Boltzmann-Fokker-Planck Research Articles

Extending deterministic transport capabilities for very-high and ultra-high energy electron beams

Focused Very-High Energy Electron (VHEE, 50–300 MeV) and Ultra-High Energy Electron (UHEE, > 300 MeV) beams can accurately target both large and deeply seated human tumors with high sparing properties, while avoiding the spatial requirements and cost of proton and heavy ion facilities. Advanced testing phases are underway at the CLEAR facilities at CERN (Switzerland), NLCTA at Stanford (USA), and SPARC at INFN (Italy), aiming to accelerate the transition to clinical application. Currently, Monte Carlo (MC) transport is the sole paradigm supporting preclinical trials and imminent clinical deployment. In this paper, we propose an alternative: the first extension of the nuclear-reactor deterministic chain Njoy-Dragon for VHEE and UHEE applications. We have extended the Boltzmann-Fokker-Planck (BFP) multigroup formalism and validated it using standard radio-oncology benchmarks, complex assemblies with a wide range of atomic numbers, and comprehensive irradiation of the entire periodic table. 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Additionally, we demonstrate that at least 97%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$97\\%$$\\end{document} of voxels of bone, lung, adipose tissue, muscle, soft tissue, tumor, steel, and aluminum meet the same criterion between 50 MeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{50 MeV}$$\\end{document} and 1.5 GeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{1.5 GeV}$$\\end{document}. For water, the thorax, and the breast intra-operative benchmark, typical average BFP-MC deviations of 0.3%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.3\\%$$\\end{document} and 0.4%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.4\\%$$\\end{document} were observed at 300 MeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{300 MeV}$$\\end{document} and 1 GeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{1 GeV}$$\\end{document}, respectively. By irradiating the entire periodic table, we observed similar performance between lithium (Z=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z=3$$\\end{document}) and cerium (Z=58\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z=58$$\\end{document}). Deficiencies observed between praseodymium (Z=59\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z=59$$\\end{document}) and einsteinium (Z=99\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z=99$$\\end{document}) have been reported, analyzed, and quantified, offering critical insights for the ongoing development of the Evaluated Nuclear Data File mode in Njoy.

Feasibility of a multigroup Boltzmann–Fokker–Planck solution for electron beam dose calculations

Legacy nuclear-reactor Boltzmann solvers start clinical deployment as an alternative to Monte Carlo (MC) codes and Fermi–Eyges semiemprical models in radiation oncology treatment planning. Today’s certified clinical solvers are limited to photon beams. In this paper, ELECTR, a state-of-the-art multigroup electron cross sections generation module in NJOY is presented and validated against Lockwood’s calorimetric measurements, EGS-nrc and GEANT-4 for 1–20 MeV unidirectional electron beams. The nuclear-reactor DRAGON-5 solver is upgraded to access the library and solve the Boltzmann–Fokker–Planck (BFP) equation. A variety of heterogeneous radiotherapy and radiosurgery phantom configurations were used for validation purpose. Case studies include a thorax benchmark, that of a typical breast Intra-Operative Radiotherapy and a high-heterogeneity patient-like benchmark. For all beams, 100% of the water voxels satisfied the American Association of Physicists in Medicine accuracy criterion for a BFP-MC dose error below 2%. At least, 97.0% of adipose, muscle, bone, lung, tumor and breast voxels satisfied the 2% criterion. The average BFP-MC relative error was about 0.56% for all voxels, beams and materials combined. By irradiating homogeneous slabs from Z=1 (hydrogen) to Z=99 (einsteinium), we reported performance and defects of the CEPXS mode [US. Sandia National Lab., SAND-89-1685] in ELECTR for the entire periodic table. For all Lockwood’s benchmarks, NJOY-DRAGON dose predictions are within the experimental data precision for 98% of voxels.

Implementation of the ELECTR module in NJOY

The ELECTR module of NJOY is designed to produce complete and accurate multigroup electroatomic cross sections from ENDF/B-VII data[1, 2]. electr produces restricted cross sections consistent with a solution of the multigroup Boltzmann-Fokker-Planck (BFP) equation. Total, elastic, inelastic (collision and bremsstrahlung) cross sections can be averaged using a variety of group structures and weighting functions. The Legendre components of the within-group elastic and group-to-group inelastic collision cross sections are calculated using tabulated data in energy and analytic expressions of the angular deviation recovered from the CEPXS code[3]. Here, we propose an Open-Source implementation of this module, named electr in NJOY2012 and NJOY-2016.[4] electr also computes partial energy deposition and charge deposition cross sections for each reaction and sum these partial contributions. The resulting multigroup constants are written on an intermediate gendf file for later conversion to any desired format.

Anisotropic Neutron Emission Spectrum and Its Utilization for Verification of Nuclear Elastic Scattering Effect in Proton-Beam-Injected Deuterium Plasmas

A possible scenario to observe a knock-on tail using anisotropic neutron emission spectrum in a proton-beam injected deuterium plasma is presented. On the basis of the ion trajectory analysis in ITER-like magnetic configuration, Boltzmann collision integral and Fokker-Planck simulation, the knock-on tail formation in deuteron distribution function due to nuclear elastic scattering, and the resulting modification of the “double differential” neutron emission spectrum are examined. As a result of the beam injection with a specific direction, the neutron emission spectrum is modified in 2-D phase space. It is shown that the ratio of neutron emission rate with >2.8-MeV energy to the 2.45-MeV peak is increased almost 10 times compared with the 1-D modified neutron emission spectrum integrated over the emission angle, which implies the improvement of the accuracy in the neutron measurement. By using the anisotropic neutron emission adjusting the relative positions between beam-injection port and the neutron detector, the possibility to catch the knock-on tail for various plasma conditions can be increased.

About an H-theorem for systems with non-conservative interactions

We present some arguments in favor of an H-theorem for a generalization of the Boltzmann equation including non-conservative interactions and a linear Fokker–Planck-like thermostatting term. Such a non-linear equation describing the evolution of the single particle probability Pi(t) of being in state i at time t is a suitable model for granular gases and is referred to here as the Boltzmann–Fokker–Planck (BFP) equation. The conjectured H-functional, which appears to be non-increasing, is HC(t) = ∑iPi(t)lnPi(t)/Πi with Πi = limt→∞Pi(t), in analogy with the H-functional of Markov processes. The extension to continuous states is straightforward. A simple proof can be given for the elastic BFP equation. A semi-analytical proof is also offered for the BFP equation for so-called inelastic Maxwell molecules. Other evidence is obtained by solving particular BFP cases through numerical integration or through ‘particle schemes’ such as the direct simulation Monte Carlo.

A Verification Scenario of Ion-Heating Enhancement due to Nuclear Plus Interference Scattering

A scenario to verify the knock-on tail formation by energetic protons and the resulting enhancement of the fractional ion heating by using 3He-containing deuterium plasma is presented. On the basis of the Boltzmann-Fokker-Planck (BFP) model, the knock-on tail formation in deuteron distribution function due to nuclear plus interference (NI) scattering of fusion-produced energetic protons and the resulting modification of the neutron emission spectrum are examined. A recognizable change in the energetic-neutron emission rate due to the knock-on tail formation depending on the external 3He inclusion is shown. Another possible scenario to ascertain the enhancement of ion-heating by using γ-ray-generating 6Li+D reaction is also presented and discussed.

Effect of thermal 3He minorities on knock-on tail formation and the resulting neutron emission spectrum modification in deuterium–tritium plasmas

The knock-on tail formations in fuel-ion velocity distribution functions by energetic alpha particles (by the T(d,n)4He reaction) and protons (by the D(d,p)T and 3He(d,p)4He reactions) are investigated by simultaneously solving the Boltzmann–Fokker–Planck (BFP) equations for deuteron, triton, 3He, alpha particle and proton in an ITER-like (DT) plasma admixed with a small amount of 3He. As a result of the 3He inclusion, a fraction of the transferred energy from energetic ions to thermal deuterons and tritons via nuclear plus interference (NI) scattering is reduced. Owing to the NI scattering of the energetic protons by fuel ions, the latter are knocked up to higher energies. The knocking-up effect of fuel ions is enhanced with increasing 3He concentration. It is shown that if 3He with relative concentration of 4.2%, i.e. , is included in Te = 20 keV, ne = 9.5 × 1019 m−3 plasma, the magnitude of the knock-on tail in deuteron distribution function in 300 keV–3 MeV energy range is reduced by about 15% from the value when 3He is not externally supplied. Such knock-on tail reduction also results in alternation of the non-Gaussian neutron emission spectrum with energies less than ∼13 MeV and above ∼15 MeV.

Modification of Alpha-Particle Emission Spectrum and Its Verification Scenario Using 9Be+α Reaction in Beam-Injected DT Plasmas

The alpha-particle emission spectrum in beam-injected deuterium-tritium (DT) plasma is examined by solving the Boltzmann-Fokker-Planck (BFP) equations for deuteron, triton and alpha-particle simultaneously. It is shown that owing to the existence of energetic component in fuel-ion energy distribution functions due to neutral-beam injection (NBI) and/or nuclear elastic scattering (NES), the fraction of the energetic (> 3.52MeV) alpha-particle generation rate increases significantly compared with the case for Gaussian distribution. Aiming at an application to plasma diagnostics, correlation between the modification of the emission spectrum and the gamma(γ)-ray generation rate from 9Be(α,nγ)12C reaction is studied.

Distortion of bulk-ion distribution function due to nuclear elastic scattering and its effect on T(d,n)He4 reaction rate coefficient in neutral-beam-injected deuterium-tritium plasmas

An effect of nuclear elastic scattering on the rate coefficient of fusion reaction between field deuteron and triton in the presence of neutral beam injection heating is studied. Without assuming a Maxwellian for bulk-ion distribution function, the Boltzmann-Fokker-Planck (BFP) equations for field (bulk) deuteron, field (bulk) triton, α-particle, and beam deuteron are simultaneously solved in an ITER-like deuterium-tritium thermonuclear plasma [R. Aymar, Fusion Eng. Des. 55, 107 (2001)]. The BFP calculation shows that enhancement of the reaction rate coefficient due to knock-on tail formation in fuel-ion distribution functions becomes appreciable, especially in the case of low-density operations.

Use of γ-Ray-Generating 6Li+D Reaction for Verification of Boltzmann-Fokker-Planck Simulation and Knock-on Tail Diagnostic in Neutral-Beam-Injected Plasmas

The γ-ray emission rate by 6Li(d,p)7 Li*, 7Li* → 7Li+γ reaction when 50∼250 keV proton beam is injected into 6Li containing deuterium plasmas (ne ∼1019 m-3 and Te = 1∼10 keV) is evaluated by simultaneously solving the Boltzmann-Fokker-Planck (BFP) equations for deuteron and proton. A possible experiment to verify the BFP simulations, e.g. knock-on tail effect on T(d,n)4 He reaction rate coefficient and/or plasma diagnostics which utilize the tail formation in ion distribution function, is proposed.

Effect of nuclear elastic scattering on plasma heating characteristics in deuteron–triton thermonuclear plasmas

Effects of nuclear elastic scattering (NES) on the neutral beam injection (NBI) and α-particle plasma heating operations are examined by solving the Boltzmann–Fokker–Planck (BFP) equations for beam ions and α-particles in deuteron–triton thermonuclear plasmas. The BFP calculations show that the enhancement in the fraction of the NBI and α-particle heating powers deposited to ions due to NES can be appreciable in reactor-grade plasmas.

Effect of nuclear elastic scattering on ion heating characteristics in deuterium-tritium thermonuclear plasmas

An effect of nuclear elastic scattering (NES) on the energy transfer to plasma ions and electrons during (a) neutral beam injection (NBI) and (b) α-particle heating operations is examined on the basis of the Boltzmann-Fokker-Planck (BFP) equation for a beam ion and an α-particle in deuterium-tritium thermonuclear plasmas. The BFP calculations show that the enhancement in the fraction of the NBI heating power deposited to ions due to NES becomes appreciable when the beam energy is larger than 1MeV. How the NES effect is influenced by the plasma condition is discussed.

Effect of Nuclear Elastic Scattering on Neutral Beam Injection Heating in Thermonuclear Plasmas

An effect of the nuclear elastic scattering (NES) on the neutral beam injection (NBI) plasma heating was examined by solving the Boltzmann-Fokker-Planck (BFP) equation for beam ion in the deuterium-tritium (DT) thermonuclear plasmas. The BFP calculations show that the enhancement in the fraction of the NBI heating power deposited to ions due to NES becomes appreciable when beam energy is larger than 1MeV, and the enhancement is strongly influenced by plasma parameters.

Time and energy dependent recoil distributions in mixtures

An approach based on linear transport theory is described here to model the transport of incident ions and recoil particles in complex target materials. Using the time and energy dependent Boltzmann Fokker-Planck (BFP) equation, we have numerically computed particle densities for two incident ion (oxygen and barium) species as well as for all recoil particle species, self-consistently, in the superconducting material YBa 2Cu 3O 7. An advantage of our approach is that the scattering, recoil and energy deposition cross sections are computed once for each combination of incident ion and target material and a given (e.g., Universal) potential. These interaction cross sections are then used in the BFP code to study both time dependent (pulsed ion beam) and equilibrium (steady-state ion beam) scenarios. Pulsed mode results provide insight into the decay behavior of each recoil species and equilibrium spectra are obtained in steady state mode. Our numerical model also describes the partitioning of energy deposition amongst the atomic species as well as the fate of the incident ion and recoils in interstitial, substitution or replacement sites.

Diffusive formulations arising from kinetics of charged test particles in a uniform field

Abstract The transport of charged test particles in a uniform electric field for a slab is treated both to improve drift-diffusion Fick's law and to reach nonclassical-diffusive equations with associated flux boundary values as integral functionals of internal flux distribution. The path from Boltzmann and Boltzmann-Fokker-Planck (BFP) equations with a force term up to nonclassical diffusive formulations goes through an “anisotropic integral equation” for total flux derived without any bound to flux and scattering anisotropy. For BFP both the previous rigorous and a simpler Pg l approaches were considered. Pg 1 leads to a diffusion theory in whose framework Einstein's relation connects diffusion coefficient to mobility in a special case and where a generalized Fick's law integral in energy is more generally valid.

Photon Transport Studies in Media with Strongly Anisotropic Scattering with the Boltzmann-Fokker-Planck Equation

The Boltzmann-Fokker-Planck (BFP) equation has been applied to treat the photon transport in highly anisotropic scattering media. Some benchmark cases relating to haze and cloud media have been investigated. It is found that, in general, the number of Legendre moments required for the BFP equation is a factor of --5 lower compared to that for the standard (discrete ordinates S/sub n/) approach for solving the Boltzmann equation for the comparable precision on the integral quantities like the albedo and transmission factor. But BFP is much more economical if one seeks precision on angular fluxes.

Investigations of Nuclear Elastic Scattering Effects by Means of the Boltzmann-Fokker-Planck Equation

Nuclear elastic scattering (NES) has recently been recognized as an important slowing down mechanism for fast ions injected into a background plasma. The present paper proposes an improved method to include this effect into slowing down calculations. This method consists of calculating multigroup cross-section data including the transfer matrices up to a desired degree of Legendre expansion and in supplying the data to a Boltzmann-Fokker-Planck (BFP) equation solved by a discrete ordinates scheme. The physical model of the BFP equation and the accuracy of the numerical method guarantee a good representation of NES.

Treatment of Scattering Anisotropy of Neutrons Through the Boltzmann-Fokker-Planck Equation

Handling the highly anisotropic scattering of fast neutrons with conventional methods usually means that high-order Legendre expansions can be necessary to obtain correct angular fluxes. This drawback in standard transport calculations is avoided by applying the Boltzmann-Fokker-Planck (BFP) method, already used in transport of charged particles, to neutrons. Two methods are described to obtain the relevant input data for the one-dimensional BFP-1 code, one using basic differential scattering cross sections and the other using existing standard multigroup libraries. Numerical results for both methods are produced, revealing BFP as a powerful method when solving transport problems dealing with very fast neutrons. It is found that high accuracy, even for extreme cases of anisotropy, is achieved without increase of the computational effort.

Numerical Analysis of the Boltzmann Equation Including Fokker-Planck Terms

After a short presentation of the Boltzmann-Fokker-Planck (BFP) equation, which was derived in a previous work, two numerical approaches to solve this equation are investigated--the multigroup method and a diamond scheme applied in a consistent way to space and energy variables. Because of the parabolic nature of the Fokker-Planck operator, it is shown that the standard neutron transport codes cannot solve such an equation. With the one-dimensional time-dependent BFP-1 code, many numerical results have been produced. All deal with the transport of charged particles in dense plasmas because such a problem is very severe from a numerical point of view. Other applications can be imagined since the BFP formalism is quite general.

The photoelectron flux in the Earth's ionosphere at energies in the vicinity of photoionization peaks

In this paper we apply Boltzmann‐Fokker‐Planck (BFP) theory to analyze the (isotropic) photoelectron flux in the Earth's ionosphere at energies in the vicinity of photoionization peaks where the electron impact cross sections are slowly varying functions of energy. We give a Green's function solution for the photoelectron flux which shows that the energy derivative terms in the BFP method dominate the solution in spectral regions of high slope. Analytical results for the photoelectron flux and the forward and backward half widths in the vicinity of isolated photoionization peaks are given which explain the observed asymmetry and shift of the photoionization peaks with increasing altitude. At low altitudes these results indicate that the theoretical width of an isolated photoionization peak is less than the experimental resolution of recent AE‐E data. A comparison between the theoretical photoelectron flux averaged using the experimental resolution and the experimental photoelectron flux from 15 to 35 eV at 182 km shows reasonable relative agreement.