# E.6 Fission and fusion (AHL)

> IB Physics HL · IB Physics HL 2025
> Source: https://www.owlsprep.com/study/ib-physics-hl-u5-e-6-fission-and-fusion/

This sub-topic explores nuclear fission and fusion, explaining how energy is released from binding energy differences, and the physical conditions required for sustained energy production from both processes.

**Prerequisites:** [Binding energy and mass defect](https://www.owlsprep.com/study/ib-physics-hl-u5-e-4-binding-energy-per-nucleon/); [Nuclear reaction notation](https://www.owlsprep.com/study/ib-physics-hl-u5-e-1-nuclear-structure-and-reactions/)

## Learning objectives

- Distinguish fission and fusion in terms of binding energy per nucleon
- Calculate energy released from mass defect in both reactions
- Explain the conditions for sustained fission chain reactions
- Describe the requirements for controlled nuclear fusion

## Energy Release from Fission and Fusion

Both processes release energy because of the shape of the binding energy per nucleon (BE/A) curve. Intermediate mass nuclei (A ≈ 56) have the highest BE/A, so splitting heavy nuclei (fission) or joining light nuclei (fusion) produces more tightly bound products, with excess energy released.

**Mass-Energy Equivalence** — The energy released in a nuclear reaction is equal to the mass defect multiplied by the speed of light squared.

*Example:* A mass defect of 1 u corresponds to 931.5 MeV of released energy

$$E = \Delta m c^2 \quad \text{where} \quad \Delta m = m_{\text{reactants}} - m_{\text{products}}$$

**Worked example:** Calculate the energy released in the fission reaction: $^{235}_{92}\text{U} + ^1_0\text{n} \rightarrow ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3^1_0\text{n}$. Given masses: $m(U-235)=235.0439$ u, $m(n)=1.0087$ u, $m(Ba-141)=140.9144$ u, $m(Kr-92)=91.9262$ u. 1 u = 931.5 MeV/c².

1. Calculate total mass of reactants:

   $$m_{\text{reactants}} = 235.0439 + 1.0087 = 236.0526 \text{ u}$$
2. Calculate total mass of products:

   $$m_{\text{products}} = 140.9144 + 91.9262 + (3 \times 1.0087) = 235.8667 \text{ u}$$
3. Find the positive mass defect:

   $$\Delta m = 236.0526 - 235.8667 = 0.1859 \text{ u}$$
4. Convert mass defect to energy released:

   $$E = 0.1859 \times 931.5 \approx 173 \text{ MeV}$$

## Nuclear Fission and Chain Reactions

When a heavy fissile nucleus absorbs a neutron and undergoes fission, it releases multiple new neutrons. These neutrons can trigger fission in other nearby nuclei, creating a multiplying chain reaction.

**Critical Chain Reaction** — A steady chain reaction where exactly one neutron from each fission event goes on to cause another fission, maintaining constant power output.

In commercial nuclear reactors, neutron-absorbing control rods (made of boron or cadmium) are adjusted to maintain a critical state. Subcritical reactions (k < 1) die out, while supercritical reactions (k > 1) grow exponentially.

**Worked example:** A reactor starts with 1 neutron in generation 1. The neutron multiplication factor $k = 1.01$. How many neutrons are present in generation 100?

1. The number of neutrons after $n-1$ generations follows the exponential growth rule:

   $$N_n = k^{n-1}$$
2. Substitute $n=100$ and $k=1.01$:

   $$N_{100} = (1.01)^{99} \approx 2.7$$
3. Interpret the result: the number of neutrons slowly increases, so control rods must be adjusted to reduce $k$ to 1 for steady power.

## Nuclear Fusion

Nuclear fusion joins two light positively charged nuclei to form a heavier nucleus. For fusion to occur, the nuclei must overcome electrostatic Coulomb repulsion between them, which requires extremely high temperatures (≥ 10⁷ K) and high density to produce frequent energetic collisions.

**Coulomb Repulsion** — The electrostatic force that repels two positively charged atomic nuclei, which must be overcome for fusion to occur.

**Worked example:** Calculate the energy released in the deuterium-tritium fusion reaction: $^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + ^1_0\text{n}$. Given masses: $m(^2\text{H})=2.0141$ u, $m(^3\text{H})=3.0160$ u, $m(^4\text{He})=4.0026$ u, $m(n)=1.0087$ u.

1. Calculate total mass of reactants:

   $$m_{\text{reactants}} = 2.0141 + 3.0160 = 5.0301 \text{ u}$$
2. Calculate total mass of products:

   $$m_{\text{products}} = 4.0026 + 1.0087 = 5.0113 \text{ u}$$
3. Calculate energy released:

   $$E = (5.0301 - 5.0113) \times 931.5 \approx 17.5 \text{ MeV}$$

> **tip**
>
> Fusion reactors on Earth require temperatures ~100 million °C, 6x hotter than the Sun's core, to compensate for the much lower plasma pressure than exists in stars.

## Exam Command Term Guidance

**Exam command terms**

Common command terms in this topic have clear exam expectations:

- **Distinguish** — Highlight key differences between fission and fusion *(You must mention differences in reactant size, energy output per mass, and waste to get full marks.)*

- **Calculate** — Show all steps from mass defect to final energy *(Always state Δm = m_reactants - m_products to avoid losing marks for sign errors.)*

## Common pitfalls

- **Wrong:** Calculating mass defect as product mass minus reactant mass
  - Why it fails: This gives a negative Δm, which can lead to confusion and lost marks for exothermic reactions
  - Correct: Always calculate Δm = m_reactants - m_products to get a positive value for energy released
- **Wrong:** Claiming fusion releases more energy per reaction than fission
  - Why it fails: Per individual reaction, fission releases ~170-200 MeV, while fusion releases ~10-20 MeV
  - Correct: State that fusion releases more energy per unit mass of fuel, not per reaction
- **Wrong:** Confusing critical, subcritical and supercritical chain reaction states
  - Why it fails: Students often mix up the value of the multiplication factor k for each state
  - Correct: k<1 = subcritical (dies out), k=1 = critical (steady), k>1 = supercritical (grows)
- **Wrong:** Thinking the Sun's core is hotter than fusion reactors on Earth
  - Why it fails: The Sun has extremely high core pressure, so fusion proceeds at lower temperatures than Earth-based reactors
  - Correct: Remember Earth fusion reactors need higher temperatures to compensate for lower pressure

## Cheatsheet

| Property | Nuclear Fission | Nuclear Fusion |
| --- | --- | --- |
| Reactant nuclei | Heavy (A > 200) | Light (A < 20) |
| Energy per reaction | ~170-200 MeV | ~10-20 MeV |
| Energy per kg fuel | ~8 × 10¹³ J | ~3 × 10¹⁴ J (4× fission) |
| Required conditions | Critical mass of fissile material | T > 10⁷ K, plasma confinement |
| Waste | Long-lived highly radioactive waste | Low-level short-lived waste |
| Commercial use | Mature technology | Still in development |

## What's next

This topic builds on your understanding of binding energy and mass defect to explain two of the most important energy-releasing nuclear processes in physics. Mastery of fission and fusion is essential for understanding stellar evolution, nucleosynthesis, and nuclear energy production, which are common long-answer topics in IB Physics HL exams. The concepts here also connect directly to real-world energy debates and modern physics research into sustainable power. Below are related topics to explore next to deepen your understanding.

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