**the critical state**, and the reactivity of the system is zero.

In the preceding chapters, the multiplication factors have been analyzed to understand a **steady-state condition**. Also, the operational changes that influence **the criticality** of the reactor were described. It must be noted and nuclear reactors are not always critical. The **k _{eff}** must be changed for a certain period to start or shut it down, which changes the neutron population in the reactor core. The study of the behavior of the neutron population in a

**noncritical reactor**is called

**reactor kinetics**. From this point of view, the state of a reactor can be distinguished by the following criteria.

**The basic classification** of states of a reactor is according to the **multiplication factor** **as eigenvalue,** which measures the change in the fission neutron population from one neutron generation to the subsequent generation.

**k**. Suppose the multiplication factor for a multiplying system is_{eff}< 1**less than 1.0**. In that case, the**number of neutrons decreases**in time (with the mean generation time), and the chain reaction will never be self-sustaining. This condition is known as**the subcritical state**.

**k**. If the multiplication factor for a multiplying system is_{eff}= 1**equal to 1.0**, then there is**no change in neutron population**in time, and the chain reaction will be**self-sustaining**. This condition is known as**the critical state**.

**k**. If the multiplication factor for a multiplying system is_{eff}> 1**greater than 1.0**, then the multiplying system produces**more neutrons**than are needed to be self-sustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as**the supercritical state**.

The definitions described above are fully applicable to a reactor at **zero power level**, that is, at such a power level.** All thermal considerations** are not important to the chain reaction (**from zero power to 1% of rated power**).

## Criticality of a Power Reactor – Power Defect

**For power reactors,** at power conditions, the reactor can behave **differently** due to the **presence of reactivity feedbacks**. Power reactors are initially started from **hot standby mode** (a subcritical state at 0% of rated power) to **power operation mode** (100% of rated power) by withdrawing control rods and boron dilution from the primary source coolant. During the reactor startup and up to about 1% of rated power, the reactor kinetics is **exponential** as in a zero-power reactor. This is due to the fact all temperature reactivity effects are minimal.

On the other hand, **temperature reactivity effects play a very important role** during further power increase from about 1% up to 100% of rated power. As the neutron population increases, the fuel and the moderator increase their **temperature**, which results in a **decrease in reactivity** of the reactor (almost all reactors are designed to have the **temperature coefficients negative**).

See also: Operational factors that affect the multiplication in PWRs.

**The negative reactivity coefficient** acts against the initial positive reactivity insertion, and this positive reactivity is **offset** by negative reactivity from **temperature feedbacks**. **Positive reactivity must be continuously inserted** (via control rods or chemical shim) to keep the power to be increasing. After each reactivity insertion, the reactor power stabilizes itself on the power level proportionately to the reactivity inserted. The **total amount of feedback reactivity** that must be offset by control rod withdrawal or boron dilution during the power increase is known as **the power defect**. **The power defects** for PWRs, graphite-moderated reactors and sodium-cooled fast reactors are:

- about
**2500pcm**for PWRs, - about
**800pcm**for graphite-moderated reactors - about
**500pcm**for sodium-cooled fast reactors

**The power defects** slightly depend on the fuel burnup because they are determined by the **power coefficient,** which depends on the fuel burnup. **The power coefficient** combines the **Doppler**, **moderator temperature**, and** void coefficients**. It is expressed as a change in reactivity per change in percent power, **Δρ/Δ% power**. The value of the power coefficient is always negative in core life. Still, it is more negative at the end of the cycle primarily due to the **decrease in the moderator temperature coefficient**.

It is logical, as power defects act against power increase, they also act **against power decrease**. When reactor power is decreased **quickly**, as in the **reactor trip**, power defect causes a positive reactivity insertion, and the initial rod insertion must be sufficient to make the reactor **safe subcritical**. Obviously, if the power defect for **PWRs** is about **2500pcm** (about 6 βeff), the control rods must weigh **more than 2500pcm** to achieve the **subcritical condition**. The control rods must weigh more than 2500pcm plus the value of **SDM** (SHUTDOWN MARGIN) to ensure the **safe subcritical condition**. The total weight of control rods is design-specific, but, for example, it may reach about 6000pcm. They must be maintained above a minimum rod height (rods insertion limits) to ensure that the control rods can **safely shut down the reactor**.

## Prompt Criticality

**The basic classification** of states of a reactor may be in some cases **insufficient,** and a finer classification is needed. The finer classification takes into account the two groups of neutrons that are produced in fission.

It is known the fission neutrons are of importance in any chain-reacting system. The fission of fissile nuclei produces 2, 3, or more free neutrons. But **not all neutrons** are released **at the same time** following fission. Even the nature of the creation of these neutrons is different. From this point of view, we usually divide the fission neutrons into two following groups:

**Prompt Neutrons.****Prompt neutrons**are emitted directly from fission, and they are emitted within a very short time of about**10**.^{-14}seconds

**Delayed Neutrons.****Delayed neutrons**are emitted by neutron-rich fission fragments that are called delayed neutron precursors. These precursors usually undergo beta decay, but a small fraction of them are excited enough to undergo neutron emission. The neutron is produced via this type of decay, and this happens**orders of magnitude later**than the emission of the prompt neutrons, which plays an extremely important role in the control of the reactor.

The prompt neutrons are emitted within **10 ^{-14}** and have significantly shorter mean generation time (

**~ 10**) than delayed neutrons (

^{-5}s**~0.1s**) have crucial consequences. The period of ~

**10**is very short and causes a very fast response of the reactor power in case of

^{-5}s**prompt criticality**. The state of a reactor, when the chain reaction is self-sustained only by prompt neutrons, is known as the

**prompt critical state**. This state of the reactor is very

**unstable**because one neutron generation takes only ~

**10**Therefore nuclear reactors

^{-5}s.**must operate**in the

**prompt subcritical**,

**delayed critical**condition. All power reactors are designed to operate in delayed critical conditions and are provided with safety systems to prevent them from ever achieving prompt criticality.

**The prompt critical state is defined as:**

**k**, where the reactivity of a reactor is higher than the effectively delayed neutron fraction. In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the neutron population. The number of neutrons_{eff}> 1; ρ ≥ β_{eff}**exponentially increases**in time (as rapidly as the prompt neutron generation lifetime ~**10**).^{-5}s

**The prompt subcritical and delayed supercritical state is defined as:**

**k**, where the reactivity of a reactor is_{eff}> 1; 0 < ρ < β_{eff}**higher than zero**and**lower than**the effectively delayed neutron fraction. In this case, the production of prompt neutrons alone is**insufficient**to balance neutron losses, and the delayed neutrons are needed to sustain the chain reaction. The neutron population increases, but**much more slowly**(as the mean generation lifetime with delayed neutrons ~0.1 s).

**The prompt subcritical and delayed critical state is defined as:**

**k**, where the reactivity of a reactor is_{eff}= 1; ρ = 0**equal to zero**. In this case, the production of prompt neutrons alone is insufficient to balance neutron losses, and the delayed neutrons are needed to sustain the chain reaction. There is no change in neutron population in time, and the chain reaction will be self-sustaining. This state is the same state as the critical state from basic classification.

**The prompt subcritical and delayed subcritical state is defined as:**

**k**, where the reactivity of a reactor is lower than zero. In this case, the production of all neutrons is insufficient to balance neutron losses, and the chain reaction is not self-sustaining. If the reactor core contains external or internal neutron sources, the reactor is in the state that is usually referred to as the s_{eff}< 1; ρ < 0**ubcritical multiplication**.