# Geometrical and Material Buckling

## Geometrical and Material Buckling

In reactor physics, the concept of buckling is used to describe the relationship between requirements on fissile material inside a reactor core and dimensions and shape of that core. In general, criticality is achieved when the rate of neutron production is equal to the rate of neutron losses, including both neutron absorption and neutron leakage.

## Material Buckling

In earlier text, the term Bm first appeared in the following equation:

This parameter is known as the material buckling and it describes the characteristics of the fuel material in an infinite medium. For example let assume a uniform reactor (multiplying system) in the shape of a slab of physical width a in the x-direction and infinite in the y- and z-directions. This reactor is situated in the center at x=0. In this geometry the flux does not vary in y and z allowing us to eliminate the y and z derivatives from 2. The flux is then a function of x only, and therefore the Laplacian and diffusion equation can be written as:

## Geometrical Buckling

Geometrical buckling is a measure of neutron leakage, while material buckling is a measure of neutron production minus absorption. With this terminology the criticality condition may also be stated as the material and geometric buckling being equal:

Bm = Bg

The quantity Bg2 is called the geometrical buckling of the reactor and depends only on the geometry. This term is derived from the notion that the neutron flux distribution is somehow ‘‘buckled’’ in a homogeneous finite reactor. It can be derived the geometrical buckling is the negative relative curvature of the neutron flux (Bg2 = ∇2Ф(x) / Ф(x)). In a small reactor the neutron flux have more concave downward or ‘‘buckled’’ curvature (higher Bg2) than in a large one.

The value of geometrical buckling for infinite slab reactor can be derived, when the vacuum boundary condition is applied on the solution of diffusion equation. The physically acceptable solution for infinite slab reactor is:

Φ(x) =  C.cos(Bg x)

The vacuum boundary condition requires the relative neutron flux near the boundary to have a slope of -1/d, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary. This zero flux boundary condition is more straightforward and is can be written mathematically as:

Therefore, the solution must be  Φ(ae/2) =  C.cos(Bg .ae/2) = 0 and the values of geometrical buckling, Bg, are limited to Bg = /a_e, where n is any odd integer. The only one physically acceptable odd integer is n=1, because higher values of n would give cosine functions which would become negative for some values of x. The solution of the diffusion equation is:

## Criticality Condition

The basic classification of states of a reactor is according to the multiplication factor as eigenvalue which is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.

But these three basic states may be defined also according to the material and geometrical bucklings:

• Bm < Bg. When a reactor is smaller (i.e. higher Bg and higher relative curveture) than the critical size for a given material, Bm < Bg, then the reactor is subcritical.
• Bm = Bg. When a reactor size matches the critical size for a given material, Bm = Bg, then the reactor is critical.
• Bm > Bg. When a reactor is larger than the critical size for a given material, Bm > Bg, then the reactor is supercritical.

It must be added, for any positive value of materials buckling, there is a unique critical size for each reactor geometry. For reactors of shapes other than spheres the geometrical buckling takes the form Bg = C/R, where the coefficient C is determined by the solution of diffusion equation with vacuum boundary condition. R is a characteristic dimension. Generally, the multiplication of a uniform reactor of any shape and size is given by keff = k.PNL, with the non/leakage probability written as (for large reactors):

where M2 is the migration area and the subscript is dropped from B, the geometric buckling. As can be seen the total non-leakage probability of large reactors is primarily a function of migration area and the relative curvature of the neutron flux given by the geometrical buckling.

## Example: Calculate the geometrical buckling

Let assume a bare square cylinder (i.e., height = diameter; H = 2R). Assume that the material buckling of this reactor, which is given using one-group cross sections, is:

Calculate the critical radius (Bm = Bg) using one-group diffusion theory.

The geometrical buckling with extrapolated distance is:

From this equation we can get Re= 30.5 cm.

## Geometric Buckling of Reflected Reactor

Adding a reflector to a reactor allows either the volume of the reactor or the requirements on fuel to be reduced, or some combination of the two. If the reflector savings is known, the calculation of the critical dimensions of a reflected reactor needs only the solution of the bare reactor, which is simpler problem. For example, for the cylindrical reactor, it is only necessary to determine the bare critical radius R0 and the reflected radius is simply R = R0 – δ, where R0  is critical diameter of a bare reactor. The geometrical buckling of infinite cylindrical reactor will then be:

References:
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