The neutron energy cycle of the one-group reactor-physics model to be used is shown in Fig. 3.28.

Consider a unit volume containing Nm atoms of fissile material (M3U, 233 U, 239 Pu, or 241 Pu) of thermal absorption cross section am, and Ng atoms of fertile material U or 232 Th) of thermal absorption cross section ag. For this model we shall develop expressions for the number of neutrons produced or absorbed at any point in the neutron cycle per unit volume per unit time. Assume that the fissionable material absorbs only thermal neutrons. The rate of

absorption of neutrons by fissionable material is Мтотф, where ф is the thermal-neutron flux. The resulting fissions produce fast neutrons at a rate гітМтотф.

Those fast neutrons that have energies greater than about 1 MeV may cause a limited amount of fission of fertile material. To account for this, the reactor designer usually specifies a quantity e, called the fast-fission factor, which is defined as the ratio of the net rate of production of fast neutrons to the rate of production of fast neutrons by thermal fission. The fraction e — 1 of the fast neutrons comes from fission of fertile material with fast neutrons; e — 1 may be of the order of a few hundredths in a thermal power reactor. The net production rate of fast neutrons from fission is erітМтотф.

As the fast neutrons undergo scattering collisions, they are degraded in energy and also tend to diffuse toward the outer surface of the reactor where they may escape. When those remaining in the reactor are degraded to energies of a few kilovolts, they have a good chance of being absorbed in the fertile material, which has large resonances in its absorption cross section in the kilovolt range. This resonance absorption is important in producing new fissionable material, i. e., 239Pu from 238U and 233U from 232Th.

The fraction of the fast neutrons that do not escape from the reactor as they degrade from fission to resonance energy depends on the size and moderating properties of the reactor. This fraction is denoted as Pu the fission-to-resonance nonleakage probability. Hence, the rate at which fast neutrons degrade into the resonance region is eijm. Wm<7m0f,1.

These resonance neutrons may be captured in fertile material or may escape resonance absorption by undergoing elastic collisions with the moderator, which degrades them to energies below resonance. A quantity p is defined as the fraction of the resonance neutrons that are not captured but are degraded to lower energies and is called the resonance escape probability. The fraction p is a function of the relative proportions and physical arrangement of the moderator and fertile material. Hence, erimNmom4>Px{ — p) neutrons undergo resonance absorption per unit volume per unit time, and et]mNmam<l>Pxp are degraded to lower energies.

Of the latter, some diffuse to outer surfaces and escape, but the fraction P2 remains in the reactor as thermal neutrons; P2 is called the resonance-to-therma! nonleakage probability. Finally, the neutrons complete an energy cycle as ецтМтотфР1рР2 neutrons reach thermal energy per unit volume per unit time. The product PP2 is the fission-to-thermal nonleakage probability, which we shall denote as Ль-

Thermal neutrons are consumed by (1) absorption in fissionable material at a rate Мтотф (2) absorption in nonfissionable material at a rate Nto$ (і Ф m) and (3) leakage at a rate ИВгф. The absorption in fissionable material leads to regeneration of fission neutrons, as shown in Fig. 3.28.