Asymptotically consistent reflection boundary conditions for diffusion theory

It is well known that the classic diffusion equation is an asymptotic limit of the one-speed linear transport equation. In this paper, we carry out the analysis required to obtain an asymptotically consistent boundary condition for this diffusion equation in the case of partial surface reflection. The specification of this boundary condition requires the transport theory solution to a purely scattering halfspace problem. A variational treatment is used to obtain explicit results for this halfspace problem in the general case. In the special case of purely diffuse reflection, simple considerations give exact halfspace results, and a more involved exact halfspace analysis is presented in the case of specular reflection. These exact results are used to assess the accuracy of the variational treatment.     

Computation of some transport theory expressions for a line source in an infinite half-space

In a previous paper, Williams [Williams, M.M.R., 2007. The transport and diffusion theory of a line source in an infinite half-space with internal reflection. Ann. Nucl. Energy 34, 910–921] has obtained transport theory closed form expressions for the particle density for a problem in which a filamentary source is placed within a half-space, perpendicular to the surface (this “line source” problem is closely related to the classical “searchlight” problem of radiative transfer). Some numerical results for limiting cases (surface, z → 0), where the expressions reduce to a single integral were also reported. In this paper, we report additional numerical results for arbitrary internal positions in the medium by evaluating the closed form expressions which have required efficient and accurate evaluations of the generalized H function and subsequent two fold integrations. We have verified the accuracy of the techniques and the results through comparisons with both the limiting case (z → 0) reported in Williams [Williams, M.M.R., 2007. The transport and diffusion theory of a line source in an infinite half-space with internal reflection. Ann. Nucl. Energy 34, 910–921] and also independent results obtained by numerical solutions of the related integral equation via use of the singularity subtraction technique and quadratures. The present work should further aid in progress on related problems with Fresnel reflections and other complications, as well as to the validation of standard commercial reactor physics computer codes. For completeness, we should mention the recent book by Ganapol [Ganapol, B.D., 2008. Analytical benchmarks for nuclear engineering applications: case studies in transport theory, NEA, OECD] in which a number of three-dimensional transport problems are solved with special reference to acceleration techniques.     

Analytical solution for the Feynman-Alpha formula for ADS with pulsed neutron sources

The theory of Feynman-alpha measurements is elaborated for the case of a “stochastically pulsed” subcritical system. The corresponding physical situation is when a pulsed neutron source is used, and no synchronisation between the start of the measurement time gate and the pulsing is made. This is the case in the European Community supported research project MUSE.

The solution to the Feynman-alpha formula was obtained for such a case through complex function techniques in an analytical form by Laplace transform and residue calculus. The final expression is a smoothly regular function with a simple periodic modulation. It consists of a Feynman-curve corresponding to a stationary source, plus an infinite sum of periodic sine functions squared. The series converges as 1/n⁶ with the summation index n, thus in practice two or three terms are sufficient for a high accuracy quantitative result. This few-term representation amounts to a compact closed form analysis solution. Such a solution is well suitable for use in the determination of the subcritical reactivity from measurements, in contrast to the case of deterministic pulsing (measurement start synchronized with pulsing), where no simple solution is available, and where no explicit relationship between the continuous and pulsed forms of the Feynman-alpha exists.     

Three-dimensional transport theory: An analytical solution of an internal beam searchlight problem, III

We describe a method for obtaining analytical solutions and numerical results for three-dimensional one-speed neutron transport problems in a half-space containing a variety of source shapes which emit neutrons mono-directionally in the direction away from the surface. Thus this paper is a supplement to Williams [Williams, M.M.R., 2009, Three-dimensional transport theory: an analytical solution for the internal beam searchlight problem I. Annals of Nuclear Energy 36, 767–783]. For example, we consider a point source, a ring source and a disk source, and calculate the surface scalar flux as a function of the radial co-ordinate when the source is at a fixed distance from the surface. The results are in full agreement with the work of Ganapol and Kornreich [Ganapol, B.D., Kornreich, D.E., this issue. Three-dimensional transport theory: an analytical solution for the internal beam searchlight problem II. Annals of Nuclear Energy]. Diffusion theory results are also included.     

Three-dimensional transport theory: An analytical solution for the internal beam searchlight problem, II

Multidimensional semi-analytical particle transport benchmarks to provide highly accurate standards of assessment are few and far between. Because of a well-established 1D theory for the analytical solution of the transport equation, it is sometimes possible however, to “bootstrap” 1D solutions to give more comprehensive solution representations. Here, we propose the internal searchlight problem in a half space, designated ISLP/HS, as a multidimensional benchmark to be constructed from 1D solutions. This is a variation of the usual SLP/HS where a source emits within the half space rather than striking its surface. Our primary interest is in the exiting intensity at the free surface established through a new Fn formulation. The benchmark features true 2/3D particle transport through integration of a point kernel to simulate 2/3D source emission. In this way, we accommodate a solid or hollow cylindrical source and a general line source in addition to the standard point, ring and disk sources featured in previous investigations.     

A Numerical Method for Solving the Neutron Transport Equation in Finite Cylindrical Geometry

A method is presented which provides a numerical solution to the steady state, energy dependent neutron transport equation for finite cylindrical geometry, with anisotropic treatment of elastic scattering and isotropic treatment of inelastic scattering. The main characteristic features of the method are the use of quasi-cartesian coordinates and the application of discrete ordinate numerical integration. A difference form of the Boltzmann equation is derived as the final expression for machine computation.

Comparisons are given of the numerical solutions with an analytical solution for a constant source distribution, and with NIOBE calculations and experimental spectra for neutron transport in water, with good agreement obtained between them.     

LIGHT ION REFLECTION STUDIED BY MONTE CARLO SIMULATION AND TRANSPORT THEORY

The reflection of light ions, such as H+,3He+ and 4He+, with energies of 0.1- 10 keV, from Cu and Ni surface has been studied by Monte Carlo simulation and transport theory. The Monte Carlo simulation gives the detail energy spectra for the reflected particles and their angular distribution for different incident angles. It shows that the reflected particle energy spectra can be approximately described by an analytical formula for the whole energy range, all the incident angles and different ion- target combination studied here. The reflected particle energy vs its average reflection angle to the surface normal can almost be expressed by a universal curve for all cases studied here. The reflection energy spectra are used for the calculation of the reflection coefficient by transport theory including the realistic surface correction. The present work is compared with both experimental measurement and other simulation codes.     

Three-dimensional transport theory: An analytical solution of an internal beam searchlight problem-I

We describe a number of methods for obtaining analytical solutions and numerical results for three-dimensional one-speed neutron transport problems in a half-space containing a variety of source shapes which emit neutrons mono-directionally. For example, we consider an off-centre point source, a ring source and a disk source, or any combination of these, and calculate the surface scalar flux as a function of the radial and angular co-ordinates. Fourier transforms in the transverse directions are used and a Laplace transform in the axial direction. This enables the Wiener–Hopf method to be employed, followed by an inverse Fourier–Hankel transform. Some additional transformations are introduced which enable the inverse Hankel transforms involving Bessel functions to be evaluated numerically more efficiently. A hybrid diffusion theory method is also described which is shown to be a useful guide to the general behaviour of the solutions of the transport equation.     

Transition radiation in the pre-wave zone for an oblique incidence of a particle on the flat target

Backward transition radiation of an electron with an arbitrary energy is studied when an ideally-conducting target is flat and its normal is not collinear with the particle velocity (oblique incidence). A model for radiation registered with a small flat detector placed at a finite distance (including the radiation in the pre-wave zone) in the vicinity of specular reflection direction is developed. Characteristics of the radiation in the far-field zone are in complete agreement with well-known results. The calculations for pre-wave zone show that the angular distribution of the radiation intensity is distorted compared to the far-field case, and the radiation asymmetry (having a place in the far-field for moderately relativistic energies) is also preserved in the pre-wave zone. Some numerical estimations of the radiation asymmetry at a finite distance are also given. The technique developed may be used for estimations of coherent transition radiation intensity in the mm-wavelength range.     

A finite element method for neutron transport—II. Some practical considerations

It is found that the even-parity form of the neutron transport equation is well suited to a variational formulation in conjunction with the finite element approximation. A finite element technique is studied in the context of reactor physics calculations which is of particular interest in the nuclear engineering field. Multi-region slab problems for a given source distribution and critical slab problems are solved with an accuracy as good as the best problem techniques available. An exact solution is obtained for the special case of forward scattering with uniform source distribution in a slab reactor.     

Generalized Exact Solution of the Slowing Down Equation

The slowing down equation for an infinite homogeneous monoatomic medium is solved exactly. The solution is obtained in analytical form. The calculated collision density is compared with the one obtained by Teichmann and Stefanovi'c respectively. For the special case of hydrogen, the present solution reduces to Bethe's solution.     

Interaction of slow ions with bulk and surfaces

Nonlinear density functional results for the stopping power and straggling of ions moving slowly (υ ⪡ υ_f) in an electron gas are presented. A self-energy formalism is used to estimate the charge state of a proton moving at intermediate velocities in an electron gas. The contribution of the surface to the energy loss of slow ions is also analyzed, within the specular reflection model.     

Solution of the neutron transport problem with anisotropic scattering in cylindrical geometry by the decomposition method

An analytical solution has been obtained for the one-speed stationary neutron transport problem, in an infinitely long cylinder with anisotropic scattering by the decomposition method. Series expansions of the angular flux distribution are proposed in terms of suitably constructed functions, recursively obtainable from the isotropic solution, to take into account anisotropy. As for the isotropic problem, an accurate closed-form solution was chosen for the problem with internal source and constant incident radiation, obtained from an integral transformation technique and the F_N method.     

The neutron flux in a spherical void

Using the basic theory developed in our earlier work (Cassell, J.S., Williams, M.M.R., 2005. Particle flux in an annular gap about a sphere, Annals of Nuclear Energy 32, 457, we have evaluated the neutron flux across a spherical void due to a point source in a moderating and absorbing medium. 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 is obtained in the form of an infinite series. This is evaluated numerically for a number of practical cases and comparison is made with an exact transport calculation using a Monte Carlo code. The hybrid method is seen to be highly accurate.     

Multigroup multi-layer models of neutron reflection and transmission for reactor transport calculations with anisotropic scattering

In this article, we extend the one-speed multi-layer models to neutron reflection and transmission developed in our earlier work (de Abreu, M.P., 2005. Multi-layer models to neutron reflection and transmission for whole-core transport calculations, Annals of Nuclear Energy 32, 215) to multigroup transport theory. We begin by considering a two-layer boundary region, and we develop for such a region discrete ordinates models to the diffuse reflection and transmission of neutrons for multigroup nuclear reactor core problems with anisotropic scattering. We perform numerical experiments to show that our models to neutron reflection and transmission can be used to replace efficiently and accurately two nonactive boundary layers in whole-core transport calculations. We conclude this article with an inductive extension of our two-layer results to a boundary region with an arbitrary number of layers.     

The Borrmann effect in parametric X-radiation under asymmetric reflection conditions

Parametric X-radiation (PXR) of a relativistic electron traversing a single crystal plate is considered in Laue geometry. The expressions describing spectral-angular distributions of PXR formed on the atomic planes situated under arbitrary angle δ to surface of the plate (asymmetric reflection) obtained on basis of two-wave approximation of dynamic diffraction theory are used for definition of the conditions of the most pronounced manifestation of the Borrmann effect (optimal value of angle δ) are clarified. This effect leads to considerable increase of the intensity of the quasi-monochromatic tuning source of coherent X-radiation built on basis of PXR.     

Radiative Heat Deposition Analysis of Fusion Reactor First Wall by Monte Carlo Transport Code

Heat flux distribution on the first wall of a fusion reactor due to the thermal radiation from high temperature protection wall placed in front of the first wall was analyzed. With necessary modifications, a three-dimensional Monte Carlo transport code developed for neuronics calculation was successfully applied in the analysis. That is, reasonable results with sufficiently small statistical error were obtained with reasonable computational time. The heat flux distribution was found to be insensitive to the reflection characteristic of the radiation at the first wall i. e. diffusive or specular.     

Analysis of one-dimensional pressure pulse propagation in a closed fluid system

This paper describes a unique numerical method for linear inviscid fluid hammer analysis based on the method of characteristics. The uniqueness lies in that it uses the analytical solutions of the wave equation in place of the compatibility relatins of the more conventional method of characteristics. The numerical solution is obtained by a simple superposition technique for tracing the waves traveling along each characteristic and extending the solution from one constant time line to the next. Using a predetermined finite difference net of grids with equal spacings, an elimination is made of the spatial interpolation, thereby maintaining the wave amplitudes in their full strength in the numerical procedure. This is in contrast to the case of a nonlinear problem in which the pressure peaks are always flattened to some degree in the interpolation procedure.The computer program NAHAMMER is a system analysis code adequate for short-term pressure transients of most engineering problems of significance involving a moderate pressure source. It considers the simplified one-dimensional, linear, inviscid set of governing equations with an isentropic flow assumption. A closed fluid-network system is considered to be composed of a multiple of one-dimensional pipe sections and components that are connected by various joints. An analytical solution is obtained under an acoustic approximation for a simple system and the result shows good agreement with the numerical solution. As examples of the application of the method, complex problems of engineering importance are calculated and the results are presented.