The Nuclear Football
The main event: analyzing criticality of an NFL football filled with weapons-grade plutonium.
The Setup
NFL Football Specifications
An official NFL football must meet these requirements (Rule 2, Section 1):
- Long axis: 11.0 to 11.25 inches (27.9 to 28.6 cm)
- Long circumference: 28.0 to 28.5 inches
- Short circumference: 21.0 to 21.25 inches → diameter ≈ 17.8 cm
We model this as a superellipsoid:
- Half-length: 14.0 cm
- Maximum radius: 8.9 cm
- Pointiness exponent: 3.0
Material
We fill the football with δ-phase plutonium-239:
- Density: 15.61 g/cm³
- Composition: 95.45% Pu-239, 0.45% Pu-240, 4.1% Ga
This is the same material used in the Jezebel benchmark—stabilized with gallium to maintain the ductile delta phase.
Calculated Properties
| Property | Value |
|---|---|
| Football volume | 2330 cm³ |
| Plutonium mass | 36.4 kg |
| Surface area | ~1400 cm² |
For comparison, a sphere of equal mass would have:
- Radius: 8.0 cm
- Volume: 2140 cm³
- Surface area: ~800 cm²
The football has 75% more surface area than an equal-mass sphere.
The Simulation
# Create NFL football geometry
nfl = Football() # 28 cm × 17.8 cm, m=3
# Run high-statistics calculation
result = run_criticality(
nfl, δ_phase_plutonium,
n_particles = 10000,
n_inactive = 50,
n_active = 100,
show_progress = true
)
Output
╔══════════════════════════════════════════════════════════╗
║ CAN A NUCLEAR FOOTBALL GO CRITICAL? ║
╚══════════════════════════════════════════════════════════╝
Material: δ-phase Pu (Jezebel)
Density: 15.61 g/cm³
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NFL FOOTBALL DIMENSIONS
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Length: 28.0 cm (11.0 inches)
Max diameter: 17.8 cm (7.0 inches)
Volume: 2330.5 cm³
If filled with δ-Pu: 36.4 kg
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CRITICALITY OF PU-FILLED NFL FOOTBALL
============================================================
Running Monte Carlo simulation...
Criticality: 100%|████████████████████| Time: 0:01:52
k_eff = 1.2142 ± 0.0038
🏈 RESULT: YES! A NUCLEAR FOOTBALL WOULD GO CRITICAL! 🏈
The system is SUPERCRITICAL by 21.4%
Results
k-effective
$$k_{eff} = 1.214 \pm 0.004$$
This is deeply supercritical. The football would undergo a prompt critical excursion.
Reactivity
$$\rho = \frac{k-1}{k} = \frac{0.214}{1.214} = 17.6%$$
In “dollars” (where \(1 = β = 0.0022\) for Pu): $$\rho = \frac{0.176}{0.0022} = 80 \text{ dollars}$$
The system is 80 dollars supercritical—far into the prompt supercritical regime.
Time Scale
The neutron generation time in fast plutonium systems is ~10 ns. The e-folding time: $$T = \frac{\ell}{k-1} = \frac{10^{-8}}{0.214} = 47 \text{ ns}$$
The neutron population would double every 47 nanoseconds. In 1 microsecond (~20 doublings), the population increases by a factor of ~10⁶.
Comparison with Sphere
We also calculated k_eff for a sphere of equal mass:
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COMPARISON WITH OPTIMAL SPHERE
============================================================
Sphere with same mass (36.4 kg):
Radius: 7.97 cm
Running Monte Carlo simulation...
Criticality: 100%|████████████████████| Time: 0:01:48
Sphere k_eff = 1.321 ± 0.0040
Shape penalty (football vs sphere): 8.8%
| Geometry | k_eff | Reactivity |
|---|---|---|
| Football | 1.214 | +17.6% |
| Sphere | 1.321 | +24.3% |
The football shape costs about 9% in reactivity compared to an optimal sphere. The pointed ends cause extra neutron leakage.
Critical Mass Analysis
How much plutonium is actually needed for criticality in each shape?
Football Critical Mass
============================================================
CRITICAL MASS IN FOOTBALL SHAPE
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Searching for critical mass in range [5.0, 50.0] kg
Geometry: Football
Material: δ-phase Pu (Jezebel)
Iteration 1: mass = 15.81 kg, k_eff = 0.9723 ± 0.0051
Iteration 2: mass = 19.37 kg, k_eff = 1.0412 ± 0.0049
Iteration 3: mass = 17.50 kg, k_eff = 1.0124 ± 0.0048
Iteration 4: mass = 16.63 kg, k_eff = 0.9892 ± 0.0050
Iteration 5: mass = 17.05 kg, k_eff = 1.0021 ± 0.0047
Critical mass (football shape): 17.1 kg
Sphere Critical Mass
Iteration 1: mass = 15.81 kg, k_eff = 1.0124 ± 0.0049
Iteration 2: mass = 13.69 kg, k_eff = 0.9712 ± 0.0051
Iteration 3: mass = 14.72 kg, k_eff = 0.9923 ± 0.0050
Iteration 4: mass = 15.25 kg, k_eff = 1.0021 ± 0.0048
Critical mass (sphere): 15.3 kg
Summary
| Shape | Critical Mass | Excess in Football |
|---|---|---|
| Sphere | 15.3 kg | — |
| Football | 17.1 kg | +12% |
The football requires 12% more material to reach criticality. But at 36.4 kg, it contains more than twice the critical mass.
Why Is the Football Supercritical?
Several factors contribute:
1. Abundant Fissile Material
The 36.4 kg of plutonium is well above critical mass for any geometry. Even the inefficient football shape can’t waste enough neutrons to stay subcritical.
2. Favorable Density
δ-phase plutonium at 15.8 g/cm³ is dense enough that neutrons undergo multiple collisions before escaping. The mean free path (~4 cm) is much smaller than the football dimensions (~28 cm).
3. High k_∞ for Pu-239
Pure Pu-239 has k_∞ ≈ 2.9—every neutron absorbed produces nearly 3 new ones (on average). Even with significant leakage, k_eff > 1 is achievable.
Physical Implications
What Would Happen?
If you somehow assembled a Pu-filled football, several outcomes are possible:
Scenario 1: Slow Assembly If assembled gradually, the system would go critical before completion. Alpha particles from decay and spontaneous fission neutrons would trigger the chain reaction. The result: a partial nuclear excursion that disperses the material (a “fizzle”).
Scenario 2: Fast Assembly (Impossible) Even at implosion speeds (~km/s), the high spontaneous fission rate of Pu-240 (~400,000 n/s per kg of WGPu) means predetonation is certain. The system would experience a low-yield explosion before reaching optimal configuration.
Scenario 3: Hypothetical Instantaneous Assembly If magically assembled instantaneously, the 80-dollar supercritical system would produce a nuclear yield. The energy release:
$$E \approx \frac{1}{2}m_{fissioned}c^2 \cdot (efficiency)$$
With perhaps 1-5% of the material fissioning before disassembly, the yield would be in the kiloton range.
Radiation Hazards
Even subcritical, a 36 kg Pu mass would be intensely radioactive:
- Alpha particles (short range, dangerous if inhaled)
- Gamma rays from decay products
- Neutrons from spontaneous fission
Handling would require specialized facilities (glove boxes, shielding, criticality safety protocols).
Conclusions
| Question | Answer |
|---|---|
| Would a Pu-filled NFL football be critical? | YES |
| By how much? | k = 1.21, ρ = +18%, 80 dollars |
| How does shape affect it? | Football requires 12% more mass than sphere |
| Mass in football? | 36.4 kg (~2.2× critical mass) |
| Time scale? | Doubling every 47 ns |
The nuclear football is not just critical—it’s deeply supercritical. The pointed shape causes extra leakage, but there’s simply too much plutonium for that to matter.
The Final Verdict
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║ FINAL VERDICT ║
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║ ║
║ ✓ YES, a plutonium-filled NFL football would be a ║
║ supercritical nuclear device. ║
║ ║
║ The pointed shape is suboptimal but the ~36 kg of ║
║ fissile material more than compensates. ║
║ ║
║ k_eff ≈ 1.21 (prompt supercritical) ║
║ Critical mass in football shape: ~17 kg ║
║ Actual mass: 36 kg (2.1× critical) ║
║ ║
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Don’t try this at home.