The behavior of average neutrons is governed by certain probability functions called cross sections. These define the probabilities of absorption, scattering, and of fission within the given system.

Assuming an initial flux of thermal neutrons <f> in a thermal reactor core we can calculate the multiplication of neutrons as follows:

Thermal flux ф

Number absorbed into the fuel ZfUel<^/27a = /ф

Number producing fission fф ^tmon/^ш

Number of fast neutrons pro­duced by fission rj f ф

where rj is vlfWon/^fueI

Number of neutrons after fast fission enhancement єг]/ф

Now in the thermal system the neutrons slow down through the resonance absorption region:

Number which escape resonance capture РЩ/Ф = A:,*,ф

Number which escape leakage during slowing down к^ф exp(—B2t)

Having now arrived at thermal energies the neutrons diffuse until they again have a chance to be absorbed in fuel or are nonproductively absorbed in structure or poison material.

Number which escape leakage during diffusion к^ф exp(—B2r)/(l + ZAS2)

Final thermal flux, ке№ф k00^[exp(—BH)/(l + ZAS2)]

(1.1)

where keff is called the neutron multiplication factor. The system is now critical if ketf = 1.0 since the original neutron flux ф is not diminished after an average generation time. If keff is less than one, the neutron population dies away and the system is subcritical, while if k,,ff is greater than one, the system is supercritical and the population continues to grow.

In fast reactors however there is little slowing down so there is very little resonance capture and p is close to unity and there is little leakage at this stage. The fast fission enhancement factor є is of course not a useful con­cept when we are only concerned with fast neutrons. Thus for a fast system

k^vfP/Q + L2B2) (1.2)

Although keff being unity implies criticality in the steady state, a fraction
of the neutrons, delayed neutrons, only arise some time later. Thus on the

short term basis only fceff(l — /3) are immediately available. Thus the prompt criticality condition is fceff(l — /3) = 1.0.

When *efl(l — /3) > 1.0, the delayed neutrons have an insignificant effect since the system is critical on prompt neutrons alone. However, if — /3) < 1.0 and keff = 1.0, then the system is not critical until the delayed neutrons are produced; the delayed neutrons allow the control of the eventually critical system.

The cross sections 2fueI, 27flssitm, 2a, and the resonance escape probability p are all dependent on temperature. The fuel and other system temperatures are involved and their effect is given by temperature coefficients of reactivity a,- where

*etr = кет + Z ai(Ti — Tio)

І

for each temperature 7). Section 1.4 will refer to these effects in detail.