Reactor Physics & Kinetics · C-RX-005

Material and Geometric Buckling — Critical When the Curvature the Materials Can Sustain Equals the Curvature the Shape Demands

The materials set how much flux curvature they can sustain, the shape sets how much it needs; the core is critical where the two meet.

Concept C-RX-005

Definition

Buckling (B²) is the constant obtained when the one-group diffusion equation is written as ∇²φ + B²φ = 0; it is the relative curvature of the flux distribution (−∇²φ/φ). The value set by the material composition alone is the material buckling B_m² = (k∞ − 1)/L² (M² in modified one-group theory), and the value set by core shape, dimensions and boundary conditions is the geometric buckling B_g². The core is critical when the two are equal.

In short: buckling is the eigenvalue of the diffusion equation that expresses the curvature of the flux; the core is critical when the B_m² allowed by the materials equals the B_g² required by the shape. It shares only its name with buckling in structural mechanics.

Principle and mechanism

Two viewpoints — physics (materials) and mathematics (shape)

D∇²φ − Σaφ + (1/k)·νΣfφ = 0

Physical viewpoint (material buckling): at criticality (k = 1) the equation becomes ∇²φ + [(νΣf − Σa)/D]φ = 0. The bracket is B_m², and with k∞ = νΣf/Σa and L² = D/Σa, B_m² = (k∞ − 1)/L². It says how much the flux must curve for the net production per unit volume, (νΣf − Σa)φ, to leave exactly as leakage, −D∇²φ. Think of it as the curvature the materials "can sustain".
Mathematical viewpoint (geometric buckling): the same equation must satisfy φ = 0 at the (extrapolated) boundary. Solutions satisfying this condition exist only when B² takes particular eigenvalues, and the fundamental mode, with φ > 0 everywhere, corresponds to the smallest eigenvalue B_g². It is the curvature the shape "demands", independent of the materials.

The judgement is simple: B_m² = B_g² means critical, B_m² > B_g² supercritical, and B_m² < B_g² subcritical. If k∞ < 1, then B_m² < 0 and no size can make the core critical.

Relation to keff and the non-leakage probabilities

Away from criticality, keeping 1/k in the equation gives keff = k∞/(1 + L²B_g²), in which the thermal non-leakage probability P_TNL = 1/(1 + L²B_g²) appears directly. Including the fast group:

Modified one-group: keff ≈ k∞/(1 + M²B_g²), M² = L² + τ
Age-diffusion: keff = k∞·e^−τB_g²/(1 + L²B_g²), fast non-leakage probability P_FNL = e^−τB_g² (two-group approximation 1/(1 + τB_g²))

As the core gets larger, B_g² gets smaller and both non-leakage probabilities approach 1. The Fermi age τ is one sixth of the mean square distance a fast neutron travels while slowing down to thermal energy. From the Lamarsh tables, light water has τ ≈ 27 cm² and L² = 8.1 cm², heavy water 131 cm² and 9.4×10³ cm², and graphite 368 cm² and 3,500 cm². In light-water reactors τ is larger than L², so a one-group calculation that leaves out fast leakage is badly wrong.

Geometric buckling by shape

Dimensions are measured to the extrapolated boundary. The extrapolation distance is d ≈ 0.71λ_tr = 2.13D and can be neglected in large reactors.

Shape	B_g²	Flux distribution	φmax/φavg
Slab (thickness a)	(π/a)²	A cos(πx/a)	1.57
Rectangular box a×b×c	(π/a)² + (π/b)² + (π/c)²	cos·cos·cos	3.88
Finite cylinder (R, H)	(2.405/R)² + (π/H)²	A J₀(2.405r/R)cos(πz/H)	3.64
Sphere (radius R)	(π/R)²	A sin(πr/R)/r	3.29

2.405 is the first zero of J₀. For the slab the symmetry condition φ(−x) = φ(x) removes the sine term, and φ(X₀/2) = 0 gives B = π/X₀ and φ = A cos(πx/X₀). For the sphere, normalizing to the power P = E_RΣf∫φdV gives φ(r) = [P/(4R²E_RΣf)]·sin(πr/R)/r.

For the same B_m², the shape with the smallest volume is the sphere (V ≈ 130/B³); the optimum cylinder, with H ≈ 1.85R (H/D ≈ 0.92), has V ≈ 148/B³, and a cube 161/B³. The Lamarsh example rounds the optimum cylinder to H = 1.82R. The larger the volume relative to the surface area, the smaller the leakage and the smaller the critical mass.

Two stacked plots against core radius R from 10 to 45 cm with H/R fixed at 1.82: in the upper plot the geometric buckling curve B_g squared = 8.770/R squared falls and crosses the material buckling line B_m squared = (k-infinity - 1)/L squared = 0.0199 per square centimetre at a critical radius of about 21.0 cm, with the subcritical region to the left and the supercritical region to the right; in the lower plot keff = k-infinity/(1 + L squared B_g squared) rises toward the asymptote k-infinity = 1.068 and crosses keff = 1 at the same radius; a side card lists geometric buckling for a slab, box, cylinder and sphere and minimum volumes of 130, 148 and 161 over B cubed for sphere, optimum cylinder and cube — Critical radius as the intersection of B_g²(R) and B_m² for the homogeneous cylindrical reactor of Example 1 (bare, fundamental mode, one group, extrapolation distance ignored, P_f = 1 as assumed in the example).

Procedure for critical radius and critical mass

Critical-size calculations follow four steps: ① composition → k∞ and L² (M²); ② B_m² = (k∞ − 1)/M²; ③ set the shape formula B_g²(dimensions) equal to B_m² to get the dimensions; ④ volume × fuel fraction × density = mass.

Example 1 — critical radius of a homogeneous cylindrical reactor: with ε = P_f = p = 1, ν = 2.43, σ_a^H₂O = 0.664 b, σ_a^UO₂ = 23 b, σf²³⁵ = 580 b and L_th² = 8.1 cm².

f = 4×23/(4×23 + 100×0.664) = 0.581
η = 2.43×(0.03×580)/23 = 1.838 → k∞ = ηf = 1.068
Homogeneous mixture L² = L_M²(1 − f) = 8.1×0.419 = 3.40 cm²
B_m² = 0.0677/3.40 = 0.0199 cm⁻²
Since H = R/0.55, B_g² = [2.405² + (0.55π)²]/R² = 8.770/R² → R ≈ 21.0 cm (H ≈ 38.1 cm)

This value results from removing fast leakage by setting P_f = 1. With real light-water moderation, M² = 3.40 + 27 ≈ 30 cm² and R ≈ 63 cm.

Example 2 — critical mass of natural uranium in a heavy-water reactor: with B_m² = 8.6 m⁻² (1 m⁻² = 10⁻⁴ cm⁻²).

B_m² = 8.6×10⁻⁴ cm⁻², H = 2.4R → B_g² = [2.405² + (π/2.4)²]/R² = 7.497/R²
R = 93.4 cm, H = 224 cm, V = πR²H = 6.14×10⁶ cm³
Fuel volume fraction = π(1.27)²/(15.2)² = 0.0219 → fuel 1.35×10⁵ cm³
Mass = 19 g/cm³ × 1.35×10⁵ cm³ ≈ 2.56 t (extrapolation distance ignored, fuel rods assumed to run the full height)

What it means in a plant

Design: large PWR cores have a small B_g², so leakage is only a few percent. The k∞ margin (excess reactivity) is offset with boron, control rods and burnable poisons. Because the core's B_g² is fixed, reactivity changes during operation can be read as changes in B_m² (k∞, M²) with moderator temperature and boron concentration.
Power distribution: φmax/φavg = 3.64 for a bare cylinder is the starting point for peaking factors. Real cores flatten the distribution with a reflector (which reduces the critical dimensions by the reflector savings δ), low-leakage loading patterns and burnable poisons.
Criticality safety: in fuel storage and handling, keff < 1 is secured by making B_g² large (thin slabs, slender cylinders, maintained rack spacing). This is "geometry control", the physical basis for storage rack pitch and neutron absorber panel design.

Regulatory and design criteria

The criteria below are examples from the United States (US NRC, ANSI/ANS) and the IAEA.

10 CFR 50.68 (introduced 1998): new fuel storage keff ≤ 0.95 (flooded with unborated water) and ≤ 0.98 under optimum moderation; spent fuel storage keff ≤ 0.95 (< 1.0 with unborated water if credit is taken for soluble boron); all at 95/95.
GDC 62: requires criticality in fuel storage and handling to be prevented, preferably by geometrically safe configurations.
NUREG-0800 SRP 4.3, 9.1.1 and 9.1.2: review core criticality and power distribution calculation methods and storage criticality analyses.
ANSI/ANS-8.1: sets single-parameter subcritical limits such as sphere diameter, cylinder diameter and slab thickness.
IAEA SSG-27 (Rev. 1, 2022): guidance on criticality safety in the handling of fissile material. Systems that are intentionally critical, such as reactor cores, are excluded; core design requirements are in SSR-2/1 (Rev. 1, 2016).
Then versus now: the physics of buckling has not changed. IAEA SSG-27 was first issued in 2014 and revised as Rev. 1 in 2022.

Easily confused pairs

	Material buckling B_m²	Geometric buckling B_g²
Determined by	Composition, enrichment, moderator ratio, temperature, boron	Shape, dimensions, extrapolation distance
Expression	(k∞ − 1)/L², (k∞ − 1)/M²	Smallest eigenvalue of the Helmholtz equation
Viewpoint	Physics (production − absorption = leakage)	Mathematics (solution satisfying the boundary condition)
When it increases	Reactivity increases	Leakage increases (the core is smaller)

	L² (thermal diffusion area)	τ (Fermi age)
Applies to	Thermal neutrons, until absorption	Fast → thermal, during slowing down
Meaning	One sixth of the mean square distance travelled	One sixth of the mean square distance while slowing down
Non-leakage	1/(1 + L²B²)	e^−τB²

Key points at a glance

Definition: the constant of the diffusion equation ∇²φ + B²φ = 0; the flux curvature −∇²φ/φ.
Material buckling: B_m² = (νΣf − Σa)/D = (k∞ − 1)/L² (modified one-group M² = L² + τ); a function of composition only.
Geometric buckling: the smallest eigenvalue satisfying φ = 0 at the extrapolated boundary; a function of shape and dimensions only.
By shape: slab (π/a)², cylinder (2.405/R)² + (π/H)², sphere (π/R)².
Criticality condition: B_m² = B_g² critical, B_m² > B_g² supercritical, B_m² < B_g² subcritical.
keff: keff = k∞/(1 + M²B_g²), P_TNL = 1/(1 + L²B_g²), P_FNL = e^−τB_g².
Critical dimensions: found from B_g²(dimensions) = B_m²; for the same B_m² the sphere has the smallest volume (130/B³).
Uses: sizing the core, reducing critical dimensions with a reflector, geometry control in storage and handling.
Common pitfalls: watch the units (1 m⁻² = 10⁻⁴ cm⁻²), state which L² or M² is used, and note when an assumption such as P_f = 1 removes fast leakage.

Sketch in two parts: (a) the cosine flux phi = A cos(pi x / a-tilde) in a slab, falling to zero at the extrapolated boundary a-tilde/2 beyond the physical boundary a/2, with d about 2.13D, a-tilde = a + 2d and B_g squared = (pi/a-tilde) squared; (b) the falling B_g squared curve versus core size crossing the horizontal material buckling line B_m squared = (k-infinity - 1)/M squared at the critical size R_c, subcritical to the left and supercritical to the right, with M squared = L squared + tau — Diagram: slab flux with the extrapolation distance, and the critical size where B_m² = B_g².