Radioactivity & Decay Laws Radioactivity & Decay Laws Some atomic nuclei are not happy staying as they are. They are a bit like a winding toy that slowly releases energy without outside help. Such a nucleus changes itself into a more stable one by emitting particles or energy: α (helium nucleus), β (electron or positron with an antineutrino or neutrino), or γ (a high-energy photon). The key feature is randomness at the level of a single nucleus: no one can predict which specific nucleus will decay at which instant. But for a large crowd of identical nuclei, the statistics become smooth and entirely predictable. That is why we can write neat formulas and make accurate predictions about how fast the activity drops, how many nuclei remain after a given time, or how long it takes for the sample to fall to half its initial size. This is the heart of the radioactive decay law. Real life connects everywhere: in hospitals (cancer therapy beams, tracers), in archaeology (radiocarbon dating), in smoke detectors, in power plants, and in the cosmos (radioactive chains in stars and Earth’s interior). Despite these diverse settings, the same mathematics rules them all: an exponential law with a decay constant , a half-life T 1/2 that is the time to reduce the sample to half, a mean life that is the statistical average lifetime, and activity A that counts decays per second. These quantities depend on the nucleus itself, not on temperature, pressure, or chemical state (with a few subtle exceptions like electron capture). For exam problems, this unity is a blessing: once you master the core equations and their domains, most questions become substitutions and careful unit handling. At the core is a simple idea: in a tiny interval of time t , the probability that a nucleus decays is proportional to t . The proportionality constant is with units s -1 . From this, differential equations lead to the famous N(t) = N 0 e - t and A(t) = A 0 e - t , where A = N . Graphically, N or A falls smoothly; on a semilog plot, you get a straight line with slope - . Half-life and mean life are just different summaries of the same exponential: T 1/2 = ( 2)/ and = 1/ . With these, you can analyze single decays, find ages of artifacts, or handle more complex chains where a daughter grows as the parent decays. The physics also tells us what each radiation type can and cannot do. α is heavy and doubly charged, so it ionizes strongly but gets stopped quickly (paper or a few cm of air). β is lighter, more penetrating than α, and easily bent by fields. γ is neutral and extremely penetrating; you need dense shielding like lead or thick concrete. In exams, you will often match properties, predict deflection in fields, or select proper shielding. In numericals, you will connect N , A , , T 1/2 , and use exponential relations cleanly and consistently. Think of a very large crowd flipping fair coins repeatedly. You can never guess an individual flip, but you can predict the crowd’s totals. Radioactive decay is the nuclear version of this: inherently random per nucleus, beautifully predictable in bulk. remember Spontaneous transformation of an unstable nucleus into a more stable one with emission of α, β, or γ radiation. It does not require external energy input. Radioactivity Probability per unit time that a nucleus decays. Units: s -1 (or min -1 , y -1 , etc.). Larger means faster decay. Decay constant ( ) Half-life ( T 1/2 ) Time taken for the number of undecayed nuclei to reduce to half its initial value. Related to by T 1/2 = ( 2)/ . Average lifetime of nuclei in a sample. For exponential decay, = 1/ and 1.44 ,T 1/2 . Mean life ( ) Activity ( A ) Number of decays per second: A = - d N/ d t = N . SI unit: becquerel (Bq). SI unit of activity: 1 Bq = 1 decay per second. Convenient for small activities. Becquerel (Bq) Curie (Ci) Older unit of activity: 1 , Ci = 3.7 10 10 , Bq . Used for strong sources; μCi and mCi are common. The nucleus produced after a decay event from the parent nucleus. It may be stable or radioactive. Daughter nucleus Branching ratio Fraction of decays that follow a given mode or path: b i = i/ total , with b i = 1 . Radioactive equilibrium A condition in decay chains where the daughter’s activity becomes related in a simple way to the parent’s activity: secular (parent half-life ≫ daughter) or transient (parent half-life slightly larger). Secular: T 1/2, parent T 1/2, daughter , activities nearly equal after build-up. Transient: comparable half-lives; daughter activity peaks then follows parent with a fixed ratio for a time. Secular/transient equilibrium Radioactive emissions differ strongly in mass, charge, penetration, and responses to fields. α particles ( 4 2 He nuclei) are heavy and doubly positive, causing intense ionization but traveling only a short distance. β particles (electrons or positrons) are much lighter, moderately ionizing, easily deflected, and more penetrating. γ rays are photons: no mass, no charge, very penetrating, and unaffected by electric or magnetic fields. These contrasts power multiple exam items from matching columns to ray path predictions in fields. Nature He nucleus 4 2 He Electron (β−) or positron (β+), plus neutrino/antineutrino Photon (EM radiation) Charge +2e ±e Mass ≈ 4 u ≈ 1/1836 u (electron mass) Penetration Very low (stopped by paper/skin/few cm air) Moderate (aluminum mm–cm) High (lead cm or concrete tens of cm) Ionization power Very high Moderate Low Deflection in E/B fields Yes (small curvature, heavy and slow) Yes (large curvature) No Speed ≈ few % of c Up to near c Energy spectrum Discrete (monoenergetic lines) Continuous (due to neutrino) Discrete lines (de-excitation of nucleus) Property α (alpha) β (beta) γ (gamma) Radioactive Decay Modes Decay Type Particle Emitted Change in Z Change in A Penetrating Power Alpha drops 4-2, Beta flips Z by one, Gamma keeps the count while the energy is done. Alpha Decay ( ) Helium Nucleus ( 4 2He ) Decreases by 2 Decreases by 4 Low (Stopped by paper) Beta Minus Decay ( - ) Electron ( 0 -1 e / - ) Increases by 1 No change ( 0 ) Moderate (Stopped by Aluminium) Beta Plus Decay ( + ) Positron ( 0 +1 e / + ) Decreases by 1 No change ( 0 ) Moderate (Stopped by Aluminium) Gamma Decay ( ) Photon ( ) No change ( 0 ) No change ( 0 ) High (Stopped by Lead/Concrete) Electron Capture ( EC ) Neutrino (captured e - ) Decreases by 1 No change ( 0 ) Low (Secondary X -rays emitted) Neutron Emission Neutron ( 1 0n ) No change ( 0 ) Decreases by 1 Very High (High flux) Proton Emission Proton ( 1 1p ) Decreases by 1 Decreases by 1 Moderate radioactive decay modes Safety and diagnostics depend on these differences. α sources are dangerous if ingested or inhaled but relatively safe outside the body. β sources require plastic or aluminum shielding to avoid bremsstrahlung in very high-Z shields. γ sources need dense shielding (Pb, concrete) and time–distance–shielding protocols. In numericals, you typically only need to know the qualitative ordering of penetration and ionization. Definition of activity Activity is proportional to the number of undecayed nuclei. sets the decay rate per nucleus. The change in number is proportional to the present number, with proportionality - . Radioactive decay differential equation Start from the proportional loss rate. Separate variables and integrate. Evaluate the integrals. Exponentiate both sides. Activity follows the same exponential. Decay probability in a small interval is proportional to the interval: P = , t Large ensemble of identical, independent nuclei No replenishment of the parent species N(t) = N 0 e - t , A(t) = A 0 e - t Exponential decay law Interpretation: On a normal plot of N vs t , the curve falls smoothly, never touching zero in finite time. On a semilog plot (plot N vs t ), the data lie on a straight line with slope - . This is a powerful experimental test: fit a line to N vs t and read from its slope. Exponential solution Number of undecayed nuclei at time t . Calculates the number of radioactive nuclei remaining at a specific time, given the initial amount and the decay