Neutron transport

Neutron transport (also known as neutronics) is the study of the motions and interactions of neutrons with materials. The neutron transport equation models the radiative transfer of neutrons and is commonly used to determine the behavior of nuclear reactor cores and experimental or industrial neutron beams.

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-reaction nuclear reactors in the 1940s. As neutron distribution came under detailed scrutiny, elegant approximations and analytic solutions were found in simple geometries. However, as computational power increased, numerical approaches to neutron transport have become prevalent. Using massively parallel computers, neutron transport remains under active development in academia and research institutions throughout the world. It is computationally challenging since it depends on time and the three dimensions of space, and the energy variable spans about nine orders of magnitude, from thermal energies of a few tens of meV to fission-neutron energies of several MeV (up to about 10 MeV). Modern solutions use discrete ordinates, Monte Carlo methods, or 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:

${\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({\mathbf {r}},E\right)}{4\pi }}\left[1-{\tilde {\beta }}({\mathbf {r}})\right]\int _{0}^{\infty }\mathrm {d} E^{\prime }\nu _{p}\left({\mathbf {r}},E^{\prime }\right)\Sigma _{f}\left(\mathbf {r} ,E^{\prime },t\right)\phi \left(\mathbf {r} ,E^{\prime },t\right)}$

${\displaystyle \quad +\sum _{i=1}^{N}{\frac {\chi _{di}\left({\mathbf {r}},E\right)}{4\pi }}\lambda _{i}C_{i}\left(\mathbf {r} ,t\right)\quad }$

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

${\displaystyle \quad +s(\mathbf {r} ,E,\mathbf {\hat {\Omega }} ,t)}$

Where the equation for the precursors of delayed neutrons is as follows:

${\displaystyle {\frac {\partial C_{i}}{\partial t}}({\mathbf {r}},t)={\tilde {\beta }}_{i}({\mathbf {r}})\int _{0}^{\infty }dE\nu _{p}({\mathbf {r}},E)\Sigma _{f}({\mathbf {r}},E,t)\phi ({\mathbf {r}},E,t)-\lambda _{i}({\mathbf {r}})C_{i}({\mathbf {r}},t),}$

${\displaystyle \quad i=1,...,N}$

All notations are as follows:

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
${\displaystyle \psi (\mathbf {r} ,E,\mathbf {\hat {\Omega }} ,t)\mathrm {d} r\,\mathrm {d} \!E\,\mathrm {d} \Omega }$	Angular neutron flux Amount of neutron track length in a differential volume ${\displaystyle \mathrm {d} r}$ about ${\displaystyle r}$, associated with particles of a differential energy in ${\displaystyle \mathrm {d} E}$ about ${\displaystyle E}$, moving in a differential solid angle in ${\displaystyle \mathrm {d} \Omega }$ about ${\displaystyle \mathbf {\hat {\Omega }} ,}$ at time ${\displaystyle t.}$	Note integrating over all angles yields scalar neutron flux ${\displaystyle \phi \ =\ \int _{4\pi }\mathrm {d} \Omega \psi }$
${\displaystyle \phi (\mathbf {r} ,E,t)\mathrm {d} r\,\mathrm {d} E}$	Scalar neutron flux Amount of neutron track length in a differential volume ${\displaystyle \mathrm {d} r}$ about ${\displaystyle r}$, associated with particles of a differential energy in ${\displaystyle \mathrm {d} E}$ about ${\displaystyle E}$, at time ${\displaystyle t.}$
${\displaystyle \nu _{p}({\mathbf {r}},E)}$	Average number of neutrons produced per fission at point ${\displaystyle r}$ with energy E, including both prompt and delayed neutrons. E.g., it is 2.43 for thermal neutrons (0.0253 eV) in U-235 at 293 K.
${\displaystyle \chi _{p}(E)}$	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 ${\displaystyle \mathrm {d} E^{\prime }}$ about ${\displaystyle E^{\prime }}$
${\displaystyle \Sigma _{s}\!\!\left(\mathbf {r} ,E'\rightarrow E,\mathbf {\hat {\Omega }} '\rightarrow \mathbf {\hat {\Omega }} ,t\right)\mathrm {d} E^{\prime }\mathrm {d} \Omega ^{\prime }}$	Double differential scattering cross section Characterizes scattering of a neutron from an incident energy ${\displaystyle E^{\prime }}$ in ${\displaystyle \mathrm {d} E^{\prime }}$ and direction ${\displaystyle \mathbf {{\hat {\Omega }}^{\prime }} }$ in ${\displaystyle \mathrm {d} \Omega ^{\prime }}$ 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
${\displaystyle {\tilde {\beta }}_{i}({\mathbf {r}})}$	Weighted average of delayed neutrons: ${\displaystyle {\tilde {\beta }}_{i}({\mathbf {r}})={\frac {\int _{0}^{\infty }\beta _{i}({\mathbf {r}},E)\nu _{p}({\mathbf {r}},E)\Sigma _{f}({\mathbf {r}},E)\phi ({\mathbf {r}},E)dE}{\int _{0}^{\infty }\nu _{p}({\mathbf {r}},E)\Sigma _{f}({\mathbf {r}},E)\phi ({\mathbf {r}},E)dE}}}$ Where ${\displaystyle \beta _{i}({\mathbf {r}},E)}$ is the fraction of delayed neutrons emitted at ${\displaystyle \mathbf {r} }$ by precursors belonging to group ${\displaystyle i}$ produced by neutrons with energy ${\displaystyle E}$
${\displaystyle {\tilde {\beta }}({\mathbf {r}})}$	${\displaystyle \sum _{i=1}^{N}{\tilde {\beta }}_{i}({\mathbf {r}})}$

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.

Neutron diffusion equation

In nuclear reactor physics, the neutron transport equation is often approximated by the neutron diffusion equation when doing 3-dimensional core calculations. The neutron diffusion equation is derived from the neutron transport equation by making a spherical harmonics expansion of the angular neutron flux and by assuming that

• the Legendre polynomials as functions of the neutron direction ${\displaystyle {\mathbf {\hat {\Omega }}}}$ are of degree less than or equal to 1,
• the neutron source is isotropic,
• the rate of change of the current density vector ${\displaystyle {\mathbf {J}}({\mathbf {r}},E,t)}$ is much smaller than the frequency of collision and
• ${\displaystyle \int _{0}^{\infty }\Sigma _{s1}({\mathbf {r}},E'\to E,t){\mathbf {J}}({\mathbf {r}},E',t)dE'=\int _{0}^{\infty }\Sigma _{s1}({\mathbf {r}},E\to E',t){\mathbf {J}}({\mathbf {r}},E,t)dE'}$, where ${\displaystyle \Sigma _{sl}({\mathbf {r}},E'\to E,t)}$ is the Legendre polynomial expansion coefficient of order ${\displaystyle l}$ of the macroscopic scattering cross section.

Further, assuming that the neutron speed is energy independent, the one-speed neutron diffusion equation is as follows:

${\displaystyle {\frac {1}{v}}{\frac {\partial }{\partial t}}\phi ({\mathbf {r}},E,t)={\mathbf {\nabla }}\cdot D({\mathbf {r}},E){\mathbf {\nabla }}\phi ({\mathbf {r}},E,t)-\Sigma _{t}({\mathbf {r}},E,t)\phi ({\mathbf {r}},E,t)+\int _{0}^{\infty }dE'\Sigma _{s0}({\mathbf {r}},E'\to E,t)\phi ({\mathbf {r}},E',t)}$${\displaystyle \qquad +\chi _{p}\left({\mathbf {r}},E\right)\left[1-{\tilde {\beta }}({\mathbf {r}})\right]\int _{0}^{\infty }\mathrm {d} E^{\prime }\nu _{p}\left({\mathbf {r}},E^{\prime }\right)\Sigma _{f}\left(\mathbf {r} ,E^{\prime },t\right)\phi \left(\mathbf {r} ,E^{\prime },t\right)+\sum _{i=1}^{N}\chi _{di}\left({\mathbf {r}},E\right)\lambda _{i}C_{i}\left(\mathbf {r} ,t\right)+s(\mathbf {r} ,E,t)}$

Where ${\displaystyle D({\mathbf {r}},E)}$ is the diffusion coefficient, ${\displaystyle D({\mathbf {r}},E)={\frac {1}{3\Sigma _{tr}({\mathbf {r}},E)}}}$, with ${\displaystyle \Sigma _{tr}({\mathbf {r}},E)=\Sigma _{t}({\mathbf {r}},E)-{\bar {\mu }}_{s}({\mathbf {r}},E)\Sigma _{s}({\mathbf {r}},E)}$ the transport macroscopic cross section, and ${\displaystyle {\bar {\mu }}_{s}}$ the average cosine of the scattering angle. The equation for the precursors of delayed neutrons and all other notations are defined in the neutron transport equation section.

The multigroup diffusion equation can be obtained by discretizing the neutron energy domain:

${\displaystyle {\frac {1}{v_{g}}}{\frac {\partial }{\partial t}}\phi _{g}({\mathbf {r}},t)={\mathbf {\nabla }}\cdot D_{g}({\mathbf {r}}){\mathbf {\nabla }}\phi _{g}({\mathbf {r}},t)-\Sigma _{t,g}({\mathbf {r}},t)\phi _{g}({\mathbf {r}},t)+\sum _{g'=1}^{G}\Sigma _{s0,g'\to g}({\mathbf {r}},t)\phi _{g'}({\mathbf {r}},t)}$${\displaystyle \qquad +\chi _{p,g}\left({\mathbf {r}}\right)\left[1-{\tilde {\beta }}({\mathbf {r}})\right]\sum _{g'=1}^{G}\left(\nu \Sigma \right)_{f,g'}\left(\mathbf {r} ,t\right)\phi _{g'}\left(\mathbf {r} ,t\right)+\sum _{i=1}^{N}\chi _{di,g}\left({\mathbf {r}}\right)\lambda _{i}C_{i}\left(\mathbf {r} ,t\right)+s_{g}(\mathbf {r} ,t)}$

Where:

${\displaystyle \phi _{g}({\mathbf {r}},t)=\int _{E_{g}}^{E_{g-1}}dE\phi ({\mathbf {r}},E,t)}$

${\displaystyle {\frac {1}{v_{g}}}={\frac {1}{\phi _{g}({\mathbf {r}},t)}}\int _{E_{g}}^{E_{g-1}}dE{\frac {\phi ({\mathbf {r}},E,t)}{v(E)}}}$

${\displaystyle {\mathbf {\nabla }}\cdot D_{g}({\mathbf {r}}){\mathbf {\nabla }}\phi _{g}({\mathbf {r}},t)=\int _{E_{g}}^{E_{g-1}}dE{\mathbf {\nabla }}\cdot D({\mathbf {r}},E){\mathbf {\nabla }}\phi ({\mathbf {r}},E,t)}$

${\displaystyle \Sigma _{t,g}({\mathbf {r}},t)\phi _{g}({\mathbf {r}},t)=\int _{E_{g}}^{E_{g-1}}dE\Sigma _{t}({\mathbf {r}},E,t)\phi ({\mathbf {r}},E,t)}$

${\displaystyle \Sigma _{s0,g'\to g}({\mathbf {r}},t)\phi _{g'}({\mathbf {r}},t)=\int _{E_{g}}^{E_{g-1}}dE\Sigma _{s0,g'\to g}({\mathbf {r}},t)\phi _{g'}({\mathbf {r}},t)}$

${\displaystyle \chi _{p,g}({\mathbf {r}})=\int _{E_{g}}^{E_{g-1}}dE\chi _{p}({\mathbf {r}},E)}$

${\displaystyle \left(\nu \Sigma \right)_{f,g}({\mathbf {r}},t)\phi _{g}({\mathbf {r}},t)=\int _{E_{g}}^{E_{g-1}}dE\nu ({\mathbf {r}},E)\Sigma _{f}({\mathbf {r}},E,t)\phi ({\mathbf {r}},E,t)}$

${\displaystyle \chi _{di,g}({\mathbf {r}})=\int _{E_{g}}^{E_{g-1}}dE\chi _{di}({\mathbf {r}},E)}$

${\displaystyle G}$ is the number of energy groups and ${\displaystyle E_{g}}$ is energy group ${\displaystyle g}$. The energy groups are ordered such that ${\displaystyle E_{g}<E_{g-1}}$.

Boundary conditions and extrapolation distance

Because the neutron diffusion equation is a differential equation, boundary conditions must be specified for a unique solution. Since the diffusion equation is only an approximation to the transport equation, the transport-theory boundary conditions serve as a guide, but they can only be satisfied approximately, because diffusion theory provides only the scalar flux ${\displaystyle \phi }$ and the current ${\displaystyle {\mathbf {J}}}$.

At an interface between two media with different cross sections, the scalar flux and the normal component of the neutron current are continuous: ${\displaystyle \phi _{1}=\phi _{2}}$ and ${\displaystyle {\mathbf {J}}_{1}\cdot \mathbf {\hat {n}} ={\mathbf {J}}_{2}\cdot \mathbf {\hat {n}} }$

where ${\displaystyle \mathbf {\hat {n}} }$ is the unit normal to the interface; in one dimension, by Fick's law, this gives ${\displaystyle -D_{1}{\frac {d\phi _{1}}{dx}}=-D_{2}{\frac {d\phi _{2}}{dx}}}$.

At a vacuum (free) boundary, the transport-theory condition requires that no neutrons enter the region, i.e. the angular flux must vanish for all incoming directions. Diffusion theory approximates this by requiring that the inward partial current vanish at the boundary. In one dimension, with the boundary at ${\displaystyle x=x_{s}}$, this gives

${\displaystyle \left.{\frac {d\phi }{dx}}\right|_{x_{s}}=-{\frac {\phi (x_{s})}{2D}}}$

which is equivalent to saying that if the flux is extrapolated linearly beyond the boundary, it reaches zero a distance ${\displaystyle 2D={\frac {2}{3}}\lambda _{tr}}$ outside the physical boundary, where ${\displaystyle \lambda _{tr}=1/\Sigma _{tr}}$ is the transport mean free path. In practice the condition above is often replaced by the simpler zero-flux condition at this extrapolated boundary, ${\displaystyle \phi (x_{s}+2D)=0}$. More accurate transport-theory calculations give an extrapolation length ${\displaystyle z_{0}=0.7104\,\lambda _{tr}}$ (about ${\displaystyle 2.13\,D}$) for plane geometry. For curved boundaries the same ${\displaystyle 2D}$ extrapolation is usually adequate, provided the radius of curvature is large compared with a mean free path.

These approximations are reliable only well inside the medium: close to the boundary the angular flux is anisotropic, so the diffusion-theory flux deviates from the exact flux, which remains nonzero beyond the extrapolated surface.

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.

Computational methods

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.

See also

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

References

Further reading

• Duderstadt, James J.; Hamilton, Louis J. (1976). Nuclear Reactor Analysis. New York: Wiley. ISBN 0-471-22363-8. A standard reactor analysis text, covering neutron transport as one of several topics.
• Lewis, E. E.; Miller, W. F. (1984). Computational Methods of Neutron Transport. New York: Wiley-Interscience. ISBN 0-471-09245-2.
• Marchuk, G. I.; Lebedev, V. I. (1986). Numerical Methods in the Theory of Neutron Transport. Chur, Switzerland: Harwood. ISBN 978-3-7186-0182-0.

External links

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