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DOE OSTI2023

Coarse-grained methods for heterogeneous vesicles with phase-separated domains: Elastic mechanics of shape fluctuations, plate compression, and channel insertion

We develop coarse-grained particle approaches for studying the elastic mechanics of vesicles with heterogeneous membranes having phase-separated domains. We perform simulations both of passive shape fluctuations and of active systems where vesicles are subjected to compression between two plates or subjected to insertion into narrow channels. Analysis methods are developed for mapping particle configurations to continuum fields with spherical harmonics representations. Heterogeneous vesicles are found to exhibit rich behaviors where the heterogeneity can amplify surface two-point correlations, reduce resistance during compression, and augment vesicle transport times in channels. The developed methods provide general approaches for characterizing the mechanics of coarse-grained heterogeneous systems taking into account the roles of thermal fluctuations, geometry, and phase separation.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

Kilometric radiation power flux dependence on area of discrete aurora

Kilometer wavelength radiation, measured from distant positions over the North Pole and over the Earth's equator, was compared to the area of discrete aurora imaged by several low-altitude spacecraft. Through correlative studies of auroral kilometric radiation (AKR) with about two thousand auroral images, a stereoscopic view of the average auroral acceleration region was obtained. A major result is that the total AKR power increases as the area of the discrete auroral oval increases. The implications are that the regions of parallel potentials or the auroral plasma cavities, in which AKR is generated, must possess the following attributes: (1) they are shallow in altitude and their radial position depends on wavelength, (2) they thread flux tubes of small cross section, (3) the generation mechanism in them reaches a saturation limit rapidly, and (4) their distribution over the discrete auroral oval is nearly uniform. The above statistical results are true for large samples collected over a long period of time (about six months). In the short term, AKR frequently exhibits temporal variations with scales as short as three minutes (the resolution of the averaged data used). These fluctuations are explainable by rapid quenchings as well as fast starts of the electron cyclotron maser mechanism. There were times when AKR was present at substantial power levels while optical emissions were below instrument thresholds. A recent theoretical result may account for this set of observations by predicting that suprathermal electrons, of energies as low as several hundred eV, can generate second harmonic AKR. The indirect observations of second harmonic AKR require that these electrons have mirror points high above the atmosphere so as to minimize auroral light emissions. The results provide evidence supporting the electron cyclotron maser mechanism.

Saflekos, N. A.↗

NASA NTRS1987

Vortex-scalar element calculations of a diffusion flame stabilized on a plane mixing layer

The vortex-scalar element method, a scheme which utilizes vortex elements to discretize the region of high vorticity and scalar elements to represent species or temperature fields, is utilized in the numerical simulations of a two-dimensional reacting mixing layer. Computations are performed for a diffusion flame at high Reynolds and Peclet numbers without resorting to turbulence models. In the nonreacting flow, the mean and fluctuation profiles of a conserved scalar show good agreement with experimental measurements. Results for the reacting flow indicate that for temperature independent kinetics, the chemical reaction begins immediately downstream of the splitter plate where mixing starts. Results for the reacting flow with Arrhenius kinetics show an ignition delay, which depends on reactant temperature, before significant chemical reaction occurs. Harmonic forcing changes the structure of the layer, and concomitantly the rates of mixing and reaction, in accordance with experimental results. Strong stretch within the braids in the nonequilibrium kinetics case causes local flame quenching due to the temperature drop associated with the large convective fluxes.

Ghoniem, Ahmed F.↗

DOE OSTI2025

Window convolution of the galaxy clustering bispectrum

In galaxy survey analysis, the observed clustering statistics do not directly match theoretical predictions but rather have been processed by a window function that arises from the survey geometry including the sky footprint, redshift-dependent background number density and systematic weights. While window convolution of the power spectrum is well studied, for the bispectrum with a larger number of degrees of freedom, it poses a significant numerical and computational challenge. In this work, we consider the effect of the survey window in the tripolar spherical harmonic decomposition of the bispectrum and lay down a formal procedure for their convolution via a series expansion of configuration-space three-point correlation functions, which was first proposed by Sugiyama et al. (2019). We then provide a linear algebra formulation of the full window convolution, where an unwindowed bispectrum model vector can be directly premultiplied by a window matrix specific to each survey geometry. To validate the pipeline, we focus on the Dark Energy Spectroscopic Instrument (DESI) Data Release 1 (DR1) luminous red galaxy (LRG) sample in the South Galactic Cap (SGC) in the redshift bin 0.4 ≤ z ≤ 0.6. We first perform convergence checks on the measurement of the window function from discrete random catalogues, and then investigate the convergence of the window convolution series expansion truncated at a finite of number of terms as well as the performance of the window matrix. This work highlights the differences in window convolution between the power spectrum and bispectrum, and provides a streamlined pipeline for the latter for current surveys such as DESI and the Euclid mission.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1995

Sensitivity analysis for aeroacoustic and aeroelastic design of turbomachinery blades

A new method for computing the effect that small changes in the airfoil shape and cascade geometry have on the aeroacoustic and aeroelastic behavior of turbomachinery cascades is presented. The nonlinear unsteady flow is assumed to be composed of a nonlinear steady flow plus a small perturbation unsteady flow that is harmonic in time. First, the full potential equation is used to describe the behavior of the nonlinear mean (steady) flow through a two-dimensional cascade. The small disturbance unsteady flow through the cascade is described by the linearized Euler equations. Using rapid distortion theory, the unsteady velocity is split into a rotational part that contains the vorticity and an irrotational part described by a scalar potential. The unsteady vorticity transport is described analytically in terms of the drift and stream functions computed from the steady flow. Hence, the solution of the linearized Euler equations may be reduced to a single inhomogeneous equation for the unsteady potential. The steady flow and small disturbance unsteady flow equations are discretized using bilinear quadrilateral isoparametric finite elements. The nonlinear mean flow solution and streamline computational grid are computed simultaneously using Newton iteration. At each step of the Newton iteration, LU decomposition is used to solve the resulting set of linear equations. The unsteady flow problem is linear, and is also solved using LU decomposition. Next, a sensitivity analysis is performed to determine the effect small changes in cascade and airfoil geometry have on the mean and unsteady flow fields. The sensitivity analysis makes use of the nominal steady and unsteady flow LU decompositions so that no additional matrices need to be factored. Hence, the present method is computationally very efficient. To demonstrate how the sensitivity analysis may be used to redesign cascades, a compressor is redesigned for improved aeroelastic stability and two different fan exit guide vanes are redesigned for reduced downstream radiated noise. In addition, a framework detailing how the two-dimensional version of the method may be used to redesign three-dimensional geometries is presented.

Lorence, Christopher B.↗

NASA NTRS2002

Development of a High Performance Acousto-ultrasonic Scan System

Acousto-ultrasonic (AU) interrogation is a single-sided nondestructive evaluation (NDE) technique employing separated sending and receiving transducers. It is used for assessing the microstructural condition/distributed damage state of the material between the transducers. AU is complementary to more traditional NDE methods such as ultrasonic c-scan, x-ray radiography, and thermographic inspection that tend to be used primarily for discrete flaw detection. Through its history, AU has been used to inspect polymer matrix composite, metal matrix composite, ceramic matrix composite, and even monolithic metallic materials. The development of a high-performance automated AU scan system for characterizing within-sample microstructural and property homogeneity is currently in a prototype stage at NASA. In this paper, a review of essential AU technology is given. Additionally, the basic hardware and software configuration, and preliminary results with the system, are described.

Roth, D. J.↗

NASA NTRS1966

Numerical Studies of the Non-linear Vlasov Equation

The subject of this dissertation is the numerical integration of the initial-value problem for the non-linear Vlasov equation. The Vlasov equation is used to describe the dynamics of a "collisionless", one-dimensional, classical electron gas confined between two perfectly reflecting boundaries. Only the long-range Coulomb interactions of the electrons are considered; effects associated with the discrete structure are neglected. The numerical results obtained for non-linear Landau damping compare well with similar results obtained by Knorr. A general statement of the results on stable initial conditions is: As the degree of non-linearity of the initial conditions is increased, the deviation from linear Landau damping appears sooner and is more severe. In some cases damping was observed to cease. Curves showing the time dependence of the damping decrement are derived and compared with predictions of non-linear theories. New results obtained in this study include the observation that for strongly non-linear cases, the damping of the electric field causes an initially Maxwellian fo (v, O) to develop a peak in the neighborhood of the phase velocity; strong growth of the second harmonic is seen after fo (v, t) develops such a peak. Also new in this study is the interpretation of the development of a certain class of strongly unstable initial conditions as approaching an inhomogeneous equilibrium.

