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We solve the Milne, constant-source and albedo problems for isotropic scattering in a two-dimensional "Flatland" half-space via the Wiener-Hopf method. The Flatland $H$-function is derived and benchmark values and some identities unique to Flatland are presented. A number of the derivations are supported by Monte Carlo simulation., final version, accepted to JCTT

We present a novel approach for calculating the stochastic eigenvalues of a linear operator. This is an extension of our earlier work on the subject and introduces a significant simplification that reduces the amount of numerical work required when evaluating the polynomial chaos expansions, improves the accuracy and enables higher harmonics to be studied. Examples based on several types of problem involving differential equations and matrices have been analysed and detailed numerical results obtained.

A study is made of radiative transfer along a cylindrical duct using the one-dimensional approximation of Larsen. This method enables three-dimensional effects to be included in a one-dimensional equation. We have generalised Larsen's approach so that the internal reflection from the cylinder surface is governed by the Fresnel law. Two problems are considered. In the first, we assume an infinite cylinder with a plane source at the origin and, in the second, we have a semi-infinite cylinder with an incident beam on the surface. A Fourier transform is employed for the infinite cylinder and the Wiener–Hopf technique for the semi-infinite one. We are able to calculate the radiation intensity as a function of position along the cylinder and the mean square distance of travel of a photon. For the semi-infinite case, we obtain the albedo, the surface intensity and the mean distance of travel. Several features arise which demonstrate that the one-dimensional approximation preserves important features of the three-dimensional problem, thereby making the present approach a valuable tool in dealing with more complex problems.

We extend the range of the searchlight problem in radiative transfer to include internal reflection arising from Fresnel and Lambert processes. For an isotropic beam and a normal beam, we calculate the albedo, the surface intensity and the mean distance of travel of a photon in the lateral direction. Numerical and graphical results are presented for the above quantities.

The classic constant source problem for a half-space is generalized to include the effect of refraction at the boundary by inclusion of the Fresnel boundary conditions. The problem is solved using the Wiener–Hopf technique with both specular and diffuse reflection. The non-singular Fredholm integral equations that arise for the surface angular distribution are solved numerically and the solutions are illustrated by a number of results in graphical and tabular forms. The significant effect of refraction on the surface flux and current and the associated angular distributions is highlighted.

An integral equation describing the fuel distribution necessary to maintain a flat flux in a nuclear reactor in two group transport theory is reduced to the solution of a singular integral equation. The formalism developed enables the physical aspects of the problem to be better understood and its relationship with the corresponding diffusion theory model is highlighted. The integral equation is solved by reducing it to a non-singular Fredholm equation which is then evaluated numerically.

In Canada, during the autumn of 1942, a group of physicists, chemists and engineers were assembled to work on what was to become the Canadian atomic energy project. The base for this work was Montreal. This paper concentrates on the contributions of a sub-group of those scientists, namely those working on the development of nuclear reactor theory. The members of that group comprised an international mix of Canadians, Britons and Americans. A few already had international reputations as theoretical physicists, but the majority were young men and women, generally under the age of 30, who were very talented but not yet famous. They worked under conditions of the utmost secrecy, intitially with little help from the United States, and developed virtually from first principles most of the important aspects of modern reactor theory. The results of their work were issued as Canadian National Research Council reports with the prefix MT (Montreal Theory), and between 1943 and 1946 about 80 such reports wer...

A model of neutron multiplication for aggregates of randomly placed fissile spheres with random material properties in a background medium is presented in terms of the transport equation. Two distinct problems are examined: (1) small spheres in an infinite bulk medium in which the total cross section in the spheres and bulk medium are the same and (2) small spheres in a void or purely absorbing medium but with different total cross sections in sphere and medium. In both cases we consider criticality in which there are random material properties of the spheres and random positions in the container. The random sphere problem is studied statistically by calculating the multiplication factor for many thousands of cases with different positions and material properties and, from the results, constructing a probability distribution function for the multiplication factor. Some of the results are also calculated using diffusion theory and therefore we are able to give guidance on the likely errors caused by diffusion theory in this type of problem.Although the problems are restricted to the one speed approximation, they may be applicable to fast neutron problems and we apply the work to spheres composed of random proportions of {sup 235}U and {sup 238}U.more » The work also has some bearing on the physical behaviour of pebble bed reactors which are of current interest, and in the storage of fissile waste. We have also discussed some of the underlying statistical problems associated with random arrays of spheres in a uniform lattice. In formulating our problem, we use the collision probability technique and as a by-product derive some new inter-lump collision probabilities for two spheres.« less

