Delayed Neutron and Nuclear Reactor Kinetics
Abstract
Introduction Reactor power changes when the temperature and position of the control rods of a nuclear reactor are changed. This change is unique to each reactor, and its characteristics are called “nuclear reactor kinetics.”
The control rods are made of strong neutronabsorbing materials, and when they are inserted into the reactor, the reaction rate of neutron absorption increases. The reactor becomes subcritical and its power decreases. Conversely, the reaction rate of neutron absorption decreases when the control rods are withdrawn; the reactor becomes supercritical and its power increases. The reaction rate of neutron absorption changes when the reactor temperature is changed and, therefore the reactor power changes.
The reactor power is proportional to the number of fission reactions per second in the nuclear reactor. As fission reactions are caused by neutrons, the number of their reactions is proportional to the total number of neutrons in the reactor. However, the number of neutrons varies depending on the neutron production rate due to the fission reactions, the rate of neutron absorption by the nuclear fuel and reactor structure materials, and the rate of neutron leakage from the reactor.
Keywords
Fission Reaction Neutron Absorption Prompt Neutron Delayed Neutron Neutron LeakageIntroduction Reactor power changes when the temperature and position of the control rods of a nuclear reactor are changed. This change is unique to each reactor, and its characteristics are called “nuclear reactor kinetics.”
The control rods are made of strong neutronabsorbing materials, and when they are inserted into the reactor, the reaction rate of neutron absorption increases. The reactor becomes subcritical and its power decreases. Conversely, the reaction rate of neutron absorption decreases when the control rods are withdrawn; the reactor becomes supercritical and its power increases. The reaction rate of neutron absorption changes when the reactor temperature is changed and, therefore the reactor power changes.
The reactor power is proportional to the number of fission reactions per second in the nuclear reactor. As fission reactions are caused by neutrons, the number of their reactions is proportional to the total number of neutrons in the reactor. However, the number of neutrons varies depending on the neutron production rate due to the fission reactions, the rate of neutron absorption by the nuclear fuel and reactor structure materials, and the rate of neutron leakage from the reactor.
1.1 Fission Chain Reactions
If k _{eff} = 1, the number of neutrons is constant in the reactor. In other words, the fission rate is constant and the constant energy release continues. At this state, the reactor is critical. If k _{eff} < 1, the number of neutrons decreases gradually in the reactor with progression of the fission chain reaction. In this state, the reactor is subcritical. However, if “k _{eff} > 1,” the chain reaction rate increases and the reactor is supercritical.
Actually, however, it is difficult to determine the length of a generation of neutrons. This is because some neutrons trigger fission reactions immediately after they are reproduced but other neutrons trigger after their moderation to thermal neutrons. Neutron capture occurs randomly, and so does leakage of neutrons to the outside of the reactor. The following discussion uses Eq. (1.1) to define the effective multiplication factor.
When a reactor operates with a constant power, its effective multiplication factor is equal to 1. The reactor is “critical”. When the reactor is shut down, it is “subcritical”. During startup, the reactor is controlled so that it becomes “supercritical”; the neutron production rate is increased above the neutron loss rate, and the number of neutrons in the reactor is gradually increased. When the reactor reaches the required power rating, it is returned to the critical state and is operated with a constant output. If the reactor is required to change from low power to high power, it is controlled to reach the supercritical state. If the reactor is required to return to low power, it is controlled to reach the subcritical state; when it reaches the required power, the reactor is returned to critical.
The nuclear reactor kinetics usually explains an increase or decrease in the number of neutrons in the entire core. In other words, the spatial distribution of neutrons is not considered in the core. Here, a pointwise reactor approximation is used, where the core is represented by one point and no space variable is considered. Reactor kinetics that considers the spatial distribution is called space–time kinetics.
1.2 Change in Multiplication Factor and Nuclear Reactor Kinetics

Short period (from seconds to minutes): startup, shutdown, and disturbances during operation (including changes of temperature, pressure, and moderator density)

Intermediate period (from hours to days): generation and decay of fission products (xenon and samarium) having strong neutron absorption

