Neutron Transport Theory Research Articles

Particle Propagation and Electron Transport in Gases

In this review, we detail the commonality of mathematical intuitions that underlie three numerical methods used for the quantitative description of electron swarms propagating in a gas under the effect of externally applied electric and/or magnetic fields. These methods can be linked to the integral transport equation, following a common thread much better known in the theory of neutron transport than in the theory of electron transport. First, we discuss the exact solution of the electron transport problem using Monte Carlo (MC) simulations. In reality we will go even further, showing the interpretative role that the diagrams used in quantum theory and quantum field theory can play in the development of MC. Then, we present two methods, the Monte Carlo Flux and the Propagator method, which have been developed at this moment. The first one is based on a modified MC method, while the second shows the advantage of explicitly applying the mathematical idea of propagator to the transport problem.

Mathematical formulae for neutron self-shielding properties of media in an isotropic neutron field

In the current study, an ab initio derivation of the neutron self-shielding factor to solve the complex neutron transport problem of the decrease of the neutron flux as it penetrates into a material placed in an isotropic neutron field having equal flux in all directions. The theory of steady-state neutron transport was employed, starting from Stuart’s formula, to derive simple analytical formulae based on the integral cross-section parameters. The formulae could be adopted by the user according to various variables, such as the neutron flux distribution and geometry of the simulation at hand. The concluded formulae of the self-shielding factors comprise an inverted sigmoid function normalized with a weight representing the ratio between the macroscopic total and scattering cross-sections of the medium. The general convex volume geometries are reduced to a set of chord lengths, while the neutron interaction probabilities within the volume are parameterized to the epithermal and thermal neutron energies. The arguments of the inverted-sigmoid function were derived from a simplified version of neutron transport formulation. The derived analytic formulae agreed greatly with the experimental observations for different elements and geometries.

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Comparative study of neutronic, mechanical and thermodynamic properties of accident tolerant cladding materials: SiC, TiC and ZrC

In the framework of the using carbide materials as alternative nuclear fuel cladding material, an investigation has been performed to obtain neutronic, mechanical and thermodynamic behaviors under different conditions and high temperature. To achieve this goal, we have used neutron transport theory methods to study neutronic performance and density functional theory methods for mechanical and thermodynamic kinetics of carbides, SiC, ZrC and TiC. In the neutronic calculations, we have compared the results of carbides with widely used cladding materials, such as baseline Zircaloy, 304SS, 310SS, FeCrAl and APMT. On the other hand, the carbides have been analyzed from the perspective of mechanical and thermodynamic properties. The obtained lattice parameters, elastic constants, elastic modulus, Poisson’s ratio, hardness, heat capacity, Debye temperature distribution, enthalpy, free energy and entropy have been discussed. The presented results advance underlying mechanisms of SiC, TiC and ZrC and guide future research, such as fuel-cladding systems, structural materials and new generation nuclear reactor systems.

Percolation properties of the neutron population in nuclear reactors.

Reactor physics aims at studying the neutron population in a reactor core under the influence of feedback mechanisms, such as the Doppler temperature effect. Numerical schemes to calculate macroscopic properties emerging from such coupled stochastic systems, however, require us to define intermediate quantities (e.g., the temperature field), which are bridging the gap between the stochastic neutron field and the deterministic feedback. By interpreting the branching random walk of neutrons in fissile media under the influence of a feedback mechanism as a directed percolation process and by leveraging on the statistical field theory of birth death processes, we will build a stochastic model of neutron transport theory and of reactor physics. The critical exponents of this model, combined with the analysis of the resulting field equationinvolving a fractional Laplacian, will show that the critical diffusion equationcannot adequately describe the spatial distribution of the neutron population and shifts instead to a critical superdiffusion equation. The analysis of this equationwill reveal that nonnegligible departure from mean-field behavior might develop in reactor cores, questioning the attainable accuracy of the numerical schemes currently used by the nuclear industry.

A Stochastic Transport Model for the Cumulative Number of Fissions and Deposited Fission Energy

The stochastic theory of neutron transport is extended to describe the cumulative distribution of fission numbers and deposited fission energy in a subvolume of a multiplying assembly. Solutions for the probability distributions are obtained using analytical approximations and Monte Carlo simulation in lumped geometry and in symmetric homogeneous and heterogeneous spheres. The results show the development of a power-law tail in the steady-state fission number and deposited energy distributions when the medium is critical, independent of the fission neutron multiplicity distribution and domain heterogeneity. In contrast, the asymptotic decay is faster than exponential in subcritical media due to rapid chain extinction and in supercritical media due to the increasing probability of chain divergence. A formal asymptotic analysis of the problem in lumped geometry with an arbitrary fission neutron multiplicity confirms the existence of power-law tails at critical.

Development of Uncertainty Analysis Techniques for the Fission Matrix–Based Neutron Transport Code RAPID

The combined fission matrix (CFM) method is a newly developed neutron transport theory. This method estimates the fission matrix of the reactor core or spent fuel pool by combining a set of database fission matrices. The RAPID neutron transport code based on the CFM routine was developed originally for the spent fuel storage system and has been applied to the reactor core calculation in recent years. It can perform high-fidelity whole-core transport calculations within minutes. However, since the fission matrix database is obtained from Monte Carlo calculations, the uncertainty in the fission matrix will inevitably pass to its eigenvalue and eigenvector. The RAPID code also uses the fission matrix homogenization and interpolation techniques to further improve the calculation efficiency. Therefore, it is difficult to establish a relationship between the fission matrix elements’ uncertainty and the resulting eigenvalue and eigenvector uncertainties. This paper proposes two uncertainty analysis methods to obtain the eigenvalue and eigenvector uncertainties. The fission matrix resampling method resamples the database fission matrix elements according to each individual uncertainty. This method could generate many fission matrix databases at little additional costs and analyze the eigenvalue and eigenvector uncertainties from these resampled fission matrix coefficients. The analog uncertainty analysis method predicts the eigenvalue uncertainty from the uncertainty of the total fission rate in a fixed-source calculation, which yields a fission matrix column. Both uncertainty analysis methods have been validated against the reference brute-force calculations on a single-pin model and the BEAVRS whole-core model. It shows that the fission matrix resampling method could well estimate the uncertainties in the fission matrix eigenvalue and eigenvector. The analog uncertainty analysis method can accurately predict the eigenvalue uncertainty, which provides a guideline for the number of neutron histories simulated per fixed-source calculation.

Diffusive limits of 2D well-balanced schemes for kinetic models of neutron transport

Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering S-matrix and corresponding "truly 2D well-balanced" numerical schemes. A first scheme is obtained by directly implementing truncated Fourier–Bessel series, whereas another proceeds by applying an exponential modulation to a former, conservative, one. Consistency with the asymptotic damped parabolic approximation is checked for both algorithms. A striking property of some of these schemes is that they can be proved to be both 2D well-balanced and asymptotic-preserving in the parabolic limit, even when setting up IMEX time-integrators: see Corollaries 3.4 and A.1. These findings are further confirmed by means of practical benchmarks carried out on coarse Cartesian computational grids.

Skin effect in neutron transport theory.

We identify a neutron-flux "skin effect" in the context of neutron transport theory. The skin effect, which emerges as a boundary layer at material interfaces, plays a critical role in a correct description of transport phenomena. A correct accounting of the boundary-layer structure helps bypass computational difficulties reported in the literature over the last several decades, and should lead to efficient numerical methods for neutron transport in two and three dimensions.

Simulation of the burnup in cell calculation using the wimsd-5b code considering different nuclear data libraries

This work proposes to implement the cell calculation considering the fuel burning using the WIMSD-5B code. The cell calculation procedure allows determining the nuclear parameters present in the multi-group neutron diffusion equation and for this purpose the neutron transport theory is used in a problem with dimensional reduction, but in contrast is considered a large number of groups associated with the neutron spectrum. There are a variety of reactor physics codes that determine the nuclear parameters by solving the neutron transport equation applied to an equivalent cell representing a fuel element. The WIMSD-5B code is a deterministic code that solves the transport equation using collision probability method. The simulation of fuel burning in the cell calculation took into account different nuclear data libraries. The WIMSD-5B code supports several nuclear data libraries and in the present work the following libraries were used: IAEA, ENDFB-VII.1, JENDL3.2, JEFF3.1 and JEF2.2, all formatted for 69 energy groups.

