# Geometric and Material Buckling

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Geometric and Material Buckling

In a nuclear reactor criticality is achieved. In order to determine the critical size and mass of a steady state reactor, the geometric and material buckling must be equal. Geometric buckling is derived from the diffusion equation for thermal neutrons

S+DΔ2Φ-ΣaΦ

where S is the source, and from diffusion theory D=1/Σtr and L2=D/Σa. For thermal neutrons the source is

S=klfΣaΦ.

where k is from the four factor formula and lf is the fast neutron leakage probability. Rearranging the diffusion equation becomes

2Φ/Φ=(klf-1)/L2

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

Material Buckling

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

lth1/(1+L2B2)

lth1/(1+τB2)

The effective multiplication factor then becomes

keff=klthlf=1/((1+L2B2)(1+τB2))

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

keff=1/(1+M2B2)

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

BM2=(k-1)/M2

Critical Reactor Dimensions

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

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