Criticality and the Multiplication Factor

The central question in nuclear criticality is simple: does the neutron population grow, shrink, or stay constant? The answer is captured in a single number called k, the effective multiplication factor.

The Multiplication Factor

Consider a generation of N neutrons in a fissile assembly. After they undergo various reactions (fission, scattering, absorption, leakage), a new generation of neutrons is produced. The ratio of successive generations defines k:

$$k = \frac{N_{n+1}}{N_n} = \frac{\text{neutrons in generation } n+1}{\text{neutrons in generation } n}$$

Three regimes exist:

The Six-Factor Formula

For thermal reactors, k can be decomposed into the six-factor formula:

$$k = \eta f p \epsilon P_{NL}^T P_{NL}^F$$

where:

• η (eta): Neutrons produced per thermal neutron absorbed in fuel (~2.1 for Pu-239)
• f: Thermal utilization—fraction of absorptions that occur in fuel
• p: Resonance escape probability—fraction of neutrons that avoid resonance capture while slowing down
• ε: Fast fission factor—boost from fast fissions in U-238
• P_NL^T: Thermal non-leakage probability
• P_NL^F: Fast non-leakage probability

For our bare plutonium football, this simplifies considerably:

• Pure fuel: f = 1
• No moderator: p = 1, ε ≈ 1
• No thermal neutrons: η replaced by fast-spectrum average

The remaining physics is dominated by the non-leakage probability.

k-effective vs k-infinite

Two versions of k are commonly used:

k_∞ (k-infinite)

The multiplication factor for an infinitely large system with no leakage:

$$k_\infty = \frac{\text{neutron production}}{\text{absorption}}$$

For Pu-239, using energy-averaged cross sections: $$k_\infty = \frac{\bar{\nu} \Sigma_f}{\Sigma_a} = \frac{\bar{\nu} \sigma_f}{\sigma_f + \sigma_c}$$

With ν̄ ≈ 2.9, σ_f ≈ 1.7 b, and σ_c ≈ 0.01 b: $$k_\infty \approx \frac{2.9 \times 1.7}{1.7 + 0.01} \approx 2.88$$

This is greater than 2—pure Pu-239 has ample criticality margin.

k_eff (k-effective)

The actual multiplication factor including leakage:

$$k_{eff} = k_\infty P_{NL}$$

where P_NL is the non-leakage probability. For a finite assembly:

$$P_{NL} = \frac{\text{neutrons absorbed}}{\text{neutrons absorbed + escaped}}$$

The non-leakage probability depends on geometry, size, and material. A sphere minimizes surface area (and thus leakage) for a given volume.

Criticality Calculation

For a given geometry and material, finding k_eff is an eigenvalue problem. The neutron transport equation can be written:

$$\hat{L}\phi = \frac{1}{k}\hat{F}\phi$$

where:

• \(\hat{L}\) is the loss operator (absorption + leakage)
• \(\hat{F}\) is the fission production operator
• \(\phi\) is the neutron flux
• \(k\) is the eigenvalue we seek

The physical interpretation: the fission source must be scaled by 1/k to maintain steady state.

Power Iteration

The standard numerical method is power iteration:

1. Guess an initial fission source distribution S₀(r)
2. Solve for the flux: \(\hat{L}\phi_n = S_n\)
3. Calculate the new source: \(S_{n+1} = \hat{F}\phi_n\)
4. Estimate k: \(k_n = \frac{\int S_{n+1}}{\int S_n}\)
5. Normalize: \(S_{n+1} \leftarrow S_{n+1} / k_n\)
6. Repeat until converged

Monte Carlo implements this by tracking neutrons generation by generation.

Reactivity

The reactivity ρ measures the departure from critical:

$$\rho = \frac{k - 1}{k}$$

Reactivity is often expressed in “dollars” where 1 dollar = β (the delayed neutron fraction). For Pu-239:

• 1 dollar = 0.22% = 220 pcm
• Prompt critical occurs at ρ = β (1 dollar)

Our football calculation will show ρ ≈ +17%, or about 75 dollars supercritical—deeply in the prompt supercritical regime.

Prompt vs Delayed Criticality

Delayed Critical (0 < ρ < β)

The system is critical including delayed neutrons. The neutron population grows slowly, with a period of seconds to minutes. Reactors operate in this regime.

Prompt Critical (ρ = β)

The system is critical on prompt neutrons alone. At this point, the delayed neutrons become irrelevant for population growth. The reactor period drops to milliseconds.

Prompt Supercritical (ρ > β)

Each prompt neutron generation produces more than one prompt neutron. Exponential growth occurs on a timescale of:

$$T = \frac{l}{k_p - 1}$$

where l is the prompt neutron generation time (~10 ns for fast systems) and k_p is the prompt multiplication factor.

For our football with k ≈ 1.2: $$T \approx \frac{10^{-8}}{0.2} = 50 \text{ ns}$$

The neutron population doubles every 50 nanoseconds—this is a nuclear explosion.

Critical Mass

The critical mass is the minimum mass of fissile material needed to achieve k = 1 for a given geometry.

Critical mass depends strongly on:

1. Material: Pu-239 < U-233 < U-235
2. Density: Higher density → smaller critical mass
3. Shape: Sphere is optimal
4. Reflector: Surrounding material can reduce critical mass by 30-50%
5. Enrichment: Higher fissile fraction → lower critical mass

Bare Critical Masses

For bare (unreflected) spheres:

Delta-phase plutonium has 63% higher critical mass than alpha-phase due to its lower density. But delta-phase is used in weapons because alpha-phase is brittle and pyrophoric.

The Football

Our NFL football holds about 22 kg of delta-phase plutonium—about 35% above critical mass. Even accounting for the suboptimal shape (higher leakage than a sphere), this is supercritical.

Geometry Effects

The non-leakage probability depends on the buckling, which quantifies how curved the flux distribution is:

$$B^2 = \frac{\nabla^2 \phi}{\phi}$$

For a sphere of radius R: $$B^2 = \left(\frac{\pi}{R}\right)^2$$

For an ellipsoid with semi-axes a, b, c: $$B^2 = \frac{\pi^2}{a^2} + \frac{\pi^2}{b^2} + \frac{\pi^2}{c^2}$$

The non-leakage probability is approximately: $$P_{NL} \approx \frac{1}{1 + L^2 B^2}$$

where L is the neutron diffusion length.

A sphere minimizes buckling (and thus maximizes P_NL) for a given volume. Any other shape has higher leakage and requires more mass to go critical.

For our football shape:

• Volume equivalent to a sphere of radius ~8.2 cm
• But actual half-length is 14 cm with pointed ends
• Effective buckling is ~15% higher than equivalent sphere
• Critical mass is ~15% higher than spherical configuration

Summary

The football is deeply supercritical. In the next chapter, we’ll see how Monte Carlo methods calculate k_eff by simulating individual neutrons.