Nuclear physics

• A nucleus weighs less than its parts, and the missing mass is the binding energy. Both fusion and fission climb the binding-energy-per-nucleon curve towards iron. Decay is random for one nucleus and exponential for many.
• Nucleons minus nucleus is the mass defect; c2 times it is the binding energy. Both fusion and fission climb towards iron. Random for one nucleus, exponential for many: λ = 0.693/t½, A = λN.

• 23.1.1 I can understand — understand the equivalence between energy and mass as represented by E = mc² and recall and use this equation
• 23.1.2 I can represent — represent simple nuclear reactions by nuclear equations of the form ¹⁴₇N + ⁴₂He → ¹⁷₈O + ¹₁H
• 23.1.3 I can define — define and use the terms mass defect and binding energy
• 23.1.4 I can sketch — sketch the variation of binding energy per nucleon with nucleon number
• 23.1.5 I can explain — explain what is meant by nuclear fusion and nuclear fission
• 23.1.6 I can explain — explain the relevance of binding energy per nucleon to nuclear reactions, including nuclear fusion and nuclear fission
• 23.1.7 I can calculate — calculate the energy released in nuclear reactions using E = c²Δm
• 23.2.1 I can understand — understand that fluctuations in count rate provide evidence for the random nature of radioactive decay
• 23.2.2 I can understand — understand that radioactive decay is both spontaneous and random
• 23.2.3 I can define — define activity and decay constant, and recall and use A = λN
• 23.2.4 I can define — define half-life
• 23.2.5 I can use — use λ = 0.693/t½
• 23.2.6 I can understand — understand the exponential nature of radioactive decay, and sketch and use the relationship x = x0e−λt, where x could represent activity, number of undecayed nuclei or received count rate

The largest source of uncertainty is the random fluctuation of the counts themselves, largest in proportion at late times, when each count is only about 100. Improvement: make each count larger — count for longer intervals, or bring the tube closer to the source so that it receives more of the radiation — and repeat the whole run several times and average the counts at each time. Reading the stopwatch is not the limitation: the counter itself times the 10 s interval.