constant. Activity-time relation Activity decays exponentially with the same . The activity or number of atoms remaining decreases exponentially over time, governed by the constant decay rate, . Half-life in terms of decay constant T 1/2 = 2 Pure exponential decay Half-life T 1/2 defined by N(T 1/2 ) = N 0/2 Apply definition using the exponential form. Divide by N 0 . Take natural logarithm. = 1 = T 1/2 2 1.44 ,T 1/2 Exponential decay probability density f(t) = e - t for t 0 Definition of mean life = 0 t f(t) , d t Mean life of exponentially decaying nuclei Write the average lifetime integral. Substitute to simplify the integral. Factor out constants. Standard gamma integral (2)=1! = 1 . Semilog plot of exponential decay gives a straight line. Slope magnitude yields the decay constant. Intercept ln N0 ln N0 T1/2 ln(N0/2) At half-life Slope = -λ ln N0 - λ t A straight line with negative slope representing ln N = ln N0 - λ t. control dependent Time t custom ln N Fraction N/N 0 remaining and percentage decayed. Use either N/N 0 = (1/2) t/T 1/2 or N/N 0 = e - t with = ( 2)/T 1/2 . A radioactive sample has half-life 10 days. What fraction of nuclei remain after 30 days? Also find the percentage decayed. easy Half-life T 1/2 = 10 , d Time t = 30 , d Both half-life steps and exponential forms are equivalent. For multiples of T 1/2 , the power-of-2 method is quickest. For arbitrary times, e - t is universal. Always track units of time consistently. medium A = 1.0 , Ci , 1 , Ci = 3.7 10 10 , Bq T 1/2 = 5.27 , y Molar mass 60 , g ,mol -1 Avogadro constant N A = 6.022 10 23 , mol -1 A Co-60 source has activity A = 1.0 , Ci . The half-life of Co-60 is 5.27 y. Find (i) the number of Co-60 nuclei present and (ii) the mass of Co-60 in the source. Use A = N , = ( 2)/T 1/2 ; then m = (N/N A) M . (i) N and (ii) mass m of Co-60. Unit memory: 1 Ci = 3.7× 10 10 Bq, 1 mCi = 3.7× 10 7 Bq, 1 μCi = 3.7× 10 4 Bq. Always convert activity to SI (Bq) if time is in seconds. tip In decay chains, a parent P decays to a daughter D. If D is also radioactive, its number first grows then decays. The exact solution shows build-up over a few daughter half-lives; afterward, in secular equilibrium, daughter activity nearly equals parent activity. In transient equilibrium (comparable half-lives), the daughter activity peaks at a finite time and then follows the parent downwards with a fixed ratio. Successive decay (parent → daughter) For parent decay constant 1 and daughter 2 ; valid if no initial daughter and no branching. This formula describes the concentration of a daughter nuclide ( N 2 ) formed from a parent nuclide ( N 1 ) undergoing radioactive decay, as Time t eq when A 2 = A 1 . hard T 1/2,1 = 6.0 , h T 1/2,2 = 2.0 , h A 1(t) = 1 N 1(t) , A 2(t) = 2 N 2(t) For zero initial daughter, the equality A 2(t) = A 1(t) occurs at t eq = 1 2- 1 ! ( 2 1 ) (when 2 > 1 ). A parent P has T 1/2,1 = 6.0 , h and its daughter D has T 1/2,2 = 2.0 , h . Initially, N 2(0)=0 . At what time does the daughter’s activity equal the parent’s activity (transient equilibrium time)? In secular equilibrium ( T 1/2,1 T 1/2,2 ), after an initial build-up phase of a few daughter half-lives, A 2 A 1 and this near-equality persists while the parent supply changes very slowly. In transient equilibrium (comparable half-lives with T 1/2,1 > T 1/2,2 ), A 2 rises to a peak at t eq and then falls, tracking the parent with a constant ratio for some interval. neet-alert In chain-decay problems, do not equate N 2 to N 1 . Equilibrium is about equal activities ( A 2 = A 1 ), not equal numbers. Also, the formula for N 2(t) has ( 2 - 1) in the denominator: watch the sign and the case 2 = 1 (rare, handle by limit). Secular equilibrium: T 1/2,1 T 1/2,2 . After build-up, A 2 A 1 ; N 2/N 1 1/ 2 (very small). Transient equilibrium: T 1/2,1 > T 1/2,2 but comparable. A 2 peaks at t eq = 1 2- 1 ! ( 2 1 ) . No equilibrium: T 1/2,1 < T 1/2,2 (daughter decays more slowly than parent). Equilibrium conditions at a glance For α, β−, β+, γ decays, half-life is a nuclear property and