Some of the neutrons participating in the chain reaction are emitted at various times after the fission event When certain fission product nuclides decay by emitting beta particles, the resultant nuclides are unstable, and each of the nuclides emits a neutron immediately after the beta decay T he rate of neutron emission therefore is the same as the rate of beta decay of these “precursor” nuclides, і e, a rate that decreases exponentially with time

I ables 1 1 and 1 2 list the half lives and decay constants [X = (In 2)/T^ = 0 693/T^l of the delayed neutron emitters resulting from the fissioning of 2 3 3 U, 2 3 5 U, and 2 3 9 Pu by thermal neutrons and by fast neutrons, respectively The tables also list the absolute yields of delayed neutrons (number of delayed neutrons per fission emitted by each precursor type) and the relative abundances of the delayed neutrons (number of delayed neutrons emitted by each precursor tvpe divided by the total number of delayed neutrons emitted in a fission process)
fraction can be expressed as the sum of the delayed-neutron fractions for each group of the delayed neutron emitters

m

0 = E ft (15)

1=1

where m is the number of delayed-neutron groups

Table 1 3 lists the values of ft for the fission of 2 3 3 U,

2 3 5 U, and 239Pu by thermal and fast neutrons The table

also lists v, the number of prompt neutrons per fission Values of ft are obtained by multiplying the relative

abundance values in Tables 1 1 and 1 2 by the values of /3 in Table 1 3

The average energy of the delayed neutrons is not the same as the average energy of the prompt neutrons Thus, in any chain-reacting system, the effectiveness of the delayed neutrons in propagating the nuclear fission chain reaction differs from that of the prompt neutrons The factor (3 used in f-q. 1 4 does not take this into account

since (3 is a simple ratio of numbers of neutrons To take the neutron energies into account, replace /3 with

7(3 = effective delayed neutron fraction

= (number of fissions caused by delated neutrons)/ (number of fissions caused by delayed plus prompt neutrons) (1.6)

where 7 is the delayed-neutron effectiveness The value of 7 depends on the chain-reacting system and is gen­erally slightly greater than 1 For the power reactors discussed in this book, it is a good approximation to assume 7=1. Likewise, the delayed-neutron effectiveness for the individual delayed-neutron groups can be assumed to be 1, і e., 7 = 7i = 1.

The basic kinetic equations for a nuclear fission chain reaction in which delayed neutrons are taken into account are obtained by writing the rate of change of the neutron density (n = neutrons/cm3) as a sum of two terms

dn

— = (rate of change of prompt-neutron density)

+ (rate of change of delayed neutron density) m

= 7fk(l -0) — 1] + E A, C, (1 7)

‘ 1=1

where C, 1S the density (number/cm1) of delayed-neutron emitters of the uh group and is the decay constant (fraction decaying/sec) of the іth delayed-neutron emittei group The number of groups, m, is 6 (see 1 ables 1 1 and 1.2). The first term of f q 1 7 is obtained by substituting the multiplication factor for prompt neutrons (Eq. 1.4) for к in Eq. 1 1

The density of each of the delayed neutron-emitting groups, C,, is obtained from the equation

dC

-jjA = (rate of production of і th group of delayed- neutron emitters) — (rate of decay of і th group of delayed-neutron emitters)

kftn

/

The first term is the multiplication factor for delayed neutrons (Eq 1 4) divided by the time interval between generations, or the prompt-neutron lifetime, / If there is a source of neutrons present other than the fissionable isotopes and the fission products that emit neutrons, then a source term must be added to the right-hand side of f q 1.8

Equations 1 7 and 1 8 are the basic neutron kinetics equations They are important in the design of the instrumentation and control systems for nuclear power reactors. In Chap 6 the equations are used to show how transfer-function measurements can yield useful informa­
tion on power-reactor behavior In Chap 7 the equations are shown to be basic to the design of reactor control sy stems