Nuclear Physics

Module 8: From the Universe to the Atom

Overview

Nuclear physics deals with the properties and behaviour of atomic nuclei, including radioactive decay, nuclear fission, and nuclear fusion. These processes release enormous amounts of energy through mass-energy conversion.

Key Syllabus Points:

  • Analyse radioactive decay and properties of alpha, beta, and gamma radiation
  • Apply half-life and decay constant relationships
  • Analyse nuclear fission and fusion with conservation of mass-energy

Key Concepts

Types of Radiation

Property Alpha (α) Beta (β⁻) Gamma (γ)
Nature ⁴₂He nucleus Electron (e⁻) EM radiation
Charge +2 -1 0
Mass 4 u ~0.0005 u 0
Speed ~5% c up to 99% c c
Penetration Paper stops Few mm Al Several cm Pb
Ionising High Medium Low
Deflection in fields Yes (towards -) Yes (towards +) No

Decay Equations

Alpha decay: \[^A_Z X \rightarrow ^{A-4}_{Z-2} Y + ^4_2 \alpha\]

Beta decay (β⁻): \[^A_Z X \rightarrow ^{A}_{Z+1} Y + ^0_{-1} \beta + \bar{\nu}_e\]

Gamma decay: \[^A_Z X^* \rightarrow ^A_Z X + \gamma\]

Radioactive Decay Law

Exponential Decay

\[N_t = N_0 e^{-\lambda t}\]

Where: - \(N_t\) = number of nuclei at time \(t\) - \(N_0\) = initial number of nuclei - \(\lambda\) = decay constant (s⁻¹)

Half-life relationship: \[\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{t_{1/2}}\]

Alternative form: \[N_t = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}\]

Mass Defect and Binding Energy

Mass defect: The difference between the mass of a nucleus and the sum of its constituent nucleons.

\[\Delta m = (Zm_p + Nm_n) - m_{nucleus}\]

Binding energy: \[E_b = \Delta m \cdot c^2\]

This energy must be supplied to completely separate the nucleus into individual nucleons.

Nuclear Reactions

Nuclear Fission

Definition

Fission is the splitting of a heavy nucleus into lighter nuclei, releasing energy.

Example: Uranium-235 fission \[^{235}_{92}U + ^1_0 n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3^1_0 n + \text{energy}\]

Chain reaction: Released neutrons can cause further fissions - Controlled: Nuclear reactors (sustained, controlled) - Uncontrolled: Nuclear weapons (explosive)

Nuclear Fusion

Definition

Fusion is the combining of light nuclei into heavier nuclei, releasing energy.

Example: Proton-proton chain (in stars) \[4^1_1 H \rightarrow ^4_2 He + 2^0_1 e + 2\nu_e + \text{energy}\]

Why fusion releases energy: Light nuclei have lower binding energy per nucleon than medium nuclei.

Conditions required: - Extremely high temperature (~10⁷ K) - High pressure/density - Confinement (gravity in stars, magnetic in reactors)

Worked Examples

Example 1: Half-Life Calculation

A radioactive sample initially contains \(4.0 \times 10^{20}\) atoms with half-life 8.0 days. Calculate: (a) Number of atoms after 24 days (b) Activity after 24 days

Solution:

  1. Number of half-lives: \(n = \frac{24}{8} = 3\) \[N_t = N_0 \left(\frac{1}{2}\right)^3 = 4.0 \times 10^{20} \times \frac{1}{8} = 5.0 \times 10^{19} \text{ atoms}\]

  2. Activity \(A = \lambda N\): \[\lambda = \frac{0.693}{8 \times 24 \times 3600} = 1.0 \times 10^{-6} \text{ s}^{-1}\] \[A = 1.0 \times 10^{-6} \times 5.0 \times 10^{19} = 5.0 \times 10^{13} \text{ Bq}\]

Example 2: Mass-Energy in Fission

In the fission of U-235, the total mass of products is 0.186 u less than reactants. Calculate the energy released per fission.

Solution:

Convert mass to kg: \[\Delta m = 0.186 \times 1.66 \times 10^{-27} = 3.09 \times 10^{-28} \text{ kg}\]

Energy released: \[E = \Delta m \cdot c^2 = 3.09 \times 10^{-28} \times (3 \times 10^8)^2\] \[E = 2.78 \times 10^{-11} \text{ J} = 174 \text{ MeV}\]

Example 3: Decay Constant

Carbon-14 has a half-life of 5730 years. A sample has an activity of 400 Bq. Calculate its activity after 17,190 years.

Solution:

Number of half-lives: \(n = \frac{17190}{5730} = 3\)

Using \(A = A_0 \left(\frac{1}{2}\right)^n\): \[A = 400 \times \left(\frac{1}{2}\right)^3 = 400 \times 0.125 = 50 \text{ Bq}\]

Common Misconceptions

Avoid These Mistakes
  1. Half-life depends on amount - Half-life is constant, independent of sample size
  2. After two half-lives, all decayed - After 2 half-lives, 25% remains (not 0%)
  3. Fission vs fusion confusion - Fission splits heavy nuclei; fusion joins light nuclei
  4. Mass is created/destroyed - Mass is converted to energy, not created or destroyed
  5. Radiation travels through materials - γ rays are absorbed/attenuated, not transmitted completely

HSC Exam Analysis

Question Types

  1. Calculation questions (5-7 marks): Half-life, decay constant, mass-energy calculations
  2. Comparison questions (4-5 marks): Compare fission and fusion
  3. Explanation questions (4-6 marks): Explain radiation properties, chain reactions

Recent HSC Questions

  • 2024 Q33: Nuclear decay calculation with half-life
  • 2023 Q31: Mass-energy calculation in fusion reaction
  • 2022 Q32: Compare and contrast fission and fusion

Practice Problems

  1. A radioactive isotope has a half-life of 15 minutes. What fraction remains after 1 hour?

  2. Calculate the energy released when 1.0 kg of matter is completely converted to energy.

  3. In the fusion of deuterium and tritium, 17.6 MeV is released. Calculate the mass converted to energy.

  4. A sample initially has activity 1200 Bq. After 30 days, the activity is 150 Bq. Calculate the half-life.

  5. Write the nuclear equations for alpha decay of Radium-226 and beta decay of Carbon-14.