Topic E.3 — Radioactive Decay — anchors the IB Physics nuclear block. SL students must master radiation types, decay equations, half-life and the exponential decay law. HL students extend this to gamma attenuation, the absorption mechanisms of each radiation type, and the binding-energy curve that governs both fission and fusion. Almost every IB Physics paper contains at least one E.3 calculation, and the conceptual MCQs on randomness and the meaning of $\lambda$ recur year after year.
This cheatsheet condenses every formula, decay equation, trick and trap from Topic E.3 SL + HL onto one page. Scroll to the bottom for the printable PDF, the full notes pack and the gated tutorial library used by Photon Academy students in Singapore.
§1 — Nature of Radioactive Decay E.3 SL+HL
Core properties
- Random: cannot predict which nucleus decays next, or when.
- Spontaneous: not affected by temperature, pressure or chemical state.
- Nuclide notation: $^{A}_{Z}\text{X}$ where $A$ is the nucleon number and $Z$ is the proton number.
NoteRandomness is a property of individual nuclei. Large numbers of nuclei behave statistically — hence a well-defined half-life.
§2 — Types of Radiation E.3 SL+HL
| Type | Identity | Charge | Ionising power | Stopped by |
|---|---|---|---|---|
| $\alpha$ | $^{4}_{2}$He nucleus | $+2e$ | Highest | Paper / cm of air |
| $\beta^{-}$ | Electron $e^-$ | $-e$ | Medium | Few mm Al |
| $\beta^{+}$ | Positron $e^+$ | $+e$ | Medium | Few mm Al |
| $\gamma$ | Photon (EM) | $0$ | Lowest | Several cm Pb |
Decay equations
Alpha:$^{A}_{Z}\text{X} \rightarrow\, ^{A-4}_{Z-2}\text{Y} + \,^{4}_{2}\text{He}$
Beta-minus:$^{A}_{Z}\text{X} \rightarrow\, ^{A}_{Z+1}\text{Y} + e^- + \bar{\nu}_e$
Beta-plus:$^{A}_{Z}\text{X} \rightarrow\, ^{A}_{Z-1}\text{Y} + e^+ + \nu_e$
TrickConservation check for all decays: top numbers ($A$) must balance; bottom numbers ($Z$) must balance. Neutrinos have $A = 0$ and $Z = 0$.
Trap$\beta^-$ and $\beta^+$ energies are continuous spectra (shared with neutrino / antineutrino). Gamma energy is discrete (nuclear energy levels). $\alpha$ energy is also discrete. The continuous beta spectrum was the original evidence that the neutrino must exist.
§3 — Half-Life, Activity & Decay Laws E.3 SL+HL
Activity:$A = \lambda N$
Decay constant:$\lambda = \dfrac{\ln 2}{t_{1/2}}$
Number of nuclei:$N = N_0 e^{-\lambda t}$
Activity (time):$A = A_0 e^{-\lambda t}$
After $n$ half-lives:$N = \dfrac{N_0}{2^n}, \quad A = \dfrac{A_0}{2^n}$
Unit of activity:$1\,\text{Bq} = 1$ decay per second
Half-life from graph:$t_{1/2} = t_2 - t_1$ (choose any $N$ and $N/2$)
TrickTo find half-life from a graph: pick any starting $N$ value, halve it, read across to the curve, then down to get the time difference. Do it twice with different starting values to confirm.
TrapAlways subtract background before analysis: $A_{\text{true}} = A_{\text{measured}} - A_{\text{background}}$. Forgetting background is the most common experimental error.
Note$\lambda$ = probability of decay per unit time per nucleus. NOT the number of decays. (IB May 2022, Paper 1, Q40 — answer D.)
§4 — HL: Background Radiation & Absorption E.3 HL
Background radiation sources
- Natural (largest): radon gas (from soil/rocks), cosmic rays, $^{40}$K in food and the human body, gamma from building materials.
- Artificial: medical X-rays, nuclear power, weapons-testing fallout.
Absorption mechanisms
| Radiation | Mechanism | Stopped by |
|---|---|---|
| $\alpha$ | Direct ionisation (loses energy in air collisions) | Paper / skin |
| $\beta^{\pm}$ | Ionisation + Bremsstrahlung (braking radiation) | Al (mm) |
| $\gamma$ | Photoelectric effect, Compton scattering, pair production | Pb (cm) |
Gamma attenuation:$I = I_0 e^{-\mu x}$ ($\mu$ = linear attenuation coefficient)
§5 — HL: Binding Energy & $E = mc^2$ E.3 HL
Mass defect:$\Delta m = Z m_p + (A - Z) m_n - m_{\text{nuc}}$
Binding energy:$E_B = \Delta m \cdot c^2$
BE per nucleon:$E_B / A$
Unit conversion:$1\,\text{u}\cdot c^2 = 931.5$ MeV
Energy released:$Q = E_B(\text{products}) - E_B(\text{reactants})$
The BE/A curve — key facts
- Peak at $^{56}_{26}$Fe: $E_B/A \approx 8.79$ MeV — the most stable nucleus.
- Fission: a heavy nucleus ($A \approx 235$) splits into medium-mass fragments (higher $E_B/A$) $\Rightarrow$ energy released.
- Fusion: light nuclei ($A < 10$) combine to form heavier products (higher $E_B/A$) $\Rightarrow$ energy released.
- $^{56}$Fe can neither fission nor fuse to release energy — it sits at the peak of the curve.
TrickEnergy released $=$ (total BE of products) $-$ (total BE of reactants). If products have higher total binding energy, energy is released (not absorbed).
TrapMass defect uses nuclear mass, not atomic mass. If atomic masses are given, subtract $Z \times m_e$ from each, OR be consistent (the electron masses cancel in the subtraction if you use atomic masses on both sides).
Worked Example — Half-Life & Binding Energy
Question (HL Paper 2 style — 7 marks)
Iodine-131 ($t_{1/2} = 8.02$ days) is used as a medical tracer.
(a) Calculate the decay constant $\lambda$ in s$^{-1}$. [2]
(b) A patient is given a dose with initial activity $A_0 = 6.0 \times 10^7$ Bq. Find the activity after 24 hours. [2]
(c) The atomic mass of $^4_2$He is 4.002602 u. Given $m_p = 1.007276$ u and $m_n = 1.008665$ u, calculate the binding energy per nucleon of helium-4 in MeV. [3]
Solution
- Convert: $t_{1/2} = 8.02 \times 24 \times 3600 = 6.93 \times 10^5$ s. (M1)
- $\lambda = \ln 2 / t_{1/2} = 0.6931 / 6.93 \times 10^5 = 1.00 \times 10^{-6}$ s$^{-1}$. (A1)
- After 24 h $= 86400$ s: $A = A_0 e^{-\lambda t} = 6.0 \times 10^7 \cdot e^{-(1.00 \times 10^{-6})(86400)} = 6.0 \times 10^7 \cdot e^{-0.0864}$. (M1)
- $A \approx 6.0 \times 10^7 \times 0.917 \approx 5.5 \times 10^7$ Bq. (A1)
- Mass defect (using atomic masses — electron masses cancel): $\Delta m = 2(1.007276) + 2(1.008665) - 4.002602 = 0.029280$ u. (M1)
- Binding energy: $E_B = 0.029280 \times 931.5 = 27.27$ MeV. (A1)
- Per nucleon: $E_B / A = 27.27 / 4 = 6.82$ MeV per nucleon. (A1)
Examiner's note: Two classic errors. (i) Forgetting to convert the half-life into seconds when $\lambda$ is wanted in s$^{-1}$ — students often leave $t_{1/2}$ in days and end up with a $\lambda$ that is wrong by a factor of $\sim 10^5$. (ii) Mixing nuclear and atomic masses in the same calculation — pick one convention and stay with it.
Common Student Questions
What does it mean that radioactive decay is random and spontaneous?
Random: you cannot predict which nucleus will decay next or when. Spontaneous: the rate is unaffected by temperature, pressure or chemical state. Randomness applies to individual nuclei; large numbers of nuclei behave statistically, which is why a sample has a well-defined half-life.
Why is the beta decay energy spectrum continuous?
Because the energy released in $\beta$ decay is shared between three particles — the daughter nucleus, the beta particle and the (anti)neutrino. The beta gets a continuous range of energies up to a maximum, depending on how the energy is split. This continuous spectrum (versus the discrete alpha and gamma spectra) was the original evidence that the neutrino must exist.
What does the decay constant $\lambda$ actually represent?
$\lambda$ is the probability that a given nucleus decays per unit time. It is NOT the number of decays. Mark schemes use the phrase "probability per unit time per nucleus." (IB May 2022 Paper 1 Q40 — answer D.) The activity $A = \lambda N$ follows directly: if each nucleus has probability $\lambda$ of decaying per second, the expected total decays per second is $\lambda N$.
Why must I subtract background radiation in experiments?
Because the GM counter always picks up background counts from radon, cosmic rays, $^{40}$K in food and building materials. The true sample activity is $A_{\text{true}} = A_{\text{measured}} - A_{\text{background}}$. Forgetting this is the most common experimental error in IB practical and IA work — and a guaranteed mark loss in any data-processing question.
Should I use atomic masses or nuclear masses for binding energy calculations?
Strictly, the formula uses nuclear masses: $\Delta m = Z m_p + (A - Z) m_n - m_{\text{nuclear}}$. However, if you are consistent and use atomic masses for both the constituents and the nucleus, the $Z$ electron masses cancel in the subtraction, so atomic masses can be used directly. Just don't mix the two — that is where students lose marks.
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