## Nuclear Chain Reaction

It was pointed out in the preceding articles that the **neutron-induced fission reaction **is the reaction, in which the incident neutron enters the heavy target nucleus (fissionable nucleus), forming a compound nucleus that is excited to such a **high energy level (E _{excitation} > E_{critical})** that the

**nucleus splits**into two large fission fragments. A large amount of energy is released in the form of radiation and fragment kinetic energy. Moreover and what is for this chapter crucial, the fission process may produce

**2, 3 or more free neutrons**that are capable of inducing

**further fissions**and so on. This sequence of fission events is known as the

**fission chain reaction**and it is of importance in nuclear reactor physics.

**nuclear chain reaction**occurs when one single nuclear reaction causes an average of one or more subsequent nuclear reactions.

**The chain reaction** can take place only in the **proper** **multiplication environment** and only under **proper conditions**. It is obvious, if one neutron causes two further fissions, the number of neutrons in the multiplication system will increase in time and the reactor power (reaction rate) will also increase in time. In order to stabilize such multiplication environment, it is necessary to increase the non-fission neutron absorption in the system (e.g. to **insert control rods**). Moreover, this multiplication environment (the nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is **exponential**.

On the other hand, if one neutron causes** less than one** further fission, the number of neutrons in the multiplication system will decrease in time and the reactor power (reaction rate) will also decrease in time. In order to **sustain the chain reaction**, it is necessary to decrease the non-fission neutron absorption in the system (e.g. to **withdraw control rods**).

In fact, there is always a** competition** for the fission neutrons in the multiplication environment, some neutrons will cause further **fission reaction**, some will be **captured** by fuel materials or non-fuel materials and some will** leak out** of the system.

In order to describe the multiplication system, it is necessary to define the **infinite and finite multiplication factor** of a reactor. The method of calculations of multiplication factors has been developed **in the early years** of nuclear energy and is only applicable to **thermal reactors**, where the bulk of fission reactions occurs at thermal energies. This method well puts into the context all the processes, that are associated with the thermal reactors (e.g. the neutron thermalisation, the neutron diffusion or the fast fission), because the most important neutron-physical processes occur in energy **regions that can be clearly separated from each other**. In short, the calculation of multiplication factor gives a good insight in the processes that occur in each thermal multiplying system.

## Infinite Multiplication Factor – Four Factor Formula

In this section, **the infinite multiplication factor**, which describes all the possible events in the life of a neutron and effectively describes the state of an infinite multiplying system, will be defined.

The required condition for a **stable, self-sustained fission chain reaction** in a multiplying system (in a nuclear reactor) is that **exactly every fission initiate another fission**. The minimum condition is for each nucleus undergoing fission to produce, on the average, at least one neutron that causes fission of another nucleus. Also the number of fissions occurring per unit time (the reaction rate) within the system must be constant.

This condition can be expressed conveniently in terms of **the multiplication factor**. The infinite multiplication factor is the ratio of the **neutrons produced by fission** in one neutron generation to the number of **neutrons lost through absorption** in the preceding neutron generation. This can be expressed mathematically as shown below.

It is obvious** the infinite multiplication factor** in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.

**k**. If the multiplication factor for a multiplying system is_{∞}< 1**less than 1.0**, then the**number of neutrons is decreasing**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_{∞}= 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_{∞}> 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 infinite multiplication factor**can be defined also in terms of the most important

**neutron-physical processes**that occur in the nuclear reactor. For simplicity, we will first consider a multiplying system that is

**infinitely large**, and therefore has

**no neutron leakage**. In the infinite system. There are

**four factors**that are completely independent of the size and shape of the reactor that give the

**inherent multiplication ability**of the fuel and moderator materials without regard to leakage:

_{∞}) may be expressed mathematically in terms of these factors by following equation, usually known as the

**four factor formula**:

**k _{∞} = η.ε.p.f**

In reactor physics, **k _{∞}** or its finite form

**k**is the most significant parameter with regard to reactor control. At any specific power level or condition of the reactor,

_{eff}**k**is kept as near

_{eff}to the value of

**1.0**as possible. At this point in operation, the

**neutron balance**is kept to exactly one neutron completing the life cycle for each original neutron absorbed in the fuel.

## From infinite to finite multiplication factor

The infinite multiplication factor is derived based on the assumption of **no neutrons leak out of the reactor** (i.e. a reactor is infinitely large). But in reality, each nuclear reactor is finite and neutrons can leak out of the reactor core. The multiplication factor that takes **neutron leakage** into account is the **effective multiplication factor** – **k _{eff}**, which is defined as the ratio of the

**neutrons produced by fission**in one neutron generation to the number of

**neutrons lost through absorption and leakage**in the preceding neutron generation.

The effective multiplication factor (**k _{eff}**) may be expressed mathematically in terms of the infinite multiplication factor (k

_{∞}) and two additional factors which account for

**neutron leakage**during neutron thermalisation (

**fast non-leakage probability**) and neutron leakage during neutron diffusion (

**thermal non-leakage probability**) by following equation, usually known as the

**six factor formula**:

**k _{eff} = k_{∞} . P_{f} . P_{t}**

## Operational factors that affect the fission chain reaction in PWRs.

Detailed knowledge of all possible operational factors that may affect the multiplication factor of the system are of importance in the **reactor control**. It was stated the **k _{eff} **is during reactor operation kept as near to the value of

**1.0 as possible**.

**The criticality**of the reactor is influenced by many factors. For illustration, in an extreme case also the presence of human (due to the water, carbon, which are good neutron moderators) near fresh uranium fuel assembly influences the multiplication properties of the assembly.

If any operational factor changes one of the contributing factors to

**k**(

_{eff}**k**), the ratio of 1.0 is not maintained and this change in

_{eff}= η.ε.p.f.P_{f}.P_{t}**k**makes the reactor either

_{eff}**subcritical**or

**supercritical**. Some examples of these operational changes, that may take place in PWRs, are below and are described below: