Types of Radioactive Decay: Study Guide (Physics 2 15.8)

Why a nucleus decays, and what decides how

Topic 15.7 described radioactive decay as random: you cannot say when one nucleus will change. This topic is about how it changes. An unstable nucleus can lose energy by emitting particles. The products have less total mass than the original nucleus, and the difference becomes kinetic energy and photons.

The type of decay depends on the isotope, not just the element. Carbon-14 decays by beta-minus decay, but carbon-11 decays by beta-plus decay. Both are carbon, with 6 protons. They differ in the number of neutrons, and that changes how they decay. As a rough pattern: nuclei with too many neutrons for their size tend to undergo beta-minus decay; nuclei with too few tend to undergo beta-plus decay; and very heavy nuclei often undergo alpha decay. You are not expected to memorise which isotope does what. A question will tell you, or give you enough to work it out.

The particles involved

Particle	Symbol	Charge (in units of e)	Nucleon number	Lepton number
Alpha particle (helium-4 nucleus)	⁴₂He or α	+2	4	0
Electron	⁰₋₁e or β⁻	−1	0	+1
Positron (antielectron)	⁰₊₁e or β⁺	+1	0	−1
Neutrino	ν	0	0	+1
Antineutrino	ν̄	0	0	−1
Gamma photon	γ	0	0	0

A few facts to know about them:

• An alpha particle is a helium-4 nucleus: two protons and two neutrons, with charge +2e. It is not a helium atom, because it has no electrons. In this course, only helium-4 counts as an alpha particle.
• A positron has the same mass as an electron but the opposite charge.
• Neutrinos and antineutrinos have no charge and almost no mass. They feel only the weak force and gravity. They do not feel the electric force or the strong force. So they hardly ever interact with ordinary matter. Most of them pass straight through the whole Earth. That makes them very hard to detect. (How the two differ, and the different types of neutrino, are beyond this course.)
• A gamma photon has no charge and no mass. It carries energy away from the nucleus.

Conservation rules for every decay

Three quantities are always conserved in a decay. Check all three for every equation you write.

1. Nucleon number. The top-left numbers balance.
2. Charge. The bottom-left numbers balance (electron −1, positron +1).
3. Lepton number. Count electrons and neutrinos as +1, and positrons and antineutrinos as −1. Before a decay of a nucleus, the lepton number is 0, so it must be 0 afterwards. That is why an electron always comes out with an antineutrino, and a positron with a neutrino.

Energy and momentum are conserved too, as in Topic 15.7. They decide how fast the products move.

The four types of decay

Alpha decay. The nucleus ejects an alpha particle:

ᴬ_Z X → ᴬ⁻⁴_(Z−2) Y + ⁴₂He

Example: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. Check: 238 = 234 + 4 and 92 = 90 + 2. The element changes from uranium to thorium, because Z changed.

Beta-minus decay (β⁻). Inside the nucleus, a neutron changes into a proton and emits an electron and an antineutrino: ¹₀n → ¹₁p + ⁰₋₁e + ν̄. For the whole nucleus:

ᴬ_Z X → ᴬ_(Z+1) Y + ⁰₋₁e + ν̄

Example: tritium, ³₁H → ³₂He + ⁰₋₁e + ν̄. Check: A is 3 = 3 + 0. Charge is 1 = 2 + (−1). Lepton number is 0 = (+1) + (−1).

Beta-plus decay (β⁺). Inside the nucleus, a proton changes into a neutron and emits a positron and a neutrino: ¹₁p → ¹₀n + ⁰₊₁e + ν. For the whole nucleus:

ᴬ_Z X → ᴬ_(Z−1) Y + ⁰₊₁e + ν

Example: ¹⁸₉F → ¹⁸₈O + ⁰₊₁e + ν. Check: A is 18 = 18. Charge is 9 = 8 + 1. Lepton number is 0 = (−1) + (+1).

Gamma decay (γ). After an alpha or beta decay, the new nucleus is often left in an excited state, shown with an asterisk. It drops to a lower energy state by emitting a photon, just as an atom does when an electron drops to a lower level (Topic 15.3), but with much more energy:

ᴬ_Z Y* → ᴬ_Z Y + γ

Example: after cobalt-60 undergoes beta-minus decay, the nickel-60 nucleus formed is excited. ⁶⁰₂₈Ni* → ⁶⁰₂₈Ni + γ. A and Z do not change, so the element and the isotope stay the same. Only the energy of the nucleus falls.

Decay on a chart of nuclei

Plot each nucleus as a square on a grid: neutron number N across, proton number Z up. Each type of decay is a fixed move on this chart.

Decay	Change in A	Change in Z	Change in N	Emitted
α	−4	−2	−2	⁴₂He
β⁻	0	+1	−1	e⁻ and ν̄
β⁺	0	−1	+1	e⁺ and ν
γ	0	0	0	photon

Worked example 1: identifying an unknown

Question. (a) Nitrogen-13 (Z = 7) decays to carbon-13 (Z = 6) plus two other particles. Identify the decay and write the equation. (b) A nucleus ²²²₈₆Rn decays to ²¹⁸₈₄Po and one other particle X. Identify X.

1. (a) A stays at 13, and Z falls from 7 to 6. A proton has become a neutron, so this is beta-plus decay.
2. Charge: 7 = 6 + (charge of the rest), so the emitted particles carry +1 in total. That is a positron, ⁰₊₁e.
3. Lepton number: the positron has −1, so a particle with +1 and no charge must also be emitted: a neutrino.
4. Equation: ¹³₇N → ¹³₆C + ⁰₊₁e + ν.
5. (b) Nucleon number: 222 = 218 + A_X, so A_X = 4. Charge: 86 = 84 + Z_X, so Z_X = 2. X is ⁴₂He, an alpha particle.

Check. In (a), all three conservation rules hold: 13 = 13; 7 = 6 + 1; 0 = −1 + 1. In (b), no lepton is emitted, so lepton number is 0 on both sides.

Worked example 2: counting decays in a chain

Question. Radium-228 (²²⁸₈₈Ra) eventually becomes bismuth-212 (²¹²₈₃Bi) through a series of alpha and beta-minus decays. How many of each?

1. Only alpha decay changes A. A falls by 228 − 212 = 16, so there are 16 ÷ 4 = 4 alpha decays.
2. Four alpha decays alone would lower Z by 8, from 88 to 80.
3. The final Z is 83, which is 3 higher. Each beta-minus decay raises Z by 1, so there are 3 beta-minus decays.

Check with neutrons. N starts at 228 − 88 = 140 and ends at 212 − 83 = 129, a fall of 11. Four alphas remove 8 neutrons and three beta-minus decays turn 3 more neutrons into protons: 8 + 3 = 11. The order of the decays cannot be found from this count alone.

Worked example 3: sharing the energy of a decay

Question. A uranium-238 nucleus at rest undergoes alpha decay. The decay releases 4.27 MeV. Taking each mass as proportional to its nucleon number, find the kinetic energy of the alpha particle and of the thorium-234 nucleus.

1. Total momentum is zero before, so the alpha particle and the thorium nucleus have equal and opposite momenta.
2. For equal momentum, K = p²/2m is inversely proportional to mass. The alpha particle’s share is 234/(234 + 4) = 0.983.
3. K_α = 0.983 × 4.27 MeV = 4.20 MeV. K_Th = 4.27 − 4.20 = 0.072 MeV.

Interpretation. With only two products, momentum and energy conservation fix the share. Every alpha particle from this decay (to the same final state) gets the same energy. In beta decay, three particles share the energy: the daughter nucleus, the electron (or positron) and the antineutrino (or neutrino). They can share it in many ways, so the emitted electrons have a range of energies, from almost zero up to a maximum. That spread was historical evidence that an unseen third particle, the neutrino, was carrying energy away.

Common misconceptions

• “The beta-minus electron comes from the atom’s electron shells.” It is created in the nucleus when a neutron becomes a proton.
• “Gamma decay changes the element.” Gamma decay changes only the energy of the nucleus. A and Z stay the same.
• “A positron is a proton.” A positron has the tiny mass of an electron and a charge of +e. It is not a nucleon.
• “An alpha particle is a helium atom.” It is a helium-4 nucleus with charge +2e and no electrons.
• “Beta decay changes the nucleon number.” In both beta decays A stays the same: a neutron and a proton swap roles.
• “Electron with neutrino” in beta-minus decay. Lepton number would be +2 after and 0 before. Beta-minus emits an antineutrino; beta-plus emits a neutrino.
• “Every isotope of an element decays the same way.” The decay type depends on the isotope. Carbon-14 and carbon-11 decay differently.

Where this leads

This is the last topic of the course. Practise first with the practice questions, then use the revision notes and the topic checklist. For half-life, the decay constant and the energy released in nuclear reactions, look back at Topic 15.7: Fission, Fusion and Nuclear Decay. Then return to the course roadmap to plan your review of the whole course.