Time-dependent Neutron Transport Research Articles

Solution of the time-dependent monoenergetic neutron transport equation in a semi-infinite medium

The multiple collision technique, also known as the method of successive approximation, is used in conjunction with generalized functions and the principle of invariance for reflection to solve the monoenergetic time-dependent neutron transport equation in a semi-infinite medium. General plane geometry with azimuthal dependence and anistropic scattering expressed as a truncated series of spherical harmonics is assumed.

The Generation of Time-Dependent Neutron Transport Solutions in Infinite Media

The multiple collision technique as applied to the monoenergetic time-dependent neutron transport equation for pulsed plane source emission in an infinite medium is used to obtain the flux due to a pulsed point source in the same medium. This result is then integrated to determine the flux due to the corresponding pulsed line source problem. The semi-infinite albedo problem is also shown to be solvable using the multiple collision approach. A generalization to include delayed neutrons follows directly from the multiple collision treatment, as does an equivalence between a monoenergetic time-dependent problem and a particular stationary slowing down problem in infinite geometry. Results are tabulated and comparisons are made to provide benchmark solutions to the fundamental time-dependent transport problems considered and thus bridge the gap between theory and practice.

Time dependent neutron transport theory in multigroup-P L-approximation

A method is developed for solving the time dependent neutron transport equation in multigroupPL approximation for one-dimensional geometries. The partial differential equations in time and space are solved by means of a power series expansion in the spatial variable. The resulting ordinary differential equations are solved up to theNth order, and the last spatial coefficients are used to satisfy the boundary conditions. Integration of the purely time dependent differential equations is carried out by means of Lie series.

Prediction of Time-Dependent Neutron Fluxes Encountered in Pulsed-Neutron Uranium Logging Experiments

One-dimensional time-dependent neutron transport calculations were performed to predict the signals that would be observed in pulsed-neutron uranium logging experiments. Although the actual experimental geometry had to be idealized somewhat to conform to the one-dimensional nature of the calculations, the results clearly show the essential features of the logging signal. These calculations are economical enough that parameter studies are feasible.

Positive solution of a time and energy dependent neutron transport problem

The purpose of this paper is to give a constructive method for the determination of a solution and an existence–uniqueness theorem for some nonlinear time and energy dependent neutron transport problems, including the linear transport system. The geometry of the medium under consideration is allowed to be either bounded or unbounded which includes the geometry of a finite or infinite cylinder, a half-space and the whole space Rm (m=1,2,⋅⋅⋅). Our approach to the problem is by successive approximation which leads to various recursion formulas for the approximations in terms of explicit integrations. It is shown under some Lipschitz conditions on the nonlinear functions, which describe the process of neutrons absorption, fission, and scattering, that the sequence of approximations converges to a unique positive solution. Since these conditions are satisfied by the linear transport equation, all the results for the nonlinear system are valid for the linear transport problem. In the general nonlinear problem, the existence of both local and global solutions are discussed, and an iterative process for the construction of the solution is given.

The Extended PL Approximation of the Multidimensional Neutron Transport Equation

A modified truncation of the P1 equations for the treatment of multidimensional time-dependent neutron transport is presented that avoids some inconvenient features of the usual PL approximation, such as the nonuniqueness of the stationary equations in vacuum and the discontinuity of certain moments at material interfaces. The mathematical properties of the original (PL) and modified (EPL) approximations, together with interface and vacuum boundary conditions, are compared. An approximate solution method for both types of equations is derived from a variational principle, and numerical results are given for time-dependent P1 and EP1 calculations in two-dimensional cylindrical geometry.

A Numerical Method for Calculation of Time-Dependent Neutron Transport Equation

Click to increase image sizeClick to decrease image sizeKEYWORDS: numerical methodtime dependenceneutron transport equationinhomogeneous ordinary differential equationfree gasicesMarshak's exact solutiontransient time regionasymptotic time regionneutron flux

A Numerical Method for Calculation of Time-Dependent Neutron Transport Equation

A Numerical Method for Calculation of Time-Dependent Neutron Transport Equation

On nonlinear neutron transport equations

The purpose of this paper is to investigate a class of time-dependent neutron transport equations in which the total and differential scattering cross sections are nonlinear functions of neutron density function. Sufficient conditions on the nonlinear cross sections are given to insure the existence, uniqueness and asymptotic stability of a solution in one, two, or three-dimensional space domains under various boundary and initial conditions. The approach to the problem is based on abstract analysis on nonlinear evolution equations which are closely related to nonlinear semigroup theory.

Solutions of Monoenergetic Time Dependent Neutron Transport Equation in Slab Geometry

Approximate solutions to the one-velocity time dependent neutron transport equation are derived by means of the eigenfunction expansion method of Case for a slab geometry. A finite set of discrete time eigenvalues is determined by a simple formula. This formula, which is derived for the case of large slab thickness, is found to be valid also for fairly thin slabs. Each time eigenvalue generates a pair of spatial modes. The expansion coefficients to these modes are examined by numerical evaluation for various slab thicknesses. Higher spatial modes are found to play an important role in the transient time evolution of neutrons.

Solutions of Monoenergetic Time Dependent Neutron Transport Equation in Slab Geometry

Approximate solutions to the one-velocity time dependent neutron transport equation are derived by means of the eigenfunction expansion method of Case for a slab geometry. A finite set of discrete time eigenvalues is determined by a simple formula. This formula, which is derived for the case of large slab thickness, is found to be valid also for fairly thin slabs. Each time eigenvalue generates a pair of spatial modes. The expansion coefficients to these modes are examined by numerical evaluation for various slab thicknesses. Higher spatial modes are found to play an important role in the transient time evolution of neutrons.

Multigroup Neutron Transport

We investigate the nature of the approximations involved in the multigroup treatment of the time-dependent neutron transport equation by using the method of approximating sequences of Banach spaces. We prove that solutions of the multigroup system converge, in a suitable sense, to the corresponding solutions of the exact transport equation. Moreover, we indicate the order of magnitude of the rate of convergence.

Solution to the Linear Time-Dependent Neutron Transport Equation with Time-Dependent Source and Cross Sections

Systems that are below prompt critical are considered, and the linear time-dependent neutron transport equation in a quite general setting is studied. Both source and cross sections are allowed to depend on space, energy, and time. The method of matched asymptotic expansions is used to find an asymptotic solution uniformly valid in time. This solution is written in the form of a sum of solutions to simpler problems and for most practical problems is essentially exact. After a short initial time period, the transport equation (with delayed neutrons neglected) may be solved at a given time by a single inversion of the steady-state transport operator; i.e., with a steady-state code.

The decay of neutron pulse in a multiplying assembly using a simple kernel

Using the Laplace transform technique, we have solved the time dependent neutron transport equation in a multiplying assembly using a simple scattering kernel. The complete discrete and continuum solutions have been obtained. Numerical calculations have been performed for water moderator which give only one discrete mode. The results have been compared with the exact calculations performed using free gas kernel.

The Integral Transform Method for Neutron Transport Problems

A new method for the solution of the Boltzmann equation for time-dependent or stationary neutron transport is presented. The essentials of the method are a Fourier transform of the transport integral equation and the decomposition of the new kernel into a bilinear form. In the monoenergetic case, the transport problem is reduced to the solution of an (infinite) linear system of equations. It is important that the matrix elements in this linear system be determined only once for any geometry and that this be done in an analytical way. In the present paper, the matrix elements for plane, spherical, and cylindrical geometries are constructed explicitly and are presented in a form suitable for numerical applications. To illustrate the efficiency of the method, the critical dimensions for these geometries have been determined in various integral transform ITN approximations, N being the order of the linear system after truncation. For samples of a thickness up to ∼10 mean-free-paths, the results for the critical dimension in stationary problems, or for the fundamental time-decay constants in nonstationary problems, have in the IT4 approximation, at least, the accuracy of the corresponding S16 results.The convergence of the ITN results, with respect to the order N, is very fast. The results for N = 4 can be considered as exact within one or two units in the fifth digit. The convergence behavior of the SN and the PN method is decidedly slower. In contrast to the SN, PN or Case’s method, the smaller the system, the better the accuracy of the ITN calculations for fixed N is. The increase in accuracy by increasing N requires less computational effort for the ITN approximation than for SN or PN approximation. Apart from providing an analytical standard for the check of numerical approximations, the present method is a simple and rigorous tool for the calculation of neutron distributions, particularly in small systems.

An Analysis of the Time-Dependent Neutron Transport Equation with Delayed Neutrons by the Method of Matched Asymptotic Expansions

The time-dependent neutron transport equation is treated as a problem in singular perturbation theory. The method of matched asymptotic expansions is used to find equations yielding approximate sol...

The expected leakage estimator applied to time-dependent neutron transport

The method of Expected Leakage Probability, applied to time-dependent Monte Carlo neutron transport calculations, is discussed in detail. It is shown that the efficiency of this method is much higher than that of the usual Analogue Monte Carlo technique, especially for the larger time-scale phenomena in small non-multiplying assemblies. The behaviour of the time-dependent neutron leakage is calculated for monoenergetic neutrons and bare spheres and the decay constants are compared with those obtained by other methods. The method of Expected Leakage Probability is also tested by calculating the leakage spectra and decay constants for a lead block exposed to a neutron pulse and comparing the results with the experimental values.

Time Behavior of a Reactor and Ergodic Theory of Semigroups

The solution of the time-dependent neutron transport equation is a semigroup of linear transformations acting on a Banach space. There are some ergodic theorems that can be used to describe the asymptotic behavior of the solution under very general conditions on the semigroup. The results are compared with Wing's famous approach.

On the Initial-Value Problem of the Transport Equation for a Moderator Slab

A mathematical analysis of the time-dependent neutron transport equation is presented for the case of a moderator slab. The scattering of neutrons is treated on the basis of the Van Hove theory and is assumed to be isotropic. First, the spectrum of the corresponding transport operator is investigated and it is shown that the complex λ-plane is decomposed by the operator into a band spectrum Re(λ) ≤ — (v∑t)min, a discrete spectrum on the real axis λ≤— (v∑t), min a resolvent set Re(λ)> — (v∑t)min deleted by the discrete spectrum. The discrete spectrum is carefully studied and is found to consist of isolated eigenvalues or otherwise empty. For all categories of moderator, there exists an upper limit to the thickness of a slab having empty discrete spectrum. A slab exceeding this limit has a finite number of eigenvalues if the moderator material is gas or solid. This is also true for a certain class of liquid moderators. For other liquids, there is another critical thickness for the slab thickness such that i...

The Initial-Value Problem for a Slab. II. Nonuniform Slab

This paper deals with the time-dependent neutron transport in a plane slab of material with variable nuclear properties. The one-velocity theory and the isotropic scattering are assumed. The spectral properties of the corresponding Boltzmann operator are found, and the initial-value problem is solved.