Cross Section Physics

Cross sections quantify the probability of nuclear reactions. Getting them right is essential for accurate Monte Carlo simulation.

Microscopic Cross Sections

The microscopic cross section σ(E) is the effective target area presented by a nucleus to a neutron of energy E. It has units of area (barns, where 1 b = 10⁻²⁴ cm²).

Conceptually: $$\sigma = \frac{\text{reaction probability per target nucleus}}{\text{neutron fluence}}$$

A nucleus is not a hard sphere—cross sections can be much larger (resonance capture) or smaller (high-energy scattering) than the geometric area.

Types of Cross Sections

For neutron reactions, we need several partial cross sections:

The total is the sum of all partials: $$\sigma_t = \sigma_{el} + \sigma_{in} + \sigma_f + \sigma_\gamma + …$$

Energy Dependence

Cross sections vary dramatically with energy. Three regimes:

Thermal Region (E < 1 eV)

Many cross sections follow the 1/v law: $$\sigma(E) \propto \frac{1}{v} \propto \frac{1}{\sqrt{E}}$$

Physically: slower neutrons spend more time near the nucleus.

Resonance Region (1 eV - 10 keV)

Sharp peaks occur when the neutron energy matches excited states of the compound nucleus. The Breit-Wigner formula:

$$\sigma(E) = \sigma_0 \frac{\Gamma^2/4}{(E-E_0)^2 + \Gamma^2/4}$$

where:

• E₀ is the resonance energy
• Γ is the total width (inverse lifetime)
• σ₀ is the peak cross section

Heavy nuclei like Pu-239 have thousands of resonances.

Fast Region (> 10 keV)

Resonances overlap and average out. Cross sections become relatively smooth:

• Fission: ~1-2 barns for Pu-239
• Elastic scatter: ~3-5 barns
• Capture: Falls to ~0.01 barns

Pu-239 Cross Section Data

Our code uses tabulated values derived from ENDF/B-VIII.0:

Key observations:

• Fission cross section relatively flat in fast region (~1.5-2.5 b)
• Elastic scatter dominates at low energy
• Inelastic scatter has threshold (~40 keV)
• Capture is negligible for fast neutrons

Interpolation

Between tabulated points, we interpolate. Standard practice is log-log interpolation:

$$\log(\sigma) = \log(\sigma_1) + \frac{\log(E) - \log(E_1)}{\log(E_2) - \log(E_1)}[\log(\sigma_2) - \log(\sigma_1)]$$

This works well because cross sections often follow power laws.

function get_σ(xs::CrossSectionData, E::Float64)
    # Clamp to data range
    E = clamp(E, xs.energy[1], xs.energy[end])
    # Log-log interpolation
    return exp(xs._interp(log(E)))
end

Macroscopic Cross Sections

For materials with number density N (atoms/cm³), the macroscopic cross section is: $$\Sigma = N \sigma$$

Units: cm⁻¹ (probability of interaction per cm traveled).

For a material with multiple nuclides: $$\Sigma_t = \sum_i N_i \sigma_{t,i}$$

Number Density Calculation

For material with density ρ and atomic mass A: $$N = \frac{\rho N_A}{A}$$

For Pu-239 at 15.8 g/cm³: $$N = \frac{15.8 \times 6.022 \times 10^{23}}{239} = 3.98 \times 10^{22} \text{ atoms/cm}^3$$

Converting to atoms/barn-cm (conventional units): $$N = 0.0398 \text{ atoms/barn-cm}$$

ν̄: Neutrons per Fission

The average number of neutrons produced per fission varies with incident energy: $$\bar{\nu}(E) = \bar{\nu}_0 + \alpha E$$

For Pu-239:

• ν̄₀ = 2.874
• α = 0.148 MeV⁻¹

At E = 1 MeV: ν̄ ≈ 3.02

The quantity νΣf (neutron production cross section) is crucial for criticality: $$\nu\Sigma_f = N \bar{\nu} \sigma_f$$

The Fission Spectrum

Fission neutrons are born with a characteristic energy distribution. The Watt spectrum is:

$$\chi(E) = C e^{-E/a} \sinh(\sqrt{bE})$$

where:

• a = 0.966 MeV for Pu-239
• b = 2.842 MeV⁻¹ for Pu-239
• C is a normalization constant

Properties:

• Peak energy: ~0.7 MeV
• Average energy: ~2.0 MeV
• Extends from 0 to ~15 MeV

Sampling the Watt Spectrum

We use rejection sampling (the MCNP algorithm):

function sample_watt(a, b, rng)
    K = 1.0 + (a * b) / 8.0
    L = (K + sqrt(K^2 - 1)) / a
    M = a * L - 1.0
    
    while true
        ξ₁ = rand(rng)
        ξ₂ = rand(rng)
        
        E = -log(ξ₁) / L
        
        x = (L * E - M)^2
        y = b * E
        
        if ξ₂^2 ≤ y / x || x ≤ y
            return E
        end
    end
end

This samples from an exponential envelope and accepts with probability proportional to sinh(√bE).

Scattering Angular Distributions

Elastic Scattering

For heavy nuclei, elastic scattering in the center-of-mass frame is approximately isotropic: $$P(\mu_{cm}) = \frac{1}{2}, \quad \mu_{cm} \in [-1, 1]$$

Transformation to lab frame: $$\mu_{lab} = \frac{1 + A\mu_{cm}}{\sqrt{A^2 + 2A\mu_{cm} + 1}}$$

For A = 239 (Pu), even μ_cm = -1 gives μ_lab ≈ 0.004 (essentially isotropic in lab too).

Energy Loss in Elastic Scatter

The post-scatter energy: $$E’ = E \cdot \frac{A^2 + 2A\mu_{cm} + 1}{(A + 1)^2}$$

Define: $$\alpha = \left(\frac{A-1}{A+1}\right)^2$$

For Pu-239: α = 0.983. A neutron can lose at most 1.7% of its energy per collision.

The average logarithmic energy loss: $$\xi = 1 + \frac{\alpha \ln \alpha}{1 - \alpha} \approx \frac{2}{A + 2/3}$$

For Pu-239: ξ ≈ 0.0084.

Cross Section Library Structure

Our code organizes cross section data as:

struct CrossSectionData
    nuclide::Nuclide
    
    # Energy grid (MeV)
    energy::Vector{Float64}
    
    # Cross sections (barns)
    σ_total::Vector{Float64}
    σ_elastic::Vector{Float64}
    σ_inelastic::Vector{Float64}
    σ_fission::Vector{Float64}
    σ_capture::Vector{Float64}
    
    # Fission data
    ν_bar::Vector{Float64}
    watt_a::Float64
    watt_b::Float64
    
    # Interpolators
    _interp_total::Interpolation
    ...
end

Interpolators are built on construction for fast lookup during transport.

ENDF Data Format

Production codes read from ENDF (Evaluated Nuclear Data File):

• MF=3: Cross sections
• MF=4: Angular distributions
• MF=5: Energy distributions
• MF=6: Product energy-angle distributions

Our simplified tables capture the essential physics without the full complexity.

Summary

These numbers fully characterize fast neutron behavior in plutonium.