Cambridge O Level Physics · Syllabus 5054 · Nuclear Physics
Half-Life
What is Half-Life?
The half-life of a particular isotope is the time taken for half the nuclei of that isotope in any sample to decay. It is constant for that isotope whatever the sample size, so the count rate and the number of undecayed nuclei halve every half-life.
This definition is part of the Nuclear Physics chapter in Cambridge O Level Physics.
Half-Life in context
Alpha-particle scattering shows that an atom is almost entirely empty space with a tiny, massive, positively charged nucleus at its centre. That nucleus holds protons and neutrons; the proton number fixes which element it is, and the nucleon number fixes which nuclide. Some nuclides are unstable, and an unstable nucleus changes spontaneously and at random, emitting an alpha particle, a beta particle and/or gamma radiation. Those three emissions differ in ionising effect and penetrating power, which is exactly what determines how we detect them, which isotope suits which job, and how we protect ourselves. Because individual decays are random but a large sample is not, the sample has a fixed half-life — and that single number carries carbon dating, activity calculations and isotope selection.
The half-life of a particular isotope is the time taken for half the nuclei of that isotope in any sample to decay. Because the same fraction decays in each equal interval, after \(n\) half-lives the fraction of the original undecayed nuclei remaining is \(\left(\tfrac{1}{2}\right)^{n}\). The same halving applies to the corrected count rate, once background has been subtracted. Half-life is a fixed property of the isotope: it cannot be changed by heating, cooling or chemical treatment, and it is the same however large the sample is.
Choosing a radioisotope is an engineering decision with two inputs: the type of radiation (which fixes the penetration and the ionising effect) and the half-life (which fixes how long the source stays useful and how long it remains a hazard). A good answer names the property and connects it to the job. "Gamma, because it is the strongest" names no property and connects to nothing.
Common mistakes with Half-Life
- Error Repair Common-Mistake Clinic Twenty-three errors, each with the reason it is wrong, the model that replaces it, a sentence you can safely write in an examination, and a check. The wrong statements are quoted so that you recognise them; every quoted misconception below is false. The full misconception register for Topic 5 The false claim Why it is wrong, and the corrected model Exam-safe sentence Retrieval check 1. "Most alpha particles hit the nucleus." It contradicts the primary observation, and misdescribes the mechanism. Deflection is caused by electrostatic repulsion at a distance between two positive charges, not by contact. Almost no alpha particle comes near a nucleus. "Most alpha particles passed straight through undeflected; a very small number were deflected through large angles because they passed close to a nucleus." Name the force and its sense. Electrostatic; repulsive. 2. "The nucleus occupies most of the atom." If it did, most alpha particles would be deflected. They were not. The nucleus is roughly \(10^{4}\) to \(10^{5}\) times smaller in diameter than the atom, though it holds nearly all the mass. "Most of the atom is empty space; the nucleus is very small but contains nearly all the mass." If the nucleus were larger, would large-angle deflections be more or less common? More. 3. "Electrons and protons have equal mass." Charge and mass are independent properties. The charges are equal and opposite; the masses differ by a factor of roughly \(1840\). "The proton and electron have equal and opposite relative charges, but the electron's relative mass is only about \(1/1840\)." Why does beta deflect more than alpha in the same field? Far smaller mass, so far larger acceleration. 4. "Isotopes have different proton numbers." Different proton number means a different element. Isotopes share \(Z\) and differ in \(N\), and therefore in \(A\). "Isotopes are atoms of the same element with the same proton number but different numbers of neutrons." Are \({}^{40}_{18}\mathrm{Ar}\) and \({}^{40}_{20}\mathrm{Ca}\) isotopes? No — different elements. 5. "An ion forms by changing its nucleus." Ion formation is entirely an electron event. The nucleus is bound far too tightly for ordinary chemical or electrostatic processes to change it. \(Z\) and \(A\) are unchanged. "An atom becomes a positive ion by losing electrons and a negative ion by gaining electrons; the nucleus is unchanged." \({}^{27}_{13}\mathrm{Al}^{3+}\): protons, neutrons, electrons? 13, 14, 10. 6. "Radioactive decay can be triggered on demand." Decay is spontaneous. Heating, cooling, compressing, dissolving or chemically reacting a source does not start, stop or alter it. "Radioactive decay is spontaneous: it is not caused or controlled by anything outside the nucleus." A source is heated to \(300\ ^{\circ}\mathrm{C}\). Effect on its activity? None attributable to the heating. 7. "An individual nucleus's decay time can be predicted." Decay is random. Every nucleus of a nuclide has the same fixed chance of decaying in the next interval, regardless of how long it has already survived. Only the large sample is predictable. "The decay of an individual nucleus is random and cannot be predicted, but the behaviour of a large sample is statistically predictable." Give both halves of that sentence from memory. Individual random; population predictable. 8. "Background radiation can be ignored." Background does not decay with the source and does not vanish when the source is removed. Leaving it in adds a fixed offset to every reading, which distorts anything based on halving. "The background count rate was measured with the source removed and subtracted from each reading to give the corrected count rate." Corrected \(200\), background \(20\). Counter reading after two half-lives? \(50 + 20 = 70\). 9. "Alpha is the most penetrating radiation." It swaps the two rankings. Alpha is the most ionising, which is precisely why it is the least penetrating — it loses its energy within a few centimetres. "Alpha is the most strongly ionising and the least penetrating; it is stopped by paper or a few centimetres of air." Unaffected by paper, stopped by \(3\ \mathrm{mm}\) of aluminium. Which radiation? Beta. 10. "Gamma is completely stopped by thin lead." Gamma is absorbed gradually: each extra thickness removes a further fraction of what remains. Thin lead removes very little; even thick lead reduces rather than eliminates. "Thick lead or concrete greatly reduces the intensity of gamma radiation; it is not completely absorbed." Is a thin lead sheet adequate gamma shielding? No — only a small fraction is absorbed. 11. "Penetration and ionisation mean the same thing." They are different quantities: ionisation counts ions produced per unit distance; penetration measures how far the radiation gets before absorption. They happen to run in opposite orders. "Gamma has the weakest ionising effect but the greatest penetrating power, so it passes through packaging while alpha cannot." Which property makes alpha dangerous inside the body? Its strong ionising effect. 12. "A beta particle is an orbital electron that escaped." The nucleus contains no electrons, and beta emission changes the nucleus — \(Z\) rises by one. Losing an orbital electron would leave \(Z\) unchanged and simply make a positive ion. "In beta decay a neutron in the nucleus changes into a proton and an electron; the electron is emitted as the beta particle." If beta were an orbital electron, what would happen to \(Z\)? Nothing — contradicting the observed change of element. 13. "Gamma emission changes \(A\) or \(Z\)." Gamma radiation carries no nucleons and no charge, so there is nothing in either ledger row for it to alter. The nucleus loses energy only. "In gamma emission neither the nucleon number nor the proton number changes; the nucleus loses excess energy." Alpha then gamma: total change in \(A\) and \(Z\)? \(-4\) and \(-2\). 14. "Deflection directions in a field diagram can be assigned freely." The direction is fixed by the field, not by the person drawing. "Alpha goes left" means nothing until the plate polarity or field direction is on the diagram. "With the upper plate positive, the alpha is deflected downwards towards the negative plate and the beta upwards towards the positive plate, and further because its mass is much smaller." Swap the plate polarities. What changes? Both charged paths reverse; gamma still goes straight. 15. "The moderator absorbs neutrons to control the reactor." That is the control rods' job. If the moderator absorbed the neutrons there would be nothing left to sustain the chain reaction. "The moderator slows the neutrons so that they are more likely to cause further fission." Raise the control rods: what happens? Fewer neutrons absorbed, so the reaction rate rises. 16. "Control rods slow neutrons." The mirror image of error 15, and equally costly. Slowing neutrons would increase the fission rate — the opposite of what control rods are for. "Control rods absorb neutrons; lowering them into the core reduces the rate of the chain reaction." Match the verbs: slows / absorbs / transfers energy away. Moderator / control rods / coolant. 17. "Fission and fusion are identical." They run in opposite directions. Fission splits one heavy nucleus; fusion joins two light ones. Only "energy is released" is shared. "Fission is the splitting of a heavy nucleus after it absorbs a neutron; fusion is the joining of two small nuclei into a larger one." Which one produces the neutrons that sustain a chain reaction? Fission. 18. "Half the sample's mass disappears each half-life." Half the undecayed nuclei of that isotope decay. They become daughter nuclei, which are still in the sample. Nothing disappears. "Half-life is the time for half the nuclei of that isotope in the sample to decay; the daughter nuclei remain in the sample." After one half-life, has the sample lost half its mass? No. 19. "The raw count rate always halves." The raw reading is source plus background. Only the source part decays, so the raw reading falls towards the background level, not towards zero, and successive raw halvings take longer and longer. "Background was subtracted from each reading before the half-life was determined, because only the corrected count rate halves." Background \(30\), corrected \(240\), \(T_{1/2} = 4\ \mathrm{h}\). Counter reading at \(8\ \mathrm{h}\)? \(90\). 20. "After a few half-lives no nuclei remain." Each half-life halves what is left, and half of something is never nothing. After \(10\) half-lives about \(0.1\%\) of the original undecayed nuclei remain. "The activity continues to halve and approaches zero, but does not reach exactly zero after a finite number of half-lives." Fraction after \(5\) half-lives? \(\tfrac{1}{32}\), about \(3.1\%\). 21. "Food irradiation makes the food radioactive." The food is irradiated, not contaminated. The source stays outside; no radioactive material is added to the food. "The food is exposed to gamma radiation which kills bacteria; the food does not become radioactive, because no radioactive material is added." Is a patient radioactive after an X-ray? No — irradiated, not contaminated. 22. "External alpha radiation is always harmless." "Less penetrating" is not "harmless". Alpha can damage the eye and exposed tissue it does reach, and direct handling risks contamination, which converts a low external hazard into a high internal one. "An alpha source is a relatively low hazard from outside the body because alpha is absorbed by air and by the outer layer of skin, but alpha-emitting material inside the body is especially damaging." Why does preventing ingestion matter more than shielding for an alpha emitter? Shielding already handles the external hazard; nothing helps once it is inside. 23. "A radioactive source can be described as completely safe." Ionising radiation carries risk at any dose reaching living tissue, and the risk of effects such as cancer rises with dose. "Safe" is an absolute the physics does not support. "The precautions reduce the dose received to a very low level, so the risk is small; the source is not without risk." Rewrite "gamma is dangerous, alpha is safe". "Gamma is the greater external hazard, alpha the greater internal one; neither is without risk."
Examiner tips on Half-Life
- Command Words Examiner-Language Guidance The command word tells you the shape of the answer required. Getting the shape wrong leaves the question unanswered even when the physics is right — a beautiful explanation does not answer "State" if it never states the thing. Command words as they apply to Topic 5 Command wordWhat the answer must containWorked shape from this topic StateThe fact, expressed in clear terms. No justification is asked for, so do not spend time on one."State the relative charge of an alpha particle." → "\(+2\)." Do not write a paragraph. DefineThe formal definition, in the standard wording, including every qualifying phrase."Define half-life." → "The time taken for half the nuclei of that isotope in any sample to decay." Dropping "of that isotope" weakens it. DescribeWhat happens, in the right order. No causes required unless asked."Describe the process of fission." → a nucleus such as U-235 absorbs a neutron, becomes unstable, splits into two daughter nuclei, releasing energy and two or more neutrons. ExplainCause and mechanism, not just outcome. Look for the word "because" in your own answer — if it is missing, you have probably described rather than explained."Explain why beta deflects more than alpha." → "Because its mass is far smaller, so the same order of force produces a much larger acceleration." CalculateWorking, substitution, answer, unit. Show the relationship before the numbers."Calculate the corrected count rate." → show both rates in the same unit, then the subtraction, then the answer with its unit. DetermineUsually means "get it from the data or the graph". Show which values you read off and where."Determine the half-life from the graph." → mark the two points used and show the time interval between them. IdentifyThe name of the thing: naming, selecting or recognising it is what is asked for. Quoting the evidence you used is good practice and is what a follow-up "explain" would want."Identify the radiation." → "Beta — the corrected rate was unchanged by paper but fell to zero with \(3\ \mathrm{mm}\) of aluminium." SuggestApply known physics to a situation you have not been taught directly. There may be more than one acceptable answer, but it must be justified."Suggest why a shorter half-life is preferred for a substance given to a patient." → the activity falls away soon after the procedure, limiting the patient's continuing exposure. CompareStatements about both things, on the same property. "A is strong" is not a comparison."Compare alpha and gamma." → "Alpha is more strongly ionising than gamma, but gamma is far more penetrating than alpha." DeduceA conclusion drawn from the information available, together with the step in the data that forces it. Do not assert the conclusion on its own."Deduce the daughter nuclide." → subtract \(4\) from \(A\) and \(2\) from \(Z\), then name the element that has that proton number. Phrase bank: what to write, and what to avoid writing Replace the left column with the right column Avoid — vague or wrongWrite instead — precise and creditableWhy "Alpha is the strongest.""Alpha is the most strongly ionising and the least penetrating.""Strong" does not name a property, and the two properties run in opposite orders. "Lead stops gamma.""Thick lead greatly reduces the intensity of gamma radiation."Gamma is absorbed gradually; "stops" is an overstatement. "The count rate halves.""The corrected count rate halves."The raw reading falls towards background, not towards zero. "Be careful with the source.""Handle it with tongs to increase the distance, and return it to its shielded container immediately to reduce the exposure time."A precaution scores when it names the quantity being changed. "The atom loses a proton to become an ion.""The atom loses an electron to become a positive ion; the nucleus is unchanged."Ion formation is an electron event. "The nucleus splits into two identical halves.""The nucleus splits into two daughter nuclei, which are usually of unequal size and are not the same every time."Fission products vary between events. "Radiation is dangerous.""Ionising radiation can cause cell death, mutations and cancer, and the risk increases with the dose received."Name the effects and the dependence. "The source ran out.""The activity fell until the corrected count rate was too small to measure reliably against the background."Sources do not run out; their activity falls. "It gets deflected in the field.""With the upper plate positive, it is deflected downwards towards the negative plate."A direction is meaningless without a defined field. "Half the atoms disappear.""Half the nuclei of that isotope decay, forming daughter nuclei which remain in the sample."Nothing disappears; it changes. Before you write a single Topic 5 sentence, ask three things: which particle or emission am I talking about; which property am I using (ionising, penetrating, charge, mass, half-life); and which number changed (\(A\), \(Z\), corrected count rate, time)? An answer that names all three is almost always creditable. An answer that names none of them almost never is.
Questions students ask about Half-Life
What does it mean to say that radioactive decay is spontaneous and random?
Spontaneous means the decay is not triggered by anything outside the nucleus — it is not caused by temperature, pressure or chemical treatment. Random means it is impossible to predict which nucleus in a sample will decay next, or exactly when; only the overall behaviour of a large sample, described by its half-life, is predictable. Both words are needed together to describe radioactive decay fully.
Last reviewed Syllabus 2026-2028