Theory and Methodology

• 1. Introduction
  • 1.1. Overview of Program Flow
• 2. Geometry
  • 2.1. Constructive Solid Geometry
    • 2.1.1. Universes
    • 2.1.2. Lattices
  • 2.2. Computing the Distance to Nearest Boundary
    • 2.2.1. Plane Perpendicular to an Axis
    • 2.2.2. Generic Plane
    • 2.2.3. Cylinder Parallel to an Axis
    • 2.2.4. Sphere
    • 2.2.5. Cone Parallel to an Axis
    • 2.2.6. General Quadric
    • 2.2.7. Torus Parallel to an Axis
  • 2.3. Finding a Cell Given a Point
  • 2.4. Finding a Lattice Tile
    • 2.4.1. Rectilinear Lattice Indexing
    • 2.4.2. Hexagonal Lattice Indexing
  • 2.5. Determining if a Coordinate is in a Cell
  • 2.6. Handling Surface Crossings
  • 2.7. Building Neighbor Lists
  • 2.8. Reflective Boundary Conditions
    • 2.8.1. Plane Perpendicular to an Axis
    • 2.8.2. Generic Plane
    • 2.8.3. Cylinder Parallel to an Axis
    • 2.8.4. Sphere
    • 2.8.5. Cone Parallel to an Axis
    • 2.8.6. General Quadric
    • 2.8.7. Torus Parallel to an Axis
  • 2.9. White Boundary Conditions
• 3. Cross Section Representations
  • 3.1. Continuous-Energy Data
    • 3.1.1. Energy Grid Methods
    • 3.1.2. Windowed Multipole Representation
    • 3.1.3. Temperature Treatment
  • 3.2. NCrystal materials
  • 3.3. Multi-Group Data
• 4. Random Number Generation
  • 4.1. Linear Congruential Generators
    • 4.1.1. Skip-ahead Capability
  • 4.2. Permuted Congruential Generators
• 5. Neutron Physics
  • 5.1. Sampling Distance to Next Collision
  • 5.2. \((n,\gamma)\) and Other Disappearance Reactions
  • 5.3. Elastic Scattering
  • 5.4. Inelastic Scattering
  • 5.5. \((n,xn)\) Reactions
  • 5.6. Multi-Group Scattering
  • 5.7. Fission
  • 5.8. Secondary Angle-Energy Distributions
    • 5.8.1. Uncorrelated Angle-Energy Distributions
    • 5.8.2. Product Angle-Energy Distributions
  • 5.9. Transforming a Particle’s Coordinates
  • 5.10. Effect of Thermal Motion on Cross Sections
    • 5.10.1. Constant Cross Section Model
    • 5.10.2. Energy-Dependent Cross Section Model
  • 5.11. S(\(\alpha,\beta,T\)) Tables
    • 5.11.1. Calculating Integrated Cross Sections
    • 5.11.2. Outgoing Angle for Coherent Elastic Scattering
    • 5.11.3. Outgoing Angle for Incoherent Elastic Scattering
    • 5.11.4. Outgoing Energy and Angle for Inelastic Scattering
  • 5.12. Unresolved Resonance Region Probability Tables
  • 5.13. Variance Reduction Techniques
    • 5.13.1. Survival Biasing
    • 5.13.2. Weight Windows
• 6. Photon Physics
  • 6.1. Photon Interactions
    • 6.1.1. Coherent (Rayleigh) Scattering
    • 6.1.2. Incoherent (Compton) Scattering
    • 6.1.3. Photoelectric Effect
    • 6.1.4. Pair Production
  • 6.2. Secondary Processes
    • 6.2.1. Atomic Relaxation
    • 6.2.2. Electron-Positron Annihilation
    • 6.2.3. Bremsstrahlung
  • 6.3. Photon Production
• 7. Tallies
  • 7.1. Filters and Scores
  • 7.2. Using Maps for Filter-Matching
  • 7.3. Volume-Integrated Flux and Reaction Rates
    • 7.3.1. Analog Estimator
    • 7.3.2. Collision Estimator
    • 7.3.3. Track-length Estimator
  • 7.4. Statistics
    • 7.4.1. Law of Large Numbers
    • 7.4.2. Central Limit Theorem
    • 7.4.3. Estimating Statistics of a Random Variable
• 8. Eigenvalue Calculations
  • 8.1. Method of Successive Generations
  • 8.2. Source Convergence Issues
    • 8.2.1. Diagnosing Convergence with Shannon Entropy
  • 8.3. Uniform Fission Site Method
• 9. Depletion
  • 9.1. Numerical Integration
  • 9.2. Matrix Exponential
  • 9.3. Data Considerations
    • 9.3.1. Transmutation Reactions
    • 9.3.2. Capture Branching Ratios
    • 9.3.3. Fission Product Yields
    • 9.3.4. Power Normalization
  • 9.4. Transfer Rates
    • 9.4.1. Coupling materials
• 10. Heating and Energy Deposition
  • 10.1. Fission
  • 10.2. OpenMC Implementation
    • 10.2.1. Neutron Transport
    • 10.2.2. Coupled neutron-photon transport
  • 10.3. References
• 11. Parallelization
  • 11.1. Fission Bank Algorithms
    • 11.1.1. Master-Slave Algorithm
    • 11.1.2. Nearest Neighbors Algorithm
    • 11.1.3. Cost of Master-Slave Algorithm
    • 11.1.4. Cost of Nearest Neighbor Algorithm
• 12. Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference
  • 12.1. Notation
  • 12.2. Theory
    • 12.2.1. Calculation of Macroscopic Cross Sections
    • 12.2.2. CMFD Equations
    • 12.2.3. CMFD Feedback
  • 12.3. Implementation in OpenMC
• 13. Variance Reduction
  • 13.1. Introduction
  • 13.2. MAGIC Method
  • 13.3. FW-CADIS
• 14. Random Ray
  • 14.1. What is Random Ray?
  • 14.2. Why is a Random Ray Solver Included in OpenMC?
  • 14.3. Random Ray Numerical Derivation
    • 14.3.1. Method of Characteristics
    • 14.3.2. Random Rays
    • 14.3.3. Converting Angular Flux to Scalar Flux
    • 14.3.4. Volume Dilemma
    • 14.3.5. What Happens When a Source Region is Missed?
    • 14.3.6. Power Iteration
    • 14.3.7. Ray Starting Conditions and Inactive Length
    • 14.3.8. Ray Ending Conditions
  • 14.4. Simplified Algorithm
  • 14.5. How are Tallies Handled?
  • 14.6. Linear Sources
  • 14.7. Shannon Entropy in Random Ray
  • 14.8. Fixed Source
  • 14.9. Adjoint Flux Solver Mode
  • 14.10. Fundamental Sources of Bias