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how are they used to control the reactor
I am not an expert so if I made mistakes I apologize and I hope someone will correct me.
The idea is to be slightly subcritical with prompt neutrons and reach a steady power with delayed neutrons.
Imagine a reactor running at stable power k=1. In that case one fission will produce neutrons and this neutrons will produce exactly one other fission.
So 100 000 fissions will give you exactly an other new 100 000 fissions (k=1, steady power). These fissions give you "instantaneously" some prompt neutrons and fission products ( I note Np the number of prompt neutrons generated by 100 000 fissions). Later some of the fissions products will give you the delayed neutrons ( I note Nd the number of delayed neutrons generated ). The total amount of neutrons generated by these 100 000 fissions is N. We have N = Np + Nd
The delayed neutron fraction is defined by B = Nd / N
For example, we imagine that the reactor is a thermal reactor with only pure U235 as a fissile material. The total number of neutrons released by one fission
is v = 2.42
So the total amount of neutron released by 100 000 fissions is roughly 242 000 neutrons. For U235 you have B = 0.679 % = 679 pcm
The number of delayed neutrons produced by this 100 000 fissions is roughly Nd = B*N = 1643 delayed neutrons ( of course the precision is not at one neutron but let's forget it)
And the number of prompt neutrons : Np = N-Nd = 240 357 prompt neutrons.
The delayed neutrons are emitted with not the same energy of the prompt neutrons, so their probability of inducing fission is different than the prompt neutrons, we manage this with the "effective fraction of delayed neutron".
The effective fraction of delayed neutron is noted Beff.
Beff depends of your reactor design. Let say that we have Beff = 0.69 % = 690 pcm. The delayed neutrons will induce Beff = 0.69 % of the new 100 000 fissions. So they will induce 690 new fissions.
The prompt neutrons will induce 99.31 % of the new fissions so they will induce 99 310 fissions, this is less than 100 000 so the reactor is subcritical with only prompt neutrons, and the delayed neutrons fill the gap until 100 000 fissions.
When you change the power of your reactor you must always be subcritical with prompt neutrons in order to control the reactor.