In a recently developed method for understanding the transport of neutrons in spatially stochastic media, it is necessary to prescribe a random distribution for the fissile lumps. In most practical cases the lumps are assumed to be uniformly distributed within the container. However, the associated centre-to-centre separation distances are not uniformly distributed and this paper describes a method for calculating that distribution function. Two methods are used: an analytical approach which is exact for lumps of zero size and a numerical simulation which accounts for finite size and excludes the possibility of more than one particle occupying the same space. The numerical simulations are compared with the analytical expression and we find excellent agreement for small spheres, which in the limit will be exact. Deviations from the theoretical curve become larger as the sphere becomes larger and size effects become significant. Results are presented for a rectangular box, spherical container, square and one-dimensional rod.

The background theory of a number of problems in rarefied gas dynamics is explained, and the relevant forms of the Boltzmann equation and its associated boundary conditions are derived for flow and heat transfer between parallel plates.

A general formulation is developed for calculating the mean neutron flux in spatially random media. It is based upon Keller's first order smoothing approximation and starts from the integral form of the transport equation in which the number densities of the various nuclear species are considered as stationary random variables. The mean flux is shown to be described by a linear integral equation. In some special cases this has been solved. In particular, for a purely absorbing medium we calculate the flux in the neighborhood of point, line and plane sources and demonstrate the importance of the degree of anisotropy in the correlation function. We also obtain an analytical expression for the collision probability in a spatially random medium and compare this with its deterministic analog.An explicit solution for the mean flux in an infinite medium is obtained in terms of a general source distribution using Fourier transforms. Using image pile theory we are able to calculate the effect of randomness on the critical size of a body. We can show that, for a fissile material, spatial randomness always increases the reactivity of the mixture.

A formalism has been developed for studying the transmission of neutrons through a spatially stochastic medium. The stochastic components are represented by absorbing plates of randomly varying strength and random position. This type of geometry enables the Feinberg-Galanin-Horning method to be employed and leads to the solution of a coupled set of linear equations for the flux at the plate positions. The matrix of the coefficients contains members that are random and these are solved by simulation. That is, the strength and plate positions are sampled from uniform distributions and the equations solved many times (in this case 10{sup 5} simulations are carried out). Probability distributions for the plate transmission and reflection factors are constructed from which the mean and variance can be computed. These essentially exact solutions enable closure approximations to be assessed for accuracy. To this end, the author has compared the mean and variance obtained from the first order smoothing approximation of Keller with the exact results and have found excellent agreement for the mean values but note deviations of up to 40% for the variance. Nevertheless, for the problems considered here, first order smoothing appears to be of practical value and is very efficient numerically inmore » comparison with simulation.« less

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.

The transport of radiation through a random array of purely absorbing slabs is considered assuming that the cross section of the slabs fluctuates according to binomial statistics. An exact solution is constructed for the transmission factor through N slabs and the mean value and variance calculated. It is also shown how the complete probability distribution may be recovered. The results are extended to the case of two different types of slab material and thickness and also to the case where the slab thickness is random subject to a constant total thickness of the system. Simple results are obtained and these are compared with the solution of an analogous problem using the dichotomic Markov process method favored by Pomraning and his co-workers. No particular conclusions are drawn regarding the relative merits of these two statistics and the matter is left open for futher discussion.

