Both terms, **the neutron generation** and **the neutron population** are closely associated with the multiplication of neutrons in the reactor core. The infinite multiplication factor is the ratio of the neutrons produced by fissionin one neutron generation to the number of neutrons lost through absorption in the preceding neutron generation. This can be expressed mathematically as shown below.

## Example:

The difference may be well expressed by following example:

**Example:**

The number of neutrons (**the neutron population**) in the core at time zero is 1000 and k_{∞} = 1.001 (~100 pcm).

Calculate the number of neutrons after 100 generations. Let say, the mean generation time is ~0.1s.

**Solution:**

To calculate the neutron population after 100 neutron generations, we use following equation:

N_{n}=N_{0}. (k_{∞})^{n}

N_{1}=N_{0}.1.001 = **1001 neutrons after one generation**

N_{2}=N_{0}.1.001.1.001 = **1002 neutrons after two generations**

N_{3}=N_{0}.1.001.1.001.1.001 = **1003 neutrons after three generations**

.

.

N_{50}=N_{0}. (k_{∞})^{50} = **1051 neutrons after fifty generations**.

.

.

N_{100}=N_{0}. (k_{∞})^{100} = **1105 neutrons after hundred generations.**

If we consider the mean generation time to be ~0.1s, so the increase from 1000 neutrons the 1105 neutrons occurs within 10 seconds.

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**.