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All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Adaptively coupled phase retrieval in multi-peak Bragg coherent diffraction imaging

Recent advances in Bragg coherent diffraction imaging (BCDI) experimental techniques permit routine measurement of multiple Bragg peaks from a single crystalline grain. The resulting images contain the full lattice distortion vector field which can be differentiated to provide lattice strain and rotation. With the advent of fourth-generation synchrotron light sources, such multi-peak datasets are produced at high rates, facilitating the need for rapid phase retrieval of the multiple peaks and subsequent image analysis. Here we describe and demonstrate a new implementation of a coupled phase retrieval technique for multi-peak BCDI which simultaneously treats each Bragg peak of the dataset and produces a three-dimensional image of the crystal's morphology and lattice distortion field. In addition, this method uses the redundant information contained in the various Bragg diffraction patterns to detect and suppress spurious signal appearing on the detector in a subset of the measurements. Compared with manual data editing, adaptive coupling produces a more consistent phase profile in reciprocal space and sharper surfaces in direct space, with no significant difference in computational cost. These improvements reduce the need for manual preprocessing and enable robust high-throughput analysis of multi-peak BCDI data, supporting near-real-time strain microscopy at modern synchrotron facilities.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Concurrent multi-peak Bragg coherent x-ray diffraction imaging of 3D nanocrystal lattice displacement via global optimization

Abstract In this paper we demonstrated a method to reconstruct vector-valued lattice distortion fields within nanoscale crystals by optimization of a forward model of multi-reflection Bragg coherent diffraction imaging (MR-BCDI) data. The method flexibly accounts for geometric factors that arise when making BCDI measurements, is amenable to efficient inversion with modern optimization toolkits, and allows for globally constraining a single image reconstruction to multiple Bragg peak measurements. This is enabled by a forward model that emulates the multiple Bragg peaks of a MR-BCDI experiment from a single estimate of the 3D crystal sample. We present this forward model, we implement it within the stochastic gradient descent optimization framework, and we demonstrate it with simulated and experimental data of nanocrystals with inhomogeneous internal lattice displacement. We find that utilizing a global optimization approach to MR-BCDI affords a reliable path to convergence of data which is otherwise challenging to reconstruct.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Mapping nanocrystal orientations via scanning Laue diffraction microscopy for multi-peak Bragg coherent diffraction imaging

The recent commissioning of a movable monochromator at the 34-ID-C endstation of the Advanced Photon Source has vastly simplified the collection of Bragg coherent diffraction imaging (BCDI) data from multiple Bragg peaks of sub-micrometre scale samples. Laue patterns arising from the scattering of a polychromatic beam by arbitrarily oriented nanocrystals permit their crystal orientations to be computed, which are then used for locating and collecting several non-co-linear Bragg reflections. The volumetric six-component strain tensor is then constructed by combining the projected displacement fields that are imaged using each of the measured reflections via iterative phase retrieval algorithms. Complications arise when the sample is heterogeneous in composition and/or when multiple grains of a given lattice structure are simultaneously illuminated by the polychromatic beam. Here, a workflow is established for orienting and mapping nanocrystals on a substrate of a different material using scanning Laue diffraction microscopy. The capabilities of the developed algorithms and procedures with both synthetic and experimental data are demonstrated. The robustness is verified by comparing experimental texture maps obtained with Laue diffraction microscopy at the beamline with maps obtained from electron back-scattering diffraction measurements on the same patch of gold nanocrystals. Such tools provide reliable indexing for both isolated and densely distributed nanocrystals, which are challenging to image in three dimensions with other techniques.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Iterative Bragg peak removal on X-ray absorption spectra with automatic intensity correction

This study introduces a novel iterative Bragg peak removal with automatic intensity correction (IBR-AIC) methodology for X-ray absorption spectroscopy (XAS), specifically addressing the challenge of Bragg peak interference in the analysis of crystalline materials. The approach integrates experimental adjustments and sophisticated post-processing, including an iterative algorithm for robust calculation of the scaling factor of the absorption coefficients and efficient elimination of the Bragg peaks, a common obstacle in accurately interpreting XAS data, particularly in crystalline samples. The method was thoroughly evaluated on dilute catalysts and thin films, with fluorescence mode and large-angle rotation. The results underscore the technique's effectiveness, adaptability and substantial potential in improving the precision of XAS data analysis. While demonstrating significant promise, the method does have limitations related to signal-to-noise ratio sensitivity and the necessity for meticulous angle selection during experimentation. Overall, IBR-AIC represents a significant advancement in XAS, offering a pragmatic solution to Bragg peak contamination challenges, thereby expanding the applications of XAS in understanding complex materials under diverse experimental conditions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2016

TU-FG-BRB-09: Thermoacoustic Range Verification with Perfect Co-Registered Overlay of Bragg Peak onto Ultrasound Image

Purpose: The potential of particle therapy has not yet been fully realized due to inaccuracies in range verification. The purpose of this work was to correlate the Bragg peak location with target structure, by overlaying thermoacoustic localization of the Bragg peak onto an ultrasound image. Methods: Pulsed delivery of 50 MeV protons was accomplished by a fast chopper installed between the ion source and the inflector of the 88″ cyclotron at Lawrence Berkeley National Lab. 2 Gy were delivered in 2 µs by a beam with peak current of 2 µA. Thermoacoustic emissions were detected by a cardiac array and Verasonics V1 ultrasound system, which also generated a grayscale ultrasound image. 1024 thermoacoustic pulses were averaged before filtering and one-way beamforming focused signal onto the Bragg peak location with perfect co-registration to the ultrasound images. Data was collected in a room temperature water bath and gelatin phantom with a cavity designed to mimic the intestine, in which gas pockets can displace the Bragg peak. Experiments were performed with the cavity both empty and filled with olive oil. Results: In the waterbath overlays of the Bragg peak agreed with Monte Carlo simulations to within 800±170 µm. Agreement within 1.3 ± 0.2 mm was achieved in the gelatin phantom, although relative stopping powers were estimated only to first order from CT scans. Protoacoustic signals were detected after travel from the Bragg peak through 29 mm and 65 mm of phantom material when the cavity was empty and full of olive oil, respectively. Conclusion: Protoacoustic range verification is feasible with a commercial clinical ultrasound array, but at doses exceeding the clinical realm. Further optimization of both transducer array and injection line chopper is required to enable range verification within a 2 Gy dose limit, which would enable online adaptive treatment. This work was supported in part by a UWM Intramural Instrumentation Grant and by the Director, Office of Science, Office of Nuclear Physics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. YMQ was supported by a UWM-OUR summer fellowship.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

TU-H-CAMPUS-TeP3-03: Dose Enhancement by Gold Nanoparticles Around the Bragg Peak of Proton Beams

Purpose: To make clear the spatial distribution of dose enhancement around gold nanoparticles (GNPs) located near the proton Bragg peak, and to evaluate the potential of GNPs as a radio sensitizer. Methods: The dose enhancement by electrons emitted from GNPs under proton irradiation was estimated by Geant4 Monte Carlo simulation toolkit in two steps. In an initial macroscopic step, 100 and 195 MeV proton beams were incident on a water cube, 30 cm on a side. Energy distributions of protons were calculated at four depths, 50% and 75% proximal to the Bragg peak, 100% peak, and 75% distal to the peak (P50, P75, Peak, and D75, respectively). In a subsequent microscopic step, protons with the energy distribution calculated above were incident on a 20 nm diameter GNP in a nanometer-size water box and the spatial distribution of dose around the GNP was determined for each energy distribution. The dose enhancement factor (DEF) was also deduced. Results: The dose enhancement effect was spread to several tens of nanometers in the both depth and radial directions. The enhancement area increased in the order of P50, P75, Peak, and D75 for both cases with 100 and 195 MeV protons. In every position around the Bragg peak, the 100 MeV beam resulted in a higher dose enhancement than the 195 MeV beam. At P75, the average and maximum DEF were 3.9 and 17.0 for 100 MeV, and 3.5 and 16.2 for 195 MeV, respectively. These results indicate that lower energy protons caused higher dose enhancement in this incident proton energy range. Conclusion: The dose enhancement around GNPs spread as the position in the Bragg peak region becomes deeper and depends on proton energy. It is expected that GNPs can be used as a radio sensitizer with consideration of the location and proton beam energy.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

Exponential Increase in Relative Biological Effectiveness Along Distal Edge of a Proton Bragg Peak as Measured by Deoxyribonucleic Acid Double-Strand Breaks

Purpose: To quantify the relative biological effectiveness (RBE) of the distal edge of the proton Bragg peak, using an in vitro assay of DNA double-strand breaks (DSBs). Methods and Materials: U2OS cells were irradiated within the plateau of a spread-out Bragg peak and at each millimeter position along the distal edge using a custom slide holder, allowing for simultaneous measurement of physical dose. A reference radiation signal was generated using photons. The DNA DSBs at 3 hours (to assess for early damage) and at 24 hours (to assess for residual damage and repair) after irradiation were measured using the γH2AX assay and quantified via flow cytometry. Results were confirmed with clonogenic survival assays. A detailed map of the RBE as a function of depth along the Bragg peak was generated using γH2AX measurements as a biological endpoint. Results: At 3 hours after irradiation, DNA DSBs were higher with protons at every point along the distal edge compared with samples irradiated with photons to similar doses. This effect was even more pronounced after 24 hours, indicating that the impact of DNA repair is less after proton irradiation relative to photons. The RBE demonstrated an exponential increase as a function of depth and was measured to be as high as 4.0 after 3 hours and as high as 6.0 after 24 hours. When the RBE-corrected dose was plotted as a function of depth, the peak effective dose was extended 2-3 mm beyond what would be expected with physical measurement. Conclusions: We generated a highly comprehensive map of the RBE of the distal edge of the Bragg peak, using a direct assay of DNA DSBs in vitro. Our data show that the RBE of the distal edge increases with depth and is significantly higher than previously reported estimates.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2016

SU-F-J-208: Prompt Gamma Imaging-Based Prediction of Bragg Peak Position for Realistic Treatment Error Scenarios

Purpose: To quantify the accuracy in predicting the Bragg peak position using simulated in-room measurements of prompt gamma (PG) emissions for realistic treatment error scenarios that combine several sources of errors. Methods: Prompt gamma measurements by a knife-edge slit camera were simulated using an experimentally validated analytical simulation tool. Simulations were performed, for 143 treatment error scenarios, on an anthropomorphic phantom and a pencil beam scanning plan for nasal cavity. Three types of errors were considered: translation along each axis, rotation around each axis, and CT-calibration errors with magnitude ranging respectively, between −3 and 3 mm, −5 and 5 degrees, and between −5 and +5%. We investigated the correlation between the Bragg peak (BP) shift and the horizontal shift of PG profiles. The shifts were calculated between the planned (reference) position and the position by the error scenario. The prediction error for one spot was calculated as the absolute difference between the PG profile shift and the BP shift. Results: The PG shift was significantly and strongly correlated with the BP shift for 92% of the cases (p<0.0001, Pearson correlation coefficient R>0.8). Moderate but significant correlations were obtained for all cases that considered only CT-calibration errors and for 1 case that combined translation and CT-errors (p<0.0001, R ranged between 0.61 and 0.8). The average prediction errors for the simulated scenarios ranged between 0.08±0.07 and 1.67±1.3 mm (grand mean 0.66±0.76 mm). The prediction error was moderately correlated with the value of the BP shift (p=0, R=0.64). For the simulated scenarios the average BP shift ranged between −8±6.5 mm and 3±1.1 mm. Scenarios that considered combinations of the largest treatment errors were associated with large BP shifts. Conclusion: Simulations of in-room measurements demonstrate that prompt gamma profiles provide reliable estimation of the Bragg peak position for complex error scenarios. Yafei Xing and Luiza Bondar are funded by BEWARE grants from the Walloon Region. The work presents simulations results for a prompt gamma camera prototype developed by IBA.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2018

Experimental Validation of Low-Z Ion-Stopping Formalisms Around the Bragg Peak in High-Energy-Density Plasmas

We report on the first accurate validation of low-Z ion-stopping formalisms in the regime ranging from low-velocity ion stopping – through the Bragg peak – to high-velocity ion stopping in well-characterized High-Energy-Density Plasmas. These measurements were executed at electron temperatures and number densities in the range of 1.4 – 2.8 keV and 4x10^23 – 8x10^23 cm^-3, respectively. For these conditions, it is experimentally demonstrated that the Brown-Preston-Singleton formalism provides a better description of the ion stopping than other formalisms around the Bragg peak, except for the ion stopping at vi ~ 0.3vth, where the Brown-Preston-Singleton formalism significantly under predicts the observation. It is postulated that the inclusion of nuclear-elastic scattering and possibly non-thermal velocity distributions, due to the Knudsen reduction of the high-velocity tail, in the modeling of the ion-ion interaction may explain the discrepancy of ~20% at this velocity, which would have an impact on our understanding of the alpha energy deposition and heating of the fuel ions, and thus reduce the ignition threshold in an ignition experiment.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

Experimental Validation of Low- Z Ion-Stopping Formalisms around the Bragg Peak in High-Energy-Density Plasmas

Here, we report on the first accurate validation of low-$Z$ ion-stopping formalisms in the regime ranging from low-velocity ion stopping—through the Bragg peak—to high-velocity ion stopping in well-characterized high-energy-density plasmas. These measurements were executed at electron temperatures and number densities in the range of 1.4–2.8 keV and 4 × 10 23 –8 × 10 23 cm –3 , respectively. For these conditions, it is experimentally demonstrated that the Brown-Preston-Singleton formalism provides a better description of the ion stopping than other formalisms around the Bragg peak, except for the ion stopping at v i ~0.3 v th , where the Brown-Preston-Singleton formalism significantly underpredicts the observation. It is postulated that the inclusion of nuclear-elastic scattering, and possibly coupled modes of the plasma ions, in the modeling of the ion-ion interaction may explain the discrepancy of ~20% at this velocity, which would have an impact on our understanding of the alpha energy deposition and heating of the fuel ions, and thus reduce the ignition threshold in an ignition experiment.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Spread Out Carbon Bragg Peak at NSRL

Charged particles lose energy continuously while traversing non-vacuum medium but have a sharp peak of dE/dx near the end of the path length. This can be very beneficial for cancer radiotherapy where healthy tissues surrounding a tumor are left with little dose compared to traditional therapy methods. However, since the region of high dose is localized to such a small area, it can present challenges when attempting to treat tumors of larger volume. By utilizing a variable thickness degrader system, the treatment area can be expanded during clinical treatment applications. To accommodate these types of radiobiology experiments, NSRL has produced a spread-out Bragg peak degrader wheel which aims to widen the Bragg peak of a 108 MeV/n Carbon beam to deliver a constant linear energy transfer, in water, of 97.7 keV/µm over a range of 36.7 mm.

43 PARTICLE ACCELERATORS↗
DOE OSTI2019

BraggNet: integrating Bragg peaks using neural networks

Neutron crystallography offers enormous potential to complement structures from X-ray crystallography by clarifying the positions of low-Z elements, namely hydrogen. Macromolecular neutron crystallography, however, remains limited, in part owing to the challenge of integrating peak shapes from pulsed-source experiments. To advance existing software, this article demonstrates the use of machine learning to refine peak locations, predict peak shapes and yield more accurate integrated intensities when applied to whole data sets from a protein crystal. The artificial neural network, based on the U-Net architecture commonly used for image segmentation, is trained using about 100 000 simulated training peaks derived from strong peaks. After 100 training epochs (a round of training over the whole data set broken into smaller batches), training converges and achieves a Dice coefficient of around 65%, in contrast to just 15% for negative control data sets. Integrating whole peak sets using the neural network yields improved intensity statistics compared with other integration methods, including k-nearest neighbours. These results demonstrate, for the first time, that neural networks can learn peak shapes and be used to integrate Bragg peaks. In conclusion, it is expected that integration using neural networks can be further developed to increase the quality of neutron, electron and X-ray crystallography data.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Simultaneous Multi-Bragg Peak Coherent X-ray Diffraction Imaging

The simultaneous measurement of two Bragg reflections by Bragg coherent X-ray diffraction is demonstrated on a twinned Au crystal, which was prepared by the solid-state dewetting of a 30 nm thin gold film on a sapphire substrate. The crystal was oriented on a goniometer so that two lattice planes fulfill the Bragg condition at the same time. The Au 111 and Au 200 Bragg peaks were measured simultaneously by scanning the energy of the incident X-ray beam and recording the diffraction patterns with two two-dimensional detectors. While the former Bragg reflection is not sensitive to the twin boundary, which is oriented parallel to the crystal–substrate interface, the latter reflection is only sensitive to one part of the crystal. The volume ratio between the two parts of the twinned crystal is about 1:9, which is also confirmed by Laue microdiffraction of the same crystal. The parallel measurement of multiple Bragg reflections is essential for future in situ and operando studies, which are so far limited to either a single Bragg reflection or several in series, to facilitate the precise monitoring of both the strain field and defects during the application of external stimuli.

