Neutron Cross-Sections

Neutron cross-sections quantify the probability of interaction between neutrons and nuclei. They are fundamental to nuclear reactor physics, criticality analysis, and neutron transport calculations. Cross-sections depend strongly on neutron energy, isotope, and interaction type, making their accurate determination and treatment essential for reactor design and safety.

1. Definition and Units

A cross-section represents the effective target area a nucleus presents to an incoming neutron. The microscopic cross-section σ describes probability for a single nucleus; the macroscopic cross-section Σ accounts for number density.

Microscopic cross-section: $\sigma = \frac{\text{number of interactions}}{\text{incident neutrons per unit area}}$

Units: barns (b) = 10⁻²⁴ cm² = 10⁻²⁸ m². One barn approximates the geometric cross-section of a nucleus (10⁻¹² cm diameter).

Macroscopic cross-section: $\Sigma(E) = N \cdot \sigma(E)$

where N is number density (nuclei/cm³). Units: cm⁻¹. Represents mean free path inverse: λ = 1/Σ.

The physical interpretation: σ is the “shadow” a nucleus casts to an incoming neutron wave. Higher energy → shorter wavelength → smaller effective area for interaction in some regions.

2. Types of Neutron Interactions

Neutron interactions are classified by outcome:

• Elastic scattering (σₑₗ): Neutron bounces off nucleus; kinetic energy conserved in center-of-mass frame. Dominant at all energies; crucial for moderation.
• Inelastic scattering (σᵢₙ): Neutron excites nucleus to higher energy state; threshold ~1 MeV. Dominates at fast energies (>1 MeV).
• Radiative capture (σ_γ): Neutron absorbed; nucleus emits gamma ray. Competes with fission at thermal energies for fissile isotopes.
• Fission (σ_f): Nucleus splits; releases 2–3 neutrons + ~200 MeV kinetic energy. Only U-235, Pu-239, etc.; zero threshold.
• (n,2n) (σ_2n): Neutron absorbed; nucleus emits 2 neutrons. Threshold ~6–10 MeV; important for fast spectrum.
• (n,α) (σ_α): Neutron absorbed; nucleus emits alpha particle. Threshold dependent; relevant in shielding analysis.

Total cross-section: σ_t = σₑₗ + σᵢₙ + σ_γ + σ_f + σ_2n + σ_α + …

3. Energy Dependence

Cross-sections vary dramatically with neutron energy (0.01 eV → 20 MeV). Three regions:

Thermal Region (0.01–0.1 eV): $\sigma(E) \propto \frac{1}{v} \propto E^{-1/2}$

Valid for nuclear interactions with no internal excitation threshold. σ increases as neutrons slow—critical for reactor physics. 1/v law allows parameterization by value at 0.0253 eV (thermal).

Resonance Region (0.1 eV – 100 keV): Cross-section exhibits sharp peaks (resonances). Breit-Wigner formula for isolated resonance: $\sigma(E) = \sigma_0 \frac{\Gamma^2/4}{(E - E_r)^2 + (\Gamma/2)^2}$

where E_r is resonance energy, Γ is width. Peak value σ₀ ~ λ²/2π for s-wave scattering. Resonances arise from compound nucleus formation (neutron + nucleus → excited state). Crucial for neutron economy in thermal reactors.

Fast Region (>1 MeV): Inelastic scattering dominates. σ_t often decreases with energy. (n,2n) and (n,α) thresholds crossed. Important for fast breeder reactors and shielding.

4. Key Isotopes in Reactor Physics

Isotope	Role	σ_f (thermal, b)	σ_γ (thermal, b)	σ_t (thermal, b)
U-235	Fissile fuel	580	99	680
U-238	Fertile (in blankets)	0.0055	2.7	10.6
Pu-239	Fissile fuel (breeder)	750	270	1020
Pu-241	Fissile, in used fuel	1010	360	1370
H-1	Thermal moderator	—	0.33	20
D-2	Heavy water moderator	—	0.0005	3.4
C-12	Graphite moderator	—	0.003	4.7

U-235 fission is the primary energy source in thermal reactors. U-238 captures neutrons → Pu-239 (breeding). Pu-239 and Pu-241 sustain fast reactors. Moderators have low absorption; elastic scattering dominates.

5. Resonance Phenomena and Self-Shielding

Resolved resonances: Discrete peaks in the 1 eV – 100 keV range. U-238 has ~20,000 resolved resonances. At pile-up in resonance, neutrons are preferentially absorbed in surface layers of fuel; interior is “shielded” from high-flux resonant energies. Self-shielding factor (f_s) < 1.

Unresolved resonances: At higher energies, resonances overlap; treat statistically with average resonance parameters. Critical above 100 keV for accurate cross-section data.

Doppler broadening: Thermal motion of nuclei broadens resonance widths: Γ_eff ∝ √(T). Hotter fuel → broader resonances → more absorption. Negative temperature coefficient; important for reactor safety.

6. Cross-Section Processing and Data

ENDF (Evaluated Nuclear Structure and Decay Data) database: International repository; contains pointwise cross-section evaluations for ~400 isotopes. Formats: ENDF-6, ACE (continuous energy).

Multigroup method: Cross-sections averaged over energy groups (e.g., 33 or 238 groups). Σ_g = ∫_g σ(E) φ(E) dE / ∫_g φ(E) dE; reduces computation for transport codes.

Self-shielding correction: Accounts for flux depression in resonance regions. Iterative process: compute flux, recalculate Σ_g, repeat until convergence.

Tools: NJOY (pointwise → multigroup), GROUPR (resonance self-shielding), ERRORJ (covariance).

7. Neutron Moderation and Lethargy

Elastic scattering slows neutrons to thermal energy. Lethargy u = ln(E₀/E) measures energy loss cumulatively.

Average logarithmic energy decrement per collision: $\xi = 1 - \frac{(A-1)^2}{2A} \ln\left(\frac{A+1}{A-1}\right)$

For hydrogen (A=1): ξ = 1 (ideal moderator). For deuterium: ξ = 0.725. For carbon: ξ = 0.158.

Number of collisions to thermalize: n = u/ξ. Hydrogen moderates fastest; graphite and heavy water slow neutrons more gradually, allowing resonance absorption (beneficial in thermal reactors). Moderating ratio (ξ/σ_a) ranks moderator quality.

8. Four-Factor Formula and Criticality

Neutron multiplication in a thermal reactor:

$k = \eta \cdot f \cdot p \cdot \varepsilon$

• η (eta): Thermal fission factor = σ_f / σ_a for fuel. Higher for U-235 (≈1.3) than plutonium. Determines energy release per absorption.
• f (thermal utilization): Fraction of thermal neutrons absorbed in fuel (vs. moderator/control/structure). ~0.7–0.9 in PWR lattices.
• p (resonance escape probability): Fraction escaping resonance absorption while thermalizing. ~0.88–0.97; degraded by U-238 resonances and fuel density.
• ε (fast fission factor): Extra fissions from fast neutrons in fuel (esp. U-238). ~1.02–1.05; small but non-negligible.

Critical condition: k_eff = 1. Subcritical: k < 1; supercritical: k > 1. Cross-sections determine all factors.

9. Reaction Rate and Neutron Flux

Reaction rate density (reactions per unit volume per unit time): $R = \Sigma \phi$

where φ is neutron flux (neutrons cm⁻² s⁻¹). Related quantities:

• Power density: P/V = f · R · E_f (f: fraction of fission reactions; E_f ≈ 200 MeV per fission)
• Burnup: ∫ P(t) dt (cumulative energy per unit fuel mass)

In transport theory, φ(r,E,Ω) solves the Boltzmann equation with σ as a coefficient. Multigroup diffusion approximation: $-\nabla \cdot D_g \nabla \phi_g + \Sigma_{t,g} \phi_g = \text{source}_g$

Cross-sections determine diffusion coefficient, removal cross-sections, and scattering matrices—thus neutron population and power distribution.

Summary Table: Cross-Section Reference

Parameter	Definition	Units	Typical Range
σ (microscopic)	Single nucleus interaction probability	barns (b)	0.001–1000
Σ (macroscopic)	Material interaction rate	cm⁻¹	0.01–100
σ_f	Fission cross-section	b	0–1000 (thermal)
σ_a	Absorption (capture + fission)	b	0.001–1000
σ_s	Scattering	b	1–100
ξ	Logarithmic energy loss	dimensionless	0.16–1.0
f_s	Self-shielding factor	dimensionless	0.5–1.0

Cross-Links

• [[nuclear-reactions]]
• [[reactor-physics]]
• [[neutron-moderation]]
• [[fuel-assembly]]
• [[criticality-analysis]]
• [[neutron-transport]]
• [[burnup-and-depletion]]