NUMERICAL INTEGRATION↗

DOE OSTI2024

Full wave modeling of radio-frequency beams in tokamaks in the electron cyclotron frequency range

Simulation of full wave, without paraxial approximation, high-resolution solution of wave equations in frequency domain in the electron cyclotron resonance (ECR) frequency range for realistic Tokamak plasma parameters became possible by using recently formulated hybrid iterative algorithm [Svidzinski et al., Phys. Plasmas 25, 082509 (2018)] for numerically solving discretized wave equations. This approach combines time evolution and iterative relaxation techniques into iteration cycles. This algorithm is implemented in 2D code FullWave, solving wave equations in Tokamaks in cold and hot plasma models, and it has been tested in 3D full wave iterative RF beams simulation tool, which is presently being developed to model 3D ECRH RF beams in fusion devices using dynamic grid adaptation. The results of 2D full wave modeling, assuming specified toroidal mode number, of ECRH RF beams in DIII-D plasma, performed in the cold and hot plasma models for outboard and top launch scenarios using FullWave are presented. Nonlocal hot plasma response model, based on accurate numerical solution of linearized Vlasov equation, is used to model beam propagation and absorption in the 2nd electron cyclotron harmonic region. Demonstration of capability of the hybrid iterative algorithm to model ECRH RF beams in 3D is made by simulating a substantial part of realistic beam in DIII-D, launched from outboard side of the machine. All relevant physics of RF beam propagation, most of which is not captured in paraxial approximation, such as beam's divergence, interference between the X and O modes in the beam, X-O mode conversion, beam splitting into the X and O mode beams, transformation of beam's cross section, and absorption at the 2nd electron cyclotron harmonic, is captured in the simulations. A numerical technique to find an optimal beam polarization at the launcher to launch a nearly pure X or O mode beam in plasma is developed and tested.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2022

Combined Bluntness and Roughness Effects on Cones at Hypersonic Speeds

This computational study investigates the effects of discrete roughness elements on a blunt cone at zero degrees angle of attack in a Mach 6 flow. Motivation was provided by experiments conducted in the Air Force Research Laboratory Mach 6 High Reynolds Number facility on a 7-degree half-angle cone with a roughness array located at 45 degrees from the apex on two nose tips of different blutness but equivalent roughness Reynolds number. Transition was only affected on the blunter cone, indicating that the transition onset is associated with the combined effects of bluntness and roughness. The present study investigates the 15.24 mm nose radius, 420 azimuthal wavenumber case via Navier-Stokes computations of the laminar base flow and instability analysis. Plane-marching parabolized stability equations (PSE) and inflow-resolvent analysis based on the three-dimensional, harmonic linearized Navier-Stokes equations (HLNSE) are used to calculate the amplification of disturbances along the roughness wake as well as over the roughness nearfield. Results show that the roughness shape can have a great impact on the characteristics of the most amplified wake instabilities. For the experimental configuration with cubic roughness elements of 15 μ m height, the flow is marginally unstable. For prismatic elements of 20 μ m height, the PSE predicts a logarithmic disturbance amplification ratio of N = 5.6 along its wake, but this ratio increases to N = 9.6 when the amplification over the roughness and separation regions is included in the inflow-resolvent analysis.

Boundary-layer transition↗

NASA NTRS2022

Combined Bluntness and Roughness Effects on Cones at Hypersonic Speeds

Boundary-layer transition↗

DOE OSTI2026

pH-Driven Restructuring of Hydration Layers and Cation Adsorption at the Alumina–Water Interface

