Nuclear Energy: Complete Guide to Fission and Fusion | HRK Chapter 55

Nuclear Energy: Complete Guide to Fission and Fusion

Mastering Nuclear Reactions, Fission, Fusion, Chain Reactions, and Energy from the Nucleus

Nuclear Fission Nuclear Fusion Chain Reaction Nuclear Reactor Thermonuclear Fusion Reading Time: 30 min

📋 Table of Contents

1. Introduction to Nuclear Reactions
2. Reaction Energy and Q-Value
3. Nuclear Fission
- 3.1 Discovery of Fission
- 3.2 Fission Products and Energy
4. Bohr-Wheeler Theory (Liquid Drop Model)
5. Chain Reactions
6. Nuclear Fission Reactors
- 6.1 Reactor Components
- 6.2 Reactor Control
7. Nuclear Fusion
8. Thermonuclear Fusion in Stars
9. Controlled Nuclear Fusion
10. Requirements for Thermonuclear Reactor
Frequently Asked Questions

📜 Historical Background

The discovery and development of nuclear energy spanned the 20th century:

Ernest Rutherford (1919): First artificial nuclear reaction by bombarding nitrogen with alpha particles
Otto Hahn & Fritz Strassman (1938): Discovered nuclear fission by bombarding uranium with neutrons
Lise Meitner & Otto Frisch (1939): Correctly interpreted fission process and coined the term "fission"
Enrico Fermi (1942): Built first nuclear reactor (Chicago Pile-1)
Manhattan Project (1942-1945): Developed first atomic bombs and nuclear reactors

These discoveries fundamentally changed our understanding of atomic nuclei and opened new possibilities for energy production.

Introduction to Nuclear Reactions

🔬 What is a Nuclear Reaction?

A nuclear reaction occurs when atomic nuclei interact, resulting in the transformation of one element into another. Unlike chemical reactions that involve electrons, nuclear reactions involve changes in the nucleus itself.

Rutherford suggested in 1919 that a massive particle with sufficient kinetic energy might be able to penetrate a nucleus. The result would be either a new nucleus with greater atomic number and mass number or a decay of the original nucleus.

⚛️ Rutherford's Nuclear Reaction

Alpha Particle (²He⁴)

↓

Nitrogen Nucleus (⁷N¹⁴)

↓

Oxygen Nucleus (⁸O¹⁷) + Proton (¹H¹)

First Artificial Nuclear Reaction: Rutherford bombarded nitrogen with alpha particles and obtained oxygen and a proton:

\[ ^{4}_{2}He + ^{14}_{7}N \rightarrow ^{17}_{8}O + ^{1}_{1}H \]

Conservation Laws: Nuclear reactions obey several conservation laws:

Conservation of charge
Conservation of momentum
Conservation of angular momentum
Conservation of energy (including rest energies)
Conservation of total number of nucleons

Reaction Energy and Q-Value

⚡ What is Reaction Energy?

The difference between the masses before and after a nuclear reaction corresponds to the reaction energy, according to Einstein's mass-energy relationship \( E = mc^2 \).

🧮 Q-Value Calculation

Step 1: Mass-Energy Relationship

If initial particles A and B interact to produce final particles C and D, the reaction energy Q is defined as:

\[ Q = (m_A + m_B)c^2 - (m_C + m_D)c^2 \]

Step 2: Kinetic Energy Relationship

If 'K' represents the kinetic energy, then the reaction energy 'Q' is also given by:

\[ Q = (K_C + K_D) - (K_A + K_B) \]

Step 3: Types of Reactions

Exothermal Reaction (Q > 0): Total mass decreases and total kinetic energy increases

Endothermal Reaction (Q < 0): Mass increases and kinetic energy decreases

💡 Key Insight

The Q-value determines whether a nuclear reaction releases or absorbs energy. Exothermal reactions (Q > 0) are energetically favorable and can occur spontaneously under the right conditions, while endothermal reactions (Q < 0) require energy input.