independent of amount and usual physical conditions. Slight variations can occur for electron-capture decays (depend on electron environment), but these are small and exceptional. Half-life depends on sample size, temperature, or pressure. Exponential decay approaches zero asymptotically; it never reaches zero at any finite time. Mean life = 1/ is longer than half-life: = T 1/2 / 2 1.44 ,T 1/2 . A radioactive sample becomes exactly zero in a finite time, and mean life equals half-life. neet-alert β-decay energies are not monoenergetic. The presence of (anti)neutrinos makes the β spectrum continuous. Do not treat β particle energy lines like α lines in spectrum-based questions. Count rate and detection efficiency Measured count rate R depends on detector efficiency (0 to 1). Counting statistics often follow Poisson behavior. This relationship holds when measuring the count rate of radioactive decay, where the detection system has a constant efficiency factor. Counting statistics tip: For N counts observed in a fixed interval, the standard deviation is roughly N counts (Poisson). Increasing counting time or using a more active source reduces relative uncertainty. Applications that use decay laws Radiocarbon dating (C-14) for archaeology and forensics Medical tracers (Tc-99m) and cancer therapy (Co-60, Ir-192) Industrial thickness gauges (β sources) and radiography (γ sources) Smoke detectors (Am-241 α source) Sterilization and food irradiation Geological dating with long-lived isotopes (U-Pb chains) years Age t . A wooden artifact shows 25% of the activity of a fresh-living sample (same carbon content). If the half-life of C-14 is 5730 y, estimate the age of the artifact. Use A/A 0 = e - t with = ( 2)/T 1/2 , or note that 0.25 = (1/2) 2. Activity ratio A/A 0 = 0.25 T 1/2 = 5730 , y medium Dating problems compare present activity (or fraction of C-14 atoms) to the initial living value. They assume the initial ratio equals that of the atmosphere at the time and that the sample remained closed to exchange after death. remember Decay constant has inverse time dimension, regardless of the time unit chosen. Dimension of decay constant Half-life ↔ mean life relation “One and a half of a half-life.” Mean life is about one and a half times the half-life: 1.44 ,T 1/2 . Fast protocol for decay numericals Write what is given in symbols: N 0 , A 0 , T 1/2 , , time units. Convert all times to a single unit (usually seconds) and activities to Bq, unless otherwise specified. Compute from T 1/2 = ( 2)/ or = 1/ as needed. Use N = N 0 e - t or A = A 0 e - t consistently. For chains, use activity equality (not number equality) to discuss equilibrium. Round to 2 significant figures for NEET-style answers unless more are asked. Branching decay: If a nucleus can decay by multiple modes (e.g., α or β), each branch has a partial decay constant i . The total total is the sum of partials, total = i i . The branching ratio b i = i/ total gives the probability of that mode. Activities per branch are A i = i N = b i A total . Probability of each mode and its activity share. Branching ratios and partial activities This formula applies to any radioactive nucleus that decays via multiple competing pathways (modes), such as electron capture, alpha decay, Flow diagram with P → D (α/β), then D → D + γ; labels for activities A1 and A2. Illustration of parent-to-daughter radioactive decay with subsequent gamma emission. Schematic of a simple decay chain: Parent P decays to excited daughter D , which emits γ to reach stable D. Arrows indicate α/β emission followed by γ de-excitation. Energy in α and γ decays appears as discrete lines because they connect well-defined nuclear energy levels. In β decay, a neutrino (or antineutrino) carries away variable energy, leaving the electron (or positron) with a continuous spectrum from zero up to a maximum endpoint energy defined by the Q-value of the decay. Boundary checks: t = 0 ⇒ N = N 0 , A = A 0 . If 0 , a nucleus is effectively stable. If is very large, the sample empties rapidly (but still exponentially). Exponential decay applies only when there is no production term for the parent. tip Successive decays with production terms require care: if a species is both produced and decays, write a balance equation: d N 2 d t = + 1 N 1 - 2 N 2 . Solve with the known N 1(t) to find N 2(t) . This yields the build-up expression used for transient and secular equilibrium analysis. Time t dependent N/N0 derived A/A0 decay N and A (normalized) T1/2 0.5 Half-life point 2T1/2 0.25 Quarter remaining Exponential decay of number and activity. Each additional half-life multiplies by 1/2. Exponential decrease of N/N0 and A/A0 together, halving at each T1/2. Decay constant = 2 / T 1/2 s -1 Per minute: divide by 60 Half-life T 1/2 T 1/2 = 2 / s (often min, h, y) 1 y ≈ 3.156× 10 7 s Mean life = 1/ 1.44 ,T 1/2 Activity A = N Bq 1 Ci = 3.7× 10 10 Bq Quantity Symbol Formula SI Unit Typical Conversion Practical detector note: The instrument displays a count rate, which is proportional to activity but reduced by efficiency and geometry. If a problem gives count rate and a calibrated efficiency , recover the activity using A = R/ before proceeding with decay-time calculations. easy = 100 , s The mean life of a radionuclide is 100 s. Find its half-life. Use T 1/2 = , 2 . T 1/2 Gamma emission often follows α or β decay if the daughter is left in an excited nuclear state. The γ photon carries away the energy difference between nuclear levels. Because γ has no charge and no rest mass, it is neither deflected by fields nor stopped easily, which is why dense shielding is required in practice. Electron capture (EC) is a β-like process in which an inner atomic electron is captured by the nucleus, converting a proton to a neutron and emitting a neutrino. Because EC uses atomic electrons, its rate can show tiny dependence on chemical/physical environment. For NEET-level problems, assume half-life is constant unless EC-specific details are explicitly mentioned. Identify whether the daughter is initially absent: if yes, use the standard build-up solution. Compare half-lives to classify equilibrium: secular, transient, or none. Relate activities, not just numbers: A 2 = 2 N 2 and A 1 = 1 N 1 . For equilibrium questions, remember A 2 A 1 (secular) or compute t eq (transient). Checklist for chain-decay numericals Half-value layer (HVL) is the thickness of a material that reduces a beam’s intensity to half. For γ shielding, successive HVLs multiply the protection: after n HVLs, intensity is (1/2) n of its initial value. While detailed shield design is beyond the syllabus, recognizing the exponential halving concept aids qualitative reasoning. Units comfort “Bq is basic, Ci is colossal.” Use Bq for precise SI work; Ci is handy shorthand for very strong sources. neet-alert Distance traveled in the nth half-life is not a meaningful concept. Use fraction remaining: after n half-lives, N/N 0 = 2 -n . Do not multiply λ by nT1/2 as if the decay were linear. Advanced note (edge case): If 2 1 in a chain, the difference form has a 0/0 appearance. Take the limit 2 1 using L’Hospital’s rule to obtain N 2(t) = 1 N 1(0) , t , e - 1 t . Such degenerate cases are rare in practice but useful to know. Radiometric dating pitfalls: contamination (gain/loss of parent or daughter) skews ages; reservoir effects alter initial ratios. In exam settings, unless stated otherwise, take the system as closed and initial activity equal to the modern or stated reference value. Key terms recap Spontaneous nuclear transformation with emission of α, β, or γ. Radioactivity Probability per unit time of decay ( , units s -1 ). Decay constant T 1/2 Time for the sample to reduce to half: T 1/2 = ( 2)/ . Half-life Average lifetime: = 1/ 1.44 ,T 1/2 . Mean life Decays per second: A = N ; units Bq. Activity Fraction of decays in a given mode: b i = i/ total . Branching ratio Parent half-life ≫ daughter; eventually A 2 A 1 . Secular equilibrium Transient equilibrium Comparable half-lives with parent longer; daughter activity peaks then follows parent.