Editing Neutron transport (section)

===Criticality===
[[Nuclear fission|Fission]] is the process through which a nucleus splits into (typically two) smaller atoms. If fission is occurring, it is often of interest to know the asymptotic behavior of the system. A reactor is called “critical” if the chain reaction is self-sustaining and time-independent. If the system is not in equilibrium the asymptotic neutron distribution, or the fundamental mode, will grow or decay exponentially over time.

Criticality calculations are used to analyze steady-state multiplying media (multiplying media can undergo fission), such as a critical nuclear reactor. The loss terms (absorption, out-scattering, and leakage) and the source terms (in-scatter and fission) are proportional to the neutron flux, contrasting with fixed-source problems where the source is independent of the flux. In these calculations, the presumption of time invariance requires that neutron production exactly equals neutron loss.

Since this criticality can only be achieved by very fine manipulations of the geometry (typically via control rods in a reactor), it is unlikely that the modeled geometry will be truly critical. To allow some flexibility in the way models are set up, these problems are formulated as eigenvalue problems, where one parameter is artificially modified until criticality is reached. The most common formulations are the time-absorption and the multiplication eigenvalues, also known as the alpha and k eigenvalues. The alpha and k are the tunable quantities.

K-eigenvalue problems are the most common in nuclear reactor analysis. The number of neutrons produced per fission is multiplicatively modified by the dominant eigenvalue. The resulting value of this eigenvalue reflects the time dependence of the neutron density in a multiplying medium.
*''k<sub>eff</sub>'' < 1, subcritical: the neutron density is decreasing as time passes;
*''k<sub>eff</sub>'' = 1, critical: the neutron density remains unchanged; and
*''k<sub>eff</sub>'' > 1, supercritical: the neutron density is increasing with time.

In the case of a [[nuclear reactor]], neutron flux and power density are proportional, hence during reactor start-up ''k<sub>eff</sub>'' > 1, during reactor operation ''k<sub>eff</sub>'' = 1 and ''k<sub>eff</sub>'' < 1 at reactor shutdown.