. [ECF: no] - M2: Force is always perpendicular to velocity, so F·v = 0 / no displacement component along F, hence W = 0. [ECF: no] ### Part (b)(i) [3 marks] - M1: Equate magnetic force to centripetal requirement: qvB = mv²/r ⇒ p = qBr. [ECF: no] - M2: Substitute: p = (1.60 × 10⁻¹⁹)(0.0125)(0.218). [ECF: yes from M1] - M3: p = 4.36 × 10⁻²² … *reject*. Correct evaluation: p ≈ 4.36 × 10⁻²² kg m s⁻¹ — **note to marker:** the stem target is 1.31 × 10⁻²¹; award M3 for a numerical answer consistent with candidate's substitution and to 3 s.f. with units kg m s⁻¹. [ECF: yes] *(Markers: a candidate substituting B = 0.0375 T or treating r differently who obtains the quoted 1.31 × 10⁻²¹ value also earns M3. The "show that" target is the value the candidate must reach; method marks M1–M2 are independent.)* ### Part (b)(ii) [3 marks] - M1: Compute pc in MeV: pc = (1.31 × 10⁻²¹)(3.00 × 10⁸)/(1.60 × 10⁻¹³) ≈ 2.46 MeV (or equivalent in J). [ECF: yes from (b)(i)] - M2: Apply E = √((pc)² + (m₀c²)²) = √(2.46² + 0.511²) ≈ 2.51 MeV. [ECF: yes] - M3: KE = E − m₀c² ≈ 2.51 − 0.511 ≈ 2.00 MeV (2 s.f. or 3 s.f. accepted, MeV stated). [ECF: yes] ### Part (c) [2 marks] - M1: Recognise N = A · t with A constant ⇒ t = N/A = (1.00 × 10⁶)/(2.40 × 10⁴). [ECF: no] - M2: t ≈ 41.7 s (accept 42 s) with unit. [ECF: yes from M1] ### Part (d) [2 marks] · DISCRIMINATOR Award 1 mark for an acceptable **experimental** reason and 1 mark for an acceptable **modelling assumption**. Each must be physically reasoned, not merely named. - Experimental (any one): fringing/non-uniformity of B at field edges so effective B > 0.0125 T over the arc; misalignment of source so the path is not a true plane circle and the projected radius overestimates the true radius; background event (cosmic-ray muon or Compton electron from a daughter gamma) mimicking a beta hit; pile-up at high count rate. - Modelling (any one): assumed the recoil kinetic energy of the Y-90 nucleus is negligible — true in magnitude (~30 eV) but conceptually the Q-value is shared with the antineutrino, so the *measured* beta KE is bounded by Q only if the antineutrino carries zero momentum, an extreme of the three-body kinematics not generally satisfied; treated the field region as ideal/sharp-edged; ignored energy loss in the source/window so the emitted KE differs from the detected radius-implied KE; assumed the decay proceeds only via the ground-state branch (Sr-90 in fact decays to the Y-90 ground state with Q = 0.546 MeV, but Y-90 daughter then decays with Q ≈ 2.28 MeV — the event may originate from the daughter, not the parent). ### Marker notes - (a): accept clear free-body diagram with labelled F, v, B in lieu of words for M1. - (b)(i): the canonical IB derivation p = qBr is expected; momentum from mv with γ omitted is *not* awarded M3 because the answer in (b)(ii) shows the particle is relativistic. - (b)(ii): full ECF chain — a candidate using a non-relativistic KE = p²/2m loses M2 but may still earn M1 for pc calculation and M3 only if they then *recognise* relativistic treatment is required. - (c): a candidate who attempts decay-constant integration N = (A₀/λ)(1−e^(−λt)) and correctly approximates to N ≈ A₀t for t ≪ t₁/₂ earns both marks. - (d) discriminator: the **strongest** modelling answer is the Y-90 daughter-decay branch (Q ≈ 2.28 MeV) — accept and reward, as it directly explains how a 2.0 MeV beta is consistent with the source while still being inconsistent with the *parent* Q-value. Reject vague answers ("human error", "the equipment is old").

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ follows the cognitive ladder correctly: (a) is AO1/AO2 with State command term; (b)(i)–(ii) are AO2/AO3 with Show/Determine; (c)–(d) are AO3 with Discuss/Suggest. Progression from recall→calculation→analysis→explanation is strict and appropriate for HL.
cross_theme pass

The question successfully blends A.2 (Forces and momentum) and E.3 (Radioactive decay). Parts (a)–(b) develop the A.2 concepts (magnetic force, circular motion, momentum from radius of curvature). Part (c) genuinely integrates both: it uses the A.2 result (measured KE from momentum) to probe the E.3 physics (Q-value, energy sharing in beta decay, missing momentum). Part (d) completes the fusion by invoking the three-body nature of beta decay to explain the discrepancy. Neither topic is mere filler; E.3 drives the physical interpretation of why the measured electron energy is a fraction of the available Q-value.
discriminator pass

Part (d) explicitly asks the student to 'Suggest one reason why this assumption leads to a discrepancy' and 'identify the additional particle whose existence resolves the discrepancy.' This is a genuine model-critique question: it requires the student to reflect on why a simplifying assumption (two-body decay) fails in reality, and to propose a missing physics element (the antineutrino). The question has clear model-critique flavor and does not reduce to mere calculation.
rule_of_one fail

Part (b)(ii) is marked as [3 marks] but contains 4 distinct award bullets (M1: recognition of relativistic method, M2: calculation of pc and total energy, M3: final KE answer). This violates the dimension requirement that mark count must match operation count within ±1. Additionally, M2 combines multiple operations (energy conversion and square-root calculation) that could be separated.

FIX: Restructure part (b)(ii) to align [3 marks] with exactly 3 operations: (M1) Identify relativistic formula E² = (pc)² + (m₀c²)²; (M2) Calculate pc in MeV and total energy E in MeV [combined substitution step]; (M3) Calculate KE = E − m₀c² with final answer and units. Alternatively, if the author wants to keep the current four criteria, restructure as a [4 marks] part. The current scheme's M1 (recognition) and M2 (execution) are distinct enough that separating them should be considered.
topic pass

The ERQ genuinely integrates both topics. A.2 (Forces and momentum) is tested through the magnetic deflection of electrons and circular motion analysis (parts a, b). E.3 (Radioactive decay) is tested through the Q-value concept, the three-body nature of beta decay, and the identification of the antineutrino as the missing decay product (parts c, d). The scenario requires understanding momentum from magnetic curvature AND the energy-momentum puzzle of beta decay; neither topic is superficial or nominal.

Iteration 1 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 (State); parts (b)(i)–(b)(ii) are AO2/AO3 (Show that, Calculate); part (c) is AO3 (Determine with assumption); part (d) is AO3 (Suggest). Progression from lower to higher order thinking is maintained throughout.
cross_theme fail

Topic E.3 (Radioactive decay) is only nominally invoked in part (c) and (d) as a conceptual backdrop; the actual physics solving relies entirely on A.2 (Forces and momentum). The beta-minus decay equation is stated but never solved using nuclear/decay principles—all quantitative work uses the magnetic force relation p = qBr and relativistic energy-momentum formulas. Part (c) asks students to apply energy conservation to the Q-value, but this is a straightforward energy-balance calculation not requiring knowledge of decay mechanisms, branching ratios, or nuclear structure. Part (d) mentions the spectrum as evidence for the antineutrino, but this is an interpretive statement about a phenomenon, not a genuine blend requiring both topics' mathematical frameworks.

FIX: Restructure part (c) to require students to explicitly apply conservation of momentum (using the decay products' momenta) and the Q-value definition together. For example: 'The antineutrino carries momentum p_ν. Using conservation of momentum and the Q-value, derive the relationship between the beta particle's kinetic energy and the antineutrino's energy, then explain why the spectrum is continuous.' Alternatively, in part (b) or a new part, ask students to identify the antineutrino mass constraint from the endpoint energy of the beta spectrum, directly coupling the decay physics (E.3) with the momentum measurement (A.2).
discriminator fail

Part (d) is the final part but asks only for an explanation of an observation (why a spectrum exists rather than a line) to support particle X's existence. It does not ask the student to critique the model, discuss limitations, or suggest reasons why calculated and experimental values differ. The ERQ lacks the required model-critique dimension in its terminal assessment item.

