What it does
A reactor is critical when each generation of neutrons makes exactly as many neutrons as the last. In an infinite block of fuel that ratio is k∞; in a real, finite core some neutrons leak out of the surface, so k_eff is smaller. One-group diffusion theory gives the leakage through a single number for the shape, the geometric buckling B²:
k_eff = k∞ / (1 + L²B²), k∞ = νΣf / Σa, L² = D / Σa
This calculator, for a bare (unreflected) homogeneous core:
- gives B² for a slab, sphere, finite or infinite cylinder, or box, with the extrapolation distance (2.13 D) added if you want;
- works out k_eff, the non-leakage probability and the critical size for each shape (with the cylinder’s height-to-diameter ratio of your choice), with volumes;
- draws the flux shape (cosine, sin(Br)/Br, or the Bessel function J₀) and k_eff against size;
- solves the same problem numerically (finite volumes and power iteration) and shows how closely it matches the formula;
- offers a two-group model (fast and thermal), with k_eff = k∞ / ((1 + L₁²B²)(1 + L₂²B²)).
It is for study: real cores have reflectors, many materials and many energy groups.
Worked example
With the made-up one-group data (D = 0.9 cm, Σa = 0.0195 /cm, νΣf = 0.0245 /cm), k∞ = 1.25641 and L² = 46.154 cm². A sphere of radius 40 cm, extrapolated by 1.917 cm, has B² = 0.0056172 cm⁻², a non-leakage probability of 79.41 % and k_eff = 0.99774: just subcritical. The critical sphere has R = 40.23 cm (272.8 L). The numeric solve with 200 cells gives 0.99774 too, differing by 2.5 × 10⁻⁶. Critical sizes of the other shapes: a slab 38.31 cm thick, a cylinder with H = D of R 36.62 cm and H 73.24 cm (308.6 L), and a cube of side 69.17 cm (330.9 L): the sphere needs the least material because it leaks least.
With the made-up two-group data, k∞ = 1.31771 and M² = 59.79 cm²; the 40 cm sphere has k_eff = 0.96726 and the critical sphere has R = 42.70 cm.
FAQ
Why is the sphere the smallest critical shape? It has the least surface for its volume, so the fewest neutrons leak out.
What is the extrapolation distance? In diffusion theory the flux does not reach zero at the surface; it would reach zero about 0.71 transport mean free paths (2.13 D) outside it.
Is anything uploaded? No. Everything runs in your browser and works offline.
Where the numbers come from
- One-group and two-group bare-reactor diffusion theory (buckling, k_eff = k∞/(1 + L²B²), critical size, flux shapes) as set out in standard reactor physics textbooks (general physics); J0 by its power series.
- Checked against SciPy (Bessel zero and J0, root-finding of k_eff = 1 in the physical size, two-group roots) to 1e-10, and the tool's finite-volume power iteration against the formulas (k within 1e-5, flux within 1e-3, second-order convergence) and against a separate node-based SciPy eigenvalue solve.
- The example material data are made up for fibub; they are not cross-sections of any real fuel or from any evaluated data library.
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Last updated 2026-11-09.