Nuclear Fission and Fusion

Spec mapping: OCR H556 Module 6.4 — Nuclear and Particle Physics (induced fission of heavy nuclei such as U-235; chain reactions, the multiplication factor $k$k and critical mass; controlled chain reaction in a thermal reactor — fuel, moderator, control rods, coolant; fusion of light nuclei (D-T) and the Coulomb-barrier requirement of $\sim 10^{8}\,\text{K}$∼108K plasma temperature; both processes release energy because nucleons move toward the iron-56 binding-energy peak; comparative energy densities of nuclear vs chemical reactions). Refer to the official OCR H556 specification document for exact wording.

In the last lesson we saw that the binding-energy-per-nucleon curve rises steeply from hydrogen, peaks at iron-56, and declines gently to uranium. This single feature explains both of the great energy-releasing nuclear processes of the twentieth century: fission (splitting heavy nuclei) and fusion (joining light nuclei). Both release energy because both move nucleons toward the Fe-56 peak. Together they power the reactors that supply a significant fraction of the world's electricity and the stars that light up the night sky.

This lesson covers Module 6.4 of the OCR A-Level Physics A specification (H556), describing both processes qualitatively and quantitatively, and explaining why they are so hard to engineer in practice even though the underlying energies are so enormous.

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Nuclear Fission

Nuclear fission is the splitting of a heavy nucleus into two smaller nuclei, typically of roughly comparable mass, with the release of several neutrons and a large amount of energy. The classic fission reaction is the neutron-induced fission of uranium-235:

$^{235}_{\ 92}\text{U} + {}^{1}_{0}n \to {}^{236}_{\ 92}\text{U}^{*} \to {}^{141}_{\ 56}\text{Ba} + {}^{92}_{36}\text{Kr} + 3\, {}^{1}_{0}n + \text{energy}$ 92235​U+01​n→ 92236​U∗→ 56141​Ba+3692​Kr+301​n+energy

A slow ("thermal") neutron is absorbed by a U-235 nucleus, forming a highly excited U-236 compound nucleus. This excited nucleus is unstable and deforms into a dumbbell shape, eventually splitting into two unequal fragments — in this example, barium-141 and krypton-92. Three fast neutrons are also released, along with gamma photons and beta-decay products from the fragments. The total energy release is about 200 MeV per fission event — split into kinetic energy of the two fragments ($\sim 165\,\text{MeV}$∼165MeV), prompt neutrons ($\sim 5\,\text{MeV}$∼5MeV), prompt gammas ($\sim 7\,\text{MeV}$∼7MeV), and delayed beta + gamma decays of fragments ($\sim 25\,\text{MeV}$∼25MeV).

Key features of fission:

• Induced by a slow neutron. Fast neutrons are less effective; they tend to pass through the nucleus without being captured. The neutrons must first be moderated — slowed down — by repeated elastic collisions with light nuclei such as hydrogen, deuterium or carbon.
• Produces multiple new neutrons. On average, about 2.5 neutrons per fission event. These can go on to induce further fissions in nearby U-235 nuclei.
• Two fission fragments of unequal mass. The typical distribution peaks around $A \approx 95$A≈95 and $A \approx 140$A≈140, not at the symmetric $A \approx 118$A≈118. The asymmetry is a shell-structure effect.
• Radioactive products. The fragments are highly neutron-rich (heavy nuclei have a high $N/Z$N/Z ratio, which the fragments inherit), so they $\beta^{-}$β−-decay through long chains to reach stability. This is the source of the long-lived radioactive waste from nuclear reactors.
• Energy $\sim 200\,\text{MeV}$∼200MeV per fission. Mostly kinetic energy of the two fragments (flying apart due to Coulomb repulsion), with smaller contributions from prompt neutrons, prompt gammas, and delayed beta + gamma decays.

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Chain Reactions

Because each fission releases 2–3 new neutrons, and each new neutron can in principle induce another fission, a chain reaction becomes possible. In a block of U-235, if on average at least one of the new neutrons from each fission goes on to cause another fission, the reaction will sustain itself. If more than one does, the reaction will grow exponentially.

