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This document reviews Monte Carlo Statistical Convergence over the last 60 years, based on the real World state of computers over this time. Stress is placed on providing examples of what was and was not practically possible over this period of time. I present a few examples to illustrate convergence based not only on sample size, but also how cross sections can be modelled.
In a Monte Carlo calculation, statistical uncertainty is inevitably included in the result. The statistical uncertainty is commonly estimated by the central limit theorem without considering correlation among generations. The statistical uncertainty is usually expected to decrease at the inverse square root of the number of neutron histories. However, in a Monte Carlo eigenvalue calculation in a large geometry whose dominance ratio is close to unity, the central limit theorem would not be valid and thus the statistical uncertainty of fission source distribution can be significantly underestimated when inter-cycle correlation is not appropriately taken into account. The true statistical uncertainty can be calculated by statistical processing of the results obtained by multiple Monte Carlo runs using different initial random seeds, but this approach has a drawback of large calculation cost. Therefore, prediction of true statistical uncertainty by alternate approach is desirable. One of the solutions for this issue is an evaluation method of statistical uncertainty based on the eigenfunction expansion and the Autoregressive model (AR model). In this method, the true statistical uncertainty is theoretically derived by applying the AR model to behavior of expansion coefficients of eigenfunction in fission source distribution during cycles. Underestimation ratio of statistical uncertainty of fission source distribution can be predicted by this method. Note that the underestimation ratio of statistical uncertainty is defined by the ratio of estimated error by a Monte Carlo calculation to the true statistical error. In the previous study, the proposed method was verified in a simple condition such as one energy group, one dimensional homogeneous slab geometry. Contributions of this summary are as follows. Firstly, expression of the noise term that is appeared in the AR model is theoretically derived. Secondly, prediction on underestimation ratio of statistical uncertainty is carried out in a more general case, i.e., flux of each fuel cell in 2 x 2 heterogeneous assembly geometry taken from the C5G7 benchmark problem. Through the present study, applicability of the theoretical prediction for underestimation of statistical uncertainty in practical problems will be verified. (authors)
The analysis of line broadening in X-ray and neutron diffraction patterns using profile functions constructed on the basis of well-established physical principles and TEM observations of lattice defects has proven to be a powerful tool for characterizing microstructures in crystalline materials. These principles are applied in the convolutional multiple-whole-profile (CMWP) procedure to determine dislocation densities, crystallite size, stacking fault and twin boundary densities, and intergranular strains. The different lattice defect contributions to line broadening are separated by considering the hkl dependence of strain anisotropy, planar defect broadening and peak shifts, and the defect dependent profile shapes. The Levenberg–Marquardt (LM) peak fitting procedure can be used successfully to determine crystal defect types and densities as long as the diffraction patterns are relatively simple. However, in more complicated cases like hexagonal materials or multiple-phase patterns, using the LM procedure alone may cause uncertainties. Here, we extended the CMWP procedure by including a Monte Carlo statistical method where the LM and a Monte Carlo algorithm were combined in an alternating manner. The updated CMWP procedure eliminated uncertainties and provided global optimized parameters of the microstructure in good correlation with electron microscopy methods.
Statistical uncertainty is inherent to any Monte Carlo simulation of radiation transport problems. In space-angle-frequency independent radiative transfer calculations, the uncertainty in the solution is entirely due to random sampling of source photon emission times. We have developed a modification to the Implicit Monte Carlo algorithm that eliminates noise due to sampling of the emission time of source photons. In problems that are independent of space, angle, and energy, the new algorithm generates a smooth solution, while a standard implicit Monte Carlo solution is noisy. For space- and angle-dependent problems, the new algorithm exhibits reduced noise relative to standard implicit Monte Carlo in some cases, and comparable noise in all other cases. In conclusion, the improvements are limited to short time scales; over long time scales, noise due to random sampling of spatial and angular variables tends to dominate the noise reduction from the new algorithm.
High statistical precision is critical for Monte Carlo (MC) samples in high energy physics and is degraded by negatively weighted events. This paper investigates a procedure to learn the relationship between the negative and positive weight distributions of any sample, allowing the reduction of statistical uncertainty by reweighting kinematically equivalent events with the same sign. A robust uncertainty quantification method is required for the practical application of such method. Two methods for the estimation of the reweighting uncertainty are developed: one at the event and another one at the final observable level. The latter method is strongly favored. The gains in statistical precision are then quantified. The method is demonstrated on Sherpa vector boson plus jets samples when using all generated events and when restricted to the signal region of a mock analysis. It is demonstrated to significantly reduce stochastic behavior in sparse MC samples while decreasing the overall uncertainty with a sufficiently well-known reweighting function.
