Modeling Neutron Transport and Interactions

Abstract

It is essential to know the spatial and energy distributions of the neutrons in a field in a nuclear fission reactor, D–T (or D–D) fusion reactor, or other nuclear reactors populated with large numbers of neutrons. It is obvious why the spatial distribution should be known, and because neutron reactions vary widely with energy, the energy distribution is also a critical parameter. The neutron energy distribution is often called the neutron spectrum. The neutron distribution satisfies transport equation. It is usually difficult to solve this equation, and often approximated equation so-called diffusion equation is solved instead. In this chapter only overview of transport equation and diffusion equation of neutrons is presented, and methods for solving these equations are presented in the following sections.

Problems

Problem 2.1

If a 1 cubic centimeter section of a reactor has a macroscopic fission cross section of 0.1 cm⁻¹, and if the thermal neutron flux is 10¹³ neutrons/cm²-sec, what is the fission rate in that cubic centimeter?

Problem 2.2

A reactor operating at a flux level of 3 × 10¹³ neutrons/cm²-sec contains 1020 atoms of uranium-235 per cm3. The reaction rate is 1.29 × 10¹² fission/cm³. Calculate Σ_f and σ_f, assuming 1 barn = 10⁻²⁴ cm².

Problem 2.3

How many collisions are required to slow a neutron from energy of 2 MeV to a thermal energy of 0.025 eV and using water as the moderator. Water has a value of 0.948 for M.

Problem 2.4

If the average fractional energy loss per collision in hydrogen is 0.63, what will be the energy of a 2 MeV neutron after (a) 5 collisions and (b) 10 collisions?

Problem 2.5

A block of aluminum has a density of 2.699 g/cm³. If the gram atomic weight of aluminum is 26.9815 g, calculate the atom density of the aluminum.

Problem 2.6

How long on average for a given nuclei to suffer a neutron interaction?

Problem 2.7

Find the macroscopic thermal neutron absorption cross section for iron, which has a density of 7.86 g/cm³. The microscopic cross section for absorption of iron is 2.56 barns, and the gram atomic weight is 55.847 g.

Problem 2.8

Calculate the reproduction factor for a reactor that uses 10% enriched uranium fuel. The microscopic absorption cross section for uranium-235 is 694 barns. The cross section for uranium-238 is 2.71 barns. The microscopic fission cross section for uranium-235 is 582 barns. The atom density of uranium-235 is 4.83 × 10²¹ atoms/cm³. The atom density of uranium-238 is 4.35 × 10²² atoms/cm³. n is 2.42.

Problem 2.9

Calculate the thermal utilization factor for a homogeneous reactor. The macroscopic absorption cross section of the fuel is 0.3020 cm⁻¹, the macroscopic absorption cross section of the moderator is 0.0104 cm⁻¹, and the macroscopic absorption cross section of the poison is 0.0118 cm⁻¹.

Problem 2.10

Determine the infinite multiplication factor k_∞ for a uniform mixture of uranium-235 and beryllium oxide in the atomic or molecular ratio of 1 to 10,000. The value of absorption cross section for beryllium is σ_a = 0.010 (barn) and for uranium is σ_U = 683 (barn). Assume the resonance escape probability and the fast-fission factor may be taken to be unity and ratio of average number of neutrons liberated per neutron absorbed for uranium-235 at thermal (2200 m/sec) is η = 2.06.