Geometric and Material Buckling

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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+DΔ²-Σ_aΦ

where S is the source, and from diffusion theory D=1/Σ_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 leakage probability. Rearranging the diffusion equation becomes

-Δ²Φ/Φ=(k_∞l_f-1)/L²

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

[edit] 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.

Geometry	Geometric Buckling Bg2
Sphere of radius R	(π/R)2
Cylinder of height H and radius R	(π/H)2+(2.405/R)2
Box with lengths 2a, 2b, and 2c	(π/2a)2+(π/2b)2+(π/2c)2

[edit] 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_th1/(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

[edit] Critical Reactor Dimensions

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