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 variables of energy span several orders of magnitude (from fractions of MeV to several 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:

:<math>\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</math><math>\quad\frac{\chi_p \left( \bold{r},E \right)}{4\pi}\left[ 1-\tilde{\beta}(\bold{r}) \right]\int_0^{\infty} \mathrm dE^{\prime}\nu_p \left( \bold{r},E^{\prime} \right) \Sigma_f \left(\mathbf{r}, E^{\prime}, t \right) \phi \left( \mathbf{r}, E^{\prime}, t \right)</math>

:<math>\quad + \sum_{i=1}^N \frac{\chi_{di}\left( \bold{r},E \right)}{4\pi} \lambda_i C_i \left( \mathbf{r}, t \right)\quad</math>

:<math>\quad + \int_{4\pi}\mathrm d\Omega^\prime\int^{\infty}_{0}\mathrm dE^\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)</math>

:<math>\quad + s(\mathbf{r},E,\mathbf{\hat{\Omega,t)</math>

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

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|<math>\chi_p(E)</math>

|Probability density function for neutrons of exit energy <math>E</math> from all neutrons produced by fission

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|<math>\chi_{di}(E)</math>

|Probability density function for neutrons of exit energy <math>E</math> from all neutrons produced by delayed neutron precursors

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|<math>\Sigma_t(\mathbf{r},E,t)</math>

|Macroscopic total cross section, which includes all possible interactions

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|<math>\Sigma_f(\mathbf{r},E^{\prime},t)</math>

|Macroscopic fission cross section, which includes all fission interactions in <math>\mathrm dE^{\prime}</math> about <math>E^{\prime}</math>

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|<math>\Sigma_s\!\!\left(\mathbf{r},E'\rightarrow E,\mathbf{\hat{\Omega'\rightarrow \mathbf{\hat{\Omega,t\right)\mathrm dE^\prime \mathrm d\Omega^\prime</math>

|Double differential scattering cross section<br />Characterizes scattering of a neutron from an incident energy <math>E^\prime</math> in <math>\mathrm dE^\prime</math> and direction <math>\mathbf{\hat\Omega^\prime}</math> in <math>\mathrm d\Omega^\prime</math> to a final energy <math>E</math> and direction <math>\mathbf{\hat{\Omega.</math>

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|<math>N</math>

|Number of delayed neutron precursors

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|<math>\lambda_i</math>

|Decay constant for precursor i

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|<math>C_i \left( \mathbf{r}, t \right)</math>

|Total number of precursor i in <math>\mathbf{r}</math> at time <math>t</math>

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|<math>s(\mathbf{r},E,\mathbf{\hat{\Omega,t)</math>

|Source term

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|<math>\tilde{\beta}_i(\bold{r})</math>

|Weighted average of delayed neutrons: <math>\tilde{\beta}_i(\bold{r})=\frac{\int_0^\infty \beta_i(\bold{r},E)\nu_p(\bold{r},E)\Sigma_f(\bold{r},E)\phi(\bold{r},E)dE}{\int_0^\infty \nu_p(\bold{r},E)\Sigma_f(\bold{r},E)\phi(\bold{r},E)dE}</math>

Where <math>\beta_i(\bold{r},E)</math> is the fraction of delayed neutrons emitted at <math>\mathbf{r}</math> by precursors belonging to group <math>i</math> produced by neutrons with energy <math>E</math>

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|<math>\tilde{\beta}(\bold{r})</math>

|<math>\sum_{i=1}^N\tilde{\beta}_i(\bold{r})</math>

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The transport equation can be applied to a given part of phase space (time t, energy E, location <math>\mathbf{r},</math> and direction of travel <math>\mathbf{\hat{\Omega.</math>) 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 <math>\phi(\mathbf{r},E),</math> 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 <math>\bold{\hat{\Omega</math> are of degree less than or equal to 1,
  • the neutron source is isotropic,
  • the rate of change of the current density vector <math>\bold{J}(\bold{r},E,t)</math> is much smaller than the frequency of collision and
  • <math>\int_0^\infty \Sigma_{s1}(\bold{r},E'\to E,t)\bold{J}(\bold{r},E',t)dE'=\int_0^\infty \Sigma_{s1}(\bold{r},E\to E',t)\bold{J}(\bold{r},E,t)dE'</math>, where <math>\Sigma_{sl}(\bold{r},E'\to E,t)</math> is the Legendre polynomial expansion coefficient of order <math>l</math> of the macroscopic scattering cross section.