Methods are developed for solving the transport equation for radionuclides moving in porous rock by hydrodynamic dispersion and advection. The unique nature of the problem arises from the long time interval over which the solutions are required, e.g., 10{sup 6} yr, during which geological and climatic changes can radially alter the system properties, such as the retardation factor and the water velocity. In order to solve this problem, the authors have developed eigenfunction expansion methods which eliminate the spatial variable and thereby enable the time dependence to be incorporated explicitly. Various problems are considered, each based on two simple boundary conditions: (a) concentration is fixed at both ends of the layer and (b) a delta function impulsive source at one end. The convergence of the solutions is improved by a technique based on the Poisson sum formula which makes them readily tractable numerically over a wide range of practically interesting parameters. Some exact solutions are obtained for purely advective transport which are particularly useful as they are very general and lend themselves to a variety of analytical averaging techniques. Of considerable importance is the development of a stochastic averaging procedure to account for uncertainties in the parameters and onset ofmore » climate changes. The authors have illustrated the effects of averaging by application to a single layer with a delta input and one climatic change (switchtime). The switchtime is regarded as a random variable and averaged over lognormal and uniform distributions. They have considered the retardation factor as uniformly distributed between upper and lower bounds and give graphical results for the concentration as a function of time. Finally, they consider various developments of the method to multinuclide chains and multilayer systems. These studies are important for the design of nuclear waste repositories and also to establish guidelines for safety assessments.« less

A statistical formulation is developed for the number of particles in a given size range following a grinding action carried out over a period of time. The regeneration point method first used by Janossy in the study of cosmic rays is employed. Essentially, the method is based on the backward form of the Chapman-Kolmogoroff equation and is closely related to the theory of fluctuations in nuclear reactors. A probability balance equation is derived and converted to a more convenient form using a generating function. Some new multi-particle breakup functions are introduced and their properties discussed. It is shown that the mean value equation is identical to that conventionally used for grinding but the equations for the variance and higher moments are new. In a special case, we are able to solve the nonlinear, partial integro-differential equation for the generating function and construct the complete probability distribution of the particle number in a given size range. The method can also be employed to study fibre breakup, of interest in the paper industry, and polymer degradation; it therefore has a wide range of application.

The physical and mathematical problems associated with radioactive waste disposal have been outlined and discussed. Some of the more important relationships and equations have been derived and explained with a view to showing how techniques developed in conventional reactor physics problems can be applied with great effect to radionuclide transport. We stress in particular the problems associated with radionuclide transport through spatially random media such as fissured and porous rock. Three distinct modeling procedures are presented: (1) the classical advective dispersion equation and its interpretation as a stochastic differential equation, (2) a purely advective approach in which the groundwater velocity and the retardation factor are random functions, and (3) an analogy with neutron transport by regarding motion along fissures and subsequent branching as a pseudo-scattering process. We describe the mathematical methods needed to solve these stochastic problems and include perturbation theory...

A symmetric factorization of the velocity-dependent charge-exchange kernel (the so-called separable-kernel model) is used in the Boltzmann equation for neutral atoms to obtain an exact solution for a half-space plasma by the Wiener-Hopf method. This work generalizes earlier work employing constant, velocity-independent charge-exchange interactions to the case of an arbitrary velocity dependence of the Maxwellian averaged charge-exchange reaction rate. The effects of velocity dependence on the speed-angle distribution of escaping neutrals and the total charge-exchange rate in the half-space are shown to be significant. It is also shown how the Wiener-Hopf method can be applied to such problems with a realistic Maxwellian plasma background, without first approximating the ion distribution.

The accuracy of the self-similarity assumption employed in the study of the grinding equation is examined in detail. This is made possible by obtaining an exact solution for any homogeneous breakup function, thereby enabling the asymptotic limit as time proceeds to be examined carefully. For the Randolph-Ranjan model of breakup, we have obtained some explicit results and these have been employed to highlight the limitations of current self-similar solutions. In particular, we note that use of a self-similar solution which depends only on the zeroth and first moment of the distribution cannot give any detailed information on the higher moments. Nevertheless, at times very soon after the start of grinding, self-similarity does lead to useful and practical asymptotic results for size distributions, since it appears that higher moments are then of less importance. Thus the reason for the success of similarity is explained and the rate of approach to this condition is given. We have also introduced a new model...

Nuclear power reactors produce useful energy in the form of heat and electricity. They also produce hazardous waste in the form of fission products which may remain dangerous to life for many thousands of years. The problems associated with the disposal of this waste is an international one and a variety of techniques have been proposed for solving it (1). The basic difficulty is that of devising a containment system that will isolate the nuclear waste from the biosphere for a period of time that is long compared with the associated half-lives. This a moral as well as a technical problem since we have a responsibility to avoid exposure to future generations or indeed future civilisations. Whilst the moral aspects will not be discussed in this paper, they have to be borne in mind since they affect political thinking and hence the public funding of the waste disposal problem.

(1990). Additional Comments on “The Effect of Random Material Density on Reactor Criticality”. Nuclear Science and Engineering: Vol. 104, No. 2, pp. 198-198.

The author considers a point source emitting particles in an infinite, absorbing medium. The subsequent history of these particles is governed by the one-speed Boltzmann transport equation which is solved with a new synthetic scattering law. The modelled scattering law allows a combination of forward scattering, backward scattering and isotropic scattering and it enables an exact, closed form solution to be obtained by the method of Fourier transforms. The solution of the transport equation is evaluated numerically for a range of parameters and compared with some essentially exact results obtained by Vanmassenhove and Grosjean (1967) who employed the Henyey-Greenstein scattering law.

A procedure is described that is used to restrain the iterate estimates of a dependent variable (such as the local neutron flux) when acceleration is applied. This scheme allows negative values but prevents extrapolation from positive values to a negative result.

Using the quantum-mechanical form of the Boltzmann equation, a general expression has been deduced for the rate of energy transfer between two species of particles in different velocity distributions. The special case of a charged test particle in a degenerate electron gas is considered in the cold limit. The two-body Rutherford collision cross section is employed, and it is shown that the energy transfer differs little from that obtained when the electrons are treated in the many-body approximation characterized by a dielectric. A rigorous calculation of the Debye shielding distance is given using Fermi-Dirac statistics. The calculations are applied to a compressed deuterium-tritium plasma typical of laser fusion thermonuclear devices and the range of alpha particles is obtained.

Using a slowing down kernel employed in radiation damage studies, it is shown that the existence of Placzek discontinuities at the boundaries of successive collision intervals depends markedly on the angular distribution in the centre-of-mass system. For hard potentials, the discontinuities will exist, whilst for soft, long-range potentials they will not. Conditions for existence are discussed.

The drag on two equal spheres in rigid contact falling along their line of centres is calculated in the slip flow regime. Standard viscous flow theory is employed with the necessary slip boundary conditions. The drag is seen to depend on the ratioλ=l/a wherel is the mean free path of an atom in the surrounding gas and ‘a’ is the radius of the sphere. Two exact limiting cases are obtained for the drag, viz: smallλ and largeλ, and a useful interpolation formula is derived by means of a variational principle. The equivalent sphere approximation is found to give satisfactory results.

Using a simple, hard-sphere model of scattering we have calculated the emergent flux and current of particles emitted from a surface during bombardment with a monoenergetic beam of particles. Analogous results are obtained when the source of particles is uniformly distributed in the half-space. This situation represents essentially an idealized sputtering problem. The calculations have been made in order to test the current theories in which the half-space nature of the problem is neglected and replaced by an infinite medium with appropriately chosen sources or, in some cases, by the infinite medium solution itself. We find, in agreement with the suggestions of previous workers, that the current methods tend to overestimate the sputtered fraction at low energies. A quantitative survey is made of this overestimate. In addition, we examine in more detail than has previously been possible the solutions of the infinite medium approximation. We are able to show that a careful application of this appro...

