Nuclear stability and modes of decay (VCE Physics Unit 1)

What this dot point is asking

VCAA wants you to describe the nucleus (protons, neutrons, the strong force, stability), identify the four classical decay modes, and write balanced nuclear equations that conserve mass number and atomic number.

The nucleus

Atomic nuclei contain protons ($Z$) and neutrons ($N$), together called nucleons. The mass number $A = Z + N$. Notation: $^A_Z X$ where $X$ is the chemical symbol.

The strong nuclear force binds nucleons. It is short-range (about $1$ fm), attractive, and stronger than the electromagnetic repulsion between protons over very short distances.

Stability

Stable nuclei occupy a narrow "valley of stability" on a $N$ vs $Z$ chart. For light nuclei ($Z < 20$), stable isotopes have $N \approx Z$. For heavier nuclei, more neutrons are needed to dilute the proton-proton repulsion: stable isotopes have $N/Z \approx 1.5$ near uranium.

Unstable nuclei decay toward the valley of stability through one or more decay events.

Alpha decay

The nucleus emits an alpha particle ($^4_2$He). Common for heavy nuclei ($A > 200$).

Alpha particles are highly ionising but have low penetration (stopped by paper).

Beta-minus decay

A neutron decays into a proton plus an electron plus an electron antineutrino. The electron is emitted as the beta particle.

Common for neutron-rich nuclei. The antineutrino carries away part of the kinetic energy, producing the continuous beta energy spectrum.

Beta-plus decay

A proton decays into a neutron plus a positron plus an electron neutrino. Common for proton-rich nuclei.

The positron annihilates with an electron, producing two $0.511$ MeV gamma photons in opposite directions. This is the basis of PET imaging.

Gamma emission

A nucleus in an excited state drops to a lower state by emitting a gamma photon. $A$ and $Z$ are unchanged.

Often follows alpha or beta decay (the daughter is left in an excited state).

Conservation in nuclear equations

Always conserve:

• Mass number $A$ (total nucleons).
• Atomic number $Z$ (total charge).
• Lepton number (electron + neutrino vs positron + antineutrino).

Energy, momentum and angular momentum are also conserved, but VCE Year 11 questions focus on $A$ and $Z$.

VCAA exam style

Year 11 SAC tasks include:

• Identifying the decay mode from a parent and daughter pair.
• Writing balanced equations.
• Identifying stable vs unstable isotopes from a chart of nuclides.

Common traps

Wrong $Z$ direction in beta decay

Beta-minus increases $Z$ (extra proton). Beta-plus decreases $Z$.

Treating gamma emission as changing the element

Gamma keeps both $A$ and $Z$ unchanged.

Forgetting the (anti)neutrino

For full marks, include it. The antineutrino in beta-minus accompanies the electron; the neutrino in beta-plus accompanies the positron.

In one sentence

Atomic nuclei contain protons and neutrons bound by the short-range strong force, with stability determined by the neutron-to-proton ratio; unstable nuclei decay by alpha ($-4 A$, $-2 Z$), beta-minus ($+1 Z$, neutron $\to$ proton plus electron plus antineutrino), beta-plus ($-1 Z$, proton $\to$ neutron plus positron plus neutrino) or gamma (no $A$/$Z$ change) emission, and every decay equation must conserve mass number and atomic number.

Examples in context

Example 1. Carbon-14 beta-minus decay used at the AMS facility, ANSTO. Carbon-14 decays by beta-minus emission: $^{14}_{6}\text{C} \to {^{14}_{7}\text{N}} + e^- + \bar{\nu}_e$. ANSTO's accelerator mass spectrometer at Lucas Heights counts $^{14}$C atoms directly in samples as small as a milligram, which is how archaeologists date Lake Mungo charcoal. A $1$ mg modern carbon sample contains $5 \times 10^{19}$ carbon atoms, of which $6 \times 10^7$ are $^{14}$C. Decay rate $A = \lambda N$ with $\lambda = \ln(2)/T_{1/2} = 3.83 \times 10^{-12}$ s$^{-1}$ gives an activity of $2.3 \times 10^{-4}$ Bq (0.23 mBq), so low that decay counting would take days to collect a few counts, while AMS counts the atoms directly in minutes.

Example 2. Positron emission tomography at the Peter MacCallum Cancer Centre. Fluorine-18 used in PET scans decays by beta-plus emission: $^{18}_{9}\text{F} \to {^{18}_{8}\text{O}} + e^+ + \nu_e$ with a half-life of $110$ minutes. The emitted positron annihilates with a tissue electron, producing back-to-back $511$ keV gamma photons detected in coincidence. The Royal Melbourne and Peter Mac receive $^{18}$F from a Clayton cyclotron at start-of-day; after a $30$ min transport, activity falls to $A/A_0 = (1/2)^{30/110} = 0.83$. Balanced equation $p \to n + e^+ + \nu$ inside the nucleus reduces atomic number by one without changing mass number, consistent with the change from F ($Z=9$) to O ($Z=8$).

Try this

Q1. Write the balanced nuclear equation for the alpha decay of $^{238}$U. [2 marks]

• Cue. $^{238}_{92}\text{U} \to {^{234}_{90}\text{Th}} + {^{4}_{2}\text{He}}$. Mass number drops by 4, atomic number by 2.

Q2. A nucleus undergoes beta-minus decay followed by an alpha decay. If the parent is $^{60}$Co ($Z = 27$), determine the mass number and atomic number of the final nucleus. [4 marks]

• Cue. $^{60}_{27}\text{Co} \to {^{60}_{28}\text{Ni}} + e^- + \bar{\nu}$ then $^{60}_{28}\text{Ni} \to {^{56}_{26}\text{Fe}} + \alpha$. Final: $A = 56$, $Z = 26$.

Q3. Refer to $^{18}$F beta-plus decay used at Peter Mac. (a) Write the balanced nuclear equation. (b) Explain the origin of the $511$ keV photons detected. (c) Determine the activity of a $400$ MBq sample after $4$ hours given a $110$ min half-life. [2+2+2 marks]

• Cue. (a) $^{18}_{9}\text{F} \to {^{18}_{8}\text{O}} + e^+ + \nu_e$. (b) Positron annihilation with an electron produces two $511$ keV photons. (c) $400 \times (1/2)^{240/110} = 88$ MBq.