Fission Fusion

13.7 Nuclear Energy — Why Fission and Fusion Both Release Energy

Recall the BE/A curve: the most tightly bound nuclei sit near A ≈ 56 (iron). Lighter nuclei are loosely bound; very heavy nuclei (A > 170) are also less tightly bound because Coulomb repulsion of many protons partly cancels the strong-force attraction. Energy is released whenever loosely-bound nuclei rearrange into more tightly-bound ones.

Two routes to climb up the BE curve:

• Fission: A heavy nucleus (A ~ 235) splits into two intermediate-mass fragments (each A ~ 100–140). Each fragment is more tightly bound — energy is released.
• Fusion: Two light nuclei merge into a heavier one with greater BE per nucleon — energy is released.

13.7.1 Nuclear Fission

Nuclear fission was discovered in 1938 by Otto Hahn and Fritz Strassmann (interpretation by Lise Meitner & Otto Frisch in 1939). When \(^{235}_{92}\mathrm{U}\) absorbs a slow neutron, the resulting \(^{236}\mathrm{U}^{*}\) compound nucleus is so unstable that it splits — typically into two unequal fragments and 2–3 prompt neutrons:

The fragments are radioactive — they sit on the neutron-rich side of the stability valley and undergo successive β-decays until reaching stable nuclei. The Q-value per fission is about 200 MeV, distributed roughly as 165 MeV kinetic energy of fragments + 5 MeV neutron KE + 7 MeV prompt γ-rays + the rest carried away by β/γ from fission products and antineutrinos.

Order-of-magnitude estimate of Q

Take a parent A = 240 nucleus with E_bn ≈ 7.6 MeV splitting into two A = 120 fragments with E_bn ≈ 8.5 MeV.

This rough estimate is in excellent agreement with measurements (~200 MeV).

Chain Reaction and Critical Mass

Each fission releases 2–3 neutrons. If at least one of those neutrons triggers another fission, a self-sustaining chain reaction develops. If on average exactly 1 neutron per fission causes the next fission (the multiplication factor k = 1), the reaction is critical and steady — the basis of a power reactor. If k > 1, fissions multiply exponentially — the basis of a fission bomb. The minimum mass of fissile material that allows k ≥ 1 (given a particular geometry) is the critical mass.

Nuclear Reactor (Schematic)

A nuclear reactor turns this chain reaction into controlled, steady heat. Five essential components:

1. Fuel: Usually uranium enriched in ²³⁵U (or ²³⁹Pu). The fissile fuel is shaped into rods.
2. Moderator: Water, heavy water (D₂O) or graphite. Slows down fast fission neutrons (≈ 1 MeV) to thermal energies (≈ 0.025 eV) where they are most efficiently absorbed by ²³⁵U.
3. Control rods: Boron or cadmium absorb excess neutrons. Pushing them in slows the reaction; pulling them out speeds it up. They keep k = 1.
4. Coolant: Water (or liquid metal/gas) carries away the heat.
5. Shielding: Thick concrete and steel absorb stray neutrons and γ-rays.

Worked Example 13.7 — Energy from 1 kg of ²³⁹Pu

13.7.2 Nuclear Fusion — Energy from the Stars

Nuclear fusion is the inverse process: light nuclei combine into a heavier one whose BE per nucleon is greater. Some examples:

The Coulomb Barrier

For fusion, the two positively charged nuclei must approach within ~1 fm so the strong force can take over. Their mutual Coulomb repulsion erects a potential barrier of typical height ≈ 400 keV (for two protons). To climb this barrier thermally requires temperatures of order

Such high temperatures are why fusion is described as thermonuclear. (The sun's core is "only" 1.5 × 10⁷ K — fusion proceeds there because the high-energy tail of the Maxwell-Boltzmann distribution and quantum tunnelling combine to make protons leak through the barrier.)

Proton-Proton Cycle (energy of the Sun)

In stars like the Sun, hydrogen is "burned" into helium through the p-p cycle:

Steps (i)-(iii) must occur twice for one (iv). Net effect:

Each second, the Sun converts ~6 × 10¹¹ kg of hydrogen into helium, losing ~4 × 10⁹ kg of mass per second to radiation — yet the Sun has enough hydrogen for another ~5 billion years.

Controlled Thermonuclear Fusion

If we can replicate stellar fusion on Earth, we get an essentially unlimited, low-radioactive-waste energy source. The challenge is plasma confinement — heating fuel to 10⁸ K without a container melting. Two main approaches:

• Magnetic confinement (tokamak): toroidal magnetic fields confine the plasma. Examples: ITER (international project), India's ADITYA and SST-1.
• Inertial confinement: high-power lasers (or ion beams) compress and heat a tiny fuel pellet (e.g. NIF, USA).

Worked Example 13.8 — Lamp powered by deuterium fusion

Worked Example 13.9 — Coulomb barrier height

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