Atomic nucleus structure (protons, neutrons), isotopes, types of radioactive decay (alpha, beta, gamma), nuclear stability, half-life, fission and fusion, and applications including nuclear power

What this dot point is asking

VCAA wants you to describe the structure of the atomic nucleus, identify the three types of radioactive decay, apply the half-life formula, and explain fission and fusion with their applications.

Atomic nucleus

The nucleus contains:

• Protons. Charge $+e$, mass $1.673 \times 10^{-27}$ kg.
• Neutrons. Charge 0, mass $1.675 \times 10^{-27}$ kg.

Notation $^A_Z X$:

• $Z$ = atomic number (number of protons) = number of electrons in neutral atom.
• $A$ = mass number = protons + neutrons.
• $N = A - Z$ = number of neutrons.

Isotopes. Same $Z$ (same element) but different $N$ (and so different $A$). Examples: $^{12}_6$C, $^{13}_6$C, $^{14}_6$C are all carbon, but with different neutron counts.

Approximate masses are measured in atomic mass units (amu): 1 amu = $1.661 \times 10^{-27}$ kg.

Nuclear forces

Inside the nucleus, two forces compete:

Coulomb repulsion between positively charged protons (long-range).

Strong nuclear force between any pair of nucleons (very short-range, around $10^{-15}$ m, but ~100 times stronger than electromagnetism at this scale).

For light nuclei, strong force dominates and stable nuclei have approximately equal protons and neutrons. For heavy nuclei, more neutrons are needed to bind the larger volume against increasing Coulomb repulsion. Above $Z = 83$ (bismuth), no nuclei are stable.

Radioactive decay

Unstable nuclei spontaneously emit radiation to reach more stable configurations.

Alpha decay. Emission of a helium nucleus ($^4_2$He). Mass number decreases by 4; atomic number by 2.

Example: $^{238}_{92} \text{U} \to ^{234}_{90} \text{Th} + ^4_2 \text{He}$.

Alpha particles are heavy and slow. Range: a few cm in air; stopped by paper.

Beta-minus decay. A neutron converts to a proton plus electron plus antineutrino. Atomic number increases by 1; mass number unchanged.

Example: $^{14}_6 \text{C} \to ^{14}_7 \text{N} + ^0_{-1} e + \bar{\nu}_e$.

Beta particles are fast electrons. Range: a few metres in air; stopped by aluminium foil.

Beta-plus decay. A proton converts to a neutron plus positron plus neutrino. (Less common; not always required in Unit 1.)

$^{22}_{11} \text{Na} \to ^{22}_{10} \text{Ne} + ^0_{+1} e + \nu_e$.

Gamma decay. The nucleus, in an excited state after another decay, emits a high-energy photon. Mass number and atomic number unchanged.

Gamma rays are highly penetrating; require lead or concrete shielding.

Conservation laws

In any nuclear equation:

• Mass number is conserved.
• Charge is conserved.
• (Energy and momentum are also conserved, accounting for kinetic energy of products.)

Half-life

The half-life $T_{1/2}$ is the time for half the nuclei in a sample to decay. The decay is random for any individual nucleus, but the half-life is a well-defined statistical property.

where $N_0$ is initial number, $N$ is number after time $t$.

Equivalent activity: $A = A_0 (1/2)^{t/T_{1/2}}$.

Common half-lives:

• Carbon-14: 5,730 years. Used for carbon dating.
• Iodine-131: 8 days. Used in medicine.
• Uranium-238: 4.5 billion years.
• Polonium-214: 0.16 ms.

Fission

A heavy nucleus (typically uranium-235 or plutonium-239) splits into two roughly equal fragments, releasing energy and free neutrons.

The energy released per fission is approximately 200 MeV.

Chain reaction. The released neutrons can induce further fissions. If on average more than one neutron per fission triggers a new fission, the chain reaction is supercritical (explosive). Controlled chain reactions (one neutron per fission triggers one new fission) power nuclear reactors.

Fusion

Light nuclei (typically deuterium and tritium, $^2$H and $^3$H) fuse into a heavier nucleus (helium), releasing energy.

Fusion powers the sun. Controlled fusion for power generation has been a long-term research goal (ITER, JET, others) but has not yet been commercialised.

Fusion produces more energy per kg of fuel than fission and has fewer long-lived radioactive products. The barrier is the temperature (around $10^8$ K) needed to overcome Coulomb repulsion.

Applications

Nuclear power

Fission reactors generate about 10 percent of world electricity. Concerns: waste storage, weapons proliferation, accident risk (Three Mile Island 1979, Chernobyl 1986, Fukushima 2011).

Nuclear medicine

Diagnostic imaging (technetium-99m, fluorine-18 in PET scans). Cancer therapy (cobalt-60, iodine-131, linear accelerators).

Industrial

Thickness measurement, smoke detectors (americium-241), industrial radiography.

Carbon dating

Carbon-14 is produced in the upper atmosphere and incorporated into living things. After death, $^{14}$C content decays with half-life 5,730 years. Used to date objects up to about 50,000 years old.

Examples in context

Example 1. ANSTO OPAL reactor neutron fluence for medical isotopes. The OPAL reactor at Lucas Heights runs at $20$ MW thermal and provides a thermal-neutron flux of about $4 \times 10^{14}$ neutrons cm$^{-2}$ s$^{-1}$ at irradiation positions. Each fission of $^{235}$U yields approximately $200$ MeV. Power output divided by energy per fission gives $20 \times 10^6 \,/\, (200 \times 1.602 \times 10^{-13}) = 6.2 \times 10^{17}$ fissions s$^{-1}$. The neutrons activate $^{98}$Mo to $^{99}$Mo for technetium-99m generators that supply Australian and Pacific hospitals; about $30\%$ of global $^{99}$Mo supply comes from ANSTO during regional shortages.

Example 2. Radon mitigation in Mt Stromlo observatory bedrock. Uranium-238 decay series at Mt Stromlo bedrock produces radon-222 (half-life $3.8$ days), which can accumulate in basement instrumentation rooms. Initial activity $200$ Bq m$^{-3}$ decays as $N = N_0 (1/2)^{t/3.8}$. After one week (7 days), activity falls to $200 \times (1/2)^{7/3.8} = 200 \times 0.277 = 55$ Bq m$^{-3}$, but continuous ingrowth from soil-bound radium replenishes it. Building physics design therefore uses sub-floor ventilation to keep airborne radon below the $200$ Bq m$^{-3}$ ARPANSA action threshold, protecting astronomers and electronics from alpha activity.

Try this

Q1. Identify the three principal types of radioactive decay and state one penetration property of each. [3 marks]

• Cue. Alpha: stopped by paper. Beta: stopped by thin aluminium. Gamma: needs lead or thick concrete.

Q2. A $^{99}$mTc sample (half-life $6.0$ hours) has an initial activity of $800$ MBq. Calculate (a) the activity after $18$ hours, and (b) the time required for activity to fall below $50$ MBq. [4 marks]

• Cue. (a) $800 \times (1/2)^3 = 100$ MBq. (b) $50/800 = 1/16 = (1/2)^4$ so $t = 24$ hours.

Q3. Refer to OPAL operations. (a) Outline the role of neutron capture in producing $^{99}$Mo. (b) Calculate the number of fissions per second at $20$ MW thermal power, assuming $200$ MeV per fission. (c) Explain one safety benefit of using $^{99}$mTc rather than $^{131}$I for diagnostic imaging. [2+2+2 marks]

• Cue. (a) $^{98}$Mo + $n \to ^{99}$Mo + $\gamma$. (b) $6.2 \times 10^{17}$ s$^{-1}$. (c) Shorter half-life gives smaller cumulative dose; pure gamma emitter has no beta dose to surrounding tissue.