Neutron Physics & the Core · 093

Start With the Smallest Size That Fits — How a Reactor's Critical Size Is Decided

The recipe fixes the material buckling; the critical size is the smallest core whose geometric buckling matches it.

Series 093

A glowing blue sphere made of many small bubbles sitting inside a loop of steel measuring tape with tick marks, on a pale lavender background — Measuring a core for size. This is a conceptual illustration, not a real reactor core.

Key idea

Once the fuel and moderator recipe fixes the material buckling, there is exactly one smallest core size at which the geometric buckling equals it. That minimum dimension is the critical size. For a sphere, the extrapolated radius is simply π divided by the square root of the material buckling. [1]

The fabric is chosen; now find the size

Pick a fabric and design in a clothing store and one question is left: the smallest size you can wear. A reactor asks the same thing. Once the enrichment, the moderator and their mix are fixed, what remains is how big the core must be to reach criticality, the state in which the chain reaction holds steady.

Geometric buckling is set purely by the core's shape and size; material buckling is set purely by its composition; and criticality requires the two to be equal. Material buckling is the fabric you can't change, geometric buckling is the size you can adjust, and the critical size is the smallest size that fits.

A clothing rack with garments from small to large: the small ones dimmed and labeled too small, subcritical; a spotlit garment labeled the first size that fits, critical size; and larger ones labeled room to spare, larger — Keep the fabric (the composition) and find the smallest size that fits: that is the critical size.

Turning the condition into a number

Each shape has its own formula for geometric buckling, written in terms of extrapolated dimensions: the flux does not drop to zero exactly at the core's edge but at an imaginary boundary slightly outside it. [1]

Core shape	Geometric buckling B_g²
Slab (extrapolated thickness ã)	(π / ã)²
Sphere (extrapolated radius R̃)	(π / R̃)²
Finite cylinder (R̃, H̃)	(2.405 / R̃)² + (π / H̃)²

Set the sphere's formula equal to the material buckling and only the radius is unknown: R̃ = π / B_m, where B_m is the square root of the material buckling. Subtract the small extrapolation distance and you have the physical radius.

As an illustration only (not a design value for any real reactor): if B_m ≈ 0.05 cm⁻¹, then R̃ = π / 0.05 ≈ 63 cm. A larger material buckling, from a mix that multiplies neutrons better, gives a smaller critical radius, which is why more highly enriched fuel can go critical at a smaller size. [2]

Spheres of increasing size along an axis on a dark panel, the small blue ones labeled subcritical, not there yet, and the largest glowing gold sphere labeled critical size reached — With the same composition, a larger core lets a smaller share of neutrons leak out. The size at which geometric buckling equals material buckling is the critical size. Sizes are not to scale.

Shape changes the volume needed

Neutrons can only escape through the core's surface. For the same volume, a shape with less surface leaks less and goes critical with less material. On pure geometry the sphere has the smallest surface for its volume, so it needs the least volume; a flat slab needs far more. That makes the sphere theoretically efficient, but real power-reactor cores are close to cylinders for engineering reasons such as refueling, coolant flow and control-rod movement.

Common mix-ups

Critical size is not critical mass. Critical size is a length; critical mass is that size multiplied by density. The same critical size can correspond to a different critical mass if the density changes.

Critical does not mean dangerous. Criticality is the normal state of a reactor running at steady power. What matters for safety is how stably that state is managed with control rods and cooling systems.

Reactors are not built at the minimum size. As fuel burns up, the material-buckling-like value slowly falls, so power reactors are built larger than the theoretical minimum. The excess reactivity is held down with control rods or burnable absorber rods and released gradually as the fuel is used.

Beyond the reactor: criticality safety

The same idea protects fuel outside the core. When nuclear fuel is stored or transported, its shape and spacing are calculated in advance so that it can never accidentally reach a critical size or mass. This field is called criticality safety; specific regulatory criteria differ by country and agency and are not covered here.

Key takeaways

The critical size is the minimum dimension at which geometric buckling equals the fixed material buckling. For a sphere, R̃ = π / B_m. Shape matters through the surface-to-volume ratio, critical size differs from critical mass, and real reactors are built larger than the minimum, with control rods managing the excess.