Method Comparison

This project implements the same nuclear physics problem using two fundamentally different approaches:

1. Julia: Monte Carlo neutron transport
2. Python: Deterministic diffusion theory (1-group, 2-group, and 2D spatial)

Let’s compare them across multiple dimensions.

Algorithmic Differences

Monte Carlo (Julia)

For each generation:
    For each neutron:
        While alive:
            Sample collision distance (exponential)
            Sample reaction type (discrete)
            If fission: bank new neutrons
            If scatter: update energy and direction
            If capture or escape: kill neutron
    Estimate k from fission/source ratio
    Normalize fission bank for next generation

Philosophy: Simulate physics exactly, average over many trials.

Diffusion Theory (Python)

Build diffusion coefficient D = 1/(3Σ_tr)
Compute material buckling B_m² = (νΣ_f - Σ_a)/D
Compute geometric buckling B_g² = (π/R')²
Criticality when B_m² = B_g²

Philosophy: Approximate physics, solve analytically or with deterministic numerics.

Comparison Table

Cross Section Treatment

Monte Carlo

• Tabulated on energy grid (24 points from 10 eV to 20 MeV)
• Log-log interpolation
• Separate fission, elastic, inelastic, capture
• Watt spectrum for fission neutrons

Diffusion

• Fission-spectrum averaged (single values)
• Transport-corrected: Σ_tr = Σ_t - μ̄Σ_s
• Grouped data (1 or 2 groups)

Critical Mass Results

For bare δ-phase Pu-239:

Key observation: Monte Carlo is 6% high due to simplified cross sections. Diffusion methods are ~35-40% low due to the diffusion approximation breaking down for this optically thin system.

Why Diffusion Fails Here

The diffusion approximation requires:

1. Many mean free paths across the system
2. Weak absorption
3. Nearly isotropic scattering

For bare Pu-239:

• System is ~1.5 mean free paths across (not many)
• Absorption is significant (Σ_a/Σ_t ≈ 0.25)
• Scattering is isotropic ✓

The approximation is at its limits, explaining the ~35% error in critical mass.

Time-Dependent Comparison

Point Kinetics (Python)

Solves coupled ODEs:

• dN/dt = αN
• dE/dt = power from N
• dR/dt = hydrodynamic expansion

Results for 6.2 kg Pu at 2.5× compression:

• Yield: ~12 kt
• Efficiency: ~3%
• Time: ~0.4 μs

Monte Carlo Kinetics (Not implemented)

Would require:

• Time-dependent source iteration
• Delayed neutron precursors
• Much more computation

For prompt supercritical systems, point kinetics is adequate.

Code Complexity

Julia Monte Carlo (~1800 lines)

NuclearFootball.jl    68 lines   Module definition
materials.jl         137 lines   Nuclide/material data
cross_sections.jl    564 lines   XS lookup, interpolation, Watt spectrum
geometry.jl          482 lines   Sphere, ellipsoid, football shapes
particle.jl          339 lines   Transport physics
criticality.jl       385 lines   Power iteration, k-eigenvalue
analysis.jl          362 lines   Football analysis, validation

Python Diffusion (~2500 lines)

nuclear_physics.py   844 lines   Full coupled simulation
point_kinetics.py    467 lines   Simpler standalone version
diffusion_2d.py      394 lines   2D axisymmetric FVM
spatial_neutronics.py 516 lines  More advanced 2D FVM
two_group_physics.py  606 lines  Two-group treatment

Runtime Comparison

For k_eff calculation on the Jezebel benchmark:

*Point kinetics gives α and k_∞, not k_eff directly.

When to Use Each

Use Monte Carlo When:

• Accuracy matters (k_eff to ~1%)
• Geometry is complex
• Energy dependence is important
• Validating against benchmarks

Use Diffusion When:

• Speed matters
• Exploring parameter space
• Need physical intuition
• Coupled physics (hydrodynamics)
• Analytical solutions available

The Football Problem

Monte Carlo Approach (Julia)

• Model exact superellipsoid shape
• Track neutrons through pointed ends
• Calculate k_eff = 1.21 ± 0.004
• Find critical mass ~17 kg in football shape

Diffusion Approach (Python)

• Approximate as ellipsoid or sphere
• Use analytical buckling for prolate spheroid
• Underestimate critical mass significantly
• Still useful for compression/yield studies

Complementary Strengths

The two approaches are complementary, not competitive:

1. Python diffusion for:

   • Fast parameter sweeps
   • Compression effects
   • Yield vs mass studies
   • Physical understanding

2. Julia Monte Carlo for:

   • Final k_eff calculation
   • Football geometry
   • Benchmark validation
   • Publication-quality results

Summary

Both codebases contribute to answering our central question: Yes, a plutonium football would be supercritical—whether calculated with Monte Carlo (k ≈ 1.21) or estimated with diffusion theory (deeply supercritical based on mass >> critical mass).