In a nuclear reactor

Nuclear reactor

A nuclear reactor is a device to initiate and control a sustained nuclear chain reaction. Most commonly they are used for generating electricity and for the propulsion of ships. Usually heat from nuclear fission is passed to a working fluid , which runs through turbines that power either ship's...

, criticality

Critical mass

A critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction. The critical mass of a fissionable material depends upon its nuclear properties A critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction. The...

 is achieved when the rate of neutron production is equal to the rate of neutron losses, including both neutron absorption and neutron leakage. Geometric buckling is a measure of neutron leakage, while material buckling is a measure of neutron production minus absorption. Thus, in the simplest case of a bare, homogeneous, steady state reactor, the geometric and material buckling must be equal.

Derivation

Both buckling terms are derived from the diffusion equation:

.

where k is the criticality eigenvalue, is the neutrons per fission, is the macroscopic

Macroscopic

The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

 cross section

Cross section (physics)

A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

 for fission

Nuclear fission

In nuclear physics and nuclear chemistry, nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts , often producing free neutrons and photons , and releasing a tremendous amount of energy...

, and from diffusion theory, the diffusion coefficient is defined as:

.

In addition, the diffusion length is defined as:

.

Rearranging the terms, the diffusion equation becomes:

.

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

Geometric Buckling

The geometric buckling is an eigenvalue problem that can be solved for different geometries

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

. The table below lists the geometric buckling for some common geometries.

Geometry	Geometric Buckling Bg2
Sphere of radius R
Cylinder of height H and radius R
Parallelepided with side lengths a, b and c

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.

Geometry	Geometric Buckling Bg2
Sphere of radius R
Cylinder of height H and radius R
Parallelepided with side lengths a, b and c

Material Buckling

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

.

As stated previously, the geometric buckling is defined as:

.

Solving for k (in the fundamental mode),

;

thus,

.

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

.

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

.

Critical Reactor Dimensions

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