An investigation has been made of the validity of the perfectly conducting sphere model used in the calculation of drag forces acting on spheres moving in rarefied gases. This assumption requires that the frictional heat generated by the motion of the sphere is conducted so rapidly through the material that the surface temperature is everywhere constant. In turn this affects the energy of the atoms reflected from the surface of the body and hence the drag experienced by it. Instead, there-fore, of making this a priori assumption, we allow the sphere to have an arbitrary thermal con-ductivity. We then solve the heat conduction equation in the sphere and relate it to the external gas conditions by computing the heat transfer rate caused by gas atom collisions. The theory so developed is applicable for arbitrary speed but, for simplecity in this introductory paper, we obtain some analytical results for speeds very much less than Mach one. Our conclusions indicate that the effects of finite conduction on the drag forces are generally small, even when the sphere is a thin shell with a non-conducting interior. Indeed, it is not difficult to show that in going from a perfect thermal conductor to a perfect thermal insulator the drag force only increases by about 3%; nevertheless, in some situations this may well be important and inter-mediate cases will have to include the correction term. More significantly, however, the surface temperature on the sphere is shown to depend on the conductivity to a much greater degree, with the leading face being appreciably hotter than the trailing one. The general conclusion is that for most practical problems involving small particles in the Knudsen regime, moving at appreciably sub-sonic speeds, the assumption of the perfect thermal conductor is a good one.

Using a flexible model of scattering for atomic collisions and the continuous energy loss approximation for electronic stopping, we have obtained exact solutions of the time-energy distribution of atoms slowing down in cascade from a primary knock on. The time moments are constructed from which the slowing down time and relative variance may be calculated. Some results of Sanders and Winterbon are generalized and the connection between the backward equation of conventional damage theory and the partial adjoint of the forward equation are pointed out. We also discuss two complementary methods for calculating energy deposition by atoms as they slow down.

The one-speed transport equation is solved for a ring reactor. A complete solution is obtained for the space-time relaxation of a pulse of neutrons in a multiplying medium in which delayed neutrons are neglected. The solution consists of a fundamental mode, a finite number of harmonics, and an integral transient. A condition is deduced, which gives the maximum number of harmonics that can exist for a given ring circumferenced. The limitations of diffusion theory are pointed out with particular reference to the shortcomings of that theory in dealing with the early stages of evolution of the pulse. Delayed neutrons are included and a complete solution is obtained by means of the prompt jump approximation. The results are illustrated by numerical calculations designed to show the onset of instabilities in the harmonics when the reactor is sufficiently large.

The problem of the spatial distributions of radiation damage is studied using the Boltzmann transport equation. We assume a plane mononergetic source or foreign atoms injected into an infinite host medium. The displacement damage due to the generation of recoils is calculated by using the straight-ahead, or path-length approximation, to the spatial operator, and a modification of the Goertzel-Greuling technique to the slowing down operator. These two approximations enable the Boltzmann equation to be solved analytically in a number of important situations, and the explicit dependence of the damage on the scattering law can be examined. As special cases, we consider the way in which energy is shared between electronic stopping and nuclear stopping and obtain results that are in general agreement with those of Lindhard. In the spatial distribution we can see, for power law scattering, how the damage depends upon the initial energy and the power law index. The distribution of damage as the mass rati...

The Boltzmann equation is used to obtain an expression for nu (E0), the energy transferred to atomic motion by a fast atom of energy E0 as it slows down in a host medium. The power law cross-section is used and, for small values of E0/EL, an exact power series solution in E0/EL may be written down. For large E0/EL it is necessary to make certain analytical approximations which result in accurate values for nu (E0) over the complete energy range. Numerical values obtained from this formula agree well with the results of Lindhard and his co-workers (1963). The method based upon a solution obtained for the time-dependent radiation damage problem with electronic stopping.

Two integro-differential equations arising in particle transport theory are solved explicitly using a technique involving difference equations. The physical problems to which these equations apply concern the energy-time and energy-space distributions of fast particles (neutrons, atoms, gamma -rays, etc.) as they slow down in a host medium. One of the equations involves the first-order derivative with respect to time or space and describes particles which scatter essentially in the forward direction. The other equation assumes a diffusive motion with almost isotropic scattering and hence involves a second-order space derivative. Solutions are obtained in heterogeneous media where the number density of scatterers varies continuously in space and also for a series of contiguous slabs in which the material properties remain constant but change discontinuously from slab to slab. The slowing-down density and energy deposition functions are discussed and evaluated explicitly in some special cases.

The thermophoretic forces acting on two spheres rigidly joined at their point of contact has been calculated and also the thermophoretic velocity which the particle acquires when moving unrestrained. The problem divides into two distinct parts: (i) calculation of the temperature distribution in and around the bispherical system due to the imposed temperature gradient and (ii) the solution of the viscous flow problem when the creep effect is included in the fluid boundary conditions at the surface of the body. The author obtains explicit expressions for the force and velocity which depend only on the solution of a second-order differential equation. Two special cases can be solved exactly: namely when the thermal conductivity of the particle and the gas are the same and also when the gas thermal conductivity is very much less than that of the body. A simple approximation to the differential equation is also derived, which leads to accurate results over a range of conductivities of practical interest. The accuracy of the equivalent-sphere approximation is assessed and found to be accurate to better than 10%.

An integro-differential transport equation which describes elastically scattered phonons in a lattice is cast into an integral form incorporating general boundary conditions. An expression is obtained for the heat flux. A variational principle is constructed which leads to an accurate value for the heat flux over the whole range of optical thicknesses. The limiting case for large optical thickness is shown to be exact by comparison with some results obtained using the Weiner-Hopf technique.

A detailed investigation has been made of the number of defects produced by a primary knock on atom in a medium of similar atoms. Electronic stopping and channelling have been included and it is clearly seen that, for m>1/4, Lindhard's energy partitioning theory is valid to a high degree of accuracy. For m

For pt.I see ibid., vol.10, p.2343 (1977). The accuracy of the transport approximation in the calculation of energy deposition of fast neutrons in spheres of moderating material is assessed by comparison with some work in a previous publication. It is found to be a very convenient method of approach, which simplifies the mathematical analysis considerably and at the same time gives acceptable accuracy for most practical purposes.

The neutron flux has been studied around absorbing bodies of spheroidal, bispherical, and toroidal shapes in an infinite nonabsorbing medium. Exact solutions have been obtained by using effective boundary conditions at the surfaces of the absorbing bodies. The problems considered are as follows: 1. Neutron flux and current distributions around prolate and oblate spheroids. It is shown that an equivalent sphere approximation can lead to accurate values for the rate of absorption. 2. Neutron flux and current in a bispherical system of unequal spheres. Three separate situations arise here: (a) two absorbing spheres, (b) two spherical sources, and (c) one spherical source and one absorbing sphere. It is shown how the absorption rate in the two spheres depends on their separation. 3. Neutron flux and current in a toroidal system: (a) an absorbing toroid and (b) a toroidal source. The latter case simulates the flux distribution from a thermonuclear reactor vessel. Finally, a brief description of how these techniques can be extended to multiregion problems is given.

An approximale method for solving the Boltzmann equation for fast particles slowing down in homogeneous media is described. The Boltzmann equation is used in its straight ahead form and the method of Goertzel and Greuling, developed for neutron transport calculations, is extended to cover the problem of ion slowing down and energy deposition. Solutions are obtained in a simple analytical form and compared with exact calculations. We observe that the error passes through a maximum as the index of anisotropy, m, goes from zero to unity. The limitations of simple age theory are discussed and it is shown how these solutions become exact in the Coulomb limit. The additional problem of electronic stopping is included and its effect on the solution is assessed. Numerical examples are given to support the general conclusions; namely that the Goertzel-Greuling method provides a convenient way to obtain useful estimates of ion implantation profiles and energy deposition.

A detailed investigation of the energy deposition in, and surface reflection from, an infinite half-space has been made. Two types of source are considered: the first is a line source embedded in the medium perpendicular to the surface and the second is an incident pencil, of arbitrary direction, incident at a point. The resulting problem involves three dimensions in space and therefore requires description by a transport equation in the appropriate coordinates. The physical problem considered is that of a beam of incident ions or a line ion source in the medium. Only the fate of the foreign incident ions is considered and no attempt is made to follow the recoil atoms generated. Progress is made in the analytical solution of the problem by assuming an energy-independent mean free path and the transport approximation for the scattering kernel. The Wiener-Hopf method is used together with Fourier transforms for transverse directions. Considerable success has been achieved in obtaining exact solutions for some special limiting cases, and the numerical results which emerge are tabulated.