Chapter 11: Nuclear Physics

Overview

Nuclear physics explores the fundamental structure of matter and the immense energy stored within atomic nuclei. This chapter covers radioactivity, nuclear reactions, fission, fusion, and their applications. Understanding these concepts reveals the source of stellar energy, the power behind nuclear reactors, and the basis of medical treatments and dating techniques.

Learning Objectives

After completing this chapter, you will be able to:

• Understand radioactive decay and its types
• Apply half-life concepts to decay calculations
• Analyze nuclear reactions and energy calculations
• Explain fission and fusion processes
• Describe applications of nuclear physics in various fields

Radioactivity and Nuclear Decay

Main Concept

Radioactivity is the spontaneous emission of radiation from unstable atomic nuclei as they transform into more stable configurations.

Key Principles

• Unstable Nuclei: Some nuclei are inherently unstable due to proton-neutron imbalance
• Random Process: Decay is unpredictable for individual nuclei but follows statistical laws for large numbers
• Conservation Laws: Energy, momentum, and charge are conserved in all nuclear reactions

Types of Radioactive Decay

Important Terms

• Radioisotope: Unstable isotope that undergoes radioactive decay
• Activity: Number of decays per second (Becquerel, Bq)
• Half-life: Time for half of radioactive nuclei to decay
• Radiation: Energy emitted from unstable nuclei

Alpha Decay

Process:

• Heavy nucleus emits alpha particle (²⁴He)
• Daughter nucleus has mass number reduced by 4
• Atomic number reduced by 2

Example:

$$^{238}_{92}U \rightarrow ^{234}_{90}Th + ^{4}_{2}He$$

Beta Decay

Beta-Minus Decay:

• Neutron converts to proton + electron + antineutrino
• Mass number unchanged, atomic number increases by 1

Example:

$$^{14}_{6}C \rightarrow ^{14}_{7}N + e⁻ + \bar{v_e}$$

Beta-Plus Decay:

• Proton converts to neutron + positron + neutrino
• Mass number unchanged, atomic number decreases by 1

Gamma Decay

Process:

• Nucleus emits gamma ray to reach lower energy state
• No change in mass number or atomic number
• Often follows alpha or beta decay

Half-Life

Main Concept

Half-life is the time required for half of the radioactive nuclei in a sample to undergo decay.

Key Principles

• Exponential Decay: Remaining nuclei decrease exponentially
• Constant Rate: Half-life is constant for a given radioisotope
• Statistical Process: Predictable for large numbers, random for individual nuclei

Key Formulas

Remaining Nuclei:

$$N = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}$$

Activity:

$$A = A_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}$$

Decay Constant:

$$\lambda = \frac{\ln(2)}{T_{1/2}}$$

Mean Life:

$$\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln(2)}$$

Where:

• N = Remaining nuclei
• N_0 = Initial nuclei
• t = Time elapsed
• T_{1/2} = Half-life
• A = Current activity
• A_0 = Initial activity
• λ = Decay constant
• τ = Mean life

Radioactive Decay Visualization

Half-Life Examples

Worked Example

Problem: A sample contains 1000 g of Carbon-14. Calculate the remaining mass after 11,460 years.

Solution:

• Initial mass = 1000 g
• Half-life = 5730 years
• Time = 11,460 years

Number of half-lives:

$$n = \frac{11,460}{5730} = 2$$

Remaining mass:

$$m = 1000 \times \left(\frac{1}{2}\right)^2 = 1000 \times \frac{1}{4} = 250 \text{ g}$$

Answer: 250 g remaining

Nuclear Reactions and Energy

Main Concept

Nuclear reactions involve changes in atomic nuclei, releasing enormous amounts of energy according to Einstein's mass-energy equivalence principle.

Key Principles

• Mass Defect: Mass of nucleus less than sum of individual nucleons
• Binding Energy: Energy holding nucleus together
• Energy Release: Mass converted to energy in reactions

Key Formulas

Einstein's Mass-Energy Equivalence:

$$E = mc^2$$

Binding Energy per Nucleon:

$$\text{Binding Energy} = \text{Total mass of protons and neutrons} - \text{Mass of nucleus}$$

Mass-Energy Conversion:

$$1 \text{ atomic mass unit (u)} = 931.5 \text{ MeV} = 1.66 \times 10^{-27} \text{ kg}$$

Nuclear Binding Energy Curve: Maximum binding energy per nucleon occurs around Iron-56 ($^{56}_{26}$Fe)

Where:

• E = Energy (Joules)
• m = Mass (kg)
• c = Speed of light (3 × 10⁸ m s⁻¹)

Nuclear Energy Relationships

Nuclear Binding Energy Diagram

Energy Calculations

Mass-Energy Conversion:

• 1 atomic mass unit (u) = 931.5 MeV
• 1 u = 1.66 × 10⁻²⁷ kg

Example Energy Release:

• Fission: ~200 MeV per reaction
• Fusion: ~3-4 MeV per nucleon

Nuclear Fission

Main Concept

Nuclear fission is the process where a heavy nucleus splits into smaller nuclei when bombarded with neutrons, releasing energy and more neutrons.

