As can be seen from the solution of the exact point kinetics equation, any reactivity insertion (**ρ** < ** β**) causes at first a sharp change in prompt neutrons population, and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response is a result of the small value of prompt neutron generation time in the denominator of the point kinetics equation.

Suppose we are interested in **long-term behavior** (asymptotic period) and not interested in the details of the prompt jump. In that case, we can simplify the point kinetics equations by assuming that the **prompt jump takes place instantaneously** in response to any reactivity change. This approximation is known as the **Prompt Jump Approximation (PJA)**. Due to prompt neutrons, the rapid power change is neglected, corresponding to taking ** dn/dt |_{0} = 0** in the point kinetics equations. That means the point kinetics equations are as follows:

From the equation for neutron flux and the assumption that the delayed neutron precursor population does not respond instantaneously to a change in reactivity (i.e., C_{i,1} = C_{i,2}), it can be derived that the ratio of the neutron population just after and before the reactivity change is equal to:

The prompt-jump approximation is usually valid for smaller reactivity insertion, for example, for **ρ < 0.5 β. **It is usually used with another simplification and the

**one delayed precursor group approximation**.

## Prompt Jump Approximation with One Group of Delayed Neutrons

In the previous section, we have simplified the point kinetics equation using **prompt jump approximation (PJA)**. This eliminated the fast time scale due to prompt neutrons. This section considers that delayed neutrons are produced only by **one group of precursors** with the same decay constant (averaged) and delayed neutron fraction. Point kinetics equation using PJA and one group of delayed neutrons becomes:

This simplification then leads to:

Assuming that the reactivity is constant and n_{1}/n_{0} can be determined from the prompt jump formula, this equation leads to a very simple formula: