IB Physics HLTopic 7 — Atomic, Nuclear & Particle PhysicsPaper 1 & 2balancing A & Z~16 min read
Decay Equations
When a nucleus decays, it turns into a different nucleus and spits out a particle. A decay equation is just the bookkeeping for that change — and it obeys two simple rules: the nucleon numbers must balance across the arrow, and so must the proton numbers. Master those two rules and you can predict the daughter nucleus for any alpha, beta or gamma decay, every single time.
📚 What you need to know
Four reasons a nucleus is unstable set the decay mode: too many neutrons → β−; too many protons → β+ or electron capture; too many nucleons → α; too much energy → γ
Alpha decay:Z −2, A −4 (emits a helium nucleus)
Beta-minus decay: a neutron becomes a proton; Z +1, A unchanged; emits an electron + antineutrino
Beta-plus decay: a proton becomes a neutron; Z −1, A unchanged; emits a positron + neutrino
Gamma decay:Z and A both unchanged (just releases energy)
In every equation, the top numbers balance and the bottom numbers balance
The golden rule: balance both numbers
A nuclear decay equation always has this shape, with the parent on the left and the daughter plus emitted particle on the right:
The balancing rulethe sum of the top numbers (A) is equal on both sidesthe sum of the bottom numbers (Z) is equal on both sides
This is the whole game. Nucleons aren’t created or destroyed, and neither is charge — so both totals have to match across the arrow. If your equation doesn’t balance top and bottom, you’ve made a slip somewhere. It’s a built-in self-check: always add up each row before moving on.
Alpha decay
Alpha decay happens when a nucleus has too many nucleons. It ejects a helium nucleus (2 protons, 2 neutrons), so the parent loses 2 from its proton number and 4 from its nucleon number.
General alpha decayAZX → A−4Z−2Y + 42α
Too many nucleons
emit α
Z − 2
and
A − 4
Beta-minus decay
Beta-minus happens when there are too many neutrons. One neutron transforms into a proton, releasing a fast electron (the beta particle) and an antineutrino. The proton number rises by 1, but since a neutron simply became a proton, the total nucleon number is unchanged.
General beta-minus decayAZX → AZ+1Y + 0−1β + v̄eunderlying change: n → p + e− + antineutrino
Beta-plus decay & electron capture
Beta-plus happens when there are too many protons. A proton turns into a neutron, releasing a positron and a neutrino. The proton number drops by 1, nucleon number unchanged.
General beta-plus decayAZX → AZ−1Y + 0+1β + veunderlying change: p → n + e+ + neutrino
An alternative for proton-rich nuclei is electron capture: the nucleus pulls in an orbiting electron, which combines with a proton to make a neutron and a neutrino. This has the same effect on the numbers as beta-plus (Z −1, A unchanged).
Here’s a neat way to remember beta-plus: think of the “p” in plus as proton — it’s the proton that turns into a neutron. For beta-minus, it’s the other way round: a neutron becomes a proton. Get the plus/proton link and you’ll never mix up which way the proton number moves.
Gamma decay
Gamma emission isn’t really a “transmutation” — the nucleus doesn’t change what it is. It just had too much energy (it was in an excited state after another decay) and releases that energy as a high-energy photon. Both numbers stay the same.
General gamma decayAZX* → AZX + 00γthe * means an excited nucleus — only energy is lost, no change to Z or A
Summary on an N–Z graph
Each decay mode moves the nucleus a characteristic step on a graph of neutron number N against proton number Z. This picture ties all the modes together:
Alpha steps down-left, beta-minus steps down-right, beta-plus steps up-left. Each move is exactly the change in N and Z from the decay rules.
WE 1
Plutonium-239 (94 protons) is an alpha emitter. It decays to uranium, which then alpha-decays to thorium. Write both decay equations.
Step 1 — plutonium to uranium (alpha)23994Pu → 23592U + 42α
Check: 239 = 235 + 4 ✓ and 94 = 92 + 2 ✓
Step 2 — uranium to thorium (alpha)23592U → 23190Th + 42α
Check: 235 = 231 + 4 ✓ and 92 = 90 + 2 ✓
Both equations balance top and bottomEach alpha drops A by 4 and Z by 2. Always verify by adding up both rows — if they don’t match, something’s wrong.
WE 2
A nucleus has 84 protons and 126 neutrons and undergoes alpha decay to form lead (Pb). Find the proton number and nucleon number of the lead isotope produced.
Step 1 — nucleon number of the parentA = 84 + 126 = 210Step 2 — apply the alpha rule (Z −2, A −4)new Z = 84 − 2 = 82new A = 210 − 4 = 206Lead-206: 20682PbFirst build the parent’s nucleon number from protons + neutrons, then subtract the alpha particle’s 2 and 4. Lead-206 is a real, stable end-point of a decay chain.
⚛ Working a decay equation
Identify the decay type from the emitted particle (or from why the nucleus is unstable).
Apply the rule: α (Z−2, A−4); β− (Z+1, A same); β+ (Z−1, A same); γ (no change).
Write the daughter with its new Z and A.
Add the emitted particle (α, β + neutrino/antineutrino, or γ).
Check both rows balance — top numbers equal, bottom numbers equal.
💡 Top tips
Always check both rows balance — it catches nearly every mistake.
Beta-plus → proton turns into a neutron (the “p” trick).
Alpha is the only decay that changes the nucleon number.
Beta decays leave Aunchanged (a nucleon just swaps type).
Don’t forget the neutrino/antineutrino in beta equations.
⚠ Common mistakes
Changing the nucleon number in a beta decay — it stays the same
Moving the proton number the wrong way for beta-plus vs beta-minus
Forgetting the emitted neutrino or antineutrino in beta equations
Not checking the balance of top and bottom numbers
Confusing neutron number with nucleon number when building the parent
Quick recap: A decay equation must balance top (A) and bottom (Z) on both sides. Alpha: Z−2, A−4. Beta-minus: Z+1, A same (n→p). Beta-plus: Z−1, A same (p→n). Gamma: no change, just energy. Identify the type, apply the rule, add the emitted particle, and always check both rows add up.
You can now predict what a nucleus becomes when it decays. But decay isn’t a one-off event — a whole sample decays over time, faster at first and slowing down. To describe how much is left and how fast it happens, we need two ideas: activity and half-life. Next page: Activity & Half-Life.
Decay equations not balancing for you?
Book a free meeting and we’ll drill alpha, beta-plus, beta-minus and gamma rules until balancing A and Z is second nature.