Nuclear Fission

💥 What is Nuclear Fission?

Nuclear fission is a decay process in which an unstable nucleus splits into two fragments of comparable mass. This process releases enormous amounts of energy due to the conversion of mass into energy according to \( E = mc^2 \).

Discovery of Fission

🔍 The Fission Discovery Story

Fission was discovered in 1938 through the experiments of Otto Hahn and Fritz Strassman in Germany. Pursuing earlier work by Fermi, they bombarded uranium (Z = 92) with neutrons.

The resulting radiation did not coincide with that of any known radioactive nuclide. Urged on by their colleague Lise Meitner, they used meticulous chemical analysis to reach the astonishing but inescapable conclusion that they had found a radioactive isotope of barium (Z = 56). Later, radioactive krypton (Z = 36) was also found.

Meitner and Otto Frisch correctly interpreted these results as showing that uranium nuclei were splitting into two massive fragments called fission fragments. Two or three free neutrons usually appear along with the fission fragments.

⚛️ Fission of Uranium-235

Neutron (⁰n¹)

↓

Uranium-235 (⁹²U²³⁵)

↓

Fission Fragments + Neutrons + Energy

Fission Process: Both the common isotope (99.3%) \(^{238}U\) and the uncommon isotope (0.7%) \(^{235}U\) can be split by neutron bombardment:

\(^{235}U\) splits with slow neutrons (kinetic energy < 1 eV)
\(^{238}U\) requires fast neutrons with minimum 1 MeV kinetic energy

Types of Fission:

Induced fission: Resulting from neutron absorption
Spontaneous fission: Without initial neutron absorption (quite rare)

Fission Products and Energy

🧮 Fission Energy Calculation

Step 1: Binding Energy Comparison

The total kinetic energy of fission fragments is enormous, about 200 MeV (compared to typical α and β energies of a few MeV).

Reason: Nuclides at the high end of the mass spectrum (near A = 240) are less tightly bound than those nearer the middle (A = 90 to 145).

Step 2: Binding Energy per Nucleon

\[ \text{Average binding energy at A = 240: } 7.6 \, \text{MeV/nucleon} \]

\[ \text{Average binding energy at A = 120: } 8.5 \, \text{MeV/nucleon} \]

Step 3: Energy Release Calculation

\[ \text{Increase in binding energy} = 8.5 - 7.6 = 0.9 \, \text{MeV/nucleon} \]

\[ \text{Total energy release} = 235 \times 0.9 \approx 200 \, \text{MeV} \]

📊 Fission Product Distribution

[Graph: Distribution of mass numbers for fission fragments from \(^{235}U\)]

Most fragments have mass numbers from 90 to 100 and from 135 to 145; fission into two fragments with nearly equal mass is unlikely.

Sample Fission Reactions

Two different nuclear fission reactions:

\[ ^{235}_{92}U + ^{1}_{0}n \rightarrow ^{236}_{92}U \rightarrow ^{144}_{56}Ba + ^{89}_{36}Kr + 3^{1}_{0}n \]

\[ ^{235}_{92}U + ^{1}_{0}n \rightarrow ^{236}_{92}U \rightarrow ^{140}_{54}Xe + ^{94}_{38}Sr + 2^{1}_{0}n \]

💡 Neutron Excess in Fission Fragments

Fission fragments always have too many neutrons to be stable. The neutron-proton ratio (N/Z) for stable nuclides:

About 1 for light nuclides
Almost 1.6 for the heaviest nuclides
About 1.3 at A = 100 and 1.4 at A = 150

The fragments have about the same N/Z as \(^{235}U\) (about 1.55). They usually respond to this surplus of neutrons by undergoing a series of β⁻ decays until a stable N/Z is reached.

