1. Nuclear Basics

The atomic nucleus is a dense, positively charged core containing protons (charge +e, count Z — the atomic number) and neutrons (neutral, count N). Together they are called nucleons. The mass number $A = Z + N$ gives the total nucleon count.

Nuclear notation uses the form $^A_Z\text{X}$ — for example $^{238}_{92}\text{U}$ for uranium-238. Isotopes share the same Z but differ in N.

Nuclear size and density

The nuclear radius follows an empirical $A^{1/3}$ scaling: $$r = r_0 \, A^{1/3}, \qquad r_0 \approx 1.2 \text{ fm}$$ This reflects the fact that nucleons pack at roughly constant density — nuclear matter is essentially incompressible. The resulting density is staggering: $$\rho_{\text{nuclear}} \approx 2.3 \times 10^{17}\ \text{kg/m}^3$$ A teaspoon of nuclear matter would weigh about 1 billion tonnes.

Binding energy

A nucleus is lighter than the sum of its free nucleons. This mass defect $\Delta m$ converts to binding energy via $E = mc^2$:

The Bethe–Weizsäcker semi-empirical mass formula models binding energy using five liquid-drop terms:

where $a_V \approx 15.8$, $a_S \approx 18.3$, $a_C \approx 0.714$, $a_A \approx 23.2$ (all in MeV), and the pairing term $\delta = +a_P A^{-1/2}$ for even-even, $0$ for odd-A, $-a_P A^{-1/2}$ for odd-odd nuclei ($a_P \approx 12$ MeV). The five terms represent: volume (bulk nuclear binding), surface (reduced binding at the surface), Coulomb (proton-proton repulsion), asymmetry (energy cost of proton–neutron imbalance), and pairing (extra stability for paired nucleons).

Binding energy per nucleon — B/A curve

Plotting $B/A$ versus $A$ reveals everything about why both fission and fusion release energy. The curve peaks near ⁵⁶Fe at about 8.8 MeV/nucleon; nuclei on either side are less tightly bound, meaning splitting heavy nuclei (fission) or fusing light ones (fusion) both yield net energy.

2. Radioactive Decay

Unstable nuclei spontaneously transform to reach a lower energy state. The four principal decay modes are:

Alpha (α) decay

A $^4_2\text{He}$ nucleus (alpha particle) is emitted. The parent loses 2 protons and 2 neutrons:

$^{238}_{92}\text{U} \;\rightarrow\; ^{234}_{90}\text{Th} \;+\; ^4_2\text{He} \;+\; E_k$

Alpha particles have short range (~4 cm in air, stopped by paper) but high ionising power — dangerous when inhaled or ingested.

Beta-minus (β⁻) decay

A neutron converts to a proton, emitting an electron and an electron antineutrino:

$n \;\rightarrow\; p^+ + e^- + \bar{\nu}_e$

Example: $^{14}_6\text{C} \;\rightarrow\; ^{14}_7\text{N} + e^- + \bar{\nu}_e$ (used in radiocarbon dating, $T_{1/2} = 5730$ yr)

Beta-plus (β⁺) decay

A proton converts to a neutron, emitting a positron and a neutrino:

$p^+ \;\rightarrow\; n + e^+ + \nu_e$

Example: $^{22}_{11}\text{Na} \;\rightarrow\; ^{22}_{10}\text{Ne} + e^+ + \nu_e$ (used in PET scanning)

Gamma (γ) decay

A nucleus in an excited state drops to its ground state by emitting a high-energy photon. No change in $Z$ or $A$ — gamma decay is a purely electromagnetic transition. Example: $^{99m}_{43}\text{Tc}^* \rightarrow ^{99}_{43}\text{Tc} + \gamma$.

Decay law

Radioactive decay is a first-order stochastic process. If $N_0$ nuclei are present at $t=0$ and $\lambda$ is the decay constant:

The half-life $T_{1/2}$ is the time for half the nuclei to decay:

Activity (decays per second) is $A(t) = \lambda N(t)$, measured in becquerels (Bq) or curies (Ci = $3.7 \times 10^{10}$ Bq).

Isotopes in use

U-238 decay chain (simplified)

Uranium-238 undergoes 14 decays (8α + 6β⁻) before reaching stable ²⁰⁶Pb. Key intermediates:

3. Nuclear Fission

When a heavy nucleus such as ²³⁵U absorbs a slow (thermal) neutron, it becomes highly excited, deforms into an elongated shape, and splits into two fission fragments plus additional neutrons and a large energy release — mostly as kinetic energy of the fragments.

Key reaction

A typical fission event:

For comparison, burning one carbon atom releases about 4 eV — fission releases 50 million times more energy per atom.

Q-value from binding energies

The energy released equals the difference in binding energies of products and reactants:

(The remaining ~28 MeV appears as prompt gamma radiation and neutrinos from fission-product beta decay.)

Chain reaction and criticality

Each fission releases on average 2–3 neutrons. If these cause further fissions, a chain reaction develops. The multiplication factor $k$ determines the regime:

• $k < 1$ — subcritical: chain dies out
• $k = 1$ — critical: sustained, steady reaction (nuclear reactor)
• $k > 1$ — supercritical: exponential growth (weapon/runaway)

Reactor design

A nuclear reactor sustains a controlled $k \approx 1$ chain reaction. Key components:

• Fuel: slightly enriched UO₂ pellets (3–5% U-235 for power reactors)
• Moderator: slows fast neutrons to thermal energies where fission cross-section is much higher (light water, heavy water, or graphite)
• Control rods: neutron-absorbing materials (boron, hafnium, cadmium) inserted/withdrawn to regulate $k$
• Coolant: carries heat to the steam turbine (water, CO₂, sodium)
• Containment: multi-layer structure to contain radioactive materials in any accident

Nuclear waste

Spent fuel contains fission products with half-lives ranging from seconds to thousands of years. High-level waste (HLW) — primarily actinides and fission products — requires geological disposal (deep borehole or mined repository). Finland's Onkalo repository is the world's first approved HLW site, currently under construction.

4. Nuclear Fusion

In fusion, two light nuclei overcome their mutual electrostatic repulsion (the Coulomb barrier) and merge, releasing energy because the product is more tightly bound — higher on the B/A curve. Fusion powers all stars and is the ultimate clean energy promise.

The primary reaction

The most accessible fusion reaction for terrestrial reactors uses deuterium ($^2$H) and tritium ($^3$H):

Total energy: 17.6 MeV per reaction. The neutron carries most of the energy and is captured in a lithium blanket to breed more tritium and generate heat.

The Lawson criterion

For fusion to produce more energy than it takes to sustain the plasma, the product of plasma density $n$ and energy confinement time $\tau_E$ must exceed:

This is equivalent to requiring that the fusion power output exceeds plasma energy losses. At $T \approx 150$ million K (ten times hotter than the Sun's core), the D-T cross-section is maximised.

Why fusion needs extreme conditions

Nuclei carry positive charge, so they repel each other strongly at close range (the Coulomb barrier: $\sim$1 MeV for protons at fm distances). Two mechanisms allow fusion to proceed:

1. Thermal energy: At temperatures of ~100 million K, particles in the tail of the Maxwell-Boltzmann distribution have enough kinetic energy to approach closely.
2. Quantum tunnelling: Even below the classical barrier, there is a non-zero probability that nuclei tunnel through — this is critical in stellar cores at "only" 15 million K.

Stellar fusion: the pp chain

In main-sequence stars like the Sun ($T_{\text{core}} \approx 1.5 \times 10^7$ K), hydrogen burns via the proton-proton (pp) chain:

The net reaction fuses four protons into one helium-4. In massive stars ($T > 2 \times 10^7$ K), the CNO cycle dominates, using carbon-12 as a catalyst to achieve the same net result faster.

Current state of fusion research (2024–2026)

NIF — National Ignition Facility (USA): On December 5, 2022, NIF achieved scientific ignition — the first experiment to produce more fusion energy than laser energy delivered to the target: 3.15 MJ output from 2.05 MJ laser input (gain Q ≈ 1.5). This historic milestone demonstrated that inertial confinement fusion is physically feasible.

JET — Joint European Torus (UK): In February 2022, JET set the world record for sustained fusion energy: 59 MJ over 5 seconds (≈11 MW average). JET closed permanently in December 2023 after 40 years of operation, its data feeding directly into ITER design.

ITER (France): A 35-country, €20 billion tokamak under construction in Cadarache. Design goal: Q = 10 (500 MW fusion from 50 MW input). First plasma delayed to 2025/2026, deuterium-tritium operations planned ~2035.

Private sector:

• Commonwealth Fusion Systems (SPARC): Demonstrated 20-tesla high-temperature superconducting (HTS) magnets in 2021 — a world record — enabling a much more compact tokamak. SPARC is projected for first plasma ~2025, commercial ARC plant ~2030s.
• Helion Energy: Field-reversed configuration, targets Q>1 by ~2024–2025; has a power purchase agreement with Microsoft for 2028 delivery.
• TAE Technologies: Aneutronic p-¹¹B fusion (no neutrons), field-reversed configuration.
• Timeline to commercial fusion: Optimistic ~2035 (private ventures), realistic 2040–2050 (ITER pathway).

5. Nuclear Weapons: History & Physics

The Manhattan Project (1942–1945)

In August 1939, Leo Szilárd and Albert Einstein sent a letter to President Roosevelt warning that Germany might be developing an atomic bomb using recently discovered fission. The US launched the Manhattan Project in 1942 — a secret $2 billion program employing ~130,000 people at sites including Los Alamos (NM), Oak Ridge (TN), and Hanford (WA).

Key figures: J. Robert Oppenheimer (scientific director at Los Alamos), Enrico Fermi (first self-sustained nuclear chain reaction, Chicago Pile-1, December 2, 1942), Niels Bohr, Edward Teller, and Klaus Fuchs (who later passed secrets to the Soviets). Werner Heisenberg led the parallel German program, which never achieved a reactor.

• Trinity test: July 16, 1945, Jornada del Muerto desert, NM — world's first nuclear explosion, yield ~20 kt TNT equivalent.
• Little Boy (Aug 6, 1945, Hiroshima): gun-type U-235 bomb, ~15 kt, ~70,000–80,000 immediate deaths.
• Fat Man (Aug 9, 1945, Nagasaki): implosion Pu-239 bomb, ~21 kt, ~40,000–50,000 immediate deaths.

Bomb designs

Gun-type: A subcritical mass of U-235 is fired at high speed into another subcritical mass, forming a supercritical assembly. Simple and reliable, but requires a large amount of highly enriched uranium (HEU) and is too slow for Pu-239 (which contains Pu-240 impurities that cause premature detonation). Used only once in combat (Little Boy).

Implosion design: A subcritical sphere of Pu-239 (or U-235) is surrounded by precisely shaped high-explosive lenses. Simultaneous detonation compresses the core to supercritical density in microseconds — increasing density increases $k$ above 1. This design is more compact and efficient, and is standard in modern warheads.

Thermonuclear (H-bomb) — Teller-Ulam design (1952): A fission "primary" stage produces X-rays that compress and ignite a fusion "secondary" stage containing lithium deuteride. The two stages are enclosed in a radiation case (typically uranium). Yields can reach tens of megatons — the Soviet Tsar Bomba (1961) yielded ~50 Mt, the largest ever detonated.

Critical mass physics

Critical mass is the minimum amount of fissile material in which a self-sustaining chain reaction can occur. The neutron multiplication factor:

where $\nu$ is the average neutrons per fission (~2.4 for U-235), $\Sigma_f$ is the macroscopic fission cross-section, $\Sigma_a$ is the absorption cross-section, and $\Sigma_{\text{leakage}}$ accounts for neutrons escaping the assembly. A reflector (e.g., beryllium) scatters neutrons back, drastically reducing critical mass:

• ²³⁵U bare critical mass: ~52 kg; with Be reflector: ~15 kg
• ²³⁹Pu bare critical mass: ~10 kg; with Be reflector: ~4 kg

The approximate fission energy yield:

At Hiroshima, only ~1 kg of the 64 kg U-235 charge actually fissioned (~1.5% efficiency) before the bomb blew itself apart.

Arms race & treaties

After the US, nuclear weapons were developed by: Soviet Union 1949 (Joe-1), UK 1952, France 1960, China 1964, India 1974, Pakistan 1998, North Korea 2006. Israel is widely believed to possess weapons but has never confirmed.

Key treaties: Nuclear Non-Proliferation Treaty (NPT, 1968, 191 parties); SALT I/II (1970s); START I/II/New START (strategic arms reductions); Comprehensive Nuclear-Test-Ban Treaty (CTBT, 1996 — signed but not yet in force, as key states have not ratified).

Current stockpiles (estimated, 2025):

6. Nuclear Energy Today (2026)

Nuclear power currently provides approximately 10% of global electricity from 440 operating reactors in 31 countries, with a combined capacity of about 390 GWe. It is the world's second-largest source of low-carbon electricity after hydropower.

Active developments

• China is the world's largest nuclear builder with ~20 reactors under construction. China's CANDU-derived HPR1000 and CAP1400 designs are being exported.
• UAE Barakah: The Arab world's first nuclear plant — all four Korean APR-1400 units are now operational (Units 1–4, 2020–2024).
• Finland Olkiluoto-3: Europe's first new reactor in 15 years finally reached full commercial operation in 2023 after significant delays.
• France: Plans to build 6–14 new EPR2 reactors, reversing a decade of nuclear phase-out policy.

Small Modular Reactors (SMRs)

SMRs (<300 MWe) promise factory-fabricated, standardised units that can be installed faster and at lower upfront cost than large conventional plants. Leading designs include NuScale (US, 77 MWe iPWR — though its first project at Idaho Falls was cancelled in 2023 due to cost increases), Rolls-Royce SMR (UK, 470 MWe), and GE Hitachi's BWRX-300 (under licensing review in Canada and the US).

Nuclear and climate

Lifecycle greenhouse gas emissions from nuclear power (~12 g CO₂eq/kWh) are among the lowest of any energy technology — comparable to wind and lower than solar PV. The IPCC's net-zero scenarios generally retain or expand nuclear capacity.

Major accidents

Thorium reactors

Thorium-232 is fertile (not fissile): when irradiated with neutrons it converts to U-233, which is fissile. Potential advantages: more abundant than uranium, harder to weaponise, produces less long-lived transuranic waste. Molten Salt Reactors (MSRs) are the leading concept — the fuel is dissolved in liquid fluoride salts at atmospheric pressure. India, which holds one-third of global thorium reserves, has an active thorium reactor development program (AHWR). No commercial thorium reactor currently operates, though China and the Netherlands have active research programs.