## Prompt Neutron Lifetime

**Prompt neutron lifetime, l**, is the **average time from a prompt neutron emission** to either **its absorbtion** (fission or radiative capture) or to** its escape** from the system. This parameter is defined in multiplying or also in nonmultiplying systems. In both systems the prompt neutron lifetimes depend strongly on:

- material composition of the system
- multiplying – nonmultiplying system
- system with or without thermalization
- isotopic composition of the system

- geometric configuration of the system
- homogeneous or heterogeneous system
- shape of entire system

- size of the system

In an infinite reactor (without escape) prompt neutron lifetime is the sum of **the slowing down time and the diffusion time**.

**l=t _{s} + t_{d}**

In an infinite thermal reactor **t _{s} << t_{d}** and therefore

**l ≈ t**. The typical prompt neutron lifetime

_{d}**in thermal reactors**is on the order of

**10**. Generally, the longer neutron lifetimes take place in systems in which the neutrons must be thermalized in order to be absorbed.

^{−4}secondSystems in which most of the neutrons are absorbed at higher energies and the neutron thermalization is suppressed (e.g. in fast reactors), have much shorter prompt neutron lifetimes. The typical prompt neutron lifetime **in fast reactors** is on the order of **10 ^{−7} second**.

Slowing Down and Diffision Times for Thermal Neutrons in an Infinite Medium

Source: Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.

Dependencies of asymptotic time period on the reactivity required for different reactor types with different prompt neutron lifetimes. Source: http://www.hindawi.com/journals/ijne/2014/373726/

## Example – Infinite Multiplying System Without Source and Delayed Neutrons

An equation governing the neutron kinetics of the system without source and with the absence of delayed neutrons is **the point kinetics equation** (in certain form). This equation states that the time change of the neutron population is equal to the **excess of neutron production** (by fission) **minus neutron loss** by absorption** in one prompt neutron lifetime**. The role of prompt neutron lifetime is evident. Shorter lifetimes give simply faster responses of multiplying systems.

If there are neutrons in the system at t=0, that is, if n(0) > 0, the solution of this equation gives the simplest form of point kinetics equation (without source and without delayed neutrons):Let us consider that **the prompt neutron lifetime is ~2 x 10 ^{-5}** and k (k

_{∞}– neutron multiplication factor) will be step increased

**by only 0.01%**(

**i.e. 10pcm or ~1.5 cents**), that is k

_{∞}=1.0000 will increase to k

_{∞}=1.0001.

It must be noted such reactivity insertion (10pcm)** is very small** in case of LWRs. The reactivity insertions **of the order of one pcm** are for LWRs **practically unrealizable**. In this case the reactor period will be:

**T = l / (k _{∞ }– 1) = 2 x 10^{-5 }/ (1.0001 – 1) = 0.2s**

**This is a very short period.** In one second the neutron flux (and power) in the reactor would increase by a factor of e^{5} = 2.718^{5}, in 10 seconds the reactor would pass through 50 periods and the power would increase by e^{50} = ……

Furthermore in case of fast reactors in which prompt neutron lifetimes are **of the order of 10 ^{-7} second**, the response of such a small reactivity insertion will be even more unimaginable. In case of 10

^{-7}the period will be:

**T = l / (k _{∞ }– 1) = 10-7 / (1.0001 – 1) = 0.001s**

**Reactors with such a kinetics would be very difficult to control.** **Fortunately this behaviour is not observed** in any multiplying system. Actual reactor periods are observed to be considerably longer than computed above and therefore the nuclear chain reaction can be **controlled more easily**. The longer periods are observed due to the presence of **the delayed neutrons.**

## Interactive chart – Infinite Multiplying System Without Source and Delayed Neutrons

Press the “**clear and run**” button and try to stabilize the power at 90%.

Look at the reactivity insertion you need to insert in order to stabilize the system (of the order to tenth of pcm).

Do you think that such a system is controlable?