• “Mass defect = mass of the nucleus minus the mass of its nucleons, and I can work it out with mp = 1.67 × 10−27 kg.” Correct Mass defect = total mass of the separate nucleons − mass of the nucleus, a positive quantity, and it uses supplied masses to six figures, in u. The defect is a difference in the third or fourth significant figure of numbers near 1 u per nucleon; the three-figure Data-sheet mp cannot resolve it, and the neutron mass is not on the Data sheet at all.
• “1 u = 931.5 MeV, so I use that with c = 3.00 × 108 m s−1 and 1 u = 1.66 × 10−27 kg.” Correct With the Data-sheet u, c and e, 1 u × c2 = 1.49 × 10−10 J = 934 MeV. 931.5 MeV comes from more precise constants. Use one set throughout a calculation; use 931.5 MeV only when a question gives it.
• “Binding energy is the energy stored in the nucleus that comes out when it splits.” Correct Binding energy is the energy needed to separate the nucleus into its separate nucleons. Energy is released in fission because the products have more binding energy than the uranium nucleus.
• “The nucleus with the biggest binding energy is the most stable.” Correct Compare binding energy per nucleon. Uranium-238 has a larger total than iron-56 (about 1800 MeV against 493 MeV) but a smaller value per nucleon (7.6 MeV against 8.8 MeV).
• “Decay is random: the count rate goes down.” Correct A falling count rate shows decay, not randomness. The evidence for randomness is the fluctuation of repeated counts, taken under identical conditions, about their mean.
• “x = x0e−λt and λ = 0.693/t½ must be learned; E = mc2 and A = λN are on the sheet.” Correct The other way round. x = x0e−λt and λ = 0.693/t½ are given; E = mc2 and A = λN are recall. And λ and t½ must be in reciprocal units: s−1 with s, h−1 with h.
• “I plotted ln of the count rate I measured and it came out straight.” Correct Subtract the background count rate from every reading first. Only the source's own rate decays to zero; the measured rate levels off at the background, and its ln graph curves.
• “E = mc2 only applies to nuclear reactions.” Repair It applies to every energy change. Only in nuclear reactions is the mass change large enough to measure.
• “Mass defect = mass of the nucleus − mass of the nucleons.” Repair The other way round: total mass of the separate nucleons − mass of the nucleus. It is positive.
• “I used mp = 1.67 × 10−27 kg from the Data sheet to find the mass defect.” Repair Three figures cannot resolve a difference in the third or fourth. Use the supplied masses in u, to six or more figures.
• “1 u = 931.5 MeV, and 1 u = 1.66 × 10−27 kg, in the same calculation.” Repair With the Data-sheet u, c and e, 1 u ≡ 934 MeV. Keep one consistent set; use 931.5 MeV only when the question gives it.
• “I used the atomic mass of uranium and the nuclear masses of the fragments.” Repair Mixed lists leave electron masses unbalanced — 56 of them, 0.031 u, here. Use nuclear masses throughout, or atomic masses throughout when the electrons balance.
• “There are three neutrons on the right, but I counted 1 in the nucleon total.” Repair The coefficient multiplies: 310n contributes 3 to the nucleon number and 3 neutron masses to the right-hand side.
• “Fusion releases energy because the product has less binding energy.” Repair The product has more binding energy, per nucleon and in total, and so less mass. The mass lost is released.
• “The binding-energy curve peaks at uranium, which is why uranium is used in fission.” Repair It peaks near A ≈ 56–62 (iron, nickel) at about 8.8 MeV. Uranium is on the falling side at about 7.6 MeV; fission moves its nucleons up towards the peak.
• “Heating a source makes it decay faster.” Repair Decay is spontaneous: unaffected by temperature, pressure or chemical combination.
• “Activity is the number of radioactive nuclei in the sample.” Repair Activity is the number of nuclei decaying per unit time, in Bq. The number of undecayed nuclei is N, and A = λN.
• “λ = 0.693 × t½, and I used t½ in days to get A in Bq.” Repair λ = 0.693/t½, in the reciprocal unit of t½. For A in Bq, λ must be in s−1, so convert the half-life to seconds first.
• “After two half-lives the sample has completely decayed.” Repair A quarter remains. After n half-lives the fraction remaining is (½)n; it never reaches zero.
• “Half-life is the time for the mass of the sample to halve.” Repair It is the time for the number of undecayed nuclei (or the activity) to halve. The decay products remain in the sample.
• “In the first hour of a 2.60 h half-life, 1/2.60 of a half, 19%, decays.” Repair Decay is exponential, not linear: 1 − e−0.693 × 1.0/2.60 = 23% decays.
• “I plotted ln of the measured count rate against time and the line curved at the end.” Repair Subtract the background from every reading first; the source's own rate gives a straight line.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Interleave with the chapters that use this one. When you reach topic 24, re-answer 23.1.1: the energy of each annihilation photon in PET is mec2, and the absorption of X-rays and ultrasound follows the same exponential form as 23.2.6. Looking back, compare the half-life here with the halving time 0.693RC of a discharging capacitor in topic 19. Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 23 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• There is no multiple-choice paper on A Level content. A Paper 4 question on this topic asks you to define mass defect, binding energy, activity, decay constant or half-life in the standard form; to sketch binding energy per nucleon against nucleon number, or an exponential decay curve, with its key features; and to explain why fusion and fission release energy, or what the fluctuation of a count rate shows.
• Energies released or absorbed from supplied masses in u, by E = c2Δm, in J and MeV; binding energies and binding energy per nucleon; activities from A = λN; values and times from x = x0e−λt. E = mc2 and A = λN must be recalled; x = x0e−λt and λ = 0.693/t½ are given on the formulas sheet, and c, u and e are on the Data sheet.
• The half-life of a short-lived source as a Paper 5 context: counts in fixed intervals measured with a Geiger–Müller tube and counter against time, the background measured first and subtracted, ln R plotted against t so that the gradient is −λ, error bars on the logarithms and a worst acceptable line. The largest uncertainty is the random fluctuation of the counts themselves.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.