Neutron transport

Neutron transport refers to determination of the neutron flux, observed from an efflux of neutrons from a neutron source. In relation to this, beams of free neutrons can be obtained via extraction from neutron sources.

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Background

Neutron sources

The source can be considered as a volume filled with a very low pressure neutron gas (density up to 10²⁴ n/m³), which has an energy distribution corresponding to the temperature of the volume. By inserting a beam tube into the volume, a current of neutrons can be observed flowing out through the tube, much like water flowing out from a tube that is stuck into a volume of wet sand. The current density (in n/m²s) depends on the solid angle formed by the aperture and the length of the tube, and also on the reflecting properties of the inner wall of the tube. Neutrons are indeed reflected at a wall, but usually at an extremely low rate, which is a function of the wall's nature and surface quality, the neutron energy, and the angle of incidence.

Characteristics of neutron beams

The moving neutron has a kinetic energy E = ½·m·v² (m = mass, v = velocity). The SI unit for energy is the joule (J), but for particles the energy unit of electronvolts (eV) is usually used (1 J = 6.2410¹⁸ eV). The energy can also be expressed in temperature (unit: kelvins) by the relation E = k·T (k = 1.38·10^-23 J/K). Furthermore, a moving particle can be considered a matter wave with the de Broglie wavelength λ = h/(2k·T·m)^1/2 (h = 6.6·10^-34 J·s). The corresponding wavelength for the neutron is generally expressed in nanometers (nm) or sometimes in angstroms (Å), where 1 Å = 0.1 nm.

A thermal neutron, for instance, with a temperature of 300 K (corresponding to E = 0.025 eV) travels with a speed of 2200 m/s, and has a wavelength of 0.17 nm.

Neutron transport calculations

Usually the neutron transport calculations can be divided into two categories: shielding and criticality search.

Shielding

In the case of shielding calculation, there are three important things which must be modeled: neutron source, shield, and detector. Shielding calculation usually yields the neutron flux in the region of space occupied by the detector. Based on the flux in the detector, the shield thickness or source strength are optimized.

Criticality search

In criticality calculation, a fissile or fissionable material is part of the modeled geometry. Neutron source is the part of the modeled geometry which contains the fissile or fissionable material. The spacial dependence of neutron source intensity is proportional to the flux, which dictates the fission reaction density, hence spacial dependence of neutron source is proportional to the fissions induced by the neutrons. The parameter which is the result of criticality calculation is K-effective (K_eff). This is a parameter which reflects the time dependence on neutron density in a multi-lying medium.

• K_eff < 1, if the neutron density is decreasing as time passes;
• K_eff = 1, if the neutron density remains unchanged; and
• K_eff > 1, if the neutron density is increasing with time.

In the case of a nuclear reactor, neutron density and power density are synonymous, hence during reactor start-up K_eff > 1, during reactor operation K_eff = 1 and K_eff < 1 at reactor shutdown.

Methodology

Both shielding and criticality calculations can be done using deterministic methods such as diffusion theory, or using stochastic methods such as Monte Carlo. Deterministic methods usually involve multi-group approaches while Monte Carlo can work with multi-group and continuous energy cross-section libraries. Multi-group calculations are usually iterative, because the group constants are calculated using flux-energy profiles, which are determined as the result of the neutron transport calculation.

References