Every formula, definition and exam trap for IB Physics Topic E.3 — alpha, beta and gamma radiation, half-life, decay law, gamma attenuation and binding energy. Hand-built by an IBO-experienced Singapore tutor.
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.
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.
Penetrating power. Alpha is stopped by paper, beta by a few mm of aluminium, and gamma only partly by cm of lead — the reverse order of their ionising power.
From the Photon question bank
What examiners actually test on this concept:
α/β/γ identity & chargePenetration α<β<γContinuous β spectrumConserve A, Z
Penetration: α stopped by paper/skin, β by mm of Al, γ by cm of Pb; ionising power is the reverse (α highest). very frequent
β energy is a CONTINUOUS spectrum (shared with the (anti)neutrino) — the original evidence for the neutrino; α and γ energies are discrete.
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 — a classic Paper 1 MCQ trap in the Photon question bank.
The decay curve. The number of nuclei halves over each successive half-life, following $N=N_0 e^{-\lambda t}$ with $\lambda=\ln2/t_{1/2}$.
From the Photon question bank
What examiners actually test on this concept:
A = λNλ = ln2/t½N = N₀ e^(−λt)N₀/2ⁿ after n half-lives
λ is the decay PROBABILITY per nucleus per unit time (not a count); $A=\lambda N$ and $\lambda=\ln2/t_{1/2}$. very frequent
After $n$ half-lives $N=N_0/2^n$; read $t_{1/2}$ off a graph as any $N\to N/2$ time interval.
Gamma attenuation. Intensity falls exponentially with thickness, $I=I_0 e^{-\mu x}$; each half-value thickness halves it.
From the Photon question bank
What examiners actually test on this concept:
Radon = largest sourceI = I₀ e^(−μx)Half-value thicknessγ absorption mechanisms
Natural background dominates (radon gas the largest source); subtract it before any decay analysis. frequent (HL)
Gamma attenuation $I=I_0 e^{-\mu x}$; the half-value thickness halves the intensity — a "half-life" in distance.
γ is absorbed via the photoelectric effect, Compton scattering and pair production.
§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).
Binding energy per nucleon. The curve peaks at iron-56; fusion of light nuclei and fission of heavy nuclei both move toward the peak, releasing energy.
From the Photon question bank
What examiners actually test on this concept:
Δm = Zm_p+(A−Z)m_n−m_nucE_B = Δm·c²BE/A peaks at ⁵⁶Fe1 u·c² = 931.5 MeV
Binding energy $E_B=\Delta m\,c^2$, with $1\,\text{u}\cdot c^2=931.5$ MeV. BE per nucleon peaks at ⁵⁶Fe (~8.8 MeV) — the most stable nucleus. frequent (HL)
Energy released = BE(products) − BE(reactants); if products are more tightly bound, energy is released. Fission (heavy→medium) and fusion (light→heavier) both climb toward the peak.
Mass defect uses NUCLEAR masses; if atomic masses are given, the electron masses cancel when used consistently 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]
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. Photon's worked solutions phrase it as "probability per unit time per nucleus." 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.
What's NOT in this cheatsheet
This page gives you the formulas and the traps. The full Photon Academy E.3 Radioactive Decay library (only available to enrolled students or via the resource library subscription) adds:
40-page Notes PDF — every concept worked through in full, with derivations of $A = \lambda N$ and the exponential decay law.
Tutorial booklet — 30+ IB-style questions sequenced from foundation to AHL difficulty.
Tutorial Solutions — full mark-scheme-style worked solutions with M1/A1/R1 annotations.
Practice Solutions — extra past-paper-style problems with detailed walk-throughs.
Cheatsheet PDF — print-ready, brand-formatted, the same one our students take into mock exams.