The previously developed ab initio model and the kinetic Monte Carlo method (KMCM) are used to simulate precipitation in a number of iron-copper alloys with different copper concentrations x and temperatures T. The same simulations are also made using an improved version of the previously suggested stochastic statistical method (SSM). The results obtained enable us to make a number of general conclusions about the dependences of the decomposition kinetics in Fe-Cu alloys on x and T. We also show that the SSM usually describes the precipitation kinetics in good agreement with the KMCM, and using the SSM in conjunction with the KMCM allows extending the KMC simulations to the longer evolution times. The results of simulations seem to agree with available experimental data for Fe-Cu alloys within statistical errors of simulations and the scatter of experimental results. Comparison of simulation results with experiments for some multicomponent Fe-Cu-based alloys allows making certain conclusions about the influence of alloying elements in these alloys on the precipitation kinetics at different stages of evolution.
STAT7 was written to automate many of the steady-state thermal hydraulic safety calculations for the MIT research reactor, both for conversion of the reactor from highly enriched uranium fuel to low-enriched uranium fuel and for future fuel re-loads after the conversion. A Monte-Carlo statistical propagation approach is used to treat uncertainties in important parameters in the analysis. These safety calculations are ultimately intended to protect against high fuel plate temperatures due to critical heat flux or departure from nucleate boiling or onset of flow instability; but additional margin is obtained by basing the limiting safety settings on avoiding onset of nucleate boiling. STAT7 can simultaneously analyze all of the axial nodes of all of the fuel plates and all of the coolant channels for one stripe of a fuel element. The stripes run the length of the fuel, from the bottom to the top. Power splits are calculated for each axial node of each plate to determine how much of the power goes out each face of the plate. By running STAT7 multiple times, full core analysis can be performed by analyzing the margin to ONB for each axial node of each stripe of each plate of each element in the core.
The STAT code was written to automate many of the steady-state thermal hydraulic safety calculations for the MIT research reactor, both for conversion of the reactor from high enrichment uranium fuel to low enrichment uranium fuel and for future fuel re-loads after the conversion. A Monte-Carlo statistical propagation approach is used to treat uncertainties in important parameters in the analysis. These safety calculations are ultimately intended to protect against high fuel plate temperatures due to critical heat flux or departure from nucleate boiling or onset of flow instability; but additional margin is obtained by basing the limiting safety settings on avoiding onset of nucleate boiling. STAT7 can simultaneously analyze all of the axial nodes of all of the fuel plates and all of the coolant channels for one stripe of a fuel element. The stripes run the length of the fuel, from the bottom to the top. Power splits are calculated for each axial node of each plate to determine how much of the power goes out each face of the plate. By running STAT7 multiple times, full core analysis has been performed by analyzing the margin to ONB for each axial node of each stripe of each plate of each element in the core.
Purpose: To use Monte Carlo radiation transport methods to calculate correction factors for a free-air ionization chamber in support of a national air-kerma standard for low-energy, miniature x-ray sources used for electronic brachytherapy (eBx). Methods: The NIST is establishing a calibration service for well-type ionization chambers used to characterize the strength of eBx sources prior to clinical use. The calibration approach involves establishing the well-chamber’s response to an eBx source whose air-kerma rate at a 50 cm distance is determined through a primary measurement performed using the Lamperti free-air ionization chamber. However, the free-air chamber measurements of charge or current can only be related to the reference air-kerma standard after applying several corrections, some of which are best determined via Monte Carlo simulation. To this end, a detailed geometric model of the Lamperti chamber was developed in the EGSnrc code based on the engineering drawings of the instrument. The egs-fac user code in EGSnrc was then used to calculate energy-dependent correction factors which account for missing or undesired ionization arising from effects such as: (1) attenuation and scatter of the x-rays in air; (2) primary electrons escaping the charge collection region; (3) lack of charged particle equilibrium; (4) atomic fluorescence and bremsstrahlung radiation. Results: Energy-dependent correction factors were calculated assuming a monoenergetic point source with the photon energy ranging from 2 keV to 60 keV in 2 keV increments. Sufficient photon histories were simulated so that the Monte Carlo statistical uncertainty of the correction factors was less than 0.01%. The correction factors for a specific eBx source will be determined by integrating these tabulated results over its measured x-ray spectrum. Conclusion: The correction factors calculated in this work are important for establishing a national standard for eBx which will help ensure that dose is accurately and consistently delivered to patients.
Through the Sandia National Laboratories (SNL) New Mexico Small Business Assistance (NMSBA) program, Sandia worked with the Environmental Restoration Group (ERG) Inc. to verify and validate a novel algorithm used to determine the scanning Critical Level (Lc) and Minimum Detectable Concentration (MDC) (or Minimum Detectable Areal Activity) for the 102F scanning system. Through the use of Monte Carlo statistical simulations the algorithm mathematically demonstrates accuracy in determining the Lc and MDC when a nearest-neighbor averaging (NNA) technique was used. To empirically validate this approach, SNL prepared several spiked sources and ran a test with the ERG 102F instrument on a bare concrete floor known to have no radiological contamination other than background naturally occurring radioactive material (NORM). The tests conclude that the NNA technique increases the sensitivity (decreases the Lc and MDC) for high-density data maps that are obtained by scanning radiological survey instruments. It is further described in Sandia Laboratories document number SAND2014-16522. (authors)
Efficient treatment of systematic uncertainties that depend on a large number of nuisance parameters is a persistent difficulty in particle physics and astrophysics experiments. Where low-level effects are not amenable to simple parameterization or re-weighting, analyses often rely on discrete simulation sets to quantify the effects of nuisance parameters on key analysis observables. Such methods may become computationally untenable for analyses requiring high statistics Monte Carlo with a large number of nuisance degrees of freedom, especially in cases where these degrees of freedom parameterize the shape of a continuous distribution. In this paper we present a method for treating systematic uncertainties in a computationally efficient and comprehensive manner using a single simulation set with multiple and continuously varied nuisance parameters. Finally, this method is demonstrated for the case of the depth-dependent effective dust distribution within the IceCube Neutrino Telescope.
Natural resource planning at all scales demands methods for assessing the impacts of resource development and use, and in particular it requires standardized methods that yield robust and unbiased results. Building from existing probabilistic methods for assessing the volumes of energy and mineral resources, we provide an algorithm for consistent, reproducible, quantitative assessment of resource development impacts. The approach combines probabilistic input data with Monte Carlo statistical methods to determine probabilistic outputs that convey the uncertainties inherent in the data. For example, one can utilize our algorithm to combine data from a natural gas resource assessment with maps of sage grouse leks and pinon-juniper woodlands in the same area to estimate possible future habitat impacts due to possible future gas development. As another example: one could combine geochemical data and maps of lynx habitat with data from a mineral deposit assessment in the same area to determine possible future mining impacts on water resources and lynx habitat. The approach can be applied to a broad range of positive and negative resource development impacts, such as water quantity or quality, economic benefits, or air quality, limited only by the availability of necessary input data and quantified relationships among geologic resources, development alternatives, and impacts. In conclusion, the framework enables quantitative evaluation of the trade-offs inherent in resource management decision-making, including cumulative impacts, to address societal concerns and policy aspects of resource development.
Additively manufactured (AM) components offer superior design flexibility compared to their conventionally manufactured counterparts, and optimizing processing parameters is key to achieving high-quality depositions with desirable and predictable mechanical properties. This study was focused on 4340 steel fabricated using laser powder bed fusion (LPBF), and 42 laser power and scan speed combinations have been systematically investigated to determine an optimized melt pool geometry that would ensure fully-dense parts. The AM material was compared with a wrought 4340 equivalent and studied in two customized heat treated conditions, optimized for strength and toughness, respectively. The microstructures of the as-fabricated and heat treated AM and wrought materials were characterized to assess differences introduced by the layer-by-layer fabrication process and subsequent heat treatment. Tensile properties of both materials were also evaluated and demonstrate that the AM materials offer equal or superior properties compared to the wrought equivalents. Differences in fracture surface morphologies indicate the distinct failure mechanisms associated with the materials’ characteristic microstructures, and the role of inclusions in the failures was studied to elucidate these differences. Complementary to the experimental investigations, the dataset was leveraged to make recommendations for future design of experiments to optimize AM build parameters in other material systems. A statistical Monte Carlo analysis was used to predict the interpolation error produced using reduced datasets and to enable informed processing parameters selection. These findings are discussed to make recommendations for the use of AM materials for high-integrity structural applications.