47 OTHER INSTRUMENTATION↗
NASA NTRS2011

Variable Depth Bragg Peak Method for Single Event Effects Testing

Traditionally, accelerator SEE testing is accomplished by removing the tops of packages so that the IC chips are accessible to heavy ions. However, ICs in some advanced packages cannot be de-lidded so a different approach is used that involves grinding and/or chemically etching away part of the package and the chip from the back side. The parts are then tested from the back side with ions having sufficient range to reach the sensitive volume. More recently, the entire silicon substrate in an SOI/SRAM was removed, making it possible to use low-energy ions with shorter ranges. Where removal of part of the package is not possible, facilities at Michigan State, NASA Space Radiation Laboratory, GANIL (France) and GSI (Germany) offer high-energy heavy ions with long ranges so that the ions can reach the devices' sensitive volumes without much change in the LET. Unfortunately, a run will typically involve only one ion species having a single energy and LET due to the long time it takes to tune a new energy. The Variable Depth Bragg Peak (VDBP) method is similar to the above method in that it involves the use of high-energy heavy ions that are able to pass through the packaging material and reach the device, obviating the need to remove the package. However, the method provides a broad range of LETs from a single ion by inserting degraders in the beam that modify the ion energy and, therefore, the LET. The crux of the method involves establishing a fiduciary point for degrader thickness, i.e., where the Bragg peak is located precisely at the sensitive volume in the device, for which the measured SEU cross-section and the ion LET are both also maxima and can be calculated using a Monte-Carlo program, TRIM. Once the fiduciary point has been established, calibrated high density polyethylene (HDPE) degraders are inserted into or removed from the beam to vary the ion LET at the device in a known manner. After each change of degrader thickness, the SEU cross-section is measured and the corresponding LET calculated from the change in degrader thickness. That information is used to generate a plot of cross-section as a function of ion LET. The advantages of this approach are that the part does not have to be de-lidded and a broad range of LETs is available from a single heavy ion without having to go to non-normal angles of incidence to change the "effective" LET. As we will show, it is possible to obtain an entire curve of cross-section versus LET using just two or three ions. Fig. 1 shows curves of cross-section vs LET for a Freescale 4 Mbit SOI/SRAM measured at the 88" Cyclotron at Berkeley and at NSRL. The open symbols are the data obtained from Berkeley for top-side and back-side irradiation. The solid data points are for the data obtained at NSRL using a device for which the package was intact. The data are for Iron and Gold and cover a range of LETs from 4 MeV.cm2/mg to 84 MeV.cm2/mg. The agreement between the data obtained from Berkeley and from NSRL is excellent, demonstrating that the VDBP method is capable of providing accurate values of cross-section versus LET, at least for the 4 Mbit SRAM. Details of the technique will be included in the final presentation.

Buchner, S.↗
DOE OSTI2025

End-to-end deep learning pipeline for real-time Bragg peak segmentation: from training to large-scale deployment

X-ray crystallography reconstruction, which transforms discrete X-ray diffraction patterns into three-dimensional molecular structures, relies critically on accurate Bragg peak finding for structure determination. As X-ray free electron laser (XFEL) facilities advance toward MHz data rates (1 million images per second), traditional peak finding algorithms that require manual parameter tuning or exhaustive grid searches across multiple experiments become increasingly impractical. While deep learning approaches offer promising solutions, their deployment in high-throughput environments presents significant challenges in automated dataset labeling, model scalability, edge deployment efficiency, and distributed inference capabilities. We present an end-to-end deep learning pipeline with three key components: (1) a data engine that combines traditional algorithms with our peak matching algorithm to generate high-quality training data at scale, (2) a modular architecture that scales from a few million to hundreds of million parameters, enabling us to train large expert-level models offline while deploying smaller, distilled models at the edge, and (3) a decoupled producer-consumer architecture that separates specialized data source layer from model inference, enabling flexible deployment across diverse computing environments. Using this integrated approach, our pipeline achieves accuracy comparable to traditional methods tuned by human experts while eliminating the need for experiment-specific parameter tuning. Although current throughput requires optimization for MHz facilities, our system's scalable architecture and demonstrated model compression capabilities provide a foundation for future high-throughput XFEL deployments.

