- 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+DΔ

^{2}Φ-Σ_{a}Φwhere S is the source, and from diffusion theory D=1/Σ

_{tr}and L^{2}=D/Σ_{a}. For thermal neutrons the source isS=k

_{∞}l_{f}Σ_{a}Φ.where k

_{∞}is from thefour factor formula and l_{f}is the fast neutron leakage probability. Rearranging the diffusion equation becomes-Δ

^{2}Φ/Φ=(k_{∞}l_{f}-1)/L^{2}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

l

_{th}1/(1+L^{2}B^{2})l

_{th}1/(1+τB^{2})The effective multiplication factor then becomes

k

_{eff}=k_{∞}l_{th}l_{f}=1/((1+L^{2}B^{2})(1+τB^{2}))In the case of a large reactor, the B

^{4}term can be neglected and we are left withk

_{eff}=1/(1+M^{2}B^{2})where M

^{2}=L^{2}+τ. For a critical reactor k_{eff}=1, so solving for B^{2}, the material buckling becomesB

_{M}^{2}=(k_{∞}-1)/M^{2}**Critical Reactor Dimensions**By equating the geometric and material buckling, one can determine the critical dimensions of a nuclear reactor.

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