## Effective Multiplication Factor

In this section, **the effectiveÂ multiplication factor**, which describes all the possible events in the life of a neutron and effectively describes the state of a finite 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 everyÂ 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 effectiveÂ 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 effectiveÂ multiplication factor** in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequentÂ generation.

**k**. If the multiplication factor for a multiplying system is_{eff}Â < 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_{eff}Â = 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}Â > 1**greater than 1.0**, then the multiplying system produces**more neutrons**Â 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**.

## Six Factor Formula

But **the effectiveÂ multiplication factor** can be defined also in terms of the most important **neutron-physical processes** that occur in the nuclear reactor.

There are **six factors**Â that describeÂ the **inherent multiplication ability** of the system. Four of them are completely independent of the size and shape of the reactor and these are:

See also: Resonance Escape Probability

See also: Thermal Utilization Factor

See also: Reproduction Factor

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

**four factor formula**:

**k _{âˆž}Â = Î·.Îµ.p.f**

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

_{âˆž}) 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_{âˆž} . P_{f} . P_{t}**

See also:Â Thermal Non-leakage Probability

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

_{eff}**k**Â is kept as near to the value of

_{eff}**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.

## Operational factors that affect the multiplication in PWRs.

Detailed knowledge of all possibleÂ 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}**Â (

**k**), the ratio of 1.0 is not maintained and this change in

_{eff}= Î·.Îµ.p.f.P_{f}.P_{t}**k**Â 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 in a separate article in detail.

**change in the control rods position****change in the boron concentration****change in the moderator temperature****change in the fuel temperature****change in the pressure****change in the flow rate****presence of boiling of the coolant****presence of burnable absorbers****fuel burnup**

See also: Operational changes thatÂ affect the multiplication in PWRs.