NEUTRON LIFE CYCLE
DOE-HDBK-1019/2-93
Reactor Theory (Nuclear Parameters)
Infinite Multiplication Factor, k
Not all of the neutrons produced by fission will have the opportunity to cause new fissions
because some neutrons will be absorbed by non-fissionable material. Some will be absorbed
parasitically in fissionable material and will not cause fission, and others will leak out of the
reactor. For the maintenance of a self-sustaining chain reaction, however, it is not necessary
that every neutron produced in fission initiate another fission. The minimum condition is for
each nucleus undergoing fission to produce, on the average, at least one neutron that causes
fission of another nucleus. This condition is conveniently expressed in terms of a multiplication
factor.
The number of neutrons absorbed or leaking out of the reactor will determine the value of this
multiplication factor, and will also determine whether a new generation of neutrons is larger,
smaller, or the same size as the preceding generation. Any reactor of a finite size will have
neutrons leak out of it. Generally, the larger the reactor, the lower the fraction of neutron
leakage. For simplicity, we will first consider a reactor that is infinitely large, and therefore
has no neutron leakage. A measure of the increase or decrease in neutron flux in an infinite
reactor is the infinite multiplication factor, k
. The infinite multiplication factor is the ratio of
the neutrons produced by fission in one generation to the number of neutrons lost through
absorption in the preceding generation. This can be expressed mathematically as shown below.
k
neutron production from fission in one generation
neutron absorption in the preceding generation
Four Factor Formula
A group of fast neutrons produced by fission can enter into several reactions. Some of these
reactions reduce the size of the neutron group while other reactions allow the group to increase
in size or produce a second generation. There are four factors that are completely independent
of the size and shape of the reactor that give the inherent multiplication ability of the fuel and
moderator materials without regard to leakage. This four factor formula accurately represents the
infinite multiplication factor as shown in the equation below.
k
= p f h
where:
= fast fission factor
p = resonance escape probability
f = thermal utilization factor
h = reproduction factor
Each of these four factors, which are explained in the following subsections, represents a process that
adds to or subtracts from the initial neutron group produced in a generation by fission.
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