Physics • Year 12 • Module 8 • Lesson 16
Particles and Antiparticles
Build HSC Band 5–6 extended-response technique: analyse experimental evidence for antimatter, evaluate conservation laws in multi-step interactions, and design an investigation.
1. Data + scenario: Carl Anderson’s cloud chamber discovery (Band 5–6)
8 marks Band 5–6
Scenario. In August 1932, Carl Anderson photographed a cosmic-ray track in a cloud chamber placed in a 1.5 T magnetic field. A lead plate across the centre of the chamber slowed the particle as it passed through. Anderson observed: (1) the track curved upward in a direction consistent with a positively charged particle; (2) the radius of curvature above the lead plate was smaller than below, indicating the particle slowed down after passing through the lead (it was moving upward); (3) the track density and curvature were indistinguishable from those of an electron, except for the opposite charge sign; (4) Anderson concluded the particle was a “positive electron” with the same mass as an ordinary electron.
| Observation | Anderson’s measurement |
|---|---|
| Magnetic field strength | 1.5 T |
| Radius of curvature below lead plate | ~5.0 cm (before energy loss) |
| Radius of curvature above lead plate | ~2.1 cm (after energy loss) |
| Charge sign deduced from field direction | Positive |
| Track “grain density” (ionisation per cm) | Same as electron tracks |
Illustrative data based on C.D. Anderson, Physical Review 43, 491–494 (1933).
Q1. Analyse and evaluate Anderson’s experimental data to assess the evidence that the observed particle was the positron (antielectron) predicted by Dirac. In your response you must:
- Explain how the curvature of the track provides evidence for the particle’s charge sign and momentum, using the relationship between magnetic force and circular motion.
- Use the smaller radius above the lead plate (compared to below) to determine the direction of travel of the particle, and explain why this was critical to Anderson’s identification.
- Explain how the track ionisation density was used to determine the particle’s mass, and what it indicated.
- Assess whether the data are consistent with Paul Dirac’s prediction of the positron, using at least two specific pieces of evidence.
- State one limitation of the cloud chamber method for identifying new particles and suggest how modern detectors have improved on this.
2. Experimental design — verifying the photon energy in electron–positron annihilation (Band 5–6)
7 marks Band 5–6
Research question. A Year 12 student states: “If electron–positron annihilation always produces two photons each with energy 0.511 MeV, then a scintillation detector placed at 180° to a second detector should always register a coincident event at exactly 0.511 MeV, with no coincidences at other energies or other angles.” Design a controlled investigation to test this hypothesis using a sodium-22 (²²Na) positron source, two sodium iodide (NaI) scintillation detectors, a coincidence circuit, and a multi-channel analyser (MCA).
Constraints: The investigation must be completable in a university teaching laboratory in one afternoon session (~3 hours). Safety: the radioactive source is sealed and handled by the supervising physicist.
Q2. Design the investigation and present it in the format below.
- State your hypothesis as a precise, testable prediction including independent variable (angle between detectors), dependent variable (coincident photon energy), and at least two controlled variables.
- Describe the procedure in at least four numbered steps, including how you will verify the 180° coincidence requirement and how you will measure photon energies.
- Explain what result would falsify your hypothesis.
- State two limitations of this design (consider sources of error specific to annihilation experiments) and one way to improve validity.
Q1 — Sample Band 6 response (8 marks), annotated
Charge sign and momentum from curvature: A charged particle moving through a magnetic field experiences a force F = qvB perpendicular to its velocity, causing it to travel in a circular arc. The direction of curvature reveals the sign of the charge: in Anderson’s field configuration, the track curved in the direction expected for a positive charge, not negative [1]. The radius of curvature r = mv/(qB), so a larger radius indicates higher momentum mv [1].
Direction of travel from radius change: The particle had a larger radius below the lead plate (r ≈ 5.0 cm) and a smaller radius above (r ≈ 2.1 cm). Because the lead plate slows the particle by ionisation loss, the smaller radius above the plate means the particle was moving upward (it had already lost energy). This was critical: if Anderson had not established the direction, he could not distinguish between a slow positive particle moving downward and a fast positive particle moving upward — the two would give opposite charge interpretations in some field geometries [1].
Mass from ionisation density: The density of ion tracks (droplets per unit length) in a cloud chamber depends on the particle’s charge and speed: dE/dx ∝ z²/v². Since the track density was identical to that of an electron track at the same speed, the particle’s mass and charge magnitude must match those of the electron. A heavier particle (e.g. proton) at the same momentum would be moving more slowly, creating denser ionisation. The equal ionisation density confirmed the particle’s mass was approximately equal to the electron mass, 9.11 × 10−31 kg [1].
Assessment against Dirac’s prediction: Dirac’s relativistic quantum equation predicted that for every spin-½ fermion there should exist an antiparticle with equal mass but opposite charge. The data are highly consistent with this prediction: (1) the particle’s charge is +1e (opposite to the electron’s −1e) [evidence 1, 1 mark]; (2) the mass is indistinguishable from the electron mass, as shown by the equal ionisation density [evidence 2, 1 mark]. Both criteria match Dirac’s positron exactly, providing strong experimental support for the theory of antimatter [evaluation, 1 mark].
Limitation and modern improvement: A limitation of the cloud chamber is that it is a passive detector — it cannot trigger on events, so the vast majority of cosmic-ray particles are not recorded, and the method relies on chance photographs. Additionally, cloud chambers cannot directly measure the particle’s energy without the lead plate trick, introducing uncertainty [1]. Modern silicon strip trackers combined with electromagnetic calorimeters can measure both the track curvature (momentum) and the total energy deposited, allowing precise mass determination E² = (pc)² + (mc²)² without the ambiguity of cloud chamber images [1].
Marking criteria summary (8 marks): 1 = correct explanation of charge sign from track curvature direction using F = qvB; 1 = correct use of r = mv/qB to relate radius to momentum; 1 = correctly uses radius decrease above plate to determine upward travel and explains why this mattered for identification; 1 = correctly links ionisation density to mass determination and explains why equal density implies electron mass; 1 = first specific piece of evidence consistent with Dirac’s prediction (opposite charge); 1 = second specific piece of evidence (equal mass); 1 = evaluative statement assessing overall consistency with Dirac’s prediction; 1 = valid limitation named and a specific modern improvement stated.
Q2 — Sample Band 6 response (7 marks), annotated
Hypothesis: If electron–positron annihilation produces two photons each with energy 0.511 MeV by conservation of energy, and they travel in exactly opposite directions by conservation of momentum, then: (1) the coincidence count rate will be a maximum when the two NaI detectors are placed at 180° to each other and will decrease as the angle deviates from 180°; and (2) the MCA will show a photopeak at exactly 0.511 MeV in each detector during coincidence events. Independent variable: angle θ between the two detector axes (0°–180°). Dependent variable: coincident count rate (counts/s) and MCA photopeak energy (MeV). Controlled variables: distance from source to each detector (e.g. 15 cm), coincidence time window (e.g. 10 ns), detector type and gain, measurement duration per angle (e.g. 5 min) [1 — hypothesis with IV, DV, controlled].
Procedure: (1) Set up the ²²Na source at the centre of a rotating goniometer; place detector A fixed at 0° and detector B on a rotating arm. Connect both to a coincidence circuit with a 10 ns time window and then to the MCA. (2) With detectors at 180°, collect a 5-minute spectrum. Identify the 0.511 MeV photopeak in both channels and record the coincidence count rate. (3) Rotate detector B in 15° steps from 180° to 90°; at each angle collect a 5-minute spectrum and record the coincidence rate and photopeak energy. (4) Plot coincidence rate vs angle and photopeak energy vs angle. Fit a Gaussian to each energy spectrum to determine the peak centroid and full-width at half-maximum (FWHM). [1 — four steps with angle variation and energy measurement]
Falsification: If the coincidence rate does not peak sharply at 180° but is equal at all angles, or if the photopeak energy differs significantly from 0.511 MeV, the hypothesis would be falsified — suggesting the annihilation photons are not emitted back-to-back or do not carry the expected energy [1].
Limitations: (1) Doppler broadening: positrons from ²²Na are not fully at rest when they annihilate in tissue or the source housing; the small residual momentum of the electron–positron pair causes the two photons to deviate slightly from exactly 180° (by ~0.5°) and the photopeak energy to broaden slightly around 0.511 MeV. This means the measured peak will be slightly broader than predicted by theory alone [1]. (2) Compton scattering in the detector crystals or surrounding material shifts some events to lower energies, creating a broad Compton shoulder below the 0.511 MeV peak that could be misread as lower-energy photons being produced [1].
Improvement: Use lead collimators (slits) aligned between the source and each detector to reduce detection of scattered photons; this improves the angular resolution of the coincidence measurement and reduces the Compton background [1].
Marking criteria summary (7 marks): 1 = testable hypothesis naming IV (angle), DV (coincidence rate and energy), and at least two controlled variables; 1 = four clear procedure steps including angle variation and MCA energy measurement; 1 = states what result would falsify the hypothesis (equal rate at all angles, or wrong energy); 1 = valid limitation 1 with physical explanation (Doppler broadening or detector geometry); 1 = valid limitation 2 (Compton scattering or another source of error); 1 = specific improvement addressing one limitation; 1 = precise terminology throughout (coincidence circuit, photopeak, conservation of momentum, 0.511 MeV, Doppler broadening or Compton scatter).