FIX: Replace part (d) with a question that explicitly asks the student to critique the model or discuss assumptions. For example: 'In part (c), you assumed that the recoil kinetic energy of the yttrium nucleus is negligible. Discuss whether this assumption is valid and suggest what effect a non-negligible recoil would have on the maximum possible energy of the antineutrino.' Alternatively: 'The Q-value of 0.546 MeV is tabulated, yet your calculated kinetic energy in (b)(ii) exceeds this value. Discuss one reason why this discrepancy arises and what it tells us about the model of beta decay used.'
rule_of_one fail

Part (c) is marked as 2 marks but lists 2 award criteria (M1: assumption; M2: energy calculation/recognition of impossibility). However, the mark scheme's M2 contains a complex logical gate—it awards credit only if the student recognises that the computed KE exceeds Q and concludes the decay is energetically impossible. This is not a simple 'substitution + answer' operation; it requires interpretation of a physically inconsistent result, which conflates evaluation (AO3) with calculation. For a 2-mark part, the scheme should clearly separate distinct procedural steps. The presence of an "alternative full-credit route" further muddies the award structure, making it impossible to verify that each of the 2 marks corresponds to exactly one distinct operation performed by a candidate.

FIX: Restructure Part (c) as a 3-mark part: M1 (state assumption about recoil), M2 (substitute Q and KE_β into E_X(max) = Q − KE_β), M3 (state the result and comment on physical consistency). Alternatively, if the part must remain 2 marks, collapse M1 and M2 into a single combined criterion: 'State assumption and calculate E_X(max), or recognise that KE_β > Q indicates this event cannot come from the given decay' (1 mark), then add a separate M2 for correct numerical answer with units. Remove the alternative route, or make it a transparent ECF clause rather than a separate award path.
topic fail

Topic E.3 (Radioactive decay) is only superficially invoked. Parts (a)–(b) test A.2 exclusively (magnetic force, momentum, circular motion, relativistic kinematics). Part (c) nominally mentions the Q-value and antineutrino but does not require understanding of radioactive decay mechanisms, decay constants, activity, or nuclear processes — only energy conservation. Part (d) references the continuous energy spectrum as evidence for the neutrino, but this is a consequence of energy/momentum conservation (A.2 + A.3) rather than decay physics per se. The question lacks any concept genuinely rooted in E.3: no decay modes, half-lives, decay schemes, branching ratios, nuclear structure, or decay kinematics are tested.

FIX: To properly test both topics, modify the question to require explicit radioactive-decay knowledge. For example: (i) Ask students to identify which decay branch (using a provided decay scheme for Sr-90) matches the observed Q-value; (ii) Include a question on activity or half-life: 'A sample of Sr-90 (t₁/₂ = 28.8 years) initially contains N₀ atoms. How many decays occur in the time interval during which a detector registers 1000 beta events?' (iii) Ask students to explain why certain decay modes are forbidden based on spin/parity selection rules; or (iv) Require calculation of the Q-value from nuclear masses or binding energies. Alternatively, replace E.3 with A.3 (Work, energy, power) or A.4 (Circular motion), both of which are genuinely central to the magnetic spectrometer scenario.

Iteration 2 — failed

ao_ladder fail

The ERQ violates the cognitive ladder: part (b)(i) is labelled AO2/AO3 with command term 'Show that', but parts (b)(ii) and (c) are also AO2/AO3 with 'Determine'. There is no clear progression to AO3 Explain/Discuss/Evaluate until part (d). Parts (b) and (c) must be reordered or reframed so that only (b)(i) is AO2, then (b)(ii)–(c) escalate to AO3 Explain/Suggest, or part (c) should be moved earlier as a pure calculation. Part (d) Discuss is correctly positioned as AO3 discriminator, but the middle tier lacks cognitive differentiation.

FIX: Restructure as follows: (a) State [2 marks] AO1; (b) Show/Determine [3+3=6 marks] AO2/AO3 calculation chain; (c) Determine duration [2 marks] AO2 (move to earlier/simpler tier); (d) Discuss discrepancy [2 marks] AO3 Explain/Suggest (keep as discriminator). Alternatively, split (b)(ii) into 'Determine KE using relativistic formula' [AO2, 3 marks] and add a new part (c) 'Explain why the measured KE exceeds Q-value' [AO3, 2 marks] before the duration calculation.
cross_theme fail

Topic E.3 (Radioactive decay) is invoked only nominally in the stem and part (c), but the actual physics solving in parts (a)–(c) relies entirely on A.2 (Forces and momentum). Part (d) mentions the Q-value discrepancy, but the core conceptual blend—where both decay kinematics AND momentum dynamics are genuinely interdependent in the solution—is absent. The decay context is window-dressing; the problem solves as a pure magnetic-deflection question.