Oxide−water interfaces underpin ion separations, catalysis, and electrochemical energy technologies, where the electrical double layer (EDL) controls adsorption, transport, and reactivity. However, the molecular-scale links between pHdependent surface protonation, hydration-layer structure, and counterion adsorption remain poorly defined. Here, we combine in situ crystal truncation rod (CTR) and resonant anomalous X-ray reflectivity (RAXR) with streaming potential measurements and ab initio molecular dynamics (AIMD) simulations to resolve the chemical and structural evolution of the EDL at the single-crystal alumina (012)−water interface in 10 mM Rb + over pH 3−12. CTR measurements reveal two distinct adsorbed water layers at ∼2.2 and ∼3.5 Å above the surface. Each water layer shifts toward the substrate at transition pHs near 6.5 and 10.6, respectively, reflecting changes in primary hydration layer structure in response to the deprotonation of bridging and terminal aluminol groups. RAXR shows a 10-fold increase in Rb + coverage and a decrease in mean adsorption height from ∼3.5 to ∼2.7 Å with increasing pH, indicating enhanced counterion binding accompanied by Stern layer contraction. Streaming potential measurements demonstrate that the zeta potential, i.e., the potential at the hydrodynamic shear plane, is positive at pH 3 and becomes negative at pH ≥ 3.5. The negative charge magnitude increases with increasing pH, consistent with progressive surface deprotonation at higher pH. AIMD identifies inner- and outer-sphere Rb+ complexes whose adsorption heights and coordination geometries depend sensitively on the protonation state of surface oxygens, providing atomistic support for the experimentally inferred trends. These measurements establish two discrete, site-specific pH transitions in hydration-layer structure that track aluminol (de)protonation and quantitatively link them to a pH-driven contraction of the Stern layer (increasing Rb + coverage and decreasing adsorption height). This provides a direct structural basis for connecting surface acid−base chemistry to ion binding distances at an oxide−water interface.

Adsorption↗

DOE OSTI2024

Principles of two-dimensional terahertz spectroscopy of collective excitations: The case of Josephson plasmons in layered superconductors

Two-dimensional terahertz spectroscopy (2DTS), a terahertz analog of nuclear magnetic resonance, is a new technique poised to address many open questions in complex condensed matter systems. The conventional theoretical framework used ubiquitously for interpreting multidimensional spectra of discrete quantum level systems is, however, insufficient for the continua of collective excitations in strongly correlated materials. Here, in this work, we develop a theory for 2DTS of a model collective excitation, the Josephson plasma resonance in layered superconductors. Starting from a mean-field approach at temperatures well below the superconducting phase transition, we obtain expressions for the multidimensional nonlinear responses that are amenable to intuition derived from the conventional single-mode scenario. We then consider temperatures near the superconducting critical temperature T c , where dynamics beyond mean-field become important and conventional intuition fails. As fluctuations proliferate near T c , the dominant contribution to nonlinear response comes from an optical parametric drive of counterpropagating Josephson plasmons, which gives rise to 2D spectra that are qualitatively different from the mean-field predictions. As such, and in contrast to one-dimensional spectroscopy techniques, such as third harmonic generation, 2DTS can be used to directly probe thermally excited finite-momentum plasmons and their interactions. Our theory can readily be tested in cuprates, and we discuss implications beyond the present context of Josephson plasmons.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS2008

Fast Plasma Instrument for MMS: Simulation Results

Magnetospheric Multiscale (MMS) mission will study small-scale reconnection structures and their rapid motions from closely spaced platforms using instruments capable of high angular, energy, and time resolution measurements. The Dual Electron Spectrometer (DES) of the Fast Plasma Instrument (FPI) for MMS meets these demanding requirements by acquiring the electron velocity distribution functions (VDFs) for the full sky with high-resolution angular measurements every 30 ms. This will provide unprecedented access to electron scale dynamics within the reconnection diffusion region. The DES consists of eight half-top-hat energy analyzers. Each analyzer has a 6 deg. x 11.25 deg. Full-sky coverage is achieved by electrostatically stepping the FOV of each of the eight sensors through four discrete deflection look directions. Data compression and burst memory management will provide approximately 30 minutes of high time resolution data during each orbit of the four MMS spacecraft. Each spacecraft will intelligently downlink the data sequences that contain the greatest amount of temporal structure. Here we present the results of a simulation of the DES analyzer measurements, data compression and decompression, as well as ground-based analysis using as a seed re-processed Cluster/PEACE electron measurements. The Cluster/PEACE electron measurements have been reprocessed through virtual DES analyzers with their proper geometrical, energy, and timing scale factors and re-mapped via interpolation to the DES angular and energy phase-space sampling measurements. The results of the simulated DES measurements are analyzed and the full moments of the simulated VDFs are compared with those obtained from the Cluster/PEACE spectrometer using a standard quadrature moment, a newly implemented spectral spherical harmonic method, and a singular value decomposition method. Our preliminary moment calculations show a remarkable agreement within the uncertