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 $^A_Z X$:

• IMATH_7 = atomic number (number of protons) = number of electrons in neutral atom.
• IMATH_8 = mass number = protons + neutrons.
• IMATH_9 = 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.

Common errors

Forgetting conservation in nuclear equations. Mass numbers must balance; charges must balance.

Confusing decay types. Alpha is heavy and slow; beta is fast and light; gamma is electromagnetic.

Half-life formula misuse. $N = N_0 (1/2)^{t/T_{1/2}}$. If $t$ is exactly $n$ half-lives, $N = N_0/2^n$.

Treating half-life as deterministic for individual atoms. Individual decay is random; half-life is statistical.

Wrong nucleus in fission. Common fissile materials are $^{235}$U and $^{239}$Pu, not all uranium isotopes.

In one sentence

The atomic nucleus contains protons and neutrons, with isotopes differing in neutron count; unstable nuclei undergo alpha decay (emit $^4$He), beta-minus decay (neutron to proton plus electron plus antineutrino), or gamma decay (emit photon); decay follows the half-life formula $N = N_0 (1/2)^{t/T_{1/2}}$; fission of heavy nuclei (uranium-235) and fusion of light nuclei (deuterium-tritium) release energy through the conversion of mass to energy, powering nuclear reactors and the sun respectively.