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 e5 = 2.7185, in 10 seconds the reactor would pass through 50 periods and the power would increase by e50 = ……
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.