Lecture on Neutron Diffusion and Criticality Conditions

Introduction

• The lecture is part of MIT OpenCourseWare.
• Focus on simplifying a complex neutron diffusion equation.
• Aim is to achieve a criticality condition for a homogeneous infinite reactor.

Key Concepts

Variables and Terms

• Flux (Φ): Number of neutrons per cm² per second.
• Current (J): Related to neutron flow, independent of angle.
• Neutron Diffusion Equation: Describes neutron flow using multiple variables and reaction rates.

Reaction Rates

• Reaction rates are generally some cross-section times the flux.
• Fission Terms: Include number of neutrons per fission, birth spectrum, energy considerations.
• External Source: Can be californium sources used in reactors.
• In-scattering Term: Considers neutrons scattering into the energy group from elsewhere.
• Absorption and Leakage: Losses due to reactions and neutrons leaving the control surface.

Simplification Process

Step-by-Step Reductions

• Neglect Angle: Remove angle dependence and corresponding 4π factors.
• Steady State Assumption: Eliminates time dependence; removes the external source term.
• Homogeneity Assumption: Simplifies cross-sections to constants across the reactor.
• Energy Simplification: Use one-group or two-group approximation of energy levels.
• Neutron Diffusion Approximation: Treats neutrons as diffusing like a chemical or gas.

Fick’s Law

• Relates current (J) to diffusion coefficient and concentration gradient.
• Important for transforming transport equation to diffusion equation.

Criticality Condition

Definition

• K-effective (K_eff): Ratio of neutrons produced to neutrons consumed.
• Critical Reactor: K_eff = 1, gains equal losses.
• Supercritical: K_eff > 1, more neutrons produced than consumed.
• Subcritical: K_eff < 1, more neutrons consumed than produced.

Simplified Neutron Diffusion Equation

• After simplifications, the equation describes balance between fission, reactions, absorption, and leakage.
• Losses involve absorption and leakage accounted for by diffusion terms.

Practical Implications

• Discussed how real reactors (e.g., AP1000) use these simplifications in design.
• Explained the emotional impact of terms like "criticality" in nuclear engineering.

Conclusion and Next Steps

• Next Lecture: Solve the simplified neutron diffusion equation.
• Upcoming Recitation: Electron microscope session to analyze material interactions with electrons.

Q&A Highlights

• Clarification on terms like birth spectrum and scattering kernel.
• Discussion on the impact of delayed neutrons and reactor control.
• Emphasized the importance of accurate terminologies and assumptions in nuclear engineering.

These notes summarize the key points from the lecture on the neutron diffusion equation and reactor criticality. They cover the simplification process, criticality conditions, and practical implications for reactor design. Further exploration in the next class will address solving the simplified equation and practical electron microscopy.