Example: Beta Decay Chain

A typical beta decay chain for fission products:

\[ ^{140}_{54}Xe \xrightarrow{\beta^-} ^{140}_{55}Cs \xrightarrow{\beta^-} ^{140}_{56}Ba \xrightarrow{\beta^-} ^{140}_{57}La \xrightarrow{\beta^-} ^{140}_{58}Ce \]

\[ \text{(stable)} \]

This series of β⁻ decays produces, on average, about 15 MeV of additional kinetic energy.

Bohr-Wheeler Theory (Liquid Drop Model)

💧 Liquid Drop Model of Nuclear Fission

According to Bohr's liquid drop model, a nucleus can be compared with a (charged) liquid drop. Two types of forces play their role in nuclear stability:

Force Type	Liquid Drop Analogy	Nuclear Equivalent	Effect
Cohesive Forces	Intermolecular forces, surface tension	Strong nuclear force between nucleons	Keeps nucleus intact
Destructive Forces	External forces, charge repulsion	Coulomb repulsive force between protons	Tries to disrupt nucleus

🔬 Fission Process According to Liquid Drop Model

Step 1: Initial State

A \(^{235}U\) nucleus absorbs a slow neutron. The neutron falls into the potential well associated with the strong nuclear force. Its potential energy is converted into excitation energy of the nucleus.

Step 2: Nuclear Oscillations

The nucleus, which already has surface oscillations, will acquire a dumbbell shape. This excess energy causes violent oscillations.

Step 3: Neck Formation

During oscillations, a neck between two lobes develops. The electrical repulsion of these two lobes stretches the neck further.

Step 4: Fragment Separation

Finally, two smaller fragments are formed that move rapidly apart.

Step 5: Neutron Emission

The fragments emit neutrons at the time of fission (or occasionally a few seconds later).

📈 Potential Energy Barrier in Fission

[Graph: Hypothetical potential energy function for two possible fission fragments]

If neutron absorption results in an excitation energy greater than the energy barrier height \( U_B \), fission occurs immediately. Even when there isn't quite enough energy to surmount the barrier, fission can take place by quantum-mechanical tunneling.

Chain Reactions

🔄 What is a Nuclear Chain Reaction?

A nuclear chain reaction occurs when fission of a uranium nucleus, triggered by neutron bombardment, releases other neutrons that can trigger more fissions. This creates a self-sustaining reaction that can release enormous amounts of energy.

⚡ Energy Comparison: Nuclear vs Chemical

Chemical Combustion of Uranium

\[ U + O_2 \rightarrow UO_2 \]

\[ \text{Heat of combustion} \approx 4500 \, \text{J/g} \]

\[ \approx 11 \, \text{eV per atom} \]

Nuclear Fission of Uranium

\[ \text{Energy release} \approx 200 \, \text{MeV per atom} \]

\[ \approx 200 \times 10^6 \, \text{eV per atom} \]

Energy Ratio

\[ \frac{\text{Nuclear energy}}{\text{Chemical energy}} \approx \frac{200 \times 10^6}{11} \]

\[ \approx 18 \times 10^6 \]

\[ \text{Nearly 20 million times more energy!} \]

💡 Criticality Conditions

The chain reaction may be made to proceed:

Slowly and controlled: In a nuclear reactor
Explosively: In a nuclear bomb

The energy release in a nuclear chain reaction is enormous, far greater than that in any chemical reaction. (In a sense, fire is a chemical chain reaction.)

Nuclear Fission Reactors

🏭 What is a Nuclear Reactor?

A nuclear reactor is a system in which a controlled nuclear chain reaction is used to liberate energy. In a nuclear power plant, this energy is used to generate steam, which operates a turbine and turns an electrical generator.

Reactor Components

⛽ Fuel Rods

Contain fissionable material, typically uranium enriched to 3-5% \(^{235}U\) (compared to natural abundance of 0.7%).

🎛️ Moderator

Slows down fast neutrons to thermal energies where they are more likely to cause fission in \(^{235}U\). Common moderators: water, graphite.

