## 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 B_{m} 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**:

**B**_{m}** = B**_{g}

The quantity **B _{g}^{2} **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**(

**B**).

_{g}^{2}= ∇^{2}Ф(x) / Ф(x)**In a small reactor**the neutron flux have more concave downward or ‘‘buckled’’ curvature (

**higher B**) than in a large one.

_{g}^{2}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(B**_{g }**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(a**_{e}**/2) = C.cos(B**_{g }**.a**_{e}**/2) = 0** and the values of geometrical buckling, B_{g}, are limited to **B**_{g}** = **^{nπ}**/**** _{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.

**k**. This condition is known as the subcritical state._{eff}< 1**k**. This condition is known as the critical state._{eff}= 1**k**. This condition is known as the supercritical state._{eff}> 1

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

**B**When a reactor is smaller (i.e. higher B_{m}< B_{g}._{g}and higher relative curveture) than the critical size for a given material, B_{m}< B_{g}, then the reactor is**subcritical**.**B**. When a reactor size matches the critical size for a given material, B_{m}= B_{g}_{m}= B_{g}, then the reactor is**critical**.**B**. When a reactor is larger than the critical size for a given material, B_{m}> B_{g}_{m}> B_{g}, 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 B_{g} = 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 **k _{eff} = k_{∞}.P_{NL}**, with the non/leakage probability written as (for large reactors):

where M^{2} 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** (**B**_{m}** = B**** _{g}**) using one-group diffusion theory.

The **geometrical buckling** with extrapolated distance is:

From this equation we can get R_{e}= 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 R_{0} and the reflected radius is simply R = R_{0} – δ, where R_{0} is critical diameter of a bare reactor. The geometrical buckling of infinite cylindrical reactor will then be: