Particle Transport Theory Research Articles

Rigorous iterated solutions to a nonlinear integral evolution problem in particle transport theory

After a preliminary functional study of the operator associated with the relevant Boltzmann equation, which is shown to be a contraction operator, a nonlinear integral evolution problem occurring in the diffusion of the particles of a mixture is solved by resorting to a rigorous iterative scheme, in the case without removal. According to this scheme, an explicit recursive representation for the general iterated solution of order n is developed. Structure and behavior of the solution so obtained are investigated and commented on.

The Rayleigh model: singular transport theory in one dimension

We present a comprehensive account of the special ‘Rayleigh piston’ model for the spatial and velocity relaxation of an ensemble of labelled test-particles in a one-dimensional heat-bath of particles with identical mass. This model, originally formulated by Rayleigh in 1891 but since largely neglected, is in effect a prototype for all later models in singular particle transport theory and serves to illustrate the mathematical problems associated with the occurrence of singular eigenfunctions and continuous spectra of a scattering operator. Although other idealized scattering models are known, the Rayleigh model remains a unique example of an exactly soluble singular system which, in including conservation laws and time-reversal symmetry in scattering, retains a degree of mechanical realism.

Solutions to a class of nonlinear integral evolution problems in particle transport theory

A class of nonlinear integral evolution problems that are of basic interest in particle transport theory is solved for a general scattering probability. A standard iterative technique is developed in order to generate explicit iterated solutions up to any general ordern for the distribution function of the particles considered. The exact solution follows then as the formal limti forn→∞ of the sequence of the iterated solutions so obtained. The influence of the removal as well as the main implications of the arbitrariness of the scattering probability are commented. The comparison with the solutions of the corresponding class of linearized problems resulting from neglecting the relevant nonlinear effects is also discussed.

Particle Transport Calculations with the Method of Streaming Rays

A new particle transport theory method has been developed for application in particle streaming and shielding calculations. The method is similar to the SN technique in that discrete directions are used, and the transport medium is divided into spatial mesh cells. However, in addition to the spatial mesh, the entire medium is overlaid with a series of streaming rays. Particles are assumed to travel along these rays until they suffer collisions. The collided fluxes within and at the cell surface are related using a difference approximation technique. The collided particles are then reassigned to streaming rays. Unlike the SN method, differencing approximation schemes are required only for particles that have collided in the cell of interest. Another feature of this method is that a finer angular quadrature set is used for the streaming portion of the transport calculation than is used in the determination of the scattering source. The remaining aspects of the technique parallel those of the SN method. Several test results demonstrating the capability of the method are presented.

A concise and accurate solution for Poiseuille flow in a plane channel

The recently developed FN method of solving problems in particle transport theory is used to establish a concise and accurate solution for the flow of a rarefied gas between two parallel plates. The Bhatnagar, Gross, and Krook model is used, and numerical results are given for a wide range of the Knudsen number.

General theory of anomalous particle transport in weakly space and time dependent plasmas

The particle flux induced by fluctuations of arbitrary polarization is evaluated within the framework of geometric optics accounting for the effects of both dissipation and and variations in space and time of the plasma parameters. The energy conservation law determining the space and time dependence of the modulus of the fluctuating wave amplitudes is also discussed. Furthermore, the limit in which dissipation effects are dominant over those of the slow spatial and temporal variations is considered for both a mainly time dependent and a mainly space dependent problem. General conditions for the flux to be outward (inward) are obtained for this case.

A stochastic theory of particle transport. II

For pt.I see Proc. R. Soc., A358, p.105 (1977). The authors are concerned with the development of radiation damage cascades initiated by fast incident particles. They derive a general probability balance equation for the number of vacancies and interstitials in given regions of space, subject to an initial atom of specified velocity. The scattering model is quite general but they do employ the radiation damage model due to Kinchin and Pease (1955). Introduction of a generating function enables the various moments of the vacancy and interstitial distributions to be obtained. Explicit equations for the vacancy and interstitial densities and their two particle correlation functions are derived. They introduce the straight-ahead approximation and solve the resulting density equations by Laplace transform. Explicit expressions are obtained for the two densities over the complete space and energy range. Calculations show the existence of a vacancy rich region near the point of origin of the cascade. The proportionality of the size of the vacancy rich zone to the logarithm of the incident energy is noted and a useful semi-empirical formula is obtained.

On two integro-differential equations arising in particle transport theory

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.

Transport theory for charged particles in electric and magnetic fields

An integral transport theory for charged particles which, in the presence of electric and magnetic fields, diffuse by collisions against the atoms of a host medium is proposed in the frame of the scattering Kernel formulation for the collisional term of the Boltzmann equation. Applications of the theory are developed for the purely velocity-dependent problem, and calculations of the most relevant physical quantities are performed.

Impurity control by neutral-beam injection

A recently developed generalization of neoclassical theory is applied to study the use of neutral-beam injection to drive impurities out of a tokamak plasma. The theory accounts for the direct momentum exchange between beam and plasma, which has been studied previously, and also for the effects of the beam in altering the first-order particle flows and the ambipolar potential. Conditions are established for which co-injection drives impurity fluxes outward. It is estimated that order-unity changes of the radial impurity flow should be found in current experiments and that modest levels of neutral-beam injection should suffice for impurity control in reactors. In the course of this work, the general theory of particle transport in the presence of particle and momentum sources is evaluated for a two-species plasma with arbitrary plasma geometry and beta.

Discrete and Continuous Interactions in Charged Particle Transport Theory

Discrete and Continuous Interactions in Charged Particle Transport Theory

Integral transport theory for charged particles in electric and magnetic fields

In this paper an integral transport theory for charged particles which, in the presence of electric and magnetic fields, diffuse by collisions against the atoms (or molecules) of a host medium is proposed. The combined effects of both the external fields and the mechanisms of scattering, removal and creation in building up the distribution function of the charged particles considered are investigated. The eigenvalue problem associated with the sourceless case of the given physical situation is also commented. Applications of the theory to a purely velocity-dependent problem and to a space-dependent problem, respectively, are illustrated for the case of a separable isotropic scattering kernel of synthetic type. Calculations of the distribution function, of the total current density and of the relevant electrical conductivity are then carried out for different specializations of the external fields.

A stochastic theory of particle transport

A probability balance equation is formulated for the number of particles present in a cascade resulting from multiple births at each collision. Janossy’s regeneration point method is used and it leads to an integro differential equation for the generating function from which statistical information can readily be extracted. The technique is applied to the interpretation of radiation damage cascades in a homogeneous, amorphous medium in which two particles are ‘born’ per collision. The history of a single chain is followed and equations for the mean and variance are obtained as well as for individual probabilities. It is further shown how the backward and forward forms of the Boltzmann equation are related via the Green function of the system. Additional study shows that the variance also obeys a forward type of equation although its solution is not obtained as conveniently as that of the corresponding backward equation. Several analogies are made with other branches of particle physics; in particular, cosmic rays and neutron transport.

A fluctuation problem in particle transport theory II

A balance equation is formulated for the probability that a particle injected into an infinite, amorphous medium will have suffered N collisions and have given rise to n new particles in a given energy range at time t. The method of regeneration points has been employed and this leads, in the case of two particle production, to a non-linear, integro-differential equation for the probability generating function. This equation is solved for the case of foreign particles slowing down, in which case it becomes linear and results are obtained which include the effects of electronic stopping and absorption, thus generalizing the work in part I. In the cascade problem, a single particle gives rise to two new particles in every collision and it is shown, for a simple hard-sphere model with 1/v scattering and absorption, how the non-linear equation may be solved. The probability for the number of particles and the number of collisions suffered to absorption is obtained in the case of zero absorption, the probability law is shown to obey a Furry distribution. The limitations of the method described in part I for dealing with cascades are highlighted.

Representation of the expectation of symmetric functionals in particle-transport theory

It is shown that the mathematical expectation of symmetric functionals of a stochastic proliferative particle-transport process can be expressed in terms of solutions of the standard kinetic equations.

A fluctuation problem in particle transport theory I

The fluctuation in the number of collision suffered by particles as they slow down in a moderating medium is studied via a probability balance equation. The equation describes the collision history of foreign particles slowing down in a host medium and also accounts for the recoil particles produced in the collision. The equations are solved by the introduction of a generating function from which the space and time dependent probability distributions are obtained. That is, the probability that a particle will suffer just N collisions to reach energy E at a time t after injection. In the space dependent case it is the probability that a particle suffers just N collisions to travel a given path length before coming to rest. Explicit expressions for the means and variances are obtained by solving a difference equation. From this solution it has been possible to obtain exact expressions for hard spheres and for a variety of models based on the inverse power law approximation. A number of new results are presented and some old ones rederived in a more efficient and general manner. The results are of value in the understanding of radiation damage cascades and in neutron slowing down in moderating materials.

Analysis of classical impurity transport

This work is an application of Braginskii's classical theory of particle transport to partially ionized plasmas in cylindrical geometry or toroidal geometry in the Pfirsch-Schlüter regime. Particular attention is given to the steady state, for which density and flux distributions are determined, and the re-cycling phenomenon is analysed. On the assumption of coronal equilibrium, numerical computation shows that, although the centrifugal and centripetal fluxes of ions of a particular ionization degree may be large, the electron and global ion fluxes are usually small, in contrast with experimental results. Possible anomalous fluxes are necessarily linked with violation of coronal equilibrium; in this case, a transport equation for the mean ionization degree is given as a substitute to the coronal hypothesis. Numerical simulation also shows the significance of friction between ions of the same impurity and different impurities, for common impurity concentrations.

On the metric theory of particle transport in a random velocity field (the problem of turbulent diffusion). Part II

The concept of the metric theory of particle transfer in a random field of velocities the problem of turbulent diffusion is the result of a suitable connection of a certain computational theoretical model of the endomorphism of Lebesgue's space L and the concept of the action of the field of velocities on the particle in its abstract form. The metric theory of this phenomenon is part of the general metric theory of dynamic systems, i.e. the ergodic theory. The adjective “suitable” means that we are introducing certain relations between the said concepts. The essence of these concepts is in that we require that the relations, describing the nature of the field-particle interaction, be a realistic counterpart to the appropriate endomorphism T of space L. This endomorphism is closely associated with Kolmogorov's automorphism and with the automorphism with a positive enthropy. We are pointing out the close relation between the T-representation, the equations of motion of the particle in a random field of velocities, the canonic transformations and the fundamental model of stochasticity of Hamiltonian systems.

On the metric theory of particle transport in a random velocity field (the problem of turbulent diffusion). Part I

On the metric theory of particle transport in a random velocity field (the problem of turbulent diffusion). Part I

Particle trapping during passage through a high-order nonlinear resonance

A theory of particle trapping and transport during a single passage through a high-order nonlinear resonance is developed. The main result is an expression for the trapping efficiency as a function of two scaling variables which are related to the resonance excitation width, the nonlinear detuning and the speed of passage through the resonance. The question of what phase-space region trapped particles are drawn from and the question of adiabaticity are discussed. The theory is then verified with a computer simulation for crossing a fifth order resonance.