摘要:

We consider steady-state particle transport through a stochastic mixture of two (or more) immiscible materials. Certain exact results, as well as approximate models, are described for ⟨ψ⟩, the ensemble average of the particle angular flux. In the simplest case of transport in a no-scattering medium, an exact description for ⟨ψ⟩ and higher stochastic moments is easily constructed. If the material mixing is taken as Markovian, the Liouville master equation leads to two (for a binary mixture) coupled deterministic differential transport equations describing ⟨ψ⟩, with similar results for higher stochastic moments, such as the variance. For non-Markovian mixing, renewal analysis leads to four coupled deterministic integral equations describing ⟨ψ⟩ and higher moments. In the more general problem involving scattering, no exact results (except in very special cases) are available, but reasonably accurate models can be developed. Certain asymptotic limits of these models lead to a single, renormalized, transport equation for ⟨ψ⟩ of the standard form, with effective values of the cross sections and source which account for the stochasticity of the problem. Diffusive approximations also result from asymptotic analyses.

ISSN:0041-1450    

DOI:10.1080/00411459808205637

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The basic duality relation between the direct and adjoint linear transport operators is analyzed and used to derive a duality relation between the direct and adjoint Green's functions with general albedo conditions. Convolution relations are derived for Green's functions defined on domains with non empty intersection. The duality framework is used to derive known results such as the reciprocity theorem, integrals equations and Placzek's lemma with applications to the construction of Green's functions with factored albedos and boundary perturbation problems. A new equation for the source function is also given.

ISSN:0041-1450    

DOI:10.1080/00411459808205638

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

In a preceding paper [1], fluctuations in a collision cascade were investigated. The variance of the yield as well as the energy and depth correlations in sputtering and defects were calculated with constant cross sections and a hard-sphere scattering model. In a more real approximation, atomic collision cascades obey power law scattering, which has a rather different collision kinematics. In particular, the energy transfer, and thus the energy distribution within the cascade evolution, is very different from that of a hard-sphere model. Therefore the yield and mean distribution, as well as the variance and the two-point cascade correlations were calculated in the present paper in a power law scattering model. It was found that despite the differences in the cascade dynamics, most previous findings are still valid. Notably, the relative variance of defects is low in infinite media, whereas in half-spaces it is a monotonically increasing function of the energy of the bombarding ion. The two-point energy and depth correlations are also quite similar to their hard-sphere counterparts. Calculated values of the yield, mean distribution, variance and two-point correlations are shown in the paper for several cases. All calculations were made by using a binary collision Monte-Carlo algorithm.

ISSN:0041-1450    

DOI:10.1080/00411459808205639

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The spectrum of time eigenvalues has been studied for one-speed neutrons in isotropically scattering spheres with a reflexion coefficientRin the interval −1 ≤R≤ 1. It is proved that a continuous spectrum exists in the Hilbert space. This continuum disappears if we suitably enlarge the function space. It is also proved that forR≤ 0 there are only discrete eigenvalues on the real axis. However, forR> 0 there is a continuum of eigenvalues beyond the Corngold limit. An expression is given for the lower boundary of the continuum, and it is supported through numerical calculations.

ISSN:0041-1450    

DOI:10.1080/00411459808205640

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

Steady-state monoenergetic transport problems for homogeneous domains of large optical thickness τ with smooth scattering functions and sources are considered. The asymptotic approximation to the solution is decomposed into two parts: a regular component and a singular one. Expansions in powers of ε = 1/τ* are constructed for these components on the basis of the boundary layer method and accuracy estimates of asymptotic approximations are established. For slab geometry problems some algorithms are proposed to construct the asymptotic approximation with the singular component exponentially decreasing with moving away from boundaries. Some discrete transport models for these problems are considered and a new approach to construct and investigate coarse-mesh solutions is proposed. It is based on the analysis of the regular and singular components of the mesh solution.

ISSN:0041-1450    

DOI:10.1080/00411459808205641

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The linear Boltzmann equation for an electron gas in a semiconductor is analyzed. Under the assumption that the kernel of the integral operator describes the interactions between free electrons and the crystal lattice, some properties of the collision operator are presented. For space homogeneous solution of a Cauchy problem the relaxation to equilibrium is proved.

ISSN:0041-1450    

DOI:10.1080/00411459808205642

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

Theoretical descriptions of ion-bombardment-induced collision cascades in solids widely employ the infinite-medium model ignoring, by and large, the presence of the target surface. The paper attempts to systematically consider analytical relations between cascade functions for this model and their bounded-medium counterparts. These relations represent a useful tool for analysis of collision cascades in a bounded medium. List of functions being considered includes the fluxes of sputtered and backscattered particles, the distributions of deposited energy and implanted ions, and the recoil density. Analytical consequences of the relations are discussed, focused primarily on i) the behavior of the depth profiles at and near the target surface, and ii) how known relations between various cascade functions are modified in the bounded-medium case. Spatial moments of the deposited energy distribution are presented; on this example, a technique for explicit reconstruction of the bounded-medium functions is sketched.

ISSN:0041-1450    

DOI:10.1080/00411459808205643

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The Boltzmann transport equation is employed to describe the time dependent evolution of two physical processes. (1) The grinding of solid particles and their subsequent size distribution and (2) the fragmentation of man-made satellites in earth orbit. It is shown how, by making rational assumptions about the break-up process, the Boltzmann equation provides a convenient framework in which to describe the mass and velocity distributions as a function of time. An equation is derived for the number of particles of a given size present in a homogeneous mixture after a specified grinding time. Also the number and mass of fragments in earth orbit due to satellite break-up by collisions is derived and some useful practical results obtained for finding the average lifetime of a satellite against collision.

ISSN:0041-1450    

DOI:10.1080/00411459808205644

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

A quadrature scheme that is particularly well suited for solving azimuthally dependent transport problems with the ANISN code is introduced. A brief description of the fundamental problem in the constructive theory of orthogonal polynomials is provided. The implementation of the modified Chebyshev and the linear-factor modification algorithms for computing sets of recurrence coefficients that are used to generate the required quadratures is discussed in detail. While both algorithms are very effective and yield accurate quadrature nodes and weights, the linear-factor modification algorithm has the advantage of being about 20% faster than the modified Chebyshev algorithm.

ISSN:0041-1450    

DOI:10.1080/00411459808205645

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

We consider a kinetic model describing the interactions between monochromatic photons and gas-particles with two internal energy levels. Using the moment equations we formulate so-called Milne-Chandrasekhar problem. Next, we provide the existence and uniqueness theorems for the resulting system of partial differential equations.

ISSN:0041-1450    

DOI:10.1080/00411459808205646

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor