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In a nuclear reactor, criticality is achieved when the rate of neutron production is equal to the rate of neutron losses, including both neutron absorption and neutron leakage. Geometric buckling is a measure of neutron leakage, while material buckling is a measure of neutron production minus absorption. Thus, in the simplest case of a bare, homogeneous, steady state reactor, the geometric and material buckling must be equal.

Both buckling terms are derived from the diffusion equation:

S+DΔ²Φ-Σ_aΦ = 0

where S is the source, and from diffusion theory D=1/3Σ_tr and L²=D/Σ_a. For thermal neutrons the source is:

S=k_∞l_fΣ_aΦ.

where k_∞ is from the four factor formula and l_f is the fast neutron non-leakage probability. Rearranging the diffusion equation becomes

-Δ²Φ/Φ=(k_∞l_f-1)/L²

The left side of the equation is the geometric buckling and the right side is the material buckling.

Geometric Buckling

The geometric buckling is an eigenvalue problem that can be solved for different geometries. The table below lists the geometric buckling for some common geometries.

Since the diffusion theory calculations overpredict the critical dimensions, an extrapolation distance δ must be subtracted to obtain an estimate of actual values. The buckling could also be calculated using actual dimensions and extrapolated distances using the following table.

Expressions for Geometric Buckling in Terms of Actual Dimensions and Extrapolated Distances^[1]

Material Buckling

Defining a fast diffusion area or neutron age τ then the thermal non-leakage probability and fast non-leakage probability are respectively

l_th1/(1+L²B²)

l_f1/(1+τB²)

The effective multiplication factor then becomes

k_eff=k_∞l_thl_f=1/((1+L²B²)(1+τB²))

In the case of a large reactor, the B⁴ term can be neglected and we are left with

k_eff=1/(1+M²B²)

where M²=L²+τ. For a critical reactor k_eff=1, so solving for B², the material buckling becomes

B_M²=(k_∞-1)/M²

Critical Reactor Dimensions

By equating the geometric and material buckling, one can determine the critical dimensions of a nuclear reactor.