Transport Code Research Articles

Influence of salinity gradients on the diffusion of water and ionic species in dual porosity clay samples

Most of the available data on diffusion in natural clayey rocks consider tracer diffusion in the absence of a salinity gradient despite the fact that such gradients are frequently found in natural and engineered subsurface environments. To assess the role of such gradients on the diffusion properties of clayey materials, through-diffusion experiments were carried out in the presence and absence of a salinity gradient using salt-diffusion and radioisotope tracer techniques. The experiments were carried out with vermiculite samples that contained equal proportions of interparticle and interlayer porosities so as to assess also the role played by the two types of porosities on the diffusion of water and ions. Data were interpreted using both a classical Fickian diffusion model and with a reactive transport code, CrunchClay that can handle multi-porosity diffusion processes in the presence of charged surfaces. By combining experimental and simulated data, we demonstrated that (i) the flux of water diffusing through vermiculite interlayer porosity was minor compared to that diffusing through the interparticle porosity, and (ii) a model considering at least three types of porous volumes (interlayer, interparticle diffuse layer, and bulk interparticle) was necessary to reproduce consistently the variations of neutral and charged species diffusion as a function of salinity gradient conditions.

A conservative first-collision source treatment for ray effect mitigation in discrete-ordinate radiation transport solutions

Deterministic transport codes play a fundamental role in the modeling and simulation of neutron transport. One of the most common deterministic methods is the method of discrete ordinates, also known as the Sn method. While offering significant advantages over other deterministic methods or stochastic methods like Monte Carlo, the method of discrete ordinates suffers from non-physical artifacts in its local solution due to its discretization of angle. These artifacts, referred to as ray effects because of their ray-like appearance, tend to be worse in problems with small sources in areas with little scattering. Significant effort has gone into developing methods to mitigate ray effects, such as the first-collision source treatment, which separates the angular flux into the uncollided and collided fluxes and solves them using non-traditional techniques such as ray tracing. Current ray tracing methods typically trace to a set of points inside a zone to compute an overall flux. However, this approach has significant drawbacks, such as a low potential for optimization, as well as not being conservative. A new method has been developed that traces instead to a set of points on each of a zone's surfaces and computes the currents, before using these to obtain the flux. A comparison between these two ray tracing methods shows significant advantages to the new surface method, including inherent particle conservation, similar convergence rate and increased potential for optimization through ray sharing.

Use of a commercial neutron counting module to validate neutron-particle transport code simulation of a new cosmic radiation neutron monitor design

This paper reports on one of several experiments used to validate MCNP models and processes employed to define a cosmic radiation (CR) neutron source and benchmark to optimise a new ground level neutron monitor (NM) design. The new NM design aims to be more compact, non-toxic and capable of producing comparable counting efficiencies to the NM-64; a monitor standardised in 1964 and which makes up most of the established global network. A broad-energy neutron monitor, the N50L, originally designed for nuclear material accountancy and safeguards, is used to provide primary data in support of our new NM design. Ambient count rate data from the N50L were acquired from it bare and inside a lead (Pb) sarcophagus to boost the neutron signal. Measured data were compared with calculations made using a Monte Carlo N-Particle (MCNP) transport code model of the setup. The calculated compared to the measured count rate for the monitor inside the Pb sarcophagus are in good agreement with a ratio close to one. However, overpredictions (by almost a factor of two) of the calculated rate for the bare monitor suggest inadequacies of the model for high-energy physics. Pressure corrected count rate data were also compared with data from several existing NM-64s over the same period. Our data are in good agreement with variations observed by these existing monitors on the global network. The N50L uses helium-3 (3He) counters and roughly equates to about 1/17th of the proposed final NM design in terms of detector volume. Scaled count rates from the N50L are within 73% of the rates derived from an NM-64 for a similar latitude, geomagnetic cutoff rigidity and altitude, and differs by <8% of the MCNP simulated response of the proposed design.

Application of stoichiometric CT number calibration method for dose calculation of tissue heterogeneous volumes in boron neutron capture therapy.

Monte Carlo simulation code is commonly used for the dose calculation of boron neutron capture therapy. In the past, dose calculation was performed assuming a homogeneous mass density and elemental composition inside the tissue, regardless of the patient's age or sex. Studies have shown that the mass density varies with patient to patient, particularly for those that have undergone surgery or radiotherapy. A method to convert computed tomography numbers into mass density and elemental weights of tissues has been developed and applied in the dose calculation process using Monte Carlo codes. A recent study has shown the variation in the computed tomography number between different scanners for low- and high-density materials. The aim of this study is to investigate the effect of the elemental composition inside each calculation voxel on the dose calculation and the application of the stoichiometric CT number calibration method for boron neutron capture therapy planning. Monte Carlo simulation package Particle and Heavy Ion Transport code System was used for the dose calculation. Firstly, a homogeneous cubic phantom with the material set to ICRU soft tissue (four component), muscle, fat, and brain was modelled and the NeuCure BNCT system accelerator-based neutron source was used. The central axis depth dose distribution was simulated and compared between the four materials. Secondly, a treatment plan of the brain and the head and neck region was simulated using a dummy patient dataset. Three models were generated; (1) a model where only the fundamental materials were considered (simple model), a model where each voxel was assigned a mass density and elemental weight using (2) the Nakao20 model, and (3) the Schneider00 model. The irradiation conditions were kept the same between the different models (irradiation time and irradiation field size) and the near maximum (D1%) and mean dose to the organs at risk were calculated and compared. A maximum percentage difference of approximately 5% was observed between the different materials for the homogeneous phantom. With the dummy patient plan, a large dose difference in the bone (greater than 12%) and region near the low-density material (mucosal membrane, 7%-11%) was found between the different models. A stoichiometric CT number calibration method using the newly developed Nakao20 model was applied to BNCT dose calculation. The results indicate the importance of calibrating the CT number to elemental composition for each individual CT scanner for the purpose of BNCT dose calculation along with the consideration of heterogeneity of the material composition inside the defined region of interest.

Optimizations of the Accelerated Axial Polynomial Method of Characteristics of the APOLLO3® Deterministic Neutron Transport Code

For some years now, the TDT (two- and three-dimensional transport) solver of the APOLLO3® deterministic neutron transport code has been able to perform lattice calculations on three-dimensional extruded and unstructured geometries. A polynomial expansion of the angular flux has been implemented to better describe the flux gradient axially to reduce the number of computational meshes required to reach a given accuracy. Then the polynomial approximation was extended to macroscopic cross sections to perform evolution calculations. Besides these transport schemes, synthetic acceleration has also been implemented, relying on double PN approximations of the angular flux on the boundaries of the spatial regions. The solver has already introduced several techniques to reduce the transport and memory footprint; for example, for the storage of the surfaces crossed by a trajectory or the classification of chords. In this paper, new optimizations are presented. One deals with how monomials of the polynomial basis are integrated along trajectories. Another one concerns the computation of the source term of the transmission equation in the case of polynomial cross sections. The last optimization exploits the fact that, along horizontal trajectories, the flux and the cross sections are constant to speed up the sweep algorithm. Calculations on 5 × 5 and 7 × 7 pressurized water reactor assemblies were performed to assess the gains of these recently developed strategies. The results show good improvements both in computing time and in memory footprint reductions.

Cosmic-ray exposure assessment using particle and heavy ion transport code system: case study Douala-Cameroon.

According to UNSCEAR, cosmic radiation contributes to ~16% (0.39mSv/y) of the total dose received by the public at sea level. The exposure to cosmic rays at a specific location is therefore a non-negligible parameter that contributes to the assessment of the overall public exposure to radiation. In this study, simulations were conducted with the Particle and Heavy Ion Transport code System, a Monte Carlo code, to determine the fluxes and effective dose due to cosmic rays received by the population of Douala. In minimum solar activity, the total effective dose considering the contribution of neutron, muon+, muon-, electron, positron and photon, was found to be 0.31±0.02mSv/y at the ground level. For maximum solar activity, it was found to be 0.27±0.02mSv/y at ground level. During maximum solar activity, galactic cosmic rays are reduced by solar flares and winds, resulting in an increase in the solar cosmic-ray component and a decrease in the galactic cosmic-ray component on Earth. This ultimately leads to a decrease in the total cosmic radiation on Earth. These results were found to be smaller than the UNSCEAR values, thus suggesting a good estimation for the population of Douala city located near the equatorial line. In fact, the cosmic radiation is more deflected at the equator than near the pole. Muons+ were found to be the main contributors to human exposure to cosmic radiation at ground level, with ~38% of the total effective dose due to cosmic exposure. However, electrons and positrons were found to be the less contributors to cosmic radiation exposure. As regards the obtained results, the population of Douala is not significantly exposed to cosmic radiation.

Constraining nucleon effective masses with flow and stopping observables from the SπRIT experiment

Properties of the nuclear equation of state (EoS) can be probed by measuring the dynamical properties of nucleus-nucleus collisions. In this study, we present the directed flow (v1), elliptic flow (v2) and stopping (VarXZ) measured in fixed target Sn + Sn collisions at ▪ with the SπRIT Time Projection Chamber. We perform Bayesian analyses in which EoS parameters are varied simultaneously within the Improved Quantum Molecular Dynamics-Skyrme (ImQMD-Sky) transport code to obtain a multivariate correlated constraint. The varied parameters include symmetry energy, S0, and slope of the symmetry energy, L, at saturation density, isoscalar effective mass, ms⁎/mN, isovector effective mass, mv⁎/mN and the in-medium cross-section enhancement factor η. We find that the flow and VarXZ observables are sensitive to the splitting of proton and neutron effective masses and the in-medium cross-section. Comparisons of ImQMD-Sky predictions to the SπRIT data suggest a narrow range of preferred values for ms⁎/mN, mv⁎/mN and η.

Effect of toroidal rotation on impurity transport in tokamak improved confinement

The centrifugal force effects from toroidal rotation in improved confinement plasmas are analyzed on high-Z impurities in tokamaks. Tungsten (W) transport simulations are performed using the impurity transport code developed in the integrated code TASK. The geometric factors PA and PB are introduced into the neoclassical transport coefficients to include the effects of the toroidal rotation, which come from poloidal asymmetry in the high-Z impurity profiles. Inward neoclassical particle pinch driven by the main ion density gradient is enhanced by the poloidal asymmetry to be the dominant mechanism for W accumulation in the plasma central region. Simulations with experimental plasma profiles show good agreement with the experimental results and first-principle simulation results in the H-mode. In the hybrid mode and advanced mode, the impurity accumulation is enhanced in the internal transport barrier (ITB) regions. The condition to suppress impurity accumulation is investigated by calculating dependencies on the toroidal rotation velocity and ITB position. The neoclassical transport is sufficiently small with the prospected ITER condition of the Mach number of main ions Mi ∼ 0.1. The impurity transport inside the ITB is strongly influenced by competition between the density peaking effect and the temperature screening effect, and the present simulations show suppression of the impurity accumulation with the outer ITB position to improve the plasma performance, due to the relatively larger temperature gradient of the main ion.

PUFFIn – A user friendly fast interface for calculating and visualizing the dose distribution in materials

This article describes a new graphical user interface that uses the PENELOPE Monte Carlo code to calculate dose distributions in materials. It is named PUFFIn, for the Penelope User-Friendly Fast Interface and was developed as an educational and scoping analysis tool for non-experts in radiation modeling. PUFFIn enables the user to visualize and compare the dose distributions in objects (e.g., sterilized healthcare products) irradiated with cobalt-60 gamma-rays, electron beam (E-beam) or X-rays. From such comparisons, the user can determine the most efficient product and/or packaging designs for any given radiation field – whether for conceptual or existing products and packaging.PUFFIn is distributed in a complete, self-contained package of software, including the PENELOPE radiation transport code, a standard graphics package, and a set of simple exercises. The package is available to any user at no cost and requires a minimal amount of training compared to other similar software. PUFFIn's capabilities are described, as well as validation measurements performed at Texas A&M University E-beam facility and the Aerial E-beam Facility in France.

ParaStell: parametric modeling and neutronics support for stellarator fusion power plants

The three-dimensional variation inherent to stellarator geometries and fusion sources motivates three-dimensional modeling to obtain accurate results from computational modeling in support of design and analysis of first wall, blanket, and shield (FWBS) systems. Manually constructing stellarator fusion power plant geometries in computer-aided design (CAD) and defining the corresponding fusion source can be cumbersome and challenging. The open-source parametric modeling toolset ParaStell has been developed to automate construction of such geometries in low-fidelity. Low-fidelity modeling is useful during the conceptual phase of engineering design as a means of rapidly exploring the design space of a given device. The modeling capability of ParaStell includes in-vessel components and magnets, for any given stellarator configuration, using a parametric definition and plasma equilibrium data. Furthermore, the toolset automates the generation of detailed, tetrahedral neutron source definitions and DAGMC geometries for use in neutronics modeling. ParaStell assists rapid design iteration, parametric study, and design optimization of stellarator fusion cores. As a demonstration of the design iteration capability, the effect of the three-dimensional parameter space on tritium breeding and magnet shielding is investigated, using the WISTELL-D configuration as a design basis. Blanket and shield thicknesses are varied in three dimensions, using the space available between the plasma edge and magnet coils as a constraint. The corresponding effects on tritium breeding ratio and magnet heating are tallied using the open-source Monte Carlo particle transport code OpenMC. The inclusion of additional and higher-fidelity modeling capabilities is planned for ParaStell’s future, as well as its implementation in machine-driven optimization.

Extensive analysis of several Indian and Yemeni soils' gamma-ray shielding characteristics: An experimental and simulation approach

The linear attenuation coefficients (LAC) of four soils (Black cotton (S1), Sandy (S2), Clay (S3), and Sandy (S4)) samples were measured at photon energies released from radioisotopes Co57 (122 keV), Ba133 (356 keV), 22Na (511 and 1275 keV), Cs137 (662 keV), Mn54 (840 keV), and Co60 (1330 keV) using a gamma spectrometer includes a NaI (Tl) scintillation detector. The experimental measurements were confirmed utilizing the Monte Carlo N-particle transport code. The linear attenuation coefficient values enhanced from 0.256 cm−1 to 0.296 cm−1 (at Eγ of 122 keV), from 0.126 cm−1 to 0.142 cm−1 (at Eγ of 662 keV), and from 0.0938 cm−1 to 0.105 cm−1 (at Eγ of 1275 keV), raising the (Fe + Mn) concentration from 0.912 wt% to 11.214 wt%, as well as raising the soil samples density from 1.62 g/cm3 to 1.79 g/cm3. The study also shows an enhancement in the half value thickness, transmission factor, radiation protection efficiency and lead's equivalent thickness due to the enrichment of Fe + Mn concentrations within the studied soils. The results show that the Black cotton soil exhibits better shielding properties for γ-ray than the other soils.

Calculations of backpatch factors and conversion coefficients for the personal dose equivalent for X-ray beams used in radiodiagnostic

The purpose of this study is to use the Monte Carlo N-Particle radiation transport code (MCNP) to calculate the conversion coefficients of air kerma for personal dose equivalent (Hp [Sv]) and Back Scattering Factors (BSF) for X-ray beams used in diagnostic radiography at potentials of 40, 50, 60, 70, 80, 90, 100, 120, and 150 kVp. Additionally, effective dose is determined using male and female anthropomorphic voxel-based simulators (REX and REGINA) as recommended by the International Commission on Radiological Protection (ICRP) Publication 110, along with tissue weight factors from ICRP Publication 103 to simulate whole-body irradiation. The results indicate that the relationship between operational metrics such as Hp and risk limitation metrics like effective dose varies from 0.14 to 0.71 for Radiation Qualities in Radiation (RQR) beams from the X-ray source assembly and up to 0.64 for Radiation Qualities based on a phantom with an added aluminium filter (RQA). When comparing the obtained values for air kerma conversion coefficients for personal dose equivalent with the value adopted by the National Commission for Nuclear Energy (CNEN) of 1.14 Sv/Gy for individual dose determination, it is observed that at low energies, the values are overestimated by up to 44%, while at high energies, the values are underestimated by up to 67%.

A method to predict space radiation biological effectiveness for non-cancer effects following intense Solar Particle Events

In addition to the continuous exposure to cosmic rays, astronauts in space are occasionally exposed to Solar Particle Events (SPE), which involve less energetic particles but can deliver much higher doses. The latter can exceed several Gy in a few hours for the most intense SPEs, for which non-stochastic effects are thus a major concern. To identify adequate shielding conditions that would allow respecting the dose limits established by the various space agencies, the absorbed dose in the considered organ/tissue must be multiplied by the corresponding Relative Biological Effectiveness (RBE), which is a complex quantity depending on several factors including particle type and energy, considered biological effect, level of effect (and thus absorbed dose), etc.While in several studies only the particle-type dependence of RBE is taken into account, in this work we developed and applied a new approach where, thanks to an interface between the FLUKA Monte Carlo transport code and the BIANCA biophysical model, the RBE dependence on particle energy and absorbed dose was also considered. Furthermore, we included in the considered SPE spectra primary particles heavier than protons, which in many studies are neglected. This approach was then applied to the October 2003 SPE (the most intense SPE of solar cycle 23, also known as “Halloween event”) and the January 2005 event, which was characterized by a lower fluence but a harder spectrum, i.e., with higher-energy particles. The calculation outcomes were then discussed and compared with the current dose limits established for skin and blood forming organs in case of 30-days missions.This work showed that the BIANCA model, if interfaced to a radiation transport code, can be used to calculate the RBE values associated to Solar Particle Events. More generally, this work emphasizes the importance of taking into account the RBE dependence on particle energy and dose when calculating equivalent doses.

Modeling transient edge plasma transport with dynamic recycling

The work presents numerical simulation studies of the role that dynamic plasma recycling on the main wall and divertor target surfaces plays in transient edge plasma transport phenomena, such as edge localized modes (ELMs). The studies are performed by coupling the edge plasma transport code UEDGE [Rognlien et al., J. Nucl. Mater. 196–198, 347 (1992)] and the wall reaction–diffusion transport code FACE [Smirnov et al., Fusion Sci. Technol. 71, 75 (2017)]. The two-dimensional, time-dependent, two-way coupling of the codes, in a realistic tokamak geometry, is accomplished using the Integrated Plasma Simulator framework [Elwasif et al., in 18th Euromicro Conference on Parallel, Distributed and Network-Based Processing (PDP 2010), Pisa, Italy (IEEE, 2010), pp. 419–427] for all modeled material plasma boundaries. The simulations show that dynamic plasma recycling has substantially different characteristics on the main wall and on the divertor plates. It is demonstrated that during an ELM cycle the outer wall can dynamically absorb and release a number of particles comparable to that expelled by the ELM from the core plasma, by far exceeding the dynamic retention capacity of the divertor surfaces. The resulting evolution of the edge and divertor plasma conditions during an ELM cycle is analyzed.

Optimization of nonlinear turbulence in stellarators

We present new stellarator equilibria that have been optimized for reduced turbulent transport using nonlinear gyrokinetic simulations within the optimization loop. The optimization routine involves coupling the pseudo-spectral GPU-native gyrokinetic code GX with the stellarator equilibrium and optimization code DESC. Since using GX allows for fast nonlinear simulations, we directly optimize for reduced nonlinear heat fluxes. To handle the noisy heat flux traces returned by these simulations, we employ the simultaneous perturbation stochastic approximation (SPSA) method that only uses two objective function evaluations for a simple estimate of the gradient. We show several examples that optimize for both reduced heat fluxes and good quasi-symmetry as a proxy for low neoclassical transport. Finally, we run full transport simulations using the T3D stellarator transport code to evaluate the changes in the macroscopic profiles.

Modeling gamma detectors in OpenMC: Validation of a newly implemented pulse-height tally

Gamma spectroscopy measurements can be simulated using a pulse-height tally functionality of Monte Carlo particle transport codes. Such a functionality must account for the complete simulation history of a particle’s energy deposited in a particular volume. OpenMC, an open-source application for neutron and photon particle transport which is widely used in the nuclear engineering community, previously lacked adequate simulation capabilities for gamma spectroscopy. In this paper, we outline the implementation of the photon pulse-height tally for OpenMC. Additionally, we validate the new function by comparing results to analytical calculations, other software, and experimental data.

Expert‐in‐the‐loop design of integral nuclear data experiments

AbstractNuclear data are fundamental inputs to radiation transport codes used for reactor design and criticality safety. The design of experiments to reduce nuclear data uncertainty has been a challenge for many years, but advances in the sensitivity calculations of radiation transport codes within the last two decades have made optimal experimental design possible. The design of integral nuclear experiments poses numerous challenges not emphasized in classical optimal design, in particular, constrained design spaces (in both a statistical and engineering sense), severely under‐determined systems, and optimality uncertainty. We present a design pipeline to optimize critical experiments that uses constrained Bayesian optimization within an iterative expert‐in‐the‐loop framework. We show a successfully completed experiment campaign designed with this framework that involved two critical configurations and multiple measurements that targeted compensating errors in 239Pu nuclear data.

Energetic particle acceleration and transport with the novel Icarus+PARADISE model

With the rise of satellites and mankind's growing dependence on technology, there is an increasing awareness of space weather phenomena related to high-energy particles. Shock waves driven by coronal mass ejections (CMEs) and corotating interaction regions (CIRs) occasionally act as potent particle accelerators, generating hazardous solar energetic particles (SEPs) that pose risks to satellite electronics and astronauts. Numerical simulation tools capable of modelling and predicting large SEP events are thus highly demanded. We introduce the new Icarus+PARADISE model as an advancement of the previous EUHFORIA+PARADISE model. Icarus, based on the MPI-AMRVAC framework, is a three-dimensional magnetohydrodynamic code that models solar wind configurations from 0.1 au onwards, encompassing transient structures like CMEs or CIRs. Differing from EUHFORIA's uniform-only grid, Icarus incorporates solution adaptive mesh refinement (AMR) and grid stretching. The particle transport code PARADISE propagates energetic particles as test particles through these solar wind configurations by solving the focused transport equation in a stochastic manner. We validate our new model by reproducing EUHFORIA+PARADISE results. This is done by modelling the acceleration and transport of energetic particles in a synthetic solar wind configuration containing an embedded CIR. Subsequently, we illustrate how the simulation results vary with grid resolution by employing different levels of AMR.. The resulting intensity profiles illustrate increased particle acceleration with higher levels of AMR in the shock region, better capturing the effects of the shock.

A package to bridge experimental tokamak data to modelling workflows for heating and transport

The comprehensive visual tool TRVIEW is presented, a software to read and fit experimental measurements and equilibrium reconstruction in the frame of nuclear fusion research. At the same time, TRVIEW provides a framework to setup physics workflows, especially those related to heating and transport. The TRVIEW package was originally created in order to provide file input to transport codes such as ASTRA and TRANSP. However, having a comprehensive set of python classes for the experimental plasma parameters, it allows direct coupling to several modules, such as the heating and current-drive codes RABBIT and TORBEAM and the equilibrium codes FEQIS and NEMEC. The direct coupling allows also direct visualisation of the input and output parameters and profiles in TRVIEW. The option of storing equilibrium files in the standard gEQDSK format is also enabled in TRVIEW. Special algorithms have been developed or adapted for TRVIEW: a class for fitting the plasma separatrix contour with Fourier moments, a recursive procedure for spline fits of the experimental kinetic profiles. Finally, TRVIEW includes the option to read, visualise and write integrated modelling and analysis system (IMAS) files with a comprehensive set of integrated data structures, covering most of the discharge and plasma parameters and several source profiles, calculated with the TORBEAM and RABBIT codes. This enables users to setup workflows for code validation and physics investigations using the world-wide established IMAS environment. TRVIEW can take the experimental input from the shotfile system of the ASDEX Upgrade tokamak or from IMAS files. Since the algorithms and the coupled codes are device-agnostic, TRVIEW is portable to any other fusion devices.

A new ground level neutron monitor for space weather assessment

We report on a new ground-level neutron monitor design for studying cosmic rays and fluxes of solar energetic particles at the Earth’s surface. The first-of-its-kind instrument, named the NM-2023 after the year it was standardised and following convention, will be installed at a United Kingdom Meteorological Office observatory (expected completion mid 2024) and will reintroduce such monitoring in the UK for the first time since ca. 1984. Monte Carlo radiation transport code is used for the development and application of parameterised models to investigate alternative neutron detectors, their location and bulk material geometry in a realistic cosmic ray neutron field. Benchmarked against a model of the current and most widespread design standardised in 1964 (the NM-64), two main parameterisation studies are conducted; a simplified standard model and a concept slab parameterisation. We show that the NM-64 standard is well optimised for the intended large-diameter boron trifluoride (BF3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}) proportional counters but not for multiple smaller diameter counters. The new design (based on a novel slab arrangement) produces comparable counting efficiencies to an NM-64 with six BF3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document} counters and has the added advantage of being more compact, lower cost and avoids the use of highly toxic BF3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}.