Interactions between neutrons and nuclei in nuclear reactors can be classified broadly into scattering and absorption reactions as shown in Table 2.1. Scattering is further classified into elastic scattering in which the kinetic energy is conserved before and after the reaction, and inelastic scattering in which a part of the kinetic energy is used in exciting a target nucleus. In absorption, the main reactions are capture, fission, charged-particle emission, and neutron emission. Thus, the microscopic cross sections of the total, scattering, and absorption reactions are given by

total cross section: at(E) = <rs(E) + aa(E) (2.10)

scattering cross section: а5(Е) = oe(E) + ^П(Я) (2.11)

absorption cross section:

(2 12!

tfa(£) = ay(£) + Vf(E) + op{E) + oa(E) + G(n,2n)

It is useful to discuss a general energy dependence of these microscopic cross sections. The neutron energy range to be considered in design of nuclear reactors is from the Maxwellian distribution of the thermal neutrons at room temperature to the fission spectrum of the prompt neutrons. Most nuclear design codes handle the range of 10_5 eV-10 MeV. In this energy range, the

microscopic cross sections introduced in Eqs. (2.10), (2.11), and (2.12) behave as shown in Fig. 2.2.

The elastic scattering cross section is mostly constant in all the energies except for the MeV region. Meanwhile, in inelastic scattering, the incident neutron should have sufficient kinetic energy to place the target nucleus in its excited state. Hence, the inelastic scattering cross section is zero up to some threshold energy of several MeV. Fast neutrons can be moderated by inelastic scattering with heavy nuclides, but by elastic scattering with light nuclides below threshold energies of the heavy nuclides.

Most absorption cross sections including the fission cross section appear as a straight line with a slope of —1/2 on a log-log scale. This means that the absorption cross sections are inversely proportional to the neutron speed (1/u law) and therefore increase as the neutron energy decreases. Using such large fission cross sections at low neutron energies and thermal neutrons in the Maxwellian distribution make it possible that natural or low-enrichment ura­nium fueled reactors reach a critical state. The current thermal reactors, represented by LWRs, use the characteristics of the cross section.

For heavy nuclides such as fuel materials, many resonances are observed in elastic scattering and absorption cross section as shown in Fig. 2.3. The widths of the resonances broaden as fuel temperature increases. This is called the Doppler effect. The width broadening facilitates resonance absorption of neu­trons under moderation. Most low-enrichment uranium fuel is composed of

fertile 238U and thermal neutrons escaping from the resonance of capture reaction induce fissions for the next generation.

Hence, a rise in fuel temperature leads to a decrease in resonance escape probability of moderated neutrons and then fission events in the reactor decrease with thermal neutrons. Such a mechanism is called negative temper­ature feedback. The temperature dependence is not described in the Boltzmann equation of Eq. (2.1), but reflected in the cross sections of the equation.