**The effective multiplication factor**(

**k**) may be expressed mathematically in terms of the infinite multiplication factor (k

_{eff}_{∞}) 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}** =

**= η.ε.p.f . P**

_{f}. P_{t}In this section, **the effective 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 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 effective 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** effective multiplication factor** in a multiplying system 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**.

## Six-Factor Formula – Theory

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

Six-factors describe 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: Fast Fission Factor

See also: Resonance Escape Probability

See also: Thermal Utilization Factor

See also: Reproduction Factor

These factors constitute the infinite multiplication factor (k_{∞}), which 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**

**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}**

See also: Fast Non-leakage Probability

See also: Thermal Non-leakage Probability

In reactor physics, **k _{eff}** is the most significant parameter with regard to reactor control. At any specific power level or condition of the reactor,

**k**is kept as near to the value of

_{eff}**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.

**Neutron Life Cycle with k _{eff} = 1**

## Operational factors that affect the multiplication 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 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 that affect the multiplication in PWRs.