**nuclear fission chain reaction**is a self-propagating sequence of fission reactions in which neutrons released in fission produce additional fission in at least one other nucleus.

**The chain reaction**can take place only in the

**proper**

**multiplication environment**and only under

**proper conditions**.

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 important in nuclear reactor physics.

**The chain reaction** can take place only in the **proper** **multiplication environment** and only under **proper conditions**. Suppose one neutron causes two further fissions. In that case, the number of neutrons in the multiplication system will increase in time, and the reactor power (reaction rate) will also increase in time. To stabilize such a 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 an exponential system, which 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. It is necessary to decrease the non-fission neutron absorption in the system (e.g., to **withdraw control rods**) to **sustain the chain reaction**.

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 or non-fuel materials, and some will** leak out** of the system.

It is necessary to define the **infinite and finite multiplication factors** of a reactor to describe the multiplication system. The method of calculations of multiplication factors was developed **in the early years** of nuclear energy. It is only applicable to **thermal reactors**, where the bulk of fission reactions occurs at thermal energies. This method puts into context all the processes associated with the thermal reactors (e.g., neutron thermalization, neutron diffusion, or 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 the multiplication factor gives a good insight into 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 necessary condition for a **stable, self-sustained fission chain reaction** in a multiplying system (in a nuclear reactor) is that **exactly every fission initiates another fission**. The minimum condition is for each nucleus undergoing fission to produce, on 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.

The** infinite multiplication factor** in a multiplying system measures 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**. In this 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_{∞}= 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**.

But **the infinite multiplication factor** can also be defined in terms of the most important **neutron-physical processes** 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.** Four factors** are completely independent of the size and shape of the reactor that gives the **inherent multiplication ability** of the fuel and moderator materials without regard to leakage:

The infinite multiplication factor (k_{∞}) may be expressed mathematically in terms of these factors by the 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 about 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 the 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 that **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 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 thermalization (

**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 is of importance in **reactor control**. It was stated the **k _{eff} **is during reactor operation kept as near to the value of

**1.0 as possible**. Many factors influence

**the criticality**of the reactor. For illustration, in an extreme case, the presence of humans (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: