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Neutron transport

Neutron transport refers to the physical processes governing the propagation, scattering, absorption, and production of neutrons within a material medium, fundamentally described by the neutron transport equation—a linearized form of the Boltzmann equation tailored to neutron kinetics in nuclear systems.^[1] This equation balances the neutron angular flux \phi(\mathbf{r}, \Omega, E, t), representing the number of neutrons crossing a unit area per unit time per unit solid angle per unit energy, against terms for streaming (\Omega \cdot \nabla \phi), total interaction losses (\Sigma_t \phi), scattering gains (\int \Sigma_s(\Omega' \to \Omega, E' \to E) \phi(\mathbf{r}, \Omega', E', t) \, d\Omega' dE'), fission production (\frac{\chi(E)}{4\pi} \int_0^\infty \nu(E') \Sigma_f(\mathbf{r}, E', t) \phi(\mathbf{r}, E', t) \, dE', where \phi(\mathbf{r}, E', t) = \int_{4\pi} \phi(\mathbf{r}, \Omega', E', t) \, d\Omega' is the scalar flux), and external sources Q(\mathbf{r}, \Omega, E, t), where \Sigma_t, \Sigma_s, and \Sigma_f are macroscopic cross-sections, \nu is the average number of neutrons per fission, and \chi(E) is the fission spectrum.^[1] The theory underpins nuclear reactor physics by enabling predictions of neutron density distributions in phase space (position, direction, energy, and time), essential for assessing criticality through the effective multiplication factor k_{\text{eff}} and ensuring safe operation.^[1] In practice, the full seven-dimensional transport equation is computationally intensive, often approximated by diffusion theory for homogeneous media with isotropic scattering, where the diffusion coefficient D \approx 1/(3 \Sigma_s (1 - \bar{\mu}_0)) relates current to flux gradient via Fick's law, though this simplification fails near boundaries, voids, or strong absorbers.^[2] Applications extend beyond reactors to radiation shielding, dosimetry, and fusion neutronics, with numerical methods like discrete ordinates or Monte Carlo simulations solving the equation for heterogeneous geometries.^[3]

Fundamentals

Neutron Interactions

Neutron interactions refer to the fundamental processes by which neutrons collide with atomic nuclei, leading to various outcomes that determine their transport behavior in matter. These interactions are probabilistic and governed by nuclear forces, with the likelihood quantified by cross-sections. The primary interactions include elastic and inelastic scattering, capture, fission, and charged-particle reactions, each dominant in different energy regimes and materials.^[4] Elastic scattering occurs when a neutron collides with a nucleus like a billiard ball, conserving the total kinetic energy of the system while transferring momentum; this is the most probable interaction for most nuclides across neutron energies and serves as the basis for neutron moderation. Inelastic scattering involves the neutron exciting the nucleus to a higher energy state, resulting in energy loss and subsequent gamma-ray emission from de-excitation; it requires a threshold energy (typically above 0.1 MeV) and is more common with heavy nuclei like uranium. Capture reactions encompass neutron absorption by the nucleus, forming a compound state that decays either radiatively—emitting prompt gamma rays, as in the reaction ^1H(n,γ)^2H yielding a 2.22 MeV gamma—or non-radiatively, potentially emitting particles without significant gamma production; radiative capture dominates for thermal neutrons in light elements. Fission is induced when a neutron absorption splits a heavy fissile nucleus, such as ^ {235}U, into fragments, releasing 2–4 neutrons and approximately 200 MeV of energy; thermal neutrons efficiently trigger fission in ^ {235}U, while fast neutrons are needed for ^ {238}U. Charged-particle reactions involve neutron absorption followed by emission of a charged particle, like a proton or alpha, as in ^ {10}B(n,α)^7Li with a high cross-section of 3840 barns; these are often endothermic with energy thresholds, such as 0.96 MeV for ^ {32}S(n,p)^ {32}P.^[4]^[5]^[4] The probability of these interactions is described by cross-sections, which are energy-dependent measures of interaction likelihood. The microscopic cross-section (σ) represents the effective area per nucleus for a specific reaction, typically in barns (10^{-28} m²), and varies sharply with neutron energy due to resonance regions—narrow peaks at keV energies from quantum interference—and thresholds for endothermic reactions above a few MeV. The macroscopic cross-section (Σ) extends this to a bulk material, given by Σ = N σ where N is the atomic density in atoms/cm³, with units of cm^{-1}, allowing calculation of mean free paths as 1/Σ. For example, thermal neutron capture cross-sections follow a 1/v law (v being neutron velocity), peaking for isotopes like ^ {10}B, while fast neutron interactions favor scattering over absorption.^[6]^[5]^[4] Neutrons are categorized into energy regimes that influence interaction dominance: thermal neutrons (~0.025 eV, in thermal equilibrium with matter) primarily undergo capture and elastic scattering, with high probabilities in absorbers like ^ {10}B (thousands of barns); epithermal neutrons (0.5 eV to ~100 keV) exhibit resonance capture in the keV range, reducing flux via absorption; and fast neutrons (>100 keV, up to MeV from fission) favor elastic and inelastic scattering, with lower overall cross-sections (~1–10 barns) but enabling fission in fertile materials. These regimes arise from the initial fission spectrum and subsequent moderation.^[7]^[4] Moderation is the process of slowing fast neutrons to thermal energies through repeated elastic collisions with light nuclei, maximizing energy transfer when masses are similar; hydrogen (A=1) is highly effective, requiring about 18 collisions to reduce 1 MeV neutrons to 0.025 eV, while carbon (A=12) needs over 100 collisions due to smaller fractional energy loss per interaction. This slowing-down occurs without significant absorption in good moderators like water or graphite, enabling sustained chain reactions.^[8] Key quantitative aspects include Q-values, the energy released (positive Q) or required (negative Q) in reactions, derived from mass differences; for instance, the radiative capture ^ {10}B(n,α)^7Li is exothermic with Q ≈ 2.8 MeV, while inelastic scattering thresholds reflect negative Q-values starting at ~0.1 MeV for heavy nuclei. Fission neutrons follow a spectrum approximated by the Watt distribution, P(E) ∝ sinh(√E) exp(-E) (in MeV), peaking near 1 MeV with average energy ~2 MeV, or a Maxwellian form for simpler estimates, reflecting evaporation from excited fragments.^[4]^[9]^[5]

Transport Phenomenon

Neutron transport refers to the phenomenon governing the directional propagation and interactions of neutrons within a material medium, characterized by their free streaming between successive collisions with atomic nuclei, resulting in an anisotropic angular distribution of neutron flux and the progressive removal of neutrons from their initial directional beams through absorption or scattering out of the beam path.^[10] This process captures the ballistic motion of neutrons over distances comparable to the mean free path, the average distance traveled before an interaction occurs, which typically ranges from millimeters to centimeters depending on the medium's density and neutron energy.^[11] Unlike simpler models, transport accounts for the full phase space of neutron motion, including position, velocity direction, and energy, emphasizing the non-isotropic nature of neutron populations in heterogeneous or low-density environments.^[3] The conceptual foundations of neutron transport emerged in the 1930s and 1940s amid early nuclear physics research, with key contributions from Enrico Fermi's group in Rome, where experiments on neutron slowing-down in moderators highlighted the need for directional tracking beyond isotropic assumptions.^[12] Fermi's development of the FERMIAC analog computer in the late 1940s further advanced practical studies of neutron paths, simulating streaming and collision effects to predict flux distributions in reactor designs. Gian-Carlo Wick, working with Fermi, formalized transport-based methods for calculating neutron moderation probabilities, providing a rigorous framework that influenced subsequent theory.^[13] These efforts established transport as essential for understanding neutron behavior in fission chains, distinct from the statistical averaging in gas kinetic theory analogs like the Boltzmann equation.^[10] Central to neutron transport are concepts such as the scalar flux, which integrates the angular flux over all directions to yield the total track length of neutrons per unit volume per unit time, and the angular flux itself, which specifies the directional density of neutron trajectories crossing a unit area per unit time.^[11] The neutron current density quantifies the net directional flow of neutrons across a surface, arising from imbalances in angular flux and crucial for leakage assessments near boundaries.^[10] The buildup factor extends this by representing the enhancement in total flux due to multiple scattering events, defined as the ratio of the total neutron flux (including scattered contributions) to the unattenuated direct beam flux, becoming significant after several mean free paths where scattered neutrons accumulate.^[14] These elements underscore the phenomenon's emphasis on directional fidelity over averaged behaviors. In contrast to diffusion theory, which approximates neutron motion as isotropic random walks suitable for optically thick media with frequent collisions and low absorption, transport theory is necessary in scenarios like voids where neutrons stream unimpeded without scattering, near material boundaries where flux gradients sharpen anisotropically, or in high-absorption regions where removal dominates and isotropic assumptions lead to errors exceeding 20-50% in flux predictions.^[11] Diffusion suffices for deep interiors of uniform reactors but fails in streaming-dominated cases, such as voids spanning multiple mean free paths, necessitating transport for accurate modeling.^[15] Transport effects are particularly pronounced in heterogeneous media like pebble-bed reactors, where neutrons stream through interstitial voids between fuel spheres, enhancing leakage and altering flux profiles compared to homogenized diffusion approximations; this streaming can increase effective multiplication factors by up to 5-10% if unaccounted for, impacting criticality safety.^[16] Such examples illustrate how transport captures removal and anisotropic buildup in complex geometries, informing designs for high-temperature gas-cooled systems.^[17]

Mathematical Formulation

The Neutron Transport Equation

The neutron transport equation is the fundamental integro-differential equation that governs the behavior of neutrons in a medium, describing the balance between neutron streaming, collisions, and sources across position, direction, energy, and time.^[10] It originates from the adaptation of the Boltzmann transport equation from statistical mechanics to neutron kinetics, with seminal formalization in nuclear engineering contexts.^[18] The general time-dependent form of the neutron transport equation is

\frac{1}{v} \frac{\partial \psi(\mathbf{r}, \boldsymbol{\Omega}, E, t)}{\partial t} + \boldsymbol{\Omega} \cdot \nabla \psi(\mathbf{r}, \boldsymbol{\Omega}, E, t) + \Sigma_t(\mathbf{r}, E, t) \psi(\mathbf{r}, \boldsymbol{\Omega}, E, t) = \int_{0}^{\infty} dE' \int_{4\pi} d\boldsymbol{\Omega}' \, \Sigma_s(\mathbf{r}, \boldsymbol{\Omega}' \to \boldsymbol{\Omega}, E' \to E, t) \psi(\mathbf{r}, \boldsymbol{\Omega}', E', t) + S(\mathbf{r}, \boldsymbol{\Omega}, E, t),

where \psi(\mathbf{r}, \boldsymbol{\Omega}, E, t) is the angular flux (neutrons per unit area per unit time per unit solid angle per unit energy), v is the neutron speed, \mathbf{r} is position, \boldsymbol{\Omega} is direction, E is energy, and t is time.^[18] The term \frac{1}{v} \frac{\partial \psi}{\partial t} accounts for the local rate of change of the angular flux due to time evolution.^[18] The streaming term \boldsymbol{\Omega} \cdot \nabla \psi represents the directional transport of neutrons without interactions.^[10] The removal term \Sigma_t \psi captures losses from total interactions (absorption and out-scattering), with \Sigma_t the total macroscopic cross-section.^[10] The integral term on the right-hand side describes in-scattering from all directions \boldsymbol{\Omega}' and energies E', governed by the differential scattering kernel \Sigma_s.^[18] The source term S includes external sources and fission production, where the fission contribution is typically S_f(\mathbf{r}, \boldsymbol{\Omega}, E, t) = \frac{\chi(E)}{4\pi} \int_0^\infty \nu \Sigma_f(\mathbf{r}, E', t) \phi(\mathbf{r}, E', t) \, dE' assuming isotropic emission, with \phi(\mathbf{r}, E', t) = \int_{4\pi} \psi(\mathbf{r}, \boldsymbol{\Omega}', E', t) \, d\boldsymbol{\Omega}' the scalar flux, \chi(E) the fission spectrum, and \nu the average neutrons per fission.^[10]^[18] For steady-state conditions, the time derivative is set to zero, simplifying the equation to

\boldsymbol{\Omega} \cdot \nabla \psi(\mathbf{r}, \boldsymbol{\Omega}, E) + \Sigma_t(\mathbf{r}, E) \psi(\mathbf{r}, \boldsymbol{\Omega}, E) = \int_{0}^{\infty} dE' \int_{4\pi} d\boldsymbol{\Omega}' \, \Sigma_s(\mathbf{r}, \boldsymbol{\Omega}' \to \boldsymbol{\Omega}, E' \to E) \psi(\mathbf{r}, \boldsymbol{\Omega}', E') + S(\mathbf{r}, \boldsymbol{\Omega}, E).

This form applies to continuous-wave or equilibrium neutron populations, such as in steady reactor operation.^[10] Extensions to time-dependent scenarios accommodate pulsed sources, where the full equation tracks transient behaviors like reactor transients.^[18] In multi-group approximations, energy is discretized into G groups with average cross-sections \Sigma_{t,g} and flux \psi_g, yielding a coupled set of equations:

\frac{1}{v_g} \frac{\partial \psi_g}{\partial t} + \boldsymbol{\Omega} \cdot \nabla \psi_g + \Sigma_{t,g} \psi_g = \sum_{g'=1}^G \int_{4\pi} \Sigma_{s,g' \to g}(\boldsymbol{\Omega}' \to \boldsymbol{\Omega}) \psi_{g'}(\boldsymbol{\Omega}') d\boldsymbol{\Omega}' + S_g,

which reduces computational complexity for broad energy spectra.^[18] The equation assumes continuous energy treatment unless discretized, and scattering can be isotropic ( \Sigma_s independent of \boldsymbol{\Omega}' \to \boldsymbol{\Omega}) for simple media or anisotropic (direction-dependent kernel) for more accurate modeling of forward-peaked scattering in light materials.^[18] For monoenergetic cases, energy integrals collapse, simplifying to single-energy scattering.^[10] These assumptions stem from the linearized Boltzmann framework, as detailed in foundational texts like Case and Zweifel's Linear Transport Theory.^[18]

Boundary and Initial Conditions

To solve the neutron transport equation, appropriate boundary and initial conditions must be specified to define the problem uniquely. These conditions account for the physical behavior of neutrons at system interfaces and at the start of time-dependent simulations, ensuring no unphysical influx or unresolved evolution. Vacuum boundary conditions are commonly applied at external surfaces where no neutrons enter the domain from outside. In this case, the incoming angular flux is set to zero for directions pointing into the medium, mathematically expressed as \psi(\mathbf{r}, \boldsymbol{\Omega}, E, t) = 0 for \boldsymbol{\Omega} \cdot \mathbf{n} > 0, where \mathbf{n} is the outward unit normal vector at the boundary \mathbf{r}.^[19] This condition models free streaming into vacuum without reflection or scattering back into the system.^[20] Reflective boundary conditions simulate interfaces where neutrons are partially or fully returned into the domain, often parameterized by an albedo coefficient \alpha (0 ≤ \alpha ≤ 1) representing the fraction of outgoing current reflected. For general albedo boundaries, the incoming flux is \psi(\mathbf{r}, \boldsymbol{\Omega}, E, t) = \alpha \psi(\mathbf{r}, -\boldsymbol{\Omega}, E, t) for \boldsymbol{\Omega} \cdot \mathbf{n} > 0, where -\boldsymbol{\Omega} reverses the direction relative to the normal.^[20] Specular reflection assumes mirror-like behavior, preserving the incidence angle such that \phi(0, \mu) = \phi(0, -\mu) for \mu > 0 in slab geometry, ideal for symmetric or polished surfaces.^[21] In contrast, diffuse reflection models isotropic reemission, where incoming neutrons are absorbed and reemitted uniformly in angle with probability \alpha, as \psi(0, \mu) = \alpha \int_0^1 W(\mu') \psi(0, \mu') d\mu' and W(\mu) derived from the H-function for half-space problems.^[21] Albedo \alpha = 0 recovers the vacuum case, while \alpha = 1 yields perfect reflection with zero net current.^[22] For discrete ordinates methods, vacuum boundaries are often approximated using Marshak conditions to enforce zero incoming current in an integral sense over odd Legendre moments. These are given by $2\pi \int_{\mu_{\text{in}}} P_i(\mu) \psi(\mu) d\mu = 0 for i = 1, 3, \dots, N odd, where P_i are Legendre polynomials and the expansion \psi(\mu) = \sum_{n=0}^N \frac{2n+1}{4\pi} \phi_n P_n(\mu) leads to coupled equations like \frac{1}{2} \phi_0 + \phi_1 + \frac{5}{8} \phi_2 = 0 for the P_3 case.^[23] Marshak conditions provide an optimal approximation to exact vacuum boundaries in P_N or spherical harmonics expansions, minimizing errors in flux moments.^[24] In time-dependent problems, initial conditions specify the angular flux distribution at t=0, typically as \psi(\mathbf{r}, \boldsymbol{\Omega}, E, t=0) = \psi_0(\mathbf{r}, \boldsymbol{\Omega}, E), which must be physically realizable and compatible with sources and boundaries to ensure uniqueness.^[1] This defines an initial-value problem where the evolution follows the transport equation forward in time.^[25] Source terms in the transport equation represent fixed external neutron inputs, such as from fission or spallation, with specified spatial, angular, and energy distributions Q(\mathbf{r}, \boldsymbol{\Omega}, E, t). Fission sources, like those from Cf-252 spontaneous fission, produce neutrons with a Maxwellian-like spectrum with temperature ~1.42 MeV (peaking at ~0.7 MeV, average 2.13 MeV) and isotropic angular distribution, yielding ~2% accuracy in transport modeling from 0.2–12 MeV.^[26] Spallation sources, driven by high-energy protons (500–1100 MeV) on heavy targets, generate broad "white" spectra up to ~1 GeV with yields of ~25 neutrons per proton, often modeled as isotropic in the lab frame for accelerator-driven systems.^[27] These sources drive fixed-source transport calculations, with distributions tailored to applications like reactor startups or shielding benchmarks.^[26] For infinite or semi-infinite media, extrapolation distances adjust effective boundaries to approximate transport effects in diffusion theory. The extrapolated endpoint lies beyond the physical boundary where flux extrapolates to zero, with d \approx 0.71 / \Sigma_{tr} from transport theory, where \Sigma_{tr} is the transport cross section, improving on the diffusion estimate d = 2D.^[28] This technique, with \lambda = 1/\Sigma_{tr} as the transport mean free path, ensures accurate flux matching at material-vacuum interfaces without explicit angular resolution.^[29]

Problem Types

Fixed-Source Calculations

Fixed-source calculations in neutron transport address linear problems where an external neutron source S(\mathbf{r}, E, \boldsymbol{\Omega}) is prescribed, and the angular flux \psi(\mathbf{r}, E, \boldsymbol{\Omega}) is solved for in non-multiplying systems. These calculations solve the steady-state neutron transport equation, balancing neutron production from the source against losses due to absorption, scattering, and leakage, typically in subcritical configurations. The source term represents neutrons introduced externally, such as from accelerators or isotopic emitters, and the flux quantifies the resulting neutron distribution across phase space.^[10]^[18] Such calculations find primary applications in radiation shielding, where the flux through protective materials is estimated to ensure safety; detector response analysis, evaluating how neutrons interact with sensing elements; and subcritical irradiation studies, assessing neutron exposure in materials without fission chain reactions. For instance, in shielding designs for neutron sources like ^{252}\text{Cf}, fixed-source methods compute dose rates and flux attenuation in structures such as iron or heavy water spheres. Normalization of the flux is typically performed per source neutron or relative to total source strength in particles per second, ensuring results scale appropriately for varying source intensities; for example, a ^{252}\text{Cf} source normalized to $1.26 \times 10^5 n/s yields a reference flux of 1 n/cm²/s at 100 cm distance. Boundary conditions, such as vacuum or reflective, close the problem by specifying incoming flux at domain edges.^[30]^[10]^[18] Specific examples include neutron beam attenuation in collimators, where directional sources model flux reduction through geometric filtering in radiography facilities, highlighting anisotropic effects. Adjoint transport complements forward calculations by deriving importance functions, which quantify the contribution of neutrons at a point to a response like detector tally, enhancing efficiency in importance sampling. Challenges in fixed-source setups often involve deep penetration, where neutrons traverse low-interaction media over long distances, and streaming, where uncollided neutrons propagate through voids or channels, both demanding high-fidelity angular and spatial resolutions to capture accurate flux tails.^[18]^[10]^[30]

Criticality Eigenvalue Problems

In criticality eigenvalue problems within neutron transport theory, the focus is on determining the conditions under which a self-sustaining nuclear chain reaction can occur in a multiplying medium, such as fissile material in a reactor core. These problems are formulated as time-independent eigenvalue equations where the neutron source depends on fission events, distinguishing them from fixed-source scenarios by incorporating the feedback from neutron-induced fissions. The effective multiplication factor k, serving as the eigenvalue, quantifies the average number of neutrons produced per neutron absorbed in the system, enabling assessment of whether the chain reaction will sustain, grow, or decay.^[10] The standard formulation of the time-independent neutron transport equation for the criticality problem, assuming a monoenergetic approximation and isotropic fission source for simplicity, is given by

\boldsymbol{\Omega} \cdot \nabla \psi(\mathbf{r}, \boldsymbol{\Omega}) + \Sigma_t(\mathbf{r}) \psi(\mathbf{r}, \boldsymbol{\Omega}) = \int_{4\pi} \Sigma_s(\mathbf{r}, \boldsymbol{\Omega}' \to \boldsymbol{\Omega}) \psi(\mathbf{r}, \boldsymbol{\Omega}') \, d\boldsymbol{\Omega}' + \frac{\nu \Sigma_f(\mathbf{r}) }{4\pi k} \int_{4\pi} \psi(\mathbf{r}, \boldsymbol{\Omega}') \, d\boldsymbol{\Omega}',

where \psi(\mathbf{r}, \boldsymbol{\Omega}) is the angular neutron flux, \Sigma_t is the total macroscopic cross-section, \Sigma_s is the scattering cross-section, \Sigma_f is the fission cross-section, \nu is the average number of neutrons per fission, and [k](/page/K) is the effective multiplication factor. This equation balances neutron losses due to streaming and total interactions with gains from scattering and fission, with the $1/[k](/page/K) term scaling the fission source to form an eigenvalue problem. The principal eigenvalue [k](/page/K) and its associated eigenfunction (flux distribution) are sought, typically the dominant mode with the lowest decay rate.^[10]^[31] The criticality condition is achieved when k = 1, corresponding to a steady-state chain reaction where the rate of neutron production exactly equals the rate of losses through absorption and leakage, maintaining a constant neutron population over time. For k < 1, the system is subcritical, with the neutron population decaying exponentially as each generation produces fewer neutrons than consumed; conversely, for k > 1, the system is supercritical, leading to an exponentially growing neutron population and potential runaway reaction. These interpretations are fundamental to reactor control and safety analysis, guiding the design of systems to operate near or at criticality.^[10]^[32] Historically, criticality eigenvalue problems played a pivotal role in the Manhattan Project during the 1940s, where theoretical calculations of k for uranium-graphite assemblies were essential to achieving the first controlled chain reaction in the Chicago Pile-1 experiment on December 2, 1942, validating the feasibility of sustained fission. Enrico Fermi's team used simplified diffusion approximations to these transport formulations to predict and measure k \approx 1.0006 during initial operations. In homogeneous reactors, the concept of buckling arises in diffusion theory approximations to the transport equation, where the material buckling B_m^2 = (k_\infty - 1)/L^2 (with k_\infty as the infinite-medium multiplication factor and L as the diffusion length) must equal the geometric buckling B_g^2 (determined by reactor shape) for criticality, providing a simple geometric condition for flux distribution in uniform media.^[33] For time-dependent criticality assessments, an initial value approach formulates the problem through the time-dependent transport equation, where solutions evolve as superpositions of eigenmodes, and power iteration-like methods iteratively propagate an initial neutron distribution to extract the dominant k or time eigenvalue (alpha mode) by observing asymptotic growth or decay rates. This approach is particularly useful for transient analyses near criticality, bridging steady-state eigenvalue solutions with dynamic behavior in non-equilibrium systems.^[34]^[35]

Solution Approaches

Deterministic Methods

Deterministic methods for solving the neutron transport equation involve discretizing the phase space—comprising position, energy, and direction—to transform the integro-differential equation into a system of algebraic equations that can be solved numerically.^[36] These approaches approximate the continuous angular dependence of the neutron flux using predefined grids or basis functions, enabling the computation of flux distributions in fixed-source problems or criticality eigenvalues without relying on statistical sampling. By balancing computational efficiency with accuracy, deterministic methods are particularly suited for problems with regular geometries and well-defined sources, such as reactor core analyses.^[37] The discrete ordinates (S_N) method, a cornerstone of deterministic transport calculations, approximates the angular flux by evaluating it at a finite set of discrete directions, or ordinates, using an angular quadrature scheme. In this approach, the integral over the scattering term is replaced by a weighted sum over these directions, resulting in a set of coupled partial differential equations for each discrete direction that are solved iteratively. Developed by Bengt G. Carlson in the 1950s at Los Alamos National Laboratory, the S_N method originated from efforts to solve one-dimensional transport problems efficiently, with early implementations appearing in reports like LA-1599 (1953).^[38] While computationally tractable, the method suffers from ray effects, where unphysical oscillations arise due to the finite number of directions, particularly in problems with localized sources or voids; these can be mitigated by increasing the quadrature order N or using higher-order spatial schemes.^[36] The spherical harmonics (P_N) expansion method approximates the angular flux as a truncated series of spherical harmonics up to order N, converting the transport equation into a system of coupled diffusion-like equations for the moments of the flux. This polynomial representation captures the angular variation smoothly, avoiding the directional discontinuities of the S_N method, and is especially effective for problems with isotropic or low-order anisotropic scattering. The P_N approach was adapted to neutron transport from radiative transfer theory in the mid-20th century, with foundational applications in plane geometry appearing in works like those by B. Davison in the 1950s.^[39] Higher-order expansions improve accuracy but increase computational cost due to the growing number of moment equations, typically scaling as (N+1)^2 per spatial point. Collision probability methods address the transport equation in its integral form by computing the probability that a neutron born in one volume element undergoes its first collision in another, enabling exact treatment of collision integrals within lattice or cell geometries without full phase-space discretization. These methods are particularly advantageous for heterogeneous reactor assemblies, where they facilitate the assembly of response matrices for flux calculations across fuel pins or assemblies. The approach gained prominence through R. Bonalumi's 1961 formulation, which introduced efficient tracking algorithms for cylindrical and annular systems, allowing direct computation of probabilities via analytical integrals or numerical tracking.^[40] Convergence in deterministic methods is achieved by refining the spatial mesh to resolve flux gradients and increasing the angular resolution—such as higher S_N quadrature orders or P_N expansions—to reduce approximation errors. For instance, in fixed-source problems, mesh refinement ensures that spatial discretization errors diminish, while angular order increases mitigate ray effects in S_N or improve moment closure in P_N. These criteria are typically assessed through convergence studies comparing solutions at successive refinement levels against benchmarks or analytical limits.^[41]

Monte Carlo Methods

Monte Carlo methods for neutron transport involve simulating the random histories of individual neutrons to statistically estimate solutions to the transport equation. In this approach, known as analog Monte Carlo simulation, each neutron's path is modeled by randomly sampling its flight distance, direction changes upon scattering, and interaction types based on probabilistic distributions derived from nuclear cross-sections.^[42] The method was first applied to neutronics problems in the late 1940s at Los Alamos National Laboratory, where Stanisław Ulam conceived the idea of using random sampling for diffusion and transport calculations, with key developments by Nicholas Metropolis, John von Neumann, and others.^[43] Their seminal 1949 paper formalized the Monte Carlo technique as a statistical method for solving integro-differential equations like those in neutron transport.^[44] A fundamental aspect of these simulations is the tracking of neutron trajectories through a defined geometry, where interactions such as scattering, absorption, or fission are sampled at each collision point. In heterogeneous media, where material boundaries are frequent, explicit surface crossing can be computationally expensive; to address this, the Woodcock delta-tracking algorithm introduces fictitious "delta" scattering events in virtual majorants to enable efficient ray marching without constant boundary checks.^[45] Originally developed for Monte Carlo neutronics in multiplying and non-multiplying geometries, this method homogenizes the transport process by adding a uniform majorant cross-section, allowing unbiased sampling of real interactions while rejecting null collisions.^[46] Despite its conceptual simplicity, analog Monte Carlo simulations often suffer from high statistical variance, particularly in problems with low-probability events like deep penetration or rare fissions, leading to the need for variance reduction techniques. Importance sampling modifies the sampling probabilities to favor regions or paths contributing more to the desired estimate, such as biasing neutron directions toward detectors, with weights adjusted to maintain unbiasedness.^[47] Russian roulette randomly terminates low-weight particles to reduce computational effort in unimportant regions, while splitting duplicates high-weight particles in critical areas to increase sampling there; these paired techniques preserve the overall estimate but require careful biasing functions.^[47] Weight windows further enhance efficiency by setting upper and lower bounds on particle weights within spatial zones, applying roulette to overweight particles and splitting to underweight ones, guided by adjoint importance maps. Estimators in Monte Carlo neutron transport can be unbiased, where the expected value exactly matches the true quantity (e.g., track-length estimators for flux), or biased, introducing a controlled systematic error to reduce variance and improve efficiency for practical computations.^[48] Biased estimators, such as those using exponential biasing for attenuation, trade a small bias for significantly lower statistical uncertainty in shielding or criticality problems.^[49] The primary outputs of these simulations are tallies that accumulate contributions from simulated neutron histories to estimate quantities like scalar flux, reaction rates, or multiplication factors, with statistical errors quantified by the standard deviation of the tally divided by the square root of the number of contributing histories.^[50] This error estimation provides a confidence interval, enabling assessment of simulation reliability without assuming normality for large sample sizes.^[48]

Computational Implementations

Spatial and Angular Discretizations

In deterministic methods for solving the neutron transport equation, spatial discretization approximates the continuous position variable \mathbf{r} across the domain, enabling numerical solution on a mesh. Common approaches include finite difference methods (FDM), which approximate derivatives using Taylor expansions on structured grids, finite element methods (FEM), which use basis functions to represent the solution within elements, and finite volume methods (FVM), which conserve flux by integrating over control volumes.^[51] The characteristics method addresses the streaming term by following neutron trajectories along straight-line paths, inverting the transport operator exactly for the collisionless case and integrating sources along characteristics.^[52] These spatial schemes balance accuracy, stability, and computational cost, with FDM and FVM often preferred for their simplicity in reactor geometries, while FEM excels in handling complex boundaries.^[53] Angular discretization approximates the direction variable \boldsymbol{\Omega} on the unit sphere, typically via the discrete ordinates (S_N) method, which replaces the integral over directions with a quadrature sum using discrete directions and weights. Gauss-Legendre quadrature is widely adopted for its positive weights and high-order accuracy in even-parity formulations, where the angular flux is expanded in spherical harmonics or Legendre polynomials to reduce ray effects.^[54] Even-parity formulations derive from decomposing the flux into even and odd components, leading to coupled diffusion-like and transport equations that improve convergence for optically thick media.^[55] Odd-parity methods complement this by emphasizing antisymmetric behavior, enhancing stability in heterogeneous problems.^[56] Energy discretization employs the multigroup approximation, collapsing the continuous lethargy variable u = \ln(E_0/E) into discrete groups where cross-sections are averaged, assuming neutrons within a group have similar interactions.^[57] This reduces the problem to a system of coupled equations solved iteratively across groups, with lethargy ensuring uniform spacing for slowing-down spectra.^[58] Specific techniques like diamond differencing in FDM provide second-order accuracy and preserve positivity, recovering the diffusion limit asymptotically for low-absorption regimes by balancing cell-centered and edge fluxes.^[59] Transport-corrected scattering adjusts isotropic scattering assumptions by incorporating linear anisotropy, mitigating errors in forward-peaked collisions and ensuring consistency with the transport operator.^[60] Error analysis reveals truncation errors from spatial schemes, such as O(\Delta x^2) in diamond differencing, which dominate in fine meshes but degrade in coarse ones due to numerical diffusion.^[61] In angular space, aliasing arises from quadrature undersampling, causing ray effects and oscillatory solutions, with L^2 error bounds scaling as O(1/N) for S_N order, where N is the number of directions.^[62] These errors are quantified through Fourier analysis, emphasizing the need for adaptive quadrature to minimize aliasing in multidimensional problems.^[63]

Software Codes and Tools

Several deterministic codes solve the neutron transport equation using discrete ordinates (S_N) methods for shielding and reactor analysis. DORT and TORT, developed at Oak Ridge National Laboratory, perform two- and three-dimensional multigroup neutron/photon transport calculations, respectively, with TORT capable of handling large-scale problems on high-performance computers.^[64] PARTISN, a parallelized successor to the DANTSYS system from [Los Alamos National Laboratory](/page/Los Alamos National Laboratory), addresses one-, two-, and three-dimensional time-dependent neutral particle transport using the S_N approximation.^[65] TWODANT, part of the DANTSYS code package, focuses on two-dimensional Cartesian and cylindrical geometries for diffusion-accelerated transport solutions.^[66] Monte Carlo codes provide stochastic simulations for versatile neutron transport applications, including criticality and shielding. MCNP, developed at Los Alamos National Laboratory, is a general-purpose continuous-energy code for tracking neutrons, photons, and other particles in complex geometries; its MCNP6 release in 2013 introduced advanced variance reduction techniques, such as improved weight windows and hybrid forward-adjoint methods, to enhance efficiency in deep-penetration problems, with further enhancements in versions up to MCNP 6.3.1 (September 2025).^[67]^[68] Serpent 2.2.3 (October 2025), from VTT Technical Research Centre of Finland, specializes in three-dimensional continuous-energy transport for lattice physics, fuel depletion, and group constant generation.^[69] OpenMC 0.15.1 (March 2025), an open-source code from the Massachusetts Institute of Technology, supports fixed-source, eigenvalue, and subcritical multiplication calculations with modern constructive solid geometry and parallel capabilities.^[70]^[71] Hybrid approaches combine deterministic and Monte Carlo methods to leverage their strengths, such as using deterministic codes for global flux solutions and Monte Carlo for detailed local tallies. Denovo, integrated within the SCALE system at Oak Ridge National Laboratory, employs discrete ordinates for three-dimensional transport and generates variance reduction parameters for subsequent Monte Carlo simulations like MCNP.^[72] These codes rely on evaluated nuclear data libraries for cross-sections and other parameters. The ENDF/B series, maintained by the National Nuclear Data Center, provides comprehensive neutron cross-section data evaluated for transport applications, with the latest version ENDF/B-VIII.1 (October 2024) incorporating updated standards and thermal scattering laws.^[73] Integration with depletion codes such as ORIGEN, part of the SCALE suite, enables coupled transport-burnup analyses by using flux spectra from transport codes to drive isotopic evolution calculations.^[74] Validation of these tools often involves benchmarks from the International Handbook of Evaluated Criticality Safety Benchmark Experiments (ICSBEP), where MCNP and similar codes demonstrate agreement within experimental uncertainties for critical configurations.^[75]^[76]

Applications

Nuclear Reactor Design

Neutron transport theory is fundamental to nuclear reactor design, enabling engineers to predict neutron behavior in multiplying media and optimize core configurations for efficient power generation and safety. In fission reactors, such as pressurized water reactors (PWRs) and boiling water reactors (BWRs), transport calculations inform fuel loading patterns, reactivity control, and operational margins by solving the Boltzmann transport equation to determine flux distributions and reaction rates. These simulations ensure that the effective multiplication factor k_\mathrm{eff} remains close to unity during operation, with lattice-level transport providing the infinite multiplication factor k_\infty as a key input for whole-core criticality assessments. By accounting for geometric heterogeneities and material interactions, transport methods support the design of robust cores that minimize power imbalances and enhance fuel economy. A primary application of neutron transport in reactor design is pin-level flux mapping, which is essential for evaluating fuel burnup and power peaking factors to prevent hotspots and extend cycle lengths. In BWR cores, for example, three-dimensional transport simulations using codes like MCNP compute pin-by-pin power distributions, revealing how burnup gradients lead to intra-pin power variations that can affect cladding integrity. These calculations typically show power peaking factors increasing with local enrichment and decreasing with burnup, guiding the placement of gadolinia burnable absorbers to flatten the flux profile. Similarly, in PWR fuel pins, transport models predict peaking factors as high as 1.5-2.0 in high-burnup regions, informing limits on linear heat generation rates for safety. Lattice physics calculations, a cornerstone of reactor design, employ neutron transport to resolve heterogeneities within fuel assemblies, generating accurate multi-group cross-sections for subsequent core simulations. Using methods like the Method of Characteristics (MoC), two-dimensional transport solves the Boltzmann equation across assembly geometries, such as 10x10 BWR lattices with control blades, to compute flux tilts and Dancoff factors that account for neutron shadowing between pins. In BWR designs like GE14, these calculations use 23-group structures derived from fine-group libraries (e.g., ENDF/B-VIII.1) to model water gaps and cladding effects, yielding homogenized parameters that improve predictions of assembly-wise reactivity by up to 5% over diffusion approximations.^[73] For PWR assemblies, transport-based lattice codes like DRAGON handle hexagonal or square lattices, ensuring precise infinite lattice multiplicities for burnup-dependent analysis. Neutron transport is critical for assessing control and shutdown systems, particularly in evaluating control rod worth and reflector interactions to guarantee sufficient shutdown margins. In BWR bundles, Monte Carlo transport simulations demonstrate that inserting B₄C control rods reduces k_\mathrm{eff} by absorbing thermal neutrons, with rod worth values around 0.23 Δk/k, while also depressing peripheral flux and peaking it centrally due to enhanced water moderation. Reflector regions, modeled via transport to capture streaming and albedo effects, contribute positively to core reactivity but require detailed angular solutions to avoid underestimating leakage. In PWRs, similar transport evaluations quantify rod insertion worth, ensuring compliance with reactivity hold-up criteria during scram transients. In specific reactor designs, neutron transport underpins safety parameter evaluations, such as in the AP1000 PWR where Monte Carlo methods verify core flux and reactivity profiles against design control documents, confirming burnup distributions up to 60 GWd/tU. For BWRs, transport calculations are vital for determining the void coefficient, which measures reactivity changes with steam void fraction; using codes like SCALE with 238-group libraries, these show negative coefficients (e.g., -1 to -3 pcm/%void) up to 75% void, turning positive in MOX fuels due to spectrum hardening from plutonium isotopes. Enhanced moderation via water holes can maintain negativity across full void ranges, informing void-tolerant designs. To capture realistic operational dynamics, neutron transport is coupled with thermal-hydraulics models, incorporating feedback effects where coolant density and fuel temperature alter cross-sections and thus flux. Iterative schemes, such as those using MCNP6 for transport and single-channel hydraulics, automate updates for burnup-dependent parameters, predicting xenon oscillations and power tilts with errors below 2% against benchmark data from reactors like Yonggwang-3. This coupling reveals how void-induced density reductions soften the spectrum in BWRs, enhancing negative feedback, while in PWRs, Doppler broadening from temperature rises provides inherent stability. Neutron transport also plays a key role in the design of advanced reactors, including small modular reactors (SMRs) like the NuScale VOYGR, where detailed transport simulations optimize compact cores for passive safety and load-following capabilities. As of 2025, transport methods using modern codes and libraries support licensing and deployment of SMRs by predicting neutron economy in novel fuels and coolants.^[77]

Radiation Shielding and Protection

Neutron transport calculations play a crucial role in designing radiation shielding to attenuate neutron fluxes in environments such as accelerators, space vehicles, and medical facilities, ensuring personnel safety and equipment integrity by modeling the interaction of neutrons with shielding materials.^[78] These calculations often involve fixed-source problems to predict beam attenuation and flux distribution through barriers, providing essential data for optimizing shield thickness and composition.^[79] Shielding designs typically employ multilayer configurations to address the dual needs of slowing down fast neutrons via elastic scattering and capturing thermalized neutrons through absorption. Hydrogen-rich materials like concrete, which contains water of hydration, effectively moderate fast neutrons by transferring kinetic energy through collisions, reducing their speed to thermal levels where capture is more probable.^[4] Boronated water, incorporating boric acid for its high thermal neutron absorption cross-section (approximately 3840 barns for boron-10), serves as an efficient capture layer in liquid shields, often used in pools or circulating systems to enhance attenuation without structural rigidity.^[80] Multilayer setups, such as concrete followed by boron-infused polyethylene, combine moderation and absorption to minimize secondary radiation, with studies showing up to 90% reduction in transmitted neutron flux for 1-10 MeV energies in 50 cm thick assemblies.^[81] Dose assessments in neutron shielding rely on transport-derived kerma factors, which quantify the kinetic energy transferred to charged particles per unit mass from neutron interactions, enabling estimation of absorbed dose in tissue or materials.^[82] Equivalent dose calculations further account for secondary gamma rays produced via neutron capture (e.g., in hydrogen yielding 2.2 MeV gammas) or inelastic scattering, using quality factors to weight biological effectiveness; for instance, paraffin-boron carbide composites have demonstrated reductions in both neutron and secondary gamma equivalent doses by factors of 10-20 in simulated fields.^[83] These metrics ensure shielding limits effective dose to below 1 mSv/year for occupational exposure in shielded zones.^[84] In space applications, polyethylene shields on the International Space Station (ISS) leverage high hydrogen content to moderate galactic cosmic ray neutrons, outperforming aluminum by 20-30% in dose reduction for solar particle events, as validated by onboard dosimetry.^[85] For medical isotope production, such as molybdenum-99 via accelerator-driven neutron capture on uranium targets, shielding incorporates lead-concrete layers to attenuate 1-20 MeV neutrons and associated gammas, with Monte Carlo transport simulations confirming dose rates below 0.1 μSv/h outside 2 m thick barriers.^[86] Neutron streaming, where particles bypass direct shielding through ducts or openings, is mitigated in accelerator facilities using labyrinthine mazes with multiple bends to increase path length and promote scattering losses, reducing transmitted dose by orders of magnitude compared to straight paths.^[79] Removable plugs, often boron-loaded concrete or polyethylene blocks, seal access ports during operation, further attenuating streaming fluxes by 10^4-10^6 in high-energy proton accelerators like those at CERN.^[87] Regulatory verification of neutron shielding adheres to ANSI/ANS standards, such as ANSI/ANS-6.1.1-2020 for neutron and gamma-ray flux-to-dose-rate factors, ensuring calculated doses align with ALARA principles through validated transport codes and benchmark experiments.^[88] ANSI/ANS-6.1.2-2013 (R2023) provides group-averaged cross sections for shielding analyses, facilitating compliance in design reviews for facilities handling neutron sources.^[89]

Fusion Neutronics

Neutron transport is essential in fusion reactor design for modeling neutron production from deuterium-tritium reactions, predicting damage to structural materials, and optimizing tritium breeding blankets. In projects like ITER, as of 2025 under construction, transport calculations using codes such as MCNP assess neutron fluxes up to 14 MeV, ensuring shielding effectiveness and heat deposition in complex geometries. These simulations support safety assessments and material qualification for future fusion power plants.^[90]

References

[1]
[PDF] THE NEUTRON TRANSPORT EQUATION 1.1 Introduction
Deterministic neutronics methods play a fundamental role in reactor core modeling and simulation. A first-principles treatment requires solution of the ...
[2]
[PDF] An Introduction to the Neutron Transport Phenomena - INIS-IAEA
They are usually denoted by Q(r,Q,E,t), which expresses the probability per unit time that a neutron of energy E will appear at r per unit volume per unit solid ...
[3]
[PDF] Module 6: Neutron Diffusion Dr. John H. Bickel
• Diffusion theory is widely used in reactor analysis. • SECRET: “use transport theory to make diffusion theory work in specific areas where it would be ...
[4]
[PDF] The Foundations of Neutron Transport Theory
of the fundamentals of neutron transport theory, it seems appropriate to probe a little for the limitations on what can or cannot be asserted in this matter ...
[5]
[PDF] Interaction of Neutrons with Matter.
Apr 2, 2011 · Elastic scattering is the most likely interaction for almost all nuclides and neutron energies. • The greatest amount of energy can be ...Missing: definition | Show results with:definition
[6]
[PDF] Neutron induced resonance reactions - IAEA Nuclear Data Services
May 26, 2008 · Neutron induced nuclear reactions: • elastic scattering (n,n). • inelastic scattering (n,n'). • capture (n,γ). • fission (n,f). • particle ...<|control11|><|separator|>
[7]
[PDF] Module 3: Neutron Induced Reactions Dr. John H. Bickel
Total microscopic neutron cross section is expressed as: σ = dN/dt / [(I/A) n A ∆x]. • Defining neutron flux as: φ = I/A (neutrons/sec.cm2).
[8]
None
### Summary of Neutron Energy Regimes, Interaction Probabilities, and Moderation Process
[9]
[PDF] Module 5: Neutron Thermalization Dr. John H. Bickel
Neutron thermalization is the process of 1MeV fission neutrons slowing down to thermal energies via collisions, which increases fission rate.Missing: regimes | Show results with:regimes
[10]
[PDF] The Fission Spectrum - CERN Indico
The energy distribution of these fission spectrum neutrons is often represented by the so-called Watt Distribution. Neutrons evaporating during fission. These ...
[11]
[PDF] Introduction to neutron transport - HAL
Mar 26, 2025 · Neutron transport describes the behavior of a neutron population moving in a fixed medium, using transport theory to know how many neutrons ...
[12]
Gian-Carlo Wick and neutron physics in the 1930s
May 8, 2024 · Working with Fermi's group in Rome in the 1930s, he took on several behind-the-scenes roles that resulted in important papers in neutron physics.
[13]
Gian-Carlo Wick and neutron physics in the 1930s - EPJ H
May 8, 2024 · He clarified Fermi's methodology for calculating neutron slowing down probabilities; using transport theory, he provided a comprehensive general ...
[14]
[PDF] THE ATTENUATION PROPERTIES OF CONCRETE FOR ... - OSTI
By age theory and diffusion theory, neutron "buildup factors" are calculated which connect the fast flux with the fluxes in the two other groups: where B^(E,z) ...
[15]
The neutron flux in a spherical void - ScienceDirect.com
Neutron motion in the moderator is described by diffusion theory and that in the void by the free streaming Boltzmann transport equation. An explicit solution ...
[16]
Effects of random pebble distribution on the multiplication factor in ...
In pebble beds, neutrons can stream through the void space in between the pebbles, resulting in an increase of neutron transport. When homogenizing the pebble ...
[17]
Improved Methods for Calculating the Neutron Diffusion Coefficient ...
There exists a neutron streaming effect in the high-temperature gas-cooled pebble-bed reactor (HTGR) pebble-bed core caused by the spatial heterogeneity of the ...
[18]
[PDF] General Principles of Neutron Transport - The University of Bath
Abstract: This chapter describes the basic theory underlying the neutron transport equation and the principal approximations used in this equation's ...Missing: seminal | Show results with:seminal
[19]
[PDF] LOS ALAMOS SCIENTIFIC LABORATORY of the - MCNP
A vacuum boundary condition is assumed;. i.e., the neutron flux in the incoming CtI.rectionson the boundary is zero. The space-clirectionmesh and calculation.
[20]
[PDF] Multi-Level Methods for Monte Carlo Reactor Kinetics
In order to solve the forward transport equation, both boundary and initial conditions must be specified. ... Common albedo conditions include: vacuum (α = 0), ...
[21]
None
Below is a merged summary of reflective boundary conditions in neutron transport, consolidating all information from the provided segments into a comprehensive response. To retain as much detail as possible, I will use a structured table format in CSV style for key concepts (reflective boundary conditions, specular reflection, and diffuse reflection with albedo), followed by additional narrative details and useful URLs. This approach ensures clarity, density, and completeness while avoiding redundancy.
[22]
[PDF] UWFDM-1208 Verification and Analysis of the Radiation Transport ...
any boundary condition [5]: AE − (n ·r)BD. ∂E. ∂r. = C. (19). Table 1 lists the factors A, B, and C for Dirichlet, source, vacuum, and albedo boundary.
[23]
[PDF] Efficient solution of the simplified PN equations - OSTI.GOV
Dec 30, 2014 · The SPN Marshak boundary conditions are given in Eq. (34) for vacuum boundaries. Equation (38) gives reflecting boundary conditions. Each ...
[24]
[PDF] ADVANCED REACTOR PHYSICS METHODS FOR ... - PSU-ETD
neutron transport [41], Vladimirov demonstrates that the Marshak boundary conditions are optimal approximations of the exact boundary conditions. The ...
[25]
Existence and Uniqueness Theorems for the Neutron Transport ...
INTEGRAL EQUATION FORMULATION. A. The Transport Equations. The time-dependent transport equation is aif;(r, v, t)/at. + v(a·v + IT(r, v»if;(r, v, t) = q(r, v ...<|separator|>
[26]
[PDF] PROPERTIES OF NEOTRON SOURCES - Publications
Neutron sources are widely used not only in basic research but also in a large number of applications: bio-medical, bore hole and bulk media.
[27]
[PDF] Physics and Technology of Spallation Neutron Sources - OSTI.GOV
Aug 6, 1998 · Next to fission and fusion, spallation is an efficient process for releasing neutrons from nuclei. Unlike the other two.Missing: external | Show results with:external
[28]
[PDF] Lecture Notes on Neutronics of Multiplying Media - MIT
This is what is known as the neutron transport equation for a homogeneous medium. ... Combining this with (58) we obtain for the extrapolated distance d = 2D.
[29]
[PDF] lec10.pdf - DSpace@MIT
One can use transport theory to do a better calculation of the extrapolated distance δ . We have seen that in simple diffusion theory this turns out to be ...
[30]
8.2. Monaco: A Fixed-Source Monte Carlo Transport Code for ...
Monaco is a general-purpose, fixed-source, Monte Carlo shielding code for the SCALE package, calculating neutron and photon fluxes. It is based on MORSE.
[31]
[PDF] Iterative Methods for Criticality Computations in Neutron Transport ...
Jan 5, 2011 · This thesis studies the so-called “criticality problem”, an important generalised eigenvalue problem arising in neutron transport theory.Missing: seminal | Show results with:seminal
[32]
Calculating Criticality | Los Alamos National Laboratory
Nov 1, 2023 · Effective neutron multiplication factor If k = 1, the reaction is critical, and the neutron population will remain constant. Finally, if k is ...
[33]
Manhattan Project: CP-1 Goes Critical, Met Lab, December 2, 1942
The neutron counter abruptly slowed; the chain reaction was over. The pile had achieved a reproduction factor k of 1.0006. Eugene Wigner then produced a ...
[34]
[PDF] ANALYSIS OF TIME-EIGENVALUE AND EIGENFUNCTIONS IN THE ...
Time-dependent neutron transport in non-critical state can be expressed by the natural mode equation. In order to estimate the dominant eigenvalue and ...
[35]
Calculating Time Eigenvalues of the Neutron Transport Equation ...
Oct 25, 2018 · A novel method to compute time eigenvalues of neutron transport problems is presented based on solutions to the time dependent transport equation.<|separator|>
[36]
[PDF] Deterministic Transport Methods and Codes at Los Alamos
2.1 Carlson's Original Sn Method. The original Sn method ... tional Efficiency of Numerical Methods for the Multigroup, Discrete-Ordinates Neutron Transport.
[37]
The discrete ordinate method for angular flux calculations in slab ...
The idea of the discrete ordinates approach is to approximate the angular flux at a set of spatial mesh points and in a number of fixed directions.
[38]
[PDF] The Early Days of the Sn Method* K. D. Lathrop .^
K.D. Lathrop and B.G. Carlson, “Discrete Ordinates Angular Quadrature of the. Neutron Transport Equation”, LA-3186, Los Alamos Scientific Laboratory (1965) ...
[39]
A numerically stable spherical harmonics solution for the neutron ...
The spherical harmonics method, also known as P N method when applied to azimuthally symmetric problems, first appeared in the neutron transport literature in ...Missing: PN seminal
[40]
[PDF] AUSTRALIAN ATOMIC ENERGY COMMISSION - INIS-IAEA
Collision probability methods offer some significant advantages in cluster geometry neutron flux calculations. The alternative diffusion and Sn transport ...
[41]
Review of the Discrete-Ordinates Method for Particle Transport in ...
The SN method is a cornerstone numerical technique for solving particle transport equations. Here, 'N' represents the order of the angular quadrature, ...2.5. Iterative Methods · 4. Advancements In S... · 4.3. Acceleration Schemes...
[42]
https://web.mit.edu/22.54/ClassNotes/Lec2.pdf
[43]
Hitting the Jackpot: The Birth of the Monte Carlo Method | LANL
Nov 1, 2023 · First conceived in 1946 by Stanislaw Ulam at Los Alamos† and subsequently developed by John von Neumann, Robert Richtmyer, and Nick Metropolis.Missing: 1940s | Show results with:1940s
[44]
[PDF] The Monte Carlo Method - Nicholas Metropolis; S. Ulam
Jan 25, 2006 · The method is, essentially, a statistical approach to the study of differential equations, or more generally, of integro-differential equations ...Missing: transport 1940s
[45]
Performance of Woodcock delta-tracking in lattice physics ...
This paper presents the delta-tracking based geometry routine used in the Serpent Monte Carlo reactor physics burnup calculation code.
[46]
Performance of Woodcock Delta-Tracking in Lattice Physics ...
Aug 6, 2025 · This paper presents the delta-tracking based geometry routine used in the Serpent Monte Carlo reactor physics burnup calculation code.
[47]
Variance-Reduction Methods for Monte Carlo Simulation ... - Frontiers
Popular variance-reduction techniques applicable to Monte Carlo simulations of radiation transport are described and motivated.Abstract · Introduction · Statistical Uncertainty... · VRTs in Transport Simulations
[48]
[PDF] Analysis of Error in Monte Carlo Transport Calculations - OSTI.GOV
The Monte Carlo method for neutron transport calculations suffers, in part, because of the inherent statistical errors associated with the method. Without an ...
[49]
Comparison of biased and unbiased simulations - ResearchGate
The method biases the scattering angles and distances between neutron collisions so that neutrons get scattered into more important places, which is compensated ...
[50]
[PDF] Monte Carlo methods for Neutron Transport
Estimator is unbiased in the limit, variance of estimator ~ (1 / N). Can also show: many similar methods are possible and also work… As h → 0, can estimate ...
[51]
Calculation of λ modes of the multi-group neutron transport equation ...
There are several spatial discretization methods. They are generally classified into three groups: Finite Difference Methods (FDM), Finite Volume Methods (FVM) ...
[52]
On a characteristic method for the SN neutron transport equation
Characteristic methods form a class of spatial approximations in which the streaming-plus-collision operator of the transport equation is inverted exactly ...
[53]
A Finite Element Method for the Neutron Transport Equation in an ...
We study the spatial discretization, in a fully discrete scheme, for the numerical solution of a model problem for the neutron transport equation in an ...
[54]
[PDF] PROTEUS-SN Methodology Manual - Argonne Scientific Publications
(4.1) is the time dependent neutron transport equation with delayed neutrons, which includes a source term corresponding to the production of delayed ...
[55]
[PDF] Nonlinear acceleration methods for even-parity neutron transport
Perhaps the most popular of these methods is the Sn or discrete ordinates method. This method reduces the equation by using a given set of discrete angular ...Missing: quadrature | Show results with:quadrature
[56]
Variational nodal methods for neutron transport: 40 years in review
The variational nodal method for solving the neutron transport equation has evolved over 40 years. Based on a functional form of the Boltzmann neutron ...
[57]
[PDF] 1 THE NEUTRON TRANSPORT EQUATION
4.4g Outer Iterations in the Multigroup P, Approximation. 197. 4.4h ... 4.7a The Lethargy Variable. 207. 4.76 Elastic Scattering in Terms of Lethargy.
[58]
[PDF] The Solution of the Multigroup Neutron Transport Equation Using<br ...
energy group g is approximated by a linear combination of seroth and first order spherical harmonics and it seemed reasonable to increase the solution accuracy ...
[59]
[PDF] Asymptotic Diffusion-Limit Accuracy of Sn Angular Differencing ...
Nov 17, 2009 · This unphysical behavior of the step and diamond differencing methods is caused by their inconsistency to the analytic transport equation ...
[60]
[PDF] Linear Diffusion Acceleration for Neutron Transport Problems
... equations on the coarse grid from the transport equation for fixed-source problems. The LDA equations will consist of. (i) a coarse-grid balance equation ...
[61]
Behavior of the Diamond Difference and Low-Order Nodal ...
The objective of this work is to investigate the thick diffusion limit of various spatial discretizations of the one-dimensional, steady-state, monoenergetic, ...
[62]
(PDF) L2-error estimates for the discrete ordinates method for three ...
Aug 7, 2025 · We prove L2-error estimates for the discrete ordinates method for the angular discretization of the three-dimensional neutron transport ...
[63]
A Quantitative Theory of Angular Truncation Errors in Three ...
A quantitative theory of angular truncation errors is developed for three-dimensional discrete-ordinates (SN) particle transport calculations.
[64]
Code Package CCC-650 - rsicc - Oak Ridge National Laboratory
TORT: Three-dimensional neutron/photon transport. DORT: Two-dimensional neutron/photon transport. ANISN: One-dimensional neutron/photon transport. GBANISN ...Missing: deterministic | Show results with:deterministic
[65]
ccc-0842 partisn 8.29. - Nuclear Energy Agency
Jan 24, 2019 · PARTISN solves the linear Boltzmann transport equation for neutral particles using the deterministic (SN) method.
[66]
ccc-547/dantsys 3.0 - rsicc
DANTSYS is a one-, two-, and three-dimensional, multigroup, discrete-ordinates transport code system, which includes five major codes: ONEDANT, TWODANT, TWOHEX, ...
[67]
MCNP® Website
The MCNP, Monte Carlo N-Particle, code can be used for general-purpose transport of many particles including neutrons, photons, electrons, ions, and many other ...How To Get The MCNP Code · Manual · Nuclear Data · Classes
[68]
[PDF] LA-UR-13-22900 - MCNP
May 29, 2013 · If using the eprdata with both pulse height tally variance reduction and DXTRAN spheres (which should give a fatal error), a format syntax error ...
[69]
Serpent - a Continuous-energy Monte Carlo neutron and photon ...
Serpent is a multi-purpose three-dimensional continuous-energy neutron and photon transport code, developed at VTT Technical Research Centre of Finland since ...Serpent Wiki · Discussion forum for Serpent... · Validation and verification<|separator|>
[70]
OpenMC
OpenMC is a community-developed Monte Carlo neutron and photon transport code. It is capable of performing fixed source, k-eigenvalue, and subcritical ...
[71]
9. Deterministic Transport — SCALE 6.3.2 documentation
Denovo [DTESSC10], is a parallel 3D discrete-ordinates code available in SCALE as part of two control module sequences for different applications, as described ...
[72]
ENDF/B-VIII.0 Evaluated Nuclear Data Library
ENDF/B-VIII.0 fully incorporates the new Neutron Data Standards, includes improved thermal neutron scattering data and uses new evaluated data.Download Library · How to Reference? · Errata
[73]
5.4. ORIGAMI: A Code for Computing Assembly Isotopics with ORIGEN
ORIGAMI computes detailed isotopic compositions for light water reactor assemblies using ORIGEN, performing isotopic depletion and decay calculations.
[74]
International Criticality Safety Benchmark Evaluation Project (ICSBEP)
ICSBEP work includes: Identifying a comprehensive set of critical benchmark data and, to the extent possible, verify the data by reviewing original and ...Missing: MCNP | Show results with:MCNP
[75]
[PDF] MCNP6 Criticality Calculations Verification and Validation Report
Apr 30, 2015 · This Criticality Validation Suite consists of 31 benchmarks taken from the International Criticality Safety. Benchmark Evaluation Project ( ...<|control11|><|separator|>
[76]
https://mcnp.lanl.gov/pdf_files/TechReport_2015_LANL_LA-UR-15-23266_SartorGreenfieldEtAl.pdf
[77]
[PDF] Accelerators, Targets and Irradiation Facilities –SATIF 7 - OECD
Therefore, it is essential to estimate radiation streaming through the mazes ... In order to validate the accuracy of calculation methods on neutron streaming ...
[78]
Radiologic assessment of a self-shield with boron-containing water ...
Boric acid is used as a neutron-absorbing material in the primary cooling water of a pressurized water reactor in nuclear power plants. There are only two ...
[79]
New shielding designed to put the block on neutrons - DOE Pulse
Jan 20, 2014 · It consists of a concrete layer to slow down neutrons, a material to absorb them and a thin lead layer to halt any residual radiation produced ...
[80]
Neutron kerma coefficients of compounds for shielding and dosimetry
The neutron kerma coefficients of eighteen compounds have been calculated for neutrons energy less than 30 MeV; these compounds are used as shielding and ...
[81]
The evaluation of neutron and gamma ray dose equivalent ...
Our simulation results showed that paraffin wax containing boron carbide presents enough effectiveness to reduce both neutron and secondary gamma ray doses.
[82]
[PDF] SHIELDS FOR A SPACE POWER REACTOR by Gerald P. Lahti and ...
Both materials generate secondary gamma rays by neutron capture and neutron inelastic scattering reactions ... Neutron Fluence-to-. Kerma Factors for the Human ...
[83]
Performances of Kevlar and Polyethylene as radiation shielding on ...
May 10, 2017 · Polyethylene is widely used for radiation shielding in space and therefore it is an excellent benchmark material to be used in comparative ...
[84]
Shielding Assessment and Optimization of the Target Station for ...
Feb 3, 2023 · The secondary particles generated, such as neutrons and photons, must be properly shielded for the safety of personnel and avoidance of damage ...
[85]
Neutron and capture gamma along the mazes of linear accelerator ...
In this work, the neutron and photon dose equivalents were measured along the mazes for several accelerator vaults with maze lengths up to 9 m.Missing: plugs | Show results with:plugs
[86]
[PDF] Chapter 6 - Shielding Evaluation. - Nuclear Regulatory Commission
The NRC shielding evaluation ensures adequate protection against direct radiation, covering design, radiation source, model, and analyses, to limit radiation ...
[87]
https://pmc.ncbi.nlm.nih.gov/articles/PMC5724469/