Key Principles

• Chain Reaction: One fission event triggers multiple subsequent events
• Critical Mass: Minimum mass needed for sustained chain reaction
• Neutron Economy: Balance between neutron production and absorption

Fission Process

Neutron Induced Fission:

1. Neutron strikes fissile nucleus (e.g., U-235)
2. Nucleus becomes unstable and splits
3. Releases energy and more neutrons
4. Products are typically two medium-mass nuclei

Chain Reaction:

• Critical: Sustained chain reaction (nuclear reactor)
• Subcritical: Reaction dies out
• Supercritical: Exponential growth (nuclear weapon)

Nuclear Fission Process Visualization

Nuclear Reactor Types and Components

Fission Applications and Advantages

Nuclear Reactor Components

Nuclear Fusion

Main Concept

Nuclear fusion is the process where light nuclei combine to form heavier nuclei, releasing enormous amounts of energy.

Key Principles

• Extreme Conditions: Requires very high temperature and pressure
• Net Energy Gain: Fusion of light nuclei releases energy
• Fuel Availability: Hydrogen fuel is abundant

Fusion Process

Proton-Proton Chain (Stars):

$$p + p \rightarrow ^{2}_{1}H + e⁺ + v_e$$ $$^{2}_{1}H + p \rightarrow ^{3}_{2}He + \gamma$$ $$^{3}_{2}He + ^{3}_{2}He \rightarrow ^{4}_{2}He + 2p$$

Deuterium-Tritium Reaction:

$$^{2}_{1}H + ^{3}_{1}H \rightarrow ^{4}_{2}He + n + 17.6 \text{ MeV}$$

Fusion Applications

Stellar Fusion:

• Sun and stars fuse hydrogen to helium
• Powers the universe

Terrestrial Fusion:

• Tokamak: Magnetic confinement reactor
• Laser Fusion: Inertial confinement
• ITER: International fusion project

Advantages:

• Abundant fuel (deuterium from seawater)
• No long-lived radioactive waste
• Inherent safety features

Applications of Nuclear Physics

Medical Applications

Diagnostic Imaging:

• PET Scans: Uses positron-emitting isotopes
• Gamma Cameras: Detect gamma radiation
• CT Scans: X-ray imaging

Radiotherapy:

• External Beam Radiation: Targeted X-rays/gamma rays
• Brachytherapy: Radioactive implants
• Proton Therapy: Precision treatment

Tracer Studies:

• Radioisotope Tracers: Track biological processes
• Blood Flow Studies: Use Tc-99m
• Organ Function: Use specific isotopes

Industrial Applications

Industrial Radiography:

• Non-destructive Testing: Detect cracks in materials
• Thickness Measurement: Gamma ray gauges
• Weld Inspection: Quality control

Smoke Detectors:

• Use Americium-241 alpha sources
• Ionization-type detectors

Dating Techniques:

• Carbon Dating: C-14 for organic materials
• Potassium-Argon Dating: K-40 for rocks
• Uranium-Lead Dating: U-238 for geological samples

Environmental Applications

Radiation Monitoring:

• Environmental Sensors: Detect radioactive contamination
• Radon Gas Detection: Home safety
• Nuclear Security: Radiation detection

Waste Management:

• Storage: Safe disposal of radioactive waste
• Treatment: Reducing radioactivity
• Containment: Preventing environmental release

Safety and Radiation Protection

Radiation Types and Penetration

Radiation Protection Principles

1. Time: Minimize exposure time
2. Distance: Increase distance from source
3. Shielding: Use appropriate barriers
4. Containment: Prevent contamination

Radiation Units

• Becquerel (Bq): Activity (decays per second)
• Gray (Gy): Absorbed dose (J/kg)
• Sievert (Sv): Equivalent dose (biological effect)

SPM Exam Tips

Common Mistakes to Avoid

1. Half-life: Remember it's the time for half the nuclei to decay, not all
2. Decay Types: Know characteristics of alpha, beta, gamma decay
3. Energy Calculations: Use E=m$c^2$ correctly with proper units
4. Conservation Laws: Check that mass number and atomic number balance

Problem-Solving Strategies

1. Identify Decay Type: Determine alpha, beta, or gamma process
2. Apply Half-life Formula: Use exponential decay equations
3. Balance Equations: Ensure mass and atomic numbers balance
4. Check Units: Use appropriate units for calculations

Important Formula Summary

Summary

This chapter covered essential nuclear physics concepts:

• Radioactive Decay: Spontaneous emission from unstable nuclei
• Half-life: Exponential decay characteristics
• Nuclear Reactions: Energy release from mass conversion
• Fission and Fusion: Energy production methods
• Applications: Medical, industrial, and environmental uses

Master these concepts to understand nuclear energy, radiation applications, and the fundamental nature of matter and energy.

Practice Questions

1. Explain the difference between alpha, beta, and gamma radiation.

2. A sample contains 800 g of a radioactive isotope with a half-life of 10 days. Calculate: a) The amount remaining after 30 days b) The time for the sample to decay to 100 g

3. Calculate the energy released when 1 kg of mass is converted to energy.

4. Describe the difference between nuclear fission and fusion, giving one example of each.

5. Explain three applications of nuclear physics in medicine.

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• Chapter 1: Electronics
• Chapter 3: Quantum Physics