Lecture on Radioactive Decay and Reactor Calculations

Introduction

• The lecture discusses numerical examples related to radioactive decay, particularly involving Cobalt-60 in reactors.
• Cobalt-60: A radioactive isotope often used in reactors; initially calibrated to 1 microcurie in March 2011.

Calculating Original Activity

• Significant Figures: Understanding importance in measurement precision.
  • Activity: 1 +/- 0.5 microcurie (could range from 0.5 to 1.5 microcuries).
• Half-Life of Cobalt-60:
  • 1925.4 days (approximately 5.25 years).
  • Conversion: 1.66 x 10^8 seconds.
• Decay Equation:
  • Activity formula: A(t) = A₀e^-λt.
  • Decay constant (λ): λ = log(2) / half-life.
    • λ = 0.693 / 1.66 x 10^8 = 4.17 x 10^-9 per second.
• Calculating Initial Activity:
  • Use measured activity and decay properties to find original.
  • Example: Initial activity calculated as 1.07 microcuries.

Mass and Atoms of Cobalt-60

• Conversion Factors:
  • 1 Curie = 3.7 x 10^10 Becquerels (disintegrations per second).
• Number of Atoms:
  • Use decay constant and initial activity to find atoms present initially.
  • Example: n₀ = A₀ / λ.
  • n₀ = 9.5 x 10^12 atoms.
• Calculating Mass:
  • Using Avogadro’s number (6 x 10^23 atoms per mole) and atomic mass.
  • Mass of Cobalt-60: 0.95 nanograms.

Gamma Rays from Cobalt-60

• Decay Processes:
  • Beta Decay: Produces Nickel-60, emits gamma rays.
  • Each decay results in two gamma rays.
• Disintegration Rate:
  • Current activity: 0.52 microcuries = 19,000 disintegrations/sec.
  • Gamma ray production: 38,000 gamma rays/sec.

Applications in Detectors

• Understanding how source calculations affect detector efficiency.
• Considerations for Geiger counter efficiency: Source-detector distance and gamma interactions.

Production of Cobalt-60

• Neutron Capture:
  • Reaction: Cobalt-59 + neutron → Cobalt-60.
  • Differential equations modeling production/destruction rates.
• Cross Sections:
  • Use data from Janus database for absorption cross-sections.
  • Example data: Capture cross-section for Co-59 ~ 20 barns at thermal energy.

Solving Differential Equations

• Setup and solve for changing atom quantities over time.
• Equation Forms:
  • DN₁/dt = -σφN₁ (similar to decay equation).
  • DN₂/dt = Production - Decay - Sigma flux terms.
• Numerical Calculation:
  • Use constants (flux, cross-sections, lambda) to solve.
  • Determine maximum Cobalt-60 production and profit point in reactor.

Real-World Application

• Economic Considerations:
  • Determine optimal extraction time for maximum profit.
  • Factors include cross-section data, neutron flux, and operational costs.

Questions & Further Exploration

• Addressing questions about simulation detail, cross-section calculations, and relating theoretical data to physical systems.