4.2 Radioactive equilibrium

Radioactive equilibrium is a crucial concept in nuclear physics, describing the balance between parent and daughter nuclides during decay. It comes in three types: secular, transient, and no equilibrium, each with unique characteristics and applications.

Understanding radioactive equilibrium is essential for analyzing decay chains, predicting nuclide concentrations, and applications in various fields. Quantitative methods like activity ratios and Bateman equations help scientists study these complex decay processes accurately.

Types of Radioactive Equilibrium

Secular Equilibrium Characteristics

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• Occurs when parent nuclide half-life significantly exceeds daughter nuclide half-life
• Parent activity remains nearly constant over many daughter half-lives
• Daughter activity approaches parent activity asymptotically
• Equilibrium reached when daughter decay rate equals parent decay rate
• Common in naturally occurring radioactive decay series (uranium-238 decay chain)

Transient Equilibrium Conditions

• Parent nuclide half-life moderately longer than daughter nuclide half-life
• Parent activity decreases noticeably during equilibrium period
• Daughter activity initially increases, then decreases at same rate as parent
• Equilibrium reached when daughter-to-parent activity ratio becomes constant
• Observed in medical isotope production (molybdenum-99/technetium-99m generator)

No Equilibrium Scenarios

• Parent nuclide half-life shorter than or comparable to daughter nuclide half-life
• Daughter activity never reaches equilibrium with parent activity
• Parent decays faster than daughter can accumulate
• Activity ratio constantly changes over time
• Encountered in some artificial radioactive decay chains (iodine-131 decay to xenon-131)

Quantifying Radioactive Equilibrium

Parent-Daughter Activity Ratio Analysis

• Measures relative activities of parent and daughter nuclides
• Calculated as ratio of daughter activity to parent activity
• Indicates type of equilibrium and decay chain progression
• Secular equilibrium ratio approaches 1 as time increases
• Transient equilibrium ratio stabilizes at a constant value greater than 1
• No equilibrium shows continuously changing ratio

Bateman Equations Application

• Set of differential equations describing radioactive decay chains
• Developed by Harry Bateman in 1910
• Calculates activities of parent and daughter nuclides over time
• Accounts for simultaneous decay and production of nuclides
• Enables prediction of nuclide concentrations at any given time
• Crucial for understanding complex decay chains with multiple daughters

Decay Chain Analysis Techniques

• Graphical representations of nuclide activities over time
• Logarithmic plots used to visualize long-term behavior
• Branching ratios considered for nuclides with multiple decay modes
• Equilibrium constant determination for transient equilibrium
• Half-life measurements using equilibrium conditions
• Applications in radiometric dating and nuclear forensics