Neutron transport

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 time and the 3 dimensions of space, and the variables of energy span several orders of magnitude (from fractions of meV to several MeV). Modern solutions use either discrete ordinates or Monte Carlo methods, or even a hybrid of both.

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:[1]

${\displaystyle \left({\frac {1}{v(E)}}{\frac {\partial }{\partial t}}+\mathbf {\hat {\Omega }} \cdot \nabla +\Sigma _{t}(\mathbf {r} ,E,t)\right)\psi (\mathbf {r} ,E,\mathbf {\hat {\Omega }} ,t)=\quad }$

${\displaystyle \quad {\frac {\chi _{p}\left(E\right)}{4\pi }}\int _{0}^{\infty }dE^{\prime }\nu _{p}\left(E^{\prime }\right)\Sigma _{f}\left(\mathbf {r} ,E^{\prime },t\right)\phi \left(\mathbf {r} ,E^{\prime },t\right)+\sum _{i=1}^{N}{\frac {\chi _{di}\left(E\right)}{4\pi }}\lambda _{i}C_{i}\left(\mathbf {r} ,t\right)+\quad }$

${\displaystyle \quad \int _{4\pi }d\Omega ^{\prime }\int _{0}^{\infty }dE^{\prime }\,\Sigma _{s}(\mathbf {r} ,E^{\prime }\rightarrow E,\mathbf {\hat {\Omega }} ^{\prime }\rightarrow \mathbf {\hat {\Omega }} ,t)\psi (\mathbf {r} ,E^{\prime },\mathbf {{\hat {\Omega }}^{\prime }} ,t)+s(\mathbf {r} ,E,\mathbf {\hat {\Omega }} ,t)}$

Where:

Symbol	Meaning	Comments
${\displaystyle \mathbf {r} }$	Position vector (i.e. x,y,z)
${\displaystyle E}$	Energy
${\displaystyle \mathbf {\hat {\Omega }} ={\frac {\mathbf {v} (E)}{|\mathbf {v} (E)|}}={\frac {\mathbf {v} (E)}{v(E)}}}$	Unit vector (solid angle) in direction of motion
${\displaystyle t}$	Time
${\displaystyle \mathbf {v} (E)}$	Neutron velocity vector
${\displaystyle \psi (\mathbf {r} ,E,\mathbf {\hat {\Omega }} ,t)dr\,dE\,d\Omega }$	Angular neutron flux Amount of neutron track length in a differential volume ${\displaystyle dr}$ about ${\displaystyle r}$, associated with particles of a differential energy in ${\displaystyle dE}$ about ${\displaystyle E}$, moving in a differential solid angle in ${\displaystyle d\Omega }$ about ${\displaystyle \mathbf {\hat {\Omega }} }$, at time ${\displaystyle t}$.	Note integrating over all angles yields scalar neutron flux ${\displaystyle \phi \ =\ \int _{4\pi }d\Omega \psi }$
${\displaystyle \phi (\mathbf {r} ,E,t)dr\,dE}$	Scalar neutron flux Amount of neutron track length in a differential volume ${\displaystyle dr}$ about ${\displaystyle r}$, associated with particles of a differential energy in ${\displaystyle dE}$ about ${\displaystyle E}$, at time ${\displaystyle t}$.
${\displaystyle \nu _{p}}$	Average number of neutrons produced per fission (e.g., 2.43 for U-235).[2]
${\displaystyle \chi _{p}(E)}$	Probability density function for neutrons of exit energy ${\displaystyle E}$ from all neutrons produced by fission
${\displaystyle \chi _{di}(E)}$	Probability density function for neutrons of exit energy ${\displaystyle E}$ from all neutrons produced by delayed neutron precursors
${\displaystyle \Sigma _{t}(\mathbf {r} ,E,t)}$	Macroscopic total cross section, which includes all possible interactions
${\displaystyle \Sigma _{f}(\mathbf {r} ,E^{\prime },t)}$	Macroscopic fission cross section, which includes all fission interactions in ${\displaystyle dE^{\prime }}$ about ${\displaystyle E^{\prime }}$
${\displaystyle \Sigma _{s}(\mathbf {r} ,E'\rightarrow E,\mathbf {\hat {\Omega }} '\rightarrow \mathbf {\hat {\Omega }} ,t)dE^{\prime }d\Omega ^{\prime }}$	Double differential scattering cross section Characterizes scattering of a neutron from an incident energy ${\displaystyle E^{\prime }}$ in ${\displaystyle dE^{\prime }}$ and direction ${\displaystyle \mathbf {\hat {\Omega ^{\prime }}} }$ in ${\displaystyle d\Omega ^{\prime }}$ to a final energy ${\displaystyle E}$ and direction ${\displaystyle \mathbf {\hat {\Omega }} }$.
${\displaystyle N}$	Number of delayed neutron precursors
${\displaystyle \lambda _{i}}$	Decay constant for precursor i
${\displaystyle C_{i}\left(\mathbf {r} ,t\right)}$	Total number of precursor i in ${\displaystyle \mathbf {r} }$ at time ${\displaystyle t}$
${\displaystyle s(\mathbf {r} ,E,\mathbf {\hat {\Omega }} ,t)}$	Source term

The transport equation can be applied to a given part of phase space (time t, energy E, location ${\displaystyle \mathbf {r} }$, and direction of travel ${\displaystyle \mathbf {\hat {\Omega }} }$). 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 (i.e., unstable nuclei which undergo neutron decay). 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 ${\displaystyle \phi (\mathbf {r} ,E)}$, since that will allow for the calculation of reaction rates, which are of primary interest in shielding and dosimetry studies.

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

Both fixed-source and criticality calculations can be solved using deterministic methods or stochastic methods. In deterministic methods the transport equation (or an approximation of it, such as diffusion theory) 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.

• Nuclear reactor
• Boltzmann equation
• TINTE
• Neutron scattering
• Monte Carlo N-Particle Transport Code

• Lewis, E., & Miller, W. (1993). Computational Methods of Neutron Transport. American Nuclear Society. ISBN 0-89448-452-4.
• Duderstadt, J., & Hamilton, L. (1976). Nuclear Reactor Analysis. New York: Wiley. ISBN 0-471-22363-8.
• Marchuk, G. I., & V. I. Lebedev (1986). Numerical Methods in the Theory of Neutron Transport. Taylor & Francis. p. 123. ISBN 978-3-7186-0182-0.

• ANSWERS Software Service website
• LANL MCNP6 website
• LANL MCNPX website
• VTT Serpent website
• OpenMC website
• MIT CRPG OpenMOC website
• TRIPOLI-4 website