Anil K. Prinja

Particle transport in the presence of parametric noise

Particle transport in rod and plane geometry random media is considered. The cross section is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. In rod geometry, an exact closure is constructed for semi-infinite media that yields exact equations for the ensemble-averaged scalar flux {phi} and current J. The same closure scheme yields a Fokker-Planck equation for the joint probability distribution function of {phi} and J, from which ensemble-averaged equations for higher order quantities are derived and solved exactly for an arbitrary correlation function. Finally, the penetration of a beam of charged particles in a highly forward scattering random medium is considered, and circumstances that yield a closed ensemble-averaged transport equation are determined.

An Asymptotic Model for the Spreading of a Collimated Beam

In this paper, the authors show, using asymptotics, that under conditions when the angular distribution is forward peaked, the transport equation can be reduced to an advection-diffusion equation for the scalar flux. This equation describes lateral diffusive spreading with depth of an initially collimated beam of arbitrary spatial cross section and is of particular significance when scattering is highly forward peaked. Numerical results for the scalar flux for a planar source (when lateral diffusion vanishes) and in the presence of strongly anisotropic scattering are contrasted with benchmark Monte Carlo results as well as with the scalar flux obtained from a novel modifed multiple scattering method. The authors observe that the asymptotic model is only accurate over distances small compared with the transport mean free path. It is conjectured that carrying the asymptotic expansions to higher orders or using a different asymptotic scaling might extend the accuracy of the asymptotic model to higher orders in the transport mean free path.

Evaluation and Uncertainty Quantification of Prompt Fission Neutron Spectra of Uranium and Plutonium Isotopes

Abstract The prompt fission neutron spectra (PFNS) of the low-incident-energy neutron-induced fission reactions n + 229-238U and n + 235-242Pu have been systematically evaluated using differential experimental data and the Los Alamos model (LA model). Using the first-order, linear Kalman filter, the LA model parameters are constrained using the experimental data and an evaluation of the PFNS and its uncertainties across a suite of isotopes’ results. Correlations between isotopes of each actinide are presented through the model parameter correlations, and the resulting evaluations can be used to fill in inconsistencies within the ENDF/B-VII.1 library where PFNS data are scarce or in need of an update.

Higher‐order multiple scattering theories for charged particle transport

The Fermi pencil beam formula and the higher-order multiple scattering theory due to Jette are shown to result from a perturbative treatment of the linear Boltzmann equation with Fokker-Planck scattering. Using asymptotic one-dimensional solutions for the transverse integrated (spherical) fluence as well as its variance, approximate higher-order pencil beam theories are constructed. These simple and explicit formulae are shown, by comparison with benchmark Monte Carlo results, to be significantly more accurate than the Fermi and Jette equations, particularly at large distances from the beam axis.

On the propagation of a charged particle beam in a random medium. I: Gaussian statistics

Abstract A model is presented for the transport of energetic charged particles in a medium whose density is a continuous random function of position. Using the straight-ahead continuous slowing down approximation and assuming Gaussian statistics for the density fluctuations, exact solutions for the ensemble-averaged flux and dose are obtained. It is demonstrated that the ensemble-averaged flux satisfies an exact closed equation of the Fokker-Planck type in energy. The effect of fluctuations is to introduce straggling in space and energy, resulting in the dose profile extending well beyond the ion range for a corresponding deterministic medium. Very reasonable results are obtained for a fluctuation amplitude on the order of the mean density, in spite of the negative densities admitted by the Gaussian process. However, for very large fluctuations, the Gaussian model leads to a considerable overestimate of the dose near the surface.

MONTE CARLO ELECTRON DOSE CALCULATIONS USING DISCRETE SCATTERING ANGLES AND DISCRETE ENERGY LOSSES

Abstract We present a computationally efficient single event Monte Carlo approach for calculating dose from electrons. Analog elastic scattering and inelastic energy-loss differential cross sections for electrons are converted into corresponding discrete cross sections that are constrained to exactly preserve low-order moments of the analog cross sections. While the method has been implemented and tested for the Rutherford model for scattering and energy loss, its dependence solely on cross-section moments makes our approach arbitrarily general. By comparison with analog Monte Carlo calculations, we demonstrate that few discrete angles and energies are required to achieve accurate dose distributions, and the calculations are fast. The method is capable of yielding accurate results across the entire spatial extent of the transport problem, from relatively isotropic scattering to highly forward-peaked scattering. We compare the accuracy of the angular approximation with the Goudsmit-Saunderson angular approximation commonly used by condensed history methods and similarly analyze the energy approximation. Finally, we present an investigation of the combined approximations and illustrate the accuracy of this method in the presence of a material interface. The computational efficiency of each method is explicitly compared using timing studies.

p-adaptive numerical methods for particle transport

Abstract A p-adaptive method is applied to the discrete ordinates form of the mo-noenergetic transport equation in one dimension. Discontinuous spatial discretizations of arbitrary accuracy are generated in a finite element context using the computer algebra program MAPLE. Source code for the high order methods is produced automatically by MAPLE and then implemented in a discrete ordinates neutral particle transport code in an adaptive fashion. The degree, p, of the polynomial representation used on a mesh cell is varied during the course of the source iterations and the corresponding order of accuracy increased or decreased according to some selected tolerance specification. The accuracy and other numerical properties of the method are investigated and the results of numerical computations are presented.

The Pencil Beam Problem for Screened Rutherford Scattering

We consider the problem of describing steady-state transport of a perpendicularly incident pencil beam of particles through a thin slab of material. The scattering is assumed to be described by the continuous slowing down approximation in energy and by the screened Rutherford formula in angle. For very small screening parameters, it is well known that the scalar flux, as a function of depth and radius, is described reasonably well by the classic Fermi-Eyges formula. However, realistic screening parameters, such as encountered in medical physics applications, are not small enough for this formula, which is Gaussian in radius, to be accurate. A correction to the spatial component of the Fermi-Eyges formula for screened Rutherford scattering is developed. This correction exhibits an algebraic, rather than exponential, falloff of the scalar flux with radius. Comparisons with benchmark Monte Carlo calculations confirm the inaccuracy of the scalar flux spatial distribution of the Fermi-Eyges formula for realistic screening parameters and demonstrate the good results obtained with the present formalism. Contact is made with earlier work by Moliere.

Molecular dynamics simulations of bulk displacement threshold energies in Si

Molecular dynamics (MD) calculations of the bulk threshold displacement energies in single crystal silicon are carried out using the Tersoff potential. The threshold values are angularly dependent and typically vary from 10 to 20 eV for initial primary recoil momentum vectors near open directions in the lattice. An analytic representation of the angular dependence of the threshold values about the and is developed to facilitate comparison with experiment

A unified theory of zero power and power reactor noise via backward master equations

Abstract Traditionally, zero power noise, i.e. inherent neutronic fluctuations in a steady medium, and power reactor noise are treated as two separate phenomena. They dominate at different power levels and are described via different mathematical tools (master equations and the Langevin equation, respectively). Because of these differences, there has been known no joint or unified description, based on first principles rather than empirical analogies, that treats a case when both types of noise are present concurrently. The subject of the present paper is to develop a unified theory of zero power and power reactor noise by calculating the probability distribution of the neutrons in a core with fluctuating material properties. A backward type master equation formalism is used with point kinetics, and the fluctuating cross-sections are represented by a binary pseudorandom process. A closed form solution is obtained which is significantly more complicated than the cases of zero power noise or power reactor noise separately, which are also given in the paper. It is shown that the general solution contains both the zero power and power reactor noise in the sense that the two forms can be extracted individually as limiting cases of the general solution.