Long period (from months to years): burnup (consumption) of nuclear fuel and accumulation of fission products
It is important to estimate of a change in the number of neutrons and a change in the power with time that occur when k _{eff} changes. This is covered by the reactor kinetics and nuclear plant dynamics. The nuclear reactor kinetics covers the change in the number of neutrons and the change in power due to a shortperiod change in multiplication factor. Over a long period, the change in multiplication factor is compensated for by control rods, chemical shim, and burnable poison. Details for this are presented in Part II Chaps. 3– 6 of this book. In an intermediate period, the change in multiplication factor is estimated using the generation and decay model of xenon and samarium. It is different from the nuclear reactor kinetics model.
1.3 Prompt Neutron and Delayed Neutron
Most neutrons (99.35 % for ^{235}U fission by thermal neutrons) are emitted immediately by a nuclear fission event. These are called “prompt neutrons.” A few neutrons are emitted a little after nuclear fission occurs and they are called “delayed neutrons.”
Data of a delayed neutron generated by thermal fission of uranium235
Group  Halflife (s)  Decay constant, λ _{i} (s^{–1})  Delayed neutron fraction, β _{i} 

1  55.72  0.012 4  0.000 215 
2  22.72  0.030 5  0.001 424 
3  6.22  0.111  0.001 274 
4  2.30  0.301  0.002 568 
5  0.610  1.14  0.000 748 
6  0.230  3.01  0.000 273 
Delayed neutron fraction of nuclides
Nuclide  Total delayed neutron fraction, β 

^{232}Th  0.020 3^{a} 
^{233}U  0.002 6 
^{235}U  0.006 5 
^{238}U  0.014 8^{a} 
^{239}Pu  0.002 0^{a} 
The delayed neutrons have approximately 0.4MeV average energy, which is lower than the approximate 2MeV average energy of the prompt neutrons. Therefore, the fraction of delayed neutrons that is leaked outside the reactor and lost disappear is slightly smaller than that of the prompt neutrons. The fraction of delayed neutrons that contributes to the fission chain reactions is slightly larger than that of the prompt neutrons. This effect is considered in the analysis of nuclear reactor kinetics. A slightly larger delayed neutron fraction is used than the absolute value “β” depending on the reactor and the effect is shown as “β _{eff}.” If the reactor has a large core volume, neutron leakage is very small during moderation and there is almost no difference between them. The “β _{eff}” value depends on the reactor size and neutron spectrum. Although the delayed neutron fraction is low, it slows down the transient change of the reactor and, therefore, it plays a very important role in the reactor control.
1.4 Kinetic Parameters
This section explains the parameters commonly used for description of reactor kinetics. The definition of effective multiplication factor, k _{eff}, has been given by Eq. (1.1) or (1.2).
If the reactor is supercritical, k _{eff} > 1 and the value of ρ is positive. If the reactor is subcritical, k _{eff} < 1 and the value of ρ is negative. ρ takes a value within the range of −∞ < ρ < 1.
The reactivity is expressed as a numerical value or a percentage. It is shown in the French unit of “pcm” (10^{−5}), in the English unit of “millik” (10^{−3}), or in the American unit of “dollars” ($) and “cents” (¢). A dollar is equal to the value ρ divided by generation rate β of delayed neutrons, and 1 $ is equal to 100 ¢.
If k _{eff} = 1, that is, if ρ = 0, it is strictly said to be the “delayed critical” state. The generation of neutrons in the reactor (including the generation of delayed neutrons caused by the decay of delayed neutron precursors) is equal to the loss of neutrons. When the reactivity is 1 $, the generation of prompt neutrons is equal to its loss. This is called the “prompt critical” state. The state above the prompt critical is called the “prompt supercritical.”
Diffusion time and slowingdown time of various moderators
Moderator  Diffusion time (ms)  Slowingdown time (μs) 

Light water (H_{2}O)  0.205  1.0 
Heavy water (D_{2}O)  100^{a}  8.1 
Beryllium (Be)  3.46  9.3 
Graphite  13.0  23 
Because the neutrons are not moderated to become thermal neutrons in the fast reactor, the prompt neutron lifetime is an order of 10^{−5}–10^{−7} s.
The prompt neutron generation time is equal to the prompt neutron lifetime divided by the effective multiplication factor.
Among the kinetics parameters described here, the denominator is the neutron loss rate obtained in Eqs. (1.1) and (1.5) for effective multiplication factor k _{eff} and prompt neutron lifetime ℓ and the denominator is the neutron generation rate for equation of reactivity ρ and prompt neutron generation time Λ. If k _{eff} = 1, values ℓ and Λ become equal to each other. In the kinetics equations described below, a pair of k _{eff} and ℓ values or a pair of ρ and Λ values should be used.