Wang, Cong↗
DOE OSTI2021

Experimental demonstration of coupled multi-peak Bragg coherent diffraction imaging with genetic algorithms

Bragg coherent diffraction imaging has the potential to provide significant insight into the structure-properties relationship for crystalline materials by imaging, with nanoscale resolution, three-dimensional strain fields within individual grains and nanoparticles. The capability of present-day synchrotrons to locate and measure a multiplicity of Bragg reflections from a single grain makes it possible to recover the full strain tensor with nanometer resolution. Recent methods for coupling reconstructions from several peaks to determine the strain tensor have been developed and applied to synthetic data, but have not been applied to experimental data. Here, using a coupled genetic reconstruction algorithm, we reconstruct an experimental data set and demonstrate improvements in the ability to resolve vector-valued displacement fields internal to the particle as compared to what is achieved with a noncoupled approach. The coupled approach developed in this work was also validated on simulated data sets. In both simulated and experimental data, reconstructions from our coupled Bragg peak algorithm show improvements over the noncoupled independent reconstruction method of 5% in terms of accuracy and 53% in terms of consistency.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2016

Investigating the Implications of a Variable RBE on Proton Dose Fractionation Across a Clinical Pencil Beam Scanned Spread-Out Bragg Peak

Purpose: To investigate the clinical implications of a variable relative biological effectiveness (RBE) on proton dose fractionation. Using acute exposures, the current clinical adoption of a generic, constant cell killing RBE has been shown to underestimate the effect of the sharp increase in linear energy transfer (LET) in the distal regions of the spread-out Bragg peak (SOBP). However, experimental data for the impact of dose fractionation in such scenarios are still limited. Methods and Materials: Human fibroblasts (AG01522) at 4 key depth positions on a clinical SOBP of maximum energy 219.65 MeV were subjected to various fractionation regimens with an interfraction period of 24 hours at Proton Therapy Center in Prague, Czech Republic. Cell killing RBE variations were measured using standard clonogenic assays and were further validated using Monte Carlo simulations and parameterized using a linear quadratic formalism. Results: Significant variations in the cell killing RBE for fractionated exposures along the proton dose profile were observed. RBE increased sharply toward the distal position, corresponding to a reduction in cell sparing effectiveness of fractionated proton exposures at higher LET. The effect was more pronounced at smaller doses per fraction. Experimental survival fractions were adequately predicted using a linear quadratic formalism assuming full repair between fractions. Data were also used to validate a parameterized variable RBE model based on linear α parameter response with LET that showed considerable deviations from clinically predicted isoeffective fractionation regimens. Conclusions: The RBE-weighted absorbed dose calculated using the clinically adopted generic RBE of 1.1 significantly underestimates the biological effective dose from variable RBE, particularly in fractionation regimens with low doses per fraction. Coupled with an increase in effective range in fractionated exposures, our study provides an RBE dataset that can be used by the modeling community for the optimization of fractionated proton therapy.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2017

Expected values and variances of Bragg peak intensities measured in a nanocrystalline powder diffraction experiment

A rigorous study of sampling and intensity statistics applicable for a powder diffraction experiment as a function of crystallite size is presented. Our analysis yields approximate equations for the expected value, variance and standard deviations for both the number of diffracting grains and the corresponding diffracted intensity for a given Bragg peak. The classical formalism published in 1948 by Alexander, Klug & Kummer [ J. Appl. Phys. (1948), 19 , 742–753] appears as a special case, limited to large crystallite sizes, here. It is observed that both the Lorentz probability expression and the statistics equations used in the classical formalism are inapplicable for nanocrystalline powder samples.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