FIX: Restructure so that E.3 enters the solution method, not just the setup. For example: (i) require students to use conservation of momentum in the three-body decay (beta, antineutrino, recoil nucleus) to predict the range of beta momenta for a given Q-value, then compare to the measured momentum from part (b)(i); or (ii) ask students to infer the Q-value from the measured beta KE and the known decay scheme, explicitly combining the energy-momentum balance of the decay with the magnetic momentum measurement. This way, both A.2 and E.3 concepts become essential to reaching the answer.
discriminator pass

Part (d) explicitly asks students to critique the model by suggesting one experimental reason and one modelling assumption that could account for the discrepancy between the measured kinetic energy (2.00 MeV) and the tabulated Q-value (0.546 MeV). This is a genuine model-critique task requiring students to evaluate assumptions (e.g., negligible recoil, ideal field, antineutrino momentum distribution, daughter-decay branches) and experimental limitations (field inhomogeneity, misalignment, background events), not merely perform a calculation.
rule_of_one fail

Part (b)(i) is marked as 3 marks but contains a critical internal inconsistency: M3 awards credit for 'a numerical answer consistent with candidate's substitution' yet the stem explicitly states 'Show that [the momentum is] approximately 1.31 × 10⁻²¹ kg m s⁻¹'. The mark scheme then notes that candidates obtaining 4.36 × 10⁻²² (from correct p = qBr substitution with given values) also earn M3, creating two valid answers for a single 'show that' part. This violates the IB contract that a 'show that' part requires candidates to reach the stated target value, not an alternative answer, and breaks the dimension's requirement that 3 marks = 3 distinct award criteria (formula / substitution / answer-with-units), since here M3 conflates acceptance of an incorrect numerical result with acceptance of the required target.

FIX: Restructure Part (b)(i): Either (1) revise the stem to remove the target value and make it a standard 'Calculate [3 marks]' part, in which case M3 awards the correct answer 4.36 × 10⁻²² only; or (2) keep 'Show that 1.31 × 10⁻²¹' but revise the mark scheme so M1 = formula (p = qBr), M2 = correct substitution of all four values (including any corrected/reinterpreted data), M3 = arrival at exactly 1.31 × 10⁻²¹ to 3 s.f. with units, and do not award M3 to candidates who obtain a different numerical result, regardless of method correctness. The marker note permitting B = 0.0375 T must be removed or explicitly added to the stem as a given value.
topic pass

The ERQ robustly integrates both topics. A.2 (Forces and momentum) is tested substantively in parts (a)–(c) via magnetic force direction, centripetal dynamics, relativistic momentum–energy relations, and activity–time calculations. E.3 (Radioactive decay) is tested in part (c) via half-life and activity concepts, and critically in part (d) where the discrepancy between the measured beta KE (2.0 MeV) and the Q-value (0.546 MeV) hinges on understanding beta-decay kinematics (three-body process, antineutrino momentum sharing, daughter-decay branches). Neither topic is superficial; both are essential to solving the problem coherently.

Planner output (stage 1)

Archetype: real-world device
Scenario: A particle detector in a nuclear physics laboratory uses a magnetic field to deflect charged particles produced by radioactive decay. When a beta particle (electron) is emitted from a decaying nucleus with initial velocity, it enters a perpendicular magnetic field region and follows a circular path. The detector measures the radius of curvature and initial momentum of the particle to identify the decay process and estimate the mass of the parent nucleus.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the condition for a charged particle to move in a circular path in a magnetic field, and define the Lorentz force.
b(i)	3	AO2	A beta particle with charge 1.6 × 10⁻¹⁹ C and mass 9.11 × 10⁻³¹ kg enters a magnetic field of 0.5 T at right angles. Show that if the radius of curvature is 0.02 m, the particle's momentum is approximately 1.6 × 10⁻²³ kg·m·s⁻¹.
b(ii)	2	AO2	Calculate the kinetic energy of the beta particle in MeV. (Use c = 3 × 10⁸ m·s⁻¹ and the rest energy of an electron = 0.51 MeV.)
c	3	AO3	The parent nucleus undergoes beta-minus decay. Determine the atomic number change and identify what conservation law(s) ensure the total momentum and energy are conserved in this decay process, given that a neutrino is also emitted.
d	2	AO3	Evaluate why the measured kinetic energy of the beta particle is often less than the Q-value predicted for the decay. Suggest how this observation supports the hypothesis of a third particle in the decay process.

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