⚡ Control Rods

Made of neutron-absorbing materials (boron, cadmium) to control reaction rate by absorbing excess neutrons.

💧 Coolant

Transfers heat from reactor core to steam generator. Common coolants: water, liquid sodium.

⚙️ Nuclear Reactor Operation

REACTOR CORE
Fuel Rods + Moderator
↓ Heat
Primary Coolant

↓

HEAT EXCHANGER
Primary → Secondary Loop
↓
Steam Generation

↓

TURBINE + GENERATOR
Steam → Mechanical → Electrical

Energy Flow:

Fission in reactor core produces heat
Primary coolant carries heat to steam generator
Heat exchanger transfers heat to secondary loop
Steam drives turbine connected to electrical generator
Condenser converts steam back to water

Safety Features: Multiple barriers contain radioactivity, and control systems ensure stable operation.

Reactor Control

🧮 Reproduction Constant and Criticality

Step 1: Reproduction Constant (K)

Defined as the average number of neutrons from each fission event that will cause another event.

\[ \text{Maximum possible K for uranium fission} = 2.5 \]

\[ \text{Actual K < 2.5 due to various losses} \]

Step 2: Criticality Conditions

\[ K = 1: \text{Critical (self-sustained chain reaction)} \]

\[ K < 1: \text{Subcritical (reaction dies out)} \]

\[ K > 1: \text{Supercritical (runaway reaction)} \]

Step 3: Factors Reducing K

Neutron leakage from reactor core
Neutron capture by non-fissile materials
Parasitic absorption by structural materials

💡 Neutron Moderation Importance

Neutrons released in fission events are highly energetic (about 2 MeV). However:

Slow neutrons are far more likely than fast neutrons to produce fission in \(^{235}U\)
\(^{238}U\) doesn't absorb slow neutrons
Moderator slows down neutrons, making them available for reaction with \(^{235}U\)
Slowing also decreases chances of neutrons being captured by \(^{238}U\)

Nuclear Fusion

☀️ What is Nuclear Fusion?

Nuclear fusion is the process where two light nuclei combine to form a heavier nucleus. Because the mass of the final nucleus is less than the masses of the original nuclei, there is a loss of mass, accompanied by a release of energy.

💡 Fusion vs Fission

Fission: Heavy nucleus splits → lighter nuclei + energy

Fusion: Light nuclei combine → heavier nucleus + energy

Both processes release energy because the products are more tightly bound (higher binding energy per nucleon).

⚠️ Coulomb Barrier Challenge

The process of nuclear fusion is hindered by the mutual Coulomb repulsion that tends to prevent two positively charged particles from coming within range of each other's attractive nuclear force.

Nucleons have to overcome the potential barrier to make nuclear fusion occur. For deuteron fusion, particles have to penetrate through a potential barrier of about 200 keV.

🧮 Achieving Fusion Conditions

Method 1: Particle Acceleration

Use one light particle as a target and accelerate the other using a cyclotron. However, this technique is not useful for obtaining energy in a controlled manner.

Method 2: Thermal Motion (Thermonuclear Fusion)

Raise the temperature of the material so that particles have sufficient energy to penetrate the barrier due to their thermal motion.

\[ \text{Mean thermal kinetic energy: } \overline{K} = \frac{3}{2} kT \]

\[ \text{Where k is Boltzmann's constant} \]

Thermonuclear Fusion in Stars

⭐ Stellar Fusion Processes

The sun's energy is generated by the thermonuclear fusion of hydrogen to form helium. The composition of the sun's core is about 35% hydrogen by mass, about 65% helium and about 1% other elements.

The temperature at the center of the sun is about \( 1.5 \times 10^7 \, K \). At this temperature, the light elements are essentially totally ionized.