The study of Ly{alpha} emission in the high-redshift universe is a useful probe of the epoch of reionization, as the Ly{alpha} line should be attenuated by the intergalactic medium (IGM) at low to moderate neutral hydrogen fractions. Here we present the results of a deep and wide imaging search for Ly{alpha} emitters in the Cosmological Evolution Survey field. We have used two ultra-narrowband filters (filter width of {approx}8-9 A) on the NOAO Extremely Wide-Field Infrared Mosaic camera, installed on the Mayall 4 m telescope at Kitt Peak National Observatory, in order to isolate Ly{alpha} emitters at z = 7.7; such ultra-narrowband imaging searches have proved to be excellent at detecting Ly{alpha} emitters. We found 5{sigma} detections of four candidate Ly{alpha} emitters in a survey volume of 2.8 Multiplication-Sign 10{sup 4} Mpc{sup 3} (total survey area {approx}760 arcmin{sup 2}). Each candidate has a line flux greater than 8 Multiplication-Sign 10{sup -18} erg s{sup -1} cm{sup -2}. Using these results to construct a luminosity function and comparing to previously established Ly{alpha} luminosity functions at z = 5.7 and z = 6.5, we find no conclusive evidence for evolution of the luminosity function between z = 5.7 and z = 7.7. Statistical Monte Carlo simulations suggest that half of these candidates are real z = 7.7 targets, and spectroscopic follow-up will be required to verify the redshift of these candidates. However, our results are consistent with no strong evolution in the neutral hydrogen fraction of the IGM between z = 5.7 and z = 7.7, even if only one or two of the z = 7.7 candidates are spectroscopically confirmed.
The Review summarizes much of particle physics and cosmology. Using data from previous editions, plus 3,062 new measurements from 721 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. All the particle properties and search limits are listed in Summary Tables. We also give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Among the 117 reviews are many that are new or heavily revised, including those on Pentaquarks and Inflation. The complete Review is published online in a journal and on the website of the Particle Data Group (http://pdg.lbl.gov). The printed PDG Book contains the Summary Tables and all review articles but no longer includes the detailed tables from the Particle Listings. A Booklet with the Summary Tables and abbreviated versions of some of the review articles is also available. Contents Abstract, Contributors, Highlights and Table of ContentsAcrobat PDF (150 KB) IntroductionAcrobat PDF (456 KB) Particle Physics Summary Tables Gauge and Higgs bosonsAcrobat PDF (155 KB) LeptonsAcrobat PDF (134 KB) QuarksAcrobat PDF (84 KB) MesonsAcrobat PDF (871 KB) BaryonsAcrobat PDF (300 KB) Searches (Supersymmetry, Compositeness, etc.)Acrobat PDF (91 KB) Tests of conservation lawsAcrobat PDF (330 KB) Reviews, Tables, and Plots Detailed contents for this sectionAcrobat PDF (37 KB) Constants, Units, Atomic and Nuclear PropertiesAcrobat PDF (278 KB) Standard Model and Related TopicsAcrobat PDF (7.3 MB) Astrophysics and CosmologyAcrobat PDF (2.7 MB) Experimental Methods and CollidersAcrobat PDF (3.8 MB) Mathematical Tools or Statistics, Monte Carlo, Group Theory Acrobat PDF (1.3 MB) Kinematics, Cross-Section Formulae, and PlotsAcrobat PDF (3.9 MB) Particle Listings Illustrative key and abbreviationsAcrobat PDF (235 KB) Gauge and Higgs bosonsAcrobat PDF (2 MB) LeptonsAcrobat PDF (1.5 MB) QuarksAcrobat PDF (1.2 MB) Mesons: Light unflavored and strangeAcrobat PDF (4 MB) Mesons: Charmed and bottomAcrobat PDF (7.4 MB) Mesons: OtherAcrobat PDF (3.1 MB) BaryonsAcrobat PDF (3.97 MB) Miscellaneous searchesAcrobat PDF (2.4 MB) IndexAcrobat PDF (160 KB)
Background: The evolution of nuclear magic numbers at extremes of isospin is a topic at the forefront of contemporary nuclear physics. N = 50 is a prime example, with increasing experimental data coming to light on potentially doubly magic Sn 100 and Ni 78 at the proton-rich and proton-deficient extremes, respectively; however, experimental discrepancies exist in the data for less exotic systems. Purpose: In Kr 86 the B ( E 2 ; 2 1 + → 0 1 + ) value—a key indicator of shell evolution—has been experimentally determined by two different methodologies, with the results deviating by 3 σ . Here, we report on a new high-precision measurement of this value, as well as the first measured lifetimes and hence transition strengths for the 2 2 + and 3 ( 2 ) - states in the nucleus. Methods: The Doppler-shift attenuation method was implemented using the TRIUMF-ISAC γ -ray escape-suppressed spectrometer (TIGRESS) γ -ray spectrometer and the TIGRESS integrated plunger device. High-statistics Monte Carlo simulations were utilized to extract lifetimes in accordance with state-of-the-art methodologies. Results: Lifetimes of τ ( 2 1 + ) = 336 ± 4 (stat.) ± 20 (sys.) fs, τ ( 2 2 + ) = 263 ± 9 (stat.) ± 19 (sys.) fs, and τ ( 3 ( 2 ) - ) = 73 ± 6 (stat.) ± 32 (sys.) fs were extracted. This yields a transition strength for the first-excited state of B ( E 2 ; 2 1 + → 0 1 + ) = 259 ± 3 (stat.) ± 16 (sys.) e 2 fm 4 . Conclusions: Finally, the measured lifetime disagrees with the previous Doppler-shift attenuation method measurement by more than 3 σ , while agreeing well with a previous value extracted from Coulomb excitation. The newly extracted B ( E 2 ; 2 1 + → 0 1 + ) value indicates a more significant reduction in the N = 50 isotones approaching Z = 40 .
Purpose: The Linear Boltzmann Transport Equation (LBTE) solved through statistical Monte Carlo (MC) method provides the accurate dose calculation in radiotherapy. This work is to investigate the alternative way for accurately solving LBTE using deterministic numerical method due to its possible advantage in computational speed from MC. Methods: Instead of using traditional spherical harmonics to approximate angular scattering kernel, our deterministic numerical method directly computes angular scattering weights, based on a new angular discretization method that utilizes linear finite element method on the local triangulation of unit angular sphere. As a Result, our angular discretization method has the unique advantage in positivity, i.e., to maintain all scattering weights nonnegative all the time, which is physically correct. Moreover, our method is local in angular space, and therefore handles the anisotropic scattering well, such as the forward-peaking scattering. To be compatible with image-guided radiotherapy, the spatial variables are discretized on the structured grid with the standard diamond scheme. After discretization, the improved sourceiteration method is utilized for solving the linear system without saving the linear system to memory. The accuracy of our 3D solver is validated using analytic solutions and benchmarked with Geant4, a popular MC solver. Results: The differences between Geant4 solutions and our solutions were less than 1.5% for various testing cases that mimic the practical cases. More details are available in the supporting document. Conclusion: We have developed a 3D LBTE solver based on a new angular discretization method that guarantees the positivity of scattering weights for physical correctness, and it has been benchmarked with Geant4 for photon dose calculation.
Calculations have been performed for 390 critical assemblies from the International Handbook of Evaluated Criticality Safety Benchmark Experiments using the beta2 release of ENDF/B-VII.1. The results are compared to previous results for ENDF/B-VII. Cases that changed between the two versions are highlighted, and the results are discussed. The Cross Section Evaluation Working Group (CSEWG) is working on a new release of the ENDF/B-VII library of evaluated nuclear data, and the 'beta2' set of files was recently made available by the National Nuclear Data Center (NNDC). A set of about 850 input files for the MCNP Monte Carlo code to run critical assemblies from the International Handbook of Evaluated Criticality Safety Benchmark Experiments was available from our previous data testing work for ENDF/B-VII.0. We have now run 390 of those cases using data based on the beta2 files, and those results will be presented below. The ENDF files were downloaded from the NNDC to a Mac workstation. They were then processed using NJOY10 into ACE format files for use in the MCNP Monte Carlo code. The processing was limited to materials needed for the data testing work at this point. The existing MCNP input decks were used. No checking was done to see if any of the benchmarks had been updated since the ENDF/B-VII testing was finished. Most runs used 50 million histories in order to get Monte Carlo statistical uncertainties down the 0.01% range.