Further, assuming that the neutron speed is energy independent, the one-speed neutron diffusion equation is as follows: – A Monte Carlo code for general radiation transport developed and supported by the ANSWERS Software Service.

  • MCNP – A LANL developed Monte Carlo code for general radiation transport
  • MC21 – A general-purpose, 3D Monte Carlo code developed at NNL.
  • MCS – The Monte Carlo code MCS has been developed since 2013 at Ulsan National Institute of Science and Technology (UNIST), Republic of Korea.
  • Mercury – A LLNL developed Monte Carlo particle transport code.
  • MONK – A Monte Carlo Code for criticality safety and reactor physics analyses developed and supported by the ANSWERS Software Service.
  • OpenMC – An open source, community-developed open source Monte Carlo code
  • RMC – A Tsinghua University Department of Engineering Physics developed Monte Carlo code for general radiation transport
  • SCONE – The Stochastic Calculator Of the Neutron Transport Equation, an open-source Monte Carlo code developed at the University of Cambridge.
  • Serpent – A VTT Technical Research Centre of Finland developed Monte Carlo particle transport code
  • Shift/KENO – ORNL developed Monte Carlo codes for general radiation transport and criticality analysis
  • TRIPOLI – 3D general purpose continuous energy Monte Carlo Transport code developed at CEA, France
  • UCN - Monte Carlo transport code for simulating experiments with ultracold neutrons developed at PNPI, Gatchina

Deterministic codes

  • AGREE - A coupled thermal-neutronics high-temperature gas code developed by University of Michigan
  • Ardra – A LLNL neutral particle transport code
  • Attila – A commercial transport code
  • DRAGON – An open-source lattice physics code
  • PHOENIX/ANC – A proprietary lattice-physics and global diffusion code suite from Westinghouse Electric
  • PARTISN – A LANL developed transport code based on the discrete ordinates method
  • NEWT – An ORNL developed 2-D S<sub>N</sub> code
  • DIF3D/VARIANT – An Argonne National Laboratory developed 3-D code originally developed for fast reactors
  • DENOVO – A massively parallel transport code under development by ORNL
  • Jaguar – A parallel 3-D Slice Balance Approach transport code for arbitrary polytope grids developed at NNL
  • DANTSYS
  • RAMA – A proprietary 3D method of characteristics code with arbitrary geometry modeling, developed for EPRI by TransWare Enterprises Inc.
  • RAPTOR-M3G – A proprietary parallel radiation transport code developed by Westinghouse Electric Company
  • OpenMOC – An MIT developed open source parallel method of characteristics code
  • MPACT – A parallel 3D method of characteristics code under development by Oak Ridge National Laboratory and the University of Michigan
  • DORT – Discrete Ordinates Transport
  • APOLLO – A lattice physics code used by CEA, EDF and Areva
  • CASMO/SIMULATE – A proprietary lattice-physics and diffusion code suite developed by Studsvik for LWR analysis including square and hex lattices
  • HELIOS – A proprietary lattice-physics code with generalized geometry developed by Studsvik for LWR analysis
  • milonga – A free nuclear reactor core analysis code
  • STREAM – A neutron transport analysis code, STREAM (Steady state and Transient REactor Analysis code with Method of Characteristics), has been developed since 2013 at Ulsan National Institute of Science and Technology (UNIST), Republic of Korea
  • TINTE – A two-group diffusion code for the study of nuclear and thermal behavior of high temperature reactors, developed by Forschungszentrum Jülich in Germany.

See also

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

References

  • Lewis, E., & Miller, W. (1993). Computational Methods of Neutron Transport. American Nuclear Society. .
  • Duderstadt, J., & Hamilton, L. (1976). Nuclear Reactor Analysis. New York: Wiley. .
  • Marchuk, G. I., & V. I. Lebedev (1986). Numerical Methods in the Theory of Neutron Transport. Taylor & Francis. p.&nbsp;123. .
  • ANSWERS Software Service website
  • LANL MCNP6 website
  • LANL MCNPX website
  • VTT Serpent website
  • OpenMC website
  • MIT CRPG OpenMOC website
  • TRIPOLI-4 website