Neutron transport


Neutron transport is the study of the motions and interactions of neutrons with materials. Nuclear scientists and engineers often need to know where neutrons are in an apparatus, what direction they are going, and how quickly they are moving. It is commonly used to determine the behavior of nuclear reactor cores and experimental or industrial neutron beams. Neutron transport is a type of radiative transport.

Background

Neutron transport has roots in the Boltzmann equation, which was used in the 1800s to study the kinetic theory of gases. It did not receive large-scale development until the invention of chain-reacting nuclear reactors in the 1940s. As neutron distributions came under detailed scrutiny, elegant approximations and analytic solutions were found in simple geometries. However, as computational power has increased, numerical approaches to neutron transport have become prevalent. Today, with massively parallel computers, neutron transport is still under very active development in academia and research institutions throughout the world. It remains a computationally challenging problem since it depends on 3-dimensions of space, time, and the variables of energy span several decades. Modern solutions use either discrete-ordinates or Monte Carlo methods, or even a hybrid of both.

Neutron transport equation

The neutron transport equation is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated as follows:
Where:
SymbolMeaningComments
Position vector
Energy
Unit vector in direction of motion
Time
Neutron velocity vector
Angular neutron flux
Amount of neutron track length in a differential volume about, associated with particles of a differential energy in about, moving in a differential solid angle in about, at time.		Note integrating over all angles yields scalar neutron flux
Scalar neutron flux
Amount of neutron track length in a differential volume about, associated with particles of a differential energy in about, at time.
Average number of neutrons produced per fission.
Probability density function for neutrons of exit energy from all neutrons produced by fission
Probability density function for neutrons of exit energy from all neutrons produced by delayed neutron precursors
Macroscopic total cross section, which includes all possible interactions
Macroscopic fission cross section, which includes all fission interactions in about
Double differential scattering cross section
Characterizes scattering of a neutron from an incident energy in and direction in to a final energy and direction.
Number of delayed neutron precursors
Decay constant for precursor i
Total number of precursor i in at time
Source term

The transport equation can be applied to a given part of phase space. The first term represents the time rate of change of neutrons in the system. The second terms describes the movement of neutrons into or out of the volume of space of interest. The third term accounts for all neutrons that have a collision in that phase space. The first term on the right hand side is the production of neutrons in this phase space due to fission, while the second term on the right hand side is the production of neutrons in this phase space due to delayed neutron precursors. The third term on the right hand side is in-scattering, these are neutrons that enter this area of phase space as a result of scattering interactions in another. The fourth term on the right is a generic source. The equation is usually solved to find, since that will allow for the calculation of reaction rates, which are of primary interest in shielding and dosimetry studies.

Types of neutron transport calculations

Several basic types of neutron transport problems exist, depending on the type of problem being solved.

Fixed source

A fixed source calculation involves imposing a known neutron source on a medium and determining the resulting neutron distribution throughout the problem. This type of problem is particularly useful for shielding calculations, where a designer would like to minimize the neutron dose outside of a shield while using the least amount of shielding material. For instance, a spent nuclear fuel cask requires shielding calculations to determine how much concrete and steel is needed to safely protect the truck driver who is shipping it.

Criticality

is the process through which a nucleus splits into smaller atoms. If fission is occurring, it is often of interest to know the asymptotic behavior of the system. A reactor is called “critical” if the chain reaction is self-sustaining and time-independent. If the system is not in equilibrium the asymptotic neutron distribution, or the fundamental mode, will grow or decay exponentially over time.
Criticality calculations are used to analyze steady-state multiplying media, such as a critical nuclear reactor. The loss terms and the source terms are proportional to the neutron flux, contrasting with fixed-source problems where the source is independent of the flux. In these calculations, the presumption of time invariance requires that neutron production exactly equals neutron loss.
Since this criticality can only be achieved by very fine manipulations of the geometry, it is unlikely that the modeled geometry will be truly critical. To allow some flexibility in the way models are set up, these problems are formulated as eigenvalue problems, where one parameter is artificially modified until criticality is reached. The most common formulations are the time-absorption and the multiplication eigenvalues, also known as the alpha and k eigenvalues. The alpha and k are the tunable quantities.
K-eigenvalue problems are the most common in nuclear reactor analysis. The number of neutrons produced per fission is multiplicatively modified by the dominant eigenvalue. The resulting value of this eigenvalue reflects the time dependence of the neutron density in a multiplying medium.
In the case of a nuclear reactor, neutron flux and power density are proportional, hence during reactor start-up keff > 1, during reactor operation keff = 1 and keff < 1 at reactor shutdown.

Computational methods

Both fixed-source and criticality calculations can be solved using deterministic methods or stochastic methods. In deterministic methods the transport equation is solved as a differential equation. In stochastic methods such as Monte Carlo discrete particle histories are tracked and averaged in a random walk directed by measured interaction probabilities. 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.

Discretization in deterministic methods

To numerically solve the transport equation using algebraic equations on a computer, the spatial, angular, energy, and time variables must be discretized.

Probabilistic codes