One-speed Neutron Transport Theory Research Articles

Calculation of the critical thickness for one-speed neutrons in a reflected slab with backward and forward scattering using modified T N method

Abstract The reflected critical slab problem is investigated in one-speed neutron transport theory for reflecting boundary conditions. The neutron angular flux is expanded in a series of first kind Chebyshev polynomials and isotropic, backward and forward scattering kernel is used as the scattering function. The critical sizes for one-speed neutrons in a uniform finite slab surrounded by identical reflectors from both sides are calculated for various values of the collision parameter, backward and forward scattering and reflection coefficient. Numerical values calculated for the critical slab are presented in the tables and are found to be in good agreement with literature values.

The effect of strongly anisotropic scattering on the critical size of a slab in one-speed neutron transport theory: Modified UN method

The criticality problem for one-speed neutrons in a uniform finite slab is studied in the case of a combination of forward and backward scattering with linearly anisotropic scattering using UN method based on the Chebyshev polynomials of second kind. The effect of the linear anisotropy on the critical thickness of the slab is investigated. The critical slab thicknesses are calculated by using Marshak boundary condition for various values of the anisotropy parameters and they are presented in the tables. In comparison to the results obtained by other methods, the results of this study are in compatible with the former ones.

The criticality calculations for one-speed neutrons in a reflected slab with anisotropic scattering using the modified UN method

Abstract The reflected critical slab problem is studied using Chebyshev polynomials of second kind in one-speed neutron transport theory. The variation of the critical thickness of the medium is investigated both in the case of linearly anisotropic scattering and reflecting boundary conditions. The critical half-thicknesses of the slab are calculated for selected values of the reflection coefficient and the anisotropy parameter. Numerical results obtained for the critical half-thickness are identical with the ones obtained by the PN method and the ones available in literature.

Modified UN method for the reflected critical slab problem with forward and backward scattering

Abstract The critical slab problem is investigated in one-speed neutron transport theory for reflecting boundary conditions using second kind Chebyshev polynomials approximation. The forward-backward-isotropic scattering kernel is used in a uniform homogeneous slab. The critical slab thicknesses for various values of the collision parameter, forward and backward scattering and reflection coefficient are given in the tables together with the ones available in literature and they are in good agreement.

Benchmark results for the critical slab and sphere problem in one-speed neutron transport theory

In this paper benchmark numerical results for the one-speed criticality problem with isotropic scattering for the slab and sphere are reported. The Fredholm integral equations of the second kind based on the Case eigenfunction formalism are numerically solved by Neumann iterations with the Double Exponential quadrature.

Application of the U N method to the reflected critical slab problem for one-speed neutrons with forward and backward scattering

Abstract The UN method is used to solve the critical slab problem for reflecting boundary conditions in one-speed neutron transport theory. The isotropic scattering kernel with the combination of forward and backward scattering is chosen for the neutrons in a uniform finite slab. It is shown that the method converges rapidly with easily executable equations. The presented numerical results are compared with the results available in the literature.

The flat flux problem in one-speed neutron transport theory : Williams, M. M. R. Annals of Nuclear Energy, 2003, 30, (5), 513–547

An explicit expression is obtained for the neutron flux in the one-speed backward-forward scattering model in a general homogeneous convex medium. The method can be extended to energy-dependent problems via the multigroup procedure.

The flat flux problem in one-speed neutron transport theory

The classic flat flux problem in nuclear reactor physics, first solved by Goertzel using diffusion theory, is extended to one-speed transport theory. A numerical solution is obtained for a general two region problem in which the inner region has a flat flux and the outer one a spatially variable flux. For the exact solution, plane geometry is employed with the optical path representation. This solution allows the accuracy of a more flexible but approximate method to be assessed. A feature of the problem which has not been noted before is that in the region of flat flux the current is not zero. The reason for this is explained and is used to assess the accuracy of the approximation method. Diffusion theory is shown to be a poor but not unreasonable approximation. However, to make it viable it is necessary to introduce generalised function solutions to the diffusion equation. The theory is illustrated in detail by numerical calculations. It is shown that a flat flux in one speed theory corresponds to a maximum fuel loading. This is in contrast to the multigroup case for which it is a minimum loading.

Variation of the critical slab thickness with the degree of strongly anisotropic scattering in one-speed neutron transport theory

The critical slab problem is studied in one-speed neutron transport theory using a linearly anisotropic kernel which combines forward and backward scattering. It is shown that, the recently observed non-monotonic variation of the thickness also exists in this strongly anisotropic case. In addition, the influence of the linear anisotropy on the critical thickness is analysed in detail. Numerical analysis for the critical thickness are performed using the spherical harmonics method and results are tabulated for selected illustrative cases as a function of different degrees of anisotropic scattering. Finally, some results are discussed and compared with those already obtained by other methods, the agreement is satisfactory. The spherical harmonic method gives generally accurate results in one dimensional geometry, and it is very suitable for the numerical solution of the neutron transport equation with linearly anisotropic scattering.

The reflected slab and sphere criticality problem with anisotropic scattering in one-speed neutron transport theory

The one-speed transport equation is applied to a homogenous, one-dimensional multiplying medium with linearly anisotropic scattering. Case's singular eigenfunction method is used to formulate the criticality conditions. In addition to available bi- ortogonality relations in the literature, some parallel relations are derived to obtain the solution. In a previous study dealing with only isotropic scattering, it has been shown that the normal-mode expansion of Case also serves to evaluate the complete spectrum of a critical system. As an extension, the spectrum of critical thicknesses and eigenvalues of a slab (also sphere because of its relation to the antisymmetric solution of a slab) are determined by using the formulation established here. The results of this study including some numerical data relevant to the method employed are presented.

96/03855 MACS makes maps of contamination

The critical slab problem based on the singular eigenfunction method solution is generalized using one-speed neutron transport theory. Changing vacuum boundary to reflective boundary, a slab is investigated to rederive the criticality conditions when both reflection coefficients are the same in both surfaces. A special emphasis is given to the evaluation of the eigenvalue spectrum and the results are presented. It is also shown that there exist several critical thicknesses corresponding to each eigenvalue. The results are compared with data available in the literature.

96/03847 The critical slab problem for reflecting boundary conditions in one-speed neutron transport theory

96/03847 The critical slab problem for reflecting boundary conditions in one-speed neutron transport theory

The critical slab problem for reflecting boundary conditions in one-speed neutron transport theory

The critical slab problem based on the singular eigenfunction method solution is generalized using one-speed neutron transport theory. Changing vacuum boundary to reflective boundary, a slab is investigated to rederive the criticality conditions when both reflection coefficients are the same in both surfaces. A special emphasis is given to the evaluation of the eigenvalue spectrum and the results are presented. It is also shown that there exist several critical thicknesses corresponding to each eigenvalue. The results are compared with data available in the literature.

The effect of strong anisotropic scattering on the critical sphere problem in neutron transport theory using a synthetic kernel

The effect of strong anisotropic scattering law on the variation of the critical radius in onespeed neutron transport theory is studied using a synthetic kernel. Numerical results for the critical radius are obtained and tabulated for selected illustrative cases. The results indicate that low-order Legendre approximations are sufficient to show the effect of anisotropic scattering on the variation of the critical radius.

Eigenvalues for reflecting boundary conditions in one-speed neutron transport theory

The spectrum of criticality eigenvalues for the one-speed neutron transport equation has been studied for an infinite slab with reflexion coefficients R 1 and R 2 at the surfaces. For R 1 = R 2 = −1 or +1 the problem can be solved exactly. In another case, R 1 = −1 and R 2 = 0, the problem is simplified considerably. When both reflexion coefficients are close to unity the criticality eigenvalues follow a very accurate approximation formula. For the high-order eigenvalues a semi-empirical formula is given. Some results have also been obtained for the corresponding time-dependent problems.

On pseudo-continuities in the eigenvalue continuum of one-speed neutron transport theory

We report on a new closed-form approximant to the singular eigenfunction transient solution of the one-speed, one-dimensional neutron transport equation with an anisotropic scattering kernel. It is proved that the associated eigenvalue continuum must be pointwise perforated.

A fictitious slab with vanishing critical thickness

Abstract In one-speed neutron transport theory, it is shown that a fictitious anisotropic fission and scattering kernel with an extreme forward bias would give a critical slab thickness varying non-monotonically with anisotropy. For this particular kernel, the critical thickness increases first with increasing forward bias but decreases later and vanishes at a pertain critical anisotropy. This is a geometrical effect which may be understood by considering the variation of the angular distribution with effectively increasing multiplicity.

The effect of anisotropic scattering on the critical slab problem in neutron transport theory using a synthetic kernel

The effect of widely differing scattering laws on the critical thickness of a multiplying slab in one-speed neutron transport theory is studied using a scattering kernel which consists of a linear combination of backward, forward and isotropic scattering. An extensive numerical survey is carried out for the critical thickness in order to provide benchmark results for future studies. 'Exact' solutions are obtained by two methods: the Wiener-Hopf technique and the method of elementary solutions. Although both methods lead to the same results, it is shown that the method of elementary solutions is superior, in terms of operational simplicity, for this type of problem.

The Effect of Linearly Anisotropic Neutron Scattering on Disadvantage Factor Calculations

An expression for the disadvantage factor for a two-region heterogeneous cell in plane geometry has been obtained within the structure of one-speed neutron transport theory. The scattering in the f...

Diffusion Length and Criticality Problems in Two- and Three-Dimensional, One-Speed Neutron Transport Theory. I. Rectangular Coordinates

Several problems in linear transport theory have been solved involving more than one dimension. We have considered the diffusion-length problem in a slab system and have studied how the flux in the transverse direction varies as a function of distance from the source. The solution is found to consist of an asymptotic term, composed of a finite number of harmonics, plus a transient which is nonseparable in the x and z coordinates. When the slab is sufficiently thin only the transient term survives and the concept of a diffusion length loses its value. A method for overcoming this difficulty is presented. In another problem, the thickness of the slab is allowed to become semi-infinite and the effect of decay in the z direction on the emergent angular distribution and surface flux are assessed. By approximating the flux in the y and z directions by a function of the form exp {iByy + iBzz} and solving the resulting one-dimensional problem in the x direction exactly, it has been possible to obtain a statement of the critical conditions in a bare rectangular parallelepiped system. The application of this method to the diffusion length problem in a system with a rectangular cross section is discussed. Finally, by comparison with the exact solution for the slab, we estimate the accuracy of a reduced Boltzmann equation deduced by the author in a previous publication.