Geometric and material buckling

Derivation

Both buckling terms are derived from the particular diffusion equation which is valid for neutrons:

-D \nabla^2 \Phi + \Sigma_a \Phi = \frac{1}{k} \nu \Sigma_f \Phi.

where k is the criticality eigenvalue, \nu is the neutrons per fission, \Sigma_f is the macroscopic cross section for fission, and from diffusion theory, the diffusion coefficient is defined as:

D=\frac{1}{3\Sigma_{\mathrm{tr}}}.

In addition, the diffusion length is defined as:

L=\sqrt{\frac{D}{\Sigma_a}}.

Rearranging the terms, the diffusion equation becomes:

-\frac{\nabla^2 \Phi}{\Phi} = \frac{\frac{k_{\infty}}{k}-1}{L^2} = {B_g}^2.

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

Geometric Buckling

The geometric buckling is a simple Helmholtz eigenvalue problem that is simply 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.{{cite book |last=Knief |first=Ronald A. |title= Nuclear Criticality Safety: Theory and Practice |year=1985 |publisher=American Nuclear Society

Material Buckling

Materials buckling is the buckling of a homogeneous configuration with respect to material properties only. If we redefine k_{\infty} in terms of purely material properties (and assume the fundamental mode), we have:

k_{\infty} = \frac{\nu \Sigma_f}{\Sigma_a}.

As stated previously, the geometric buckling is defined as:

{B_g}^2 = \frac{\frac{k_{\infty}}{k} - 1}{L^2} = \frac{\frac{1}{k} \nu \Sigma_f - \Sigma_a}{D}.

Solving for k (in the fundamental mode),

k = k_{\mathrm{eff}} = \frac{\nu \Sigma_f}{\Sigma_a + D {B_g}^2};

thus,

k = \frac{\frac{\nu \Sigma_f}{\Sigma_a}}{1 + L^2 {B_g}^2}.

Assuming the reactor is in a critical state (k = 1),

{B_g}^2 = \frac{\nu \Sigma_f - \Sigma_a}{D}.

This expression is in purely material properties; therefore, this is called the materials buckling:

{B_m}^2 = \frac{\nu \Sigma_f - \Sigma_a}{D}.

Critical Reactor Dimensions

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

References

References

1. Lamarsh, John R.. (2018). "Introduction to Nuclear Engineering". Pearson Education Inc..
2. Adams, Marvin L.. (2009). "Introduction to Nuclear Reactor Theory". Texas A&M University.