Stochastic Theory of Neutron Transport in Nuclear Reactor

The work is an extended version of the report presented at the conference ICTT-26. In the Part I we derive the forward and backward time-dependent linear stochastic equations for probability density of the integer number of neutrons and delayed neutron precursors in distributed model of a nuclear reactor. In the Part II we derive the distributed criticality stochastic equations and obtain the analytical solutions for asymptotic neutron number probability density function for a reactor in a close-to-critical state.

Uniqueness for branching diffusion process with absorbing boundary

With suitable assumptions, uniqueness of nontrivial nonnegative solution is established for the following nonlinear integral equation This result is applied to branching diffusion processes with absorbing boundary. In particular, a question existing in the study by Pazy and Rabinowitz [A nonlinear integral equation with applications to neutron transport theory. Arch Ration Mech Anal. 1969;32:226-246] is solved.

Stochastic methods for the neutron transport equation II: Almost sure growth

The neutron transport equation (NTE) describes the flux of neutrons across a planar cross-section in an inhomogeneous fissile medium when the process of nuclear fission is active. Classical work on the NTE emerges from the applied mathematics literature in the 1950s through the work of R. Dautray and collaborators (Méthodes Probabilistes Pour les équations de la Physique (1989) Eyrolles; Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6: Evolution Problems. II (1993) Springer; Mathematical Topics in Neutron Transport Theory: New Aspects (1997) World Scientific). The NTE also has a probabilistic representation through the semigroup of the underlying physical process when envisaged as a stochastic process (cf. Méthodes Probabilistes pour les équations de la Physique (1989) Eyrolles; Introduction to Monte-Carlo Methods for Transport and Diffusion Equations (2003) Oxford Univ. Press; IMA J. Numer. Anal. 26 (2006) 657–685; Publ. Res. Inst. Math. Sci. 7 (1971/72) 153–179). More recently, Cox et al. (J. Stat. Phys. 176 (2019) 425–455) and Cox et al. (2019) have continued the probabilistic analysis of the NTE, introducing more recent ideas from the theory of spatial branching processes and quasi-stationary distributions. In this paper, we continue in the same vein and look at a fundamental description of stochastic growth in the supercritical regime. Our main result provides a significant improvement on the last known contribution to growth properties of the physical process in (Publ. Res. Inst. Math. Sci. 7 (1971/72) 153–179), bringing neutron transport theory in line with modern branching process theory such as (Ann. Probab. 44 (2016) 235–275; Ann. Probab. 43 (2015) 2545–2610). An important aspect of the proofs focuses on the use of a skeletal path decomposition, which we derive for general branching particle systems in the new context of nonlocal branching generators.

Extension to quadratic anisotropic scattering in one-speed neutron transport equation for critical thickness calculations / Berücksichtigung der quadratischen anisotropen Streuung bei der Lösung der 1-Geschwindigkeites-Neutronentransportgleichung zur kritischen Dickenberechnung

Abstract The effect for different types of scattering on the critical half thickness in slab geometry for one speed neutron transport equation is studied for isotropic, linearly anisotropic and quadratic anisotropic scatterings. An extensive numerical survey is carried out for the critical thickness in order to provide the effect of the different scattering types. The numerical results are obtained by PN, TN and UN methods. The PN method is the Legendre polynomial solution that is accepted as the exact result for the neutron transport theory calculations and the UN and TN methods are the types of Chebyshev polynomials. Critical thickness values are calculated by using Mark boundary condition. Results are compared with the literature

Dynamic behavior in fast burst reactor with three-dimensional coupled multiphysics method

Abstract In this paper, the neutronic and thermoelastic behaviors of prompt supercritical bursts in the fast burst reactor are numerically studied by developing a 3D coupled multiphysics FEM (finite element method) code FBR-MPC based on the neutron transport theory for neutronics, thermal conduction equation for temperature and thermoelastic equations for displacement/stress. Because the feedback effect of core expansion on the neutronics can be accurately modeled by reforming the discretized element and changing the material density, the developed code is expected to be applicable for simulating the dynamic behaviors of all the bursts under controlled condition. To validate the developed code and obtain precise pictures of the dynamic behaviors, the slow, medium and fast bursts in the Godiva I reactor are investigated. The results show that the developed code can provide accurate results for all the three selected bursts. For the medium and fast bursts where the inertia effect is generated, the fission rate oscillation caused by the core vibration, which cannot be reproduced with the simplified models in early works, is first observed by using the developed FBR-MPC code. Due to the small core expansion and short burst time, the point kinetic and adiabatic approximations are suitable for the neutronic and temperature calculations for all the three selected bursts. At the time of displacement wave crest, the stress concentration in the core central regions becomes more significant for faster bursts.

On a resolvent approach for perturbed semigroups and application to $$L^1$$ L 1 -neutron transport theory

We give new sufficient and practical conditions in terms of the generators ensuring the stability of the critical or the essential type of a perturbed $$C_0$$ -semigroup in general Banach spaces. We apply our theoretical results in order to investigate the control and in particular the time asymptotic behavior of solutions to a broad class of transport equations in $$L^1$$ -spaces and higher dimension. Our results improve, complete and enrich several earlier works.

Neutron total cross section calculation within the framework of quasi-harmonic approximation

The accuracy of neutron scattering cross sections is the gauge for the realistic outcome of a neutron transport simulation. To improve the traditional harmonic physics model used in such simulations, we revisit the slow neutron transport theory in crystalline materials and aim to develop a unified model that has good performance for neutron transport problems in crystals in a wide range of temperatures and pressures. The quasi-harmonic approximation (QHA) correlates phonon evolution explicitly with unit cell volume. Therefore, it is capable of evaluating a variety of material properties at finite temperatures. In this work, we show numerically that it is a very effective tool for our application as well. Within the framework of QHA, we calculate the temperature dependent characteristics of phonons in three elemental crystals, namely Be, Mg and Al. Based on the obtained results, our calculated neutron total cross sections agree closely with experimental transmission cross sections in a large temperature range below the melting point. We show that as the harmonic cross section model ignores the effects of phonon softening in these crystals, it underestimates the total inelastic cross sections at high temperatures. In the case of Al, we observe that such underestimation is up to 7% at room temperature. In addition, we study the phonon–phonon scatterings in Al. We observe that the cross section is insensitive to the finite phonon lifetimes even at 800 K.

Milne problem for non-absorbing medium with extremely anisotropic scattering kernel in the case of specular and diffuse reflecting boundaries

The Milne problem is studied in one speed neutron transport theory using the linearly anisotropic scattering kernel which combines forward and backward scatterings (extremely anisotropic scattering) for a non-absorbing medium with specular and diffuse reflection boundary conditions. In order to calculate the extrapolated endpoint for the Milne problem, Legendre polynomial approximation ( $$P_{N}$$ method) is applied and numerical results are tabulated for selected cases as a function of different degrees of anisotropic scattering. Finally, some results are discussed and compared with the existing results in literature.

Improvement and verification of the fast reactor neutronics code NECP-SARAX

The NECP-SARAX code developed by NECP research team is used for the fast reactor analysis. In order to consider the complex neutron slowing down effects from scattering resonance of nuclides with medium mass and inelastic scattering, the Monte-Carlo method was used previously but brought big uncertainty in tallying the spectrum in low energy. Besides, the efficiency is extremely low. In the improvement, a new resonance method based on ultra-fine group and point-wise cross section was involved. It generated effective multi-group cross sections with similar accuracy of Monte-Carlo method but avoided its disadvantages. In the core calculation, the multi-group cross sections were used for the 3D transport calculation. The first-order and exact perturbation theories were used in calculating the reactivity coefficients of fast reactor. To be capable of handling the geometric distorting in fast reactor, a new pseudo density method based on neutron transport and perturbation theory was proposed. Both the point kinetic and quasi-steady method were used in the transient calculation as two options to fit the requirement of fast reactor and ADS. The OECD fast reactor benchmarks were tested to verify the SARAX code, the results show good accuracy in the fast reactor calculations, which is similar to other widely accepted fast reactor code. The relative error of keff is less than 300 pcm compared with the Monte-Carlo code. Based on the pseudo density model and three-dimensional spatially dependent kinetics model, the SARAX code has more complete functions to fit the requirements of fast reactor design better.

Exact solution of the neutron transport equation in spherical geometry

Abstract Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using PN approximation which is widely used in neutron transport theory.