# Nuclear Reactor Physics - Criticality

Criticality

In a nuclear reactor, the neutron population at any instant is a function of the rate of neutron production (due to fission processes) and the rate of neutron losses (via non-fission absorption mechanisms and leakage from the system). When a reactor’s neutron population remains steady from one generation to the next (creating as many new neutrons as are lost), the fission chain reaction is self-sustaining and the reactor's condition is referred to as "critical". When the reactor’s neutron production exceeds losses, characterized by increasing power level, it is considered "supercritical", and; when losses dominate, it is considered "subcritical" and exhibits decreasing power.

The "Six-factor formula" is the neutron life-cycle balance equation, which includes six separate factors, the product of which is equal to the ratio of the number of neutrons in any generation to that of the previous one; this parameter is called the effective multiplication factor (k), a.k.a. Keff. k = LfρLthfηЄ, where Lf = "fast non-leakage factor"; ρ = "resonance escape probability"; Lth = "thermal non-leakage factor"; f = "thermal fuel utilization factor"; η = "reproduction factor"; Є = "fast-fission factor".

k = (Neutrons produced in one generation)/(Neutrons produced in the previous generation) When the reactor is critical, k = 1. When the reactor is subcritical, k < 1. When the reactor is supercritical, k > 1.

"Reactivity" is an expression of the departure from criticality. δk = (k - 1)/k When the reactor is critical, δk = 0. When the reactor is subcritical, δk < 0. When the reactor is supercritical, δk > 0. Reactivity is also represented by the lowercase Greek letter rho (ρ). Reactivity is commonly expressed in decimals or percentages or pcm (per cent mille) of Δk/k. When reactivity ρ is expressed in units of delayed neutron fraction β, the unit is called the dollar.

If we write 'N' for the number of free neutrons in a reactor core and '' for the average lifetime of each neutron (before it either escapes from the core or is absorbed by a nucleus), then the reactor will follow differential equation (the evolution equation)

where is a constant of proportionality, and is the rate of change of the neutron count in the core. This type of differential equation describes exponential growth or exponential decay, depending on the sign of the constant, which is just the expected number of neutrons after one average neutron lifetime has elapsed:

Here, is the probability that a particular neutron will strike a fuel nucleus, is the probability that the neutron, having struck the fuel, will cause that nucleus to undergo fission, is the probability that it will be absorbed by something other than fuel, and is the probability that it will "escape" by leaving the core altogether. is the number of neutrons produced, on average, by a fission event—it is between 2 and 3 for both 235U and 239Pu.

If is positive, then the core is supercritical and the rate of neutron production will grow exponentially until some other effect stops the growth. If is negative, then the core is "subcritical" and the number of free neutrons in the core will shrink exponentially until it reaches an equilibrium at zero (or the background level from spontaneous fission). If is exactly zero, then the reactor is critical and its output does not vary in time (, from above).

Nuclear reactors are engineered to reduce and . Small, compact structures reduce the probability of direct escape by minimizing the surface area of the core, and some materials (such as graphite) can reflect some neutrons back into the core, further reducing .

The probability of fission, depends on the nuclear physics of the fuel, and is often expressed as a cross section. Reactors are usually controlled by adjusting . Control rods made of a strongly neutron-absorbent material such as cadmium or boron can be inserted into the core: any neutron that happens to impact the control rod is lost from the chain reaction, reducing . is also controlled by the recent history of the reactor core itself (see below).