\( 3.9 \times 10^{26} \, W \)

The Sun's radiation power output

\( 4.5 \times 10^9 \, \text{years} \)

How long the Sun has been radiating at this rate

🧮 Proton-Proton Cycle in Stars

Step 1: Deuterium Formation

\[ ^1H + ^1H \rightarrow ^2H + e^+ + \nu \quad (Q = 0.42 \, \text{MeV}) \]

\[ e^- + e^+ \rightarrow \gamma + \gamma \quad (Q = 1.02 \, \text{MeV}) \]

This process is extremely rare - only once in about \(10^{26}\) proton-proton collisions results in deuteron formation.

Step 2: Helium-3 Formation

\[ ^2H + ^1H \rightarrow ^3He + \gamma \quad (Q = 5.49 \, \text{MeV}) \]

Step 3: Helium-4 Formation

\[ ^3He + ^3He \rightarrow ^4He + ^1H + ^1H \quad (Q = 12.86 \, \text{MeV}) \]

Overall Proton-Proton Cycle

The complete cycle amounts to:

\[ 4^1H + 2e^- \rightarrow ^4He + 2\nu + 6\gamma \]

Adding two electrons to each side yields atomic representation:

\[ 4(^1H + e^-) \rightarrow (^4He + 2e^-) + 2\nu + 6\gamma \]

Energy released calculation:

\[ Q = \Delta m c^2 = [4m(^1H) - m(^4He)]c^2 \]

\[ = [4(1.007825u) - 4.002603u](931.5 \, \text{MeV/u}) \]

\[ = 26.7 \, \text{MeV} \]

💡 Stellar Nucleosynthesis

As stars evolve and become hotter, other elements can be formed by fusion reactions:

At about \(10^8 \, K\): Helium burning to carbon

\[ ^4He + ^4He + ^4He \rightarrow ^{12}C + \gamma \quad (Q = 7.3 \, \text{MeV}) \]

Elements beyond A = 56 cannot be manufactured by fusion (they lie beyond the peak of binding energy curve)
Elements with A = 56 (\(^{56}Fe\), \(^{56}Co\), \(^{56}Ni\)) lie near the peak of binding energy per nucleon curve

Controlled Nuclear Fusion

💣 Fusion Weapons vs Fusion Power

Thermonuclear reactions have been going on in the universe since its creation. Such reactions have been taken place on earth, however, only since October 1952, when the first fusion (hydrogen) bomb was exploded.

The high temperatures needed to initiate thermonuclear reaction in this case were provided by a fission bomb used as a trigger.

A sustained and controllable thermonuclear power source (fusion reactor) is proving much more difficult to achieve. The goal, however, is being vigorously pursued because many look to the fusion reactor as the ultimate power source of the future.

🔬 Promising Fusion Reactions

The proton-proton interaction is not suitable for terrestrial fusion reactors because this process is hopelessly slow. The most attractive reactions for terrestrial use appear to be:

Deuteron-Deuteron (d-d)

\[ ^2H + ^2H \rightarrow ^3He + n \quad (Q = 3.27 \, \text{MeV}) \]

\[ ^2H + ^2H \rightarrow ^3H + H \quad (Q = 4.03 \, \text{MeV}) \]

Deuterium natural abundance: 0.015% in normal hydrogen, available in unlimited quantities from seawater.

Deuteron-Triton (d-t)

\[ ^2H + ^3H \rightarrow ^4He + n \quad (Q = 17.59 \, \text{MeV}) \]

Tritium is radioactive and not normally found in naturally occurring hydrogen. Must be bred from lithium.

Requirements for Thermonuclear Reactor

🔬 Three Basic Requirements for Fusion Reactor

1. High Particle Density (n)

The number of interacting particles (deuterons) per unit volume must be great enough to ensure a sufficiently high deuteron-deuteron collision rate.

2. High Plasma Temperature (T)

High temperature is required so that the deuterium gas would be completely ionized into natural plasma consisting of deuterons and electrons.

The colliding deuterons should be energetic enough to penetrate the mutual Coulomb barrier. A plasma temperature of \( 2.8 \times 10^8 \, K \), corresponding to kinetic energy of 33 keV, has been achieved in laboratory.

3. Long Confinement Time (τ)

A major problem is containing the hot plasma to ensure that its density and temperature remains sufficiently high. No actual solid container can withstand the high temperatures, so special techniques must be employed.

🧮 Lawson's Criterion

Fusion Reactor Success Condition

\[ n\tau \geq 10^{20} \, \text{s·m}^{-3} \]

This condition is called Lawson's criterion. It tells us that we have a choice between confining a lot of particles for a relatively short time or confining fewer particles for a somewhat longer time.

Fusion Confinement Techniques

Magnetic confinement: Uses magnetic fields to confine the plasma while its temperature is increased (Tokamak, Stellarator)
Inertial confinement: A small amount of fuel is compressed and heated so rapidly that fusion occurs before the fuel can expand and cool (Laser fusion)

Frequently Asked Questions

❓ Why does nuclear fission release so much more energy than chemical reactions?

Nuclear fission releases about 20 million times more energy per atom than chemical reactions because:

Chemical reactions involve rearrangements of electrons in atomic orbitals, with energy changes on the order of electronvolts (eV)
Nuclear reactions involve changes in the nucleus itself, with energy changes on the order of millions of electronvolts (MeV)

This huge difference arises because nuclear forces are about a million times stronger than electromagnetic forces that govern chemical reactions.

\[ \frac{\text{Nuclear energy (200 MeV)}}{\text{Chemical energy (11 eV)}} \approx 18 \times 10^6 \]

❓ What is the difference between nuclear fission and nuclear fusion?

Nuclear Fission:

Heavy nucleus splits into lighter nuclei
Examples: Uranium-235, Plutonium-239
Releases energy for nuclei heavier than iron
Used in current nuclear power plants and atomic bombs

Nuclear Fusion:

Light nuclei combine to form heavier nucleus
Examples: Hydrogen fusion in stars
Releases energy for nuclei lighter than iron
Used in hydrogen bombs; being developed for power generation

Both processes release energy because the products are more tightly bound (have higher binding energy per nucleon).

❓ Why is nuclear fusion so difficult to achieve on Earth?

Nuclear fusion is extremely challenging to achieve on Earth due to several fundamental obstacles:

Coulomb Barrier: Positively charged nuclei repel each other strongly. To overcome this repulsion, nuclei need extremely high temperatures (millions of degrees) to have enough kinetic energy.
Plasma Confinement: At fusion temperatures, matter exists as plasma (ionized gas). No material container can withstand these temperatures, so magnetic or inertial confinement is needed.
Energy Balance: The energy required to heat and confine the plasma must be less than the energy produced by fusion reactions for net energy gain.
Lawson Criterion: The product of plasma density, temperature, and confinement time must exceed a critical value (\(n\tau \geq 10^{20} \, \text{s·m}^{-3}\)).

Despite these challenges, significant progress has been made in experimental fusion reactors like ITER, which aims to demonstrate net energy gain.

❓ What are the advantages of fusion power over fission power?

Fusion power offers several potential advantages over current fission power:

Abundant Fuel: Deuterium can be extracted from seawater, and tritium can be bred from lithium
Reduced Radioactive Waste: Fusion produces less long-lived radioactive waste compared to fission
No Meltdown Risk: Fusion reactions require precise conditions and will naturally terminate if those conditions are not maintained
No Greenhouse Gases: Fusion doesn't produce carbon dioxide or other greenhouse gases
Weapons Proliferation: Fusion reactors are less likely to be used for weapons production

However, fusion power is still in the development stage, while fission power has been commercially available for decades.

📚 Master Nuclear Physics

Understanding nuclear energy, fission, and fusion is fundamental to modern physics, energy production, and our understanding of stellar processes. Continue your journey into the fascinating world of nuclear physics and its applications.

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