The key quantity is the multiplication factor $k$k, defined as the average number of neutrons from each fission that go on to cause a subsequent fission. Three regimes:

• $k < 1$k<1: subcritical. The chain reaction dies out.
• $k = 1$k=1: critical. The reaction runs at a constant rate. This is the regime of a power reactor.
• $k > 1$k>1: supercritical. The reaction grows exponentially. Controlled supercriticality is used in reactor startup; uncontrolled supercriticality is a nuclear explosion.

The multiplication factor depends on the amount of fissile material (more material $\Rightarrow$⇒ fewer neutrons escape from the surface), the shape (a sphere minimises surface area per unit volume), the presence of moderators and neutron absorbers, and the enrichment of U-235 relative to U-238 (natural uranium is only 0.72 % U-235 and will not sustain a chain reaction without enrichment and moderation).

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Critical Mass

For a given geometry and purity of fissile material, there is a minimum mass below which $k < 1$k<1 and a chain reaction cannot be sustained. This is called the critical mass. For a bare sphere of pure U-235 metal, the critical mass is about $52\,\text{kg}$52kg; for pure Pu-239 it is about $10\,\text{kg}$10kg. Surrounding the fissile core with a tamper (a dense material that reflects escaping neutrons back into the core) reduces these figures significantly.

You do not need to memorise the numbers, but you should understand the qualitative reason. The fission rate scales with the volume of the sample, $\sim r^{3}$∼r3, while the neutron-leakage rate scales with the surface area, $\sim r^{2}$∼r2. The ratio surface/volume $\sim 1/r$∼1/r falls as the sample grows. In a small sample, too many neutrons escape from the surface before causing further fission and the chain dies; in a large sample, enough are retained to sustain the reaction. This same surface-to-volume argument explains why critical mass is a geometric quantity: the same amount of material can be subcritical as a thin plate (high surface area) and supercritical as a sphere (minimum surface area).

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The Reactor: Controlled Fission

A nuclear power reactor runs a controlled, self-sustaining chain reaction at $k = 1$k=1. The key components are:

• Fuel rods: zirconium-clad tubes of uranium dioxide pellets, typically enriched to $\sim 3$∼3–$5\,\%$5% U-235.
• Moderator: a material that slows fast neutrons to thermal energies without absorbing them. Water ($\text{H}_{2}\text{O}$H2​O), heavy water ($\text{D}_{2}\text{O}$D2​O), and graphite are the common choices.
• Control rods: made of strong neutron absorbers (boron, cadmium). Inserting them lowers $k$k; withdrawing them raises $k$k. Operators adjust them continuously to keep $k = 1$k=1.
• Coolant: transfers heat from the fuel to a steam-generating circuit. Often the same water that serves as moderator.
• Containment: thick concrete and steel shielding to contain radiation and radioactive material in the event of an accident.

The thermal power of a typical reactor is around $3\,\text{GW}$3GW, corresponding to about $10^{20}$1020 fissions per second. Each fission releases about $200\,\text{MeV}$200MeV, and the total thermal output is converted to electricity by a steam-turbine cycle at $\sim 33\,\%$∼33% efficiency, giving $\sim 1\,\text{GW}$∼1GW of electrical output. A gigawatt reactor burns about a tonne of U-235 per year (see Worked Example 1).

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Nuclear Fusion

Nuclear fusion is the joining of two light nuclei to form a heavier, more tightly bound nucleus, releasing energy. Fusion is how the Sun and other stars produce their energy: the net result of the proton-proton chain in the solar core is that four protons combine into one helium-4 nucleus, releasing $\approx 26.7\,\text{MeV}$≈26.7MeV per helium produced.

The simplest terrestrial fusion reaction — and the target of ITER and future commercial fusion reactors — is the deuterium-tritium (D-T) reaction:

$^{2}_{1}\text{H} + {}^{3}_{1}\text{H} \to {}^{4}_{2}\text{He} + {}^{1}_{0}n + 17.6\,\text{MeV}.$12​H+13​H→24​He+01​n+17.6MeV.

The energy release is about $17.6\,\text{MeV}$17.6MeV per reaction (computed by the binding-energy or mass-defect route in the previous two lessons). Of this, $14.1\,\text{MeV}$14.1MeV is carried by the neutron and $3.5\,\text{MeV}$3.5MeV by the alpha particle. In a reactor, the alpha (charged) stays in the plasma and heats it further (helping to sustain the reaction), while the neutron (uncharged) escapes and is captured in a surrounding lithium blanket where it deposits its energy as heat and breeds further tritium fuel.

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Why Is Fusion So Hard?

Fusion is energetically very favourable: the D-T reaction releases $17.6\,\text{MeV}$17.6MeV, far more than any chemical reaction. The difficulty is initiating it. For the two nuclei to get close enough for the short-range strong nuclear force to bind them, they must first overcome their mutual electrostatic (Coulomb) repulsion. The energy scale of this Coulomb barrier for D and T is of order an MeV per pair, requiring temperatures of order $10^{10}\,\text{K}$1010K to overcome by thermal energy alone.

In practice, quantum tunnelling allows fusion at substantially lower temperatures, and the D-T reaction has a usable cross-section around $T \sim 10^{8}\,\text{K}$T∼108K (mean particle energies of order $10\,\text{keV}$10keV). Even at this reduced temperature, maintaining a plasma at 100 million kelvin, confined densely enough for fusion to proceed at a useful rate, is an extraordinary engineering challenge. The two main approaches are:

• Magnetic confinement (tokamaks such as JET and ITER): strong magnetic fields trap the charged plasma in a torus-shaped vacuum chamber.
• Inertial confinement (laser-driven fusion, e.g. NIF): intense lasers compress and heat a tiny D-T pellet to fusion conditions in a few nanoseconds.

No fusion reactor has yet produced net energy output over extended operation, though NIF and JET have achieved scientific energy gains of order unity in brief experiments. Commercial fusion power remains several decades away.

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Fission vs Fusion: A Comparison

Feature	Fission	Fusion
Fuel	U-235, Pu-239	D, T, $^{3}\text{He}$3He, light nuclei
Reaction	Heavy nucleus splits	Light nuclei merge
Trigger	Induced by slow neutron	Thermal motion overcoming Coulomb barrier
Energy per reaction	$\sim 200\,\text{MeV}$∼200MeV	$\sim 17.6\,\text{MeV}$∼17.6MeV (D-T)
Energy per unit mass	$\sim 8 \times 10^{13}\,\text{J/kg}$∼8×1013J/kg U-235	$\sim 3 \times 10^{14}\,\text{J/kg}$∼3×1014J/kg D-T
Conditions	Moderated thermal neutrons, room temperature	Plasma at $T \sim 10^{8}\,\text{K}$T∼108K
Products	Radioactive fragments + neutrons	He (inert) + neutron
Radioactive waste	Long-lived, high-level	Short-lived (activated structural materials)
Current status	Commercial since 1950s	Experimental only
Examples	Power reactors, fission bomb	Sun, H-bomb, JET, ITER, NIF

Both fission and fusion are explained by the binding-energy-per-nucleon curve of the previous lesson: heavy nuclei moving toward Fe-56 (fission) and light nuclei moving toward Fe-56 (fusion) both increase the average $E_{B}/A$EB​/A, releasing the difference as kinetic energy of the products.

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Energy Density Compared to Chemical Reactions

Per atom (or molecule), chemical reactions release of order $1\,\text{eV}$1eV, whereas nuclear reactions release of order $1\,\text{MeV}$1MeV — a factor of $10^{6}$106. Per unit mass of fuel, the comparison is even more striking:

Fuel	Energy density
Petrol (combustion of $\text{C}_{8}\text{H}_{18}$C8​H18​)	$\sim 4.7 \times 10^{7}\,\text{J/kg}$∼4.7×107J/kg
U-235 (fission, $\sim 200\,\text{MeV/nucleus}$∼200MeV/nucleus)	$\sim 8.2 \times 10^{13}\,\text{J/kg}$∼8.2×1013J/kg
D-T (fusion, $17.6\,\text{MeV/pair}$17.6MeV/pair)	$\sim 3.4 \times 10^{14}\,\text{J/kg}$∼3.4×1014J/kg
Antimatter (annihilation, full rest energy)	$\sim 9.0 \times 10^{16}\,\text{J/kg}$∼9.0×1016J/kg

Fission of 1 kg of U-235 releases the same energy as burning roughly two million kg of petrol — a factor of order $10^{6}$106. Fusion gives another factor of $\sim 4$∼4 on top. This is why a nuclear reactor refuels every $\sim 2$∼2 years while a coal-fired power station of the same output needs trainloads of fuel every day.

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Worked Example 1 — Fissions per second in a 3 GW reactor

A nuclear reactor produces $3.0\,\text{GW}$3.0GW of thermal power. Assuming $200\,\text{MeV}$200MeV per fission, how many fissions occur per second, and how much U-235 does it consume per day?

Energy per fission: $E = 200 \times 10^{6} \times 1.60 \times 10^{-19} = 3.2 \times 10^{-11}\,\text{J}$E=200×106×1.60×10−19=3.2×10−11J.

Rate: $R = P/E = (3.0 \times 10^{9})/(3.2 \times 10^{-11}) \approx 9.4 \times 10^{19}\,\text{fissions s}^{-1}$R=P/E=(3.0×109)/(3.2×10−11)≈9.4×1019fissions s−1.

Mass consumed per second: $\dot m = R \times 235\,\text{u} \times (1\,\text{u in kg}) = 9.4 \times 10^{19} \times 235 \times 1.66 \times 10^{-27} \approx 3.7 \times 10^{-5}\,\text{kg s}^{-1}$m˙=R×235u×(1u in kg)=9.4×1019×235×1.66×10−27≈3.7×10−5kg s−1, or about $3.2\,\text{kg per day}$3.2kg per day. A gigawatt-electric reactor therefore burns about a tonne of U-235 per year (or about three tonnes of enriched UO$_{2}$2​ fuel).

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Worked Example 2 — Hydrogen burn rate of the Sun

The Sun radiates at $L = 3.8 \times 10^{26}\,\text{W}$L=3.8×1026W. The net solar fusion reaction converts 4 protons into a helium-4 nucleus, releasing $\approx 26.7\,\text{MeV}$≈26.7MeV per helium. Estimate how much hydrogen the Sun burns per second.

Energy per reaction: $E = 26.7 \times 10^{6} \times 1.60 \times 10^{-19} = 4.3 \times 10^{-12}\,\text{J}$E=26.7×106×1.60×10−19=4.3×10−12J.

Reactions per second: $N = L/E = (3.8 \times 10^{26})/(4.3 \times 10^{-12}) \approx 8.8 \times 10^{37}\,\text{s}^{-1}$N=L/E=(3.8×1026)/(4.3×10−12)≈8.8×1037s−1.

Each reaction consumes 4 protons (mass $4 m_{p} \approx 6.7 \times 10^{-27}\,\text{kg}$4mp​≈6.7×10−27kg), so

$\dot m = 8.8 \times 10^{37} \times 6.7 \times 10^{-27} \approx 5.9 \times 10^{11}\,\text{kg s}^{-1}.$m˙=8.8×1037×6.7×10−27≈5.9×1011kg s−1.

That is about $\mathbf{600\,million\,tonnes\,of\,hydrogen\,every\,second}$600milliontonnesofhydrogeneverysecond. Fortunately the Sun has an awful lot of hydrogen ($M_{\odot} \approx 2 \times 10^{30}\,\text{kg}$M⊙​≈2×1030kg) — this rate can be sustained for about 10 billion years, of which about 4.6 billion have already passed. The Sun is roughly halfway through its main-sequence life.

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Worked Example 3 — Critical mass scaling

A sphere of U-235 of radius $r$r has fission rate proportional to its volume ($\propto r^{3}$∝r3) and neutron leakage proportional to its surface area ($\propto r^{2}$∝r2). The chain just sustains itself when the ratio of fissions induced to neutrons leaked exceeds one. Show qualitatively that there exists a minimum critical radius $r_{c}$rc​ and hence a critical mass $m_{c} \propto r_{c}^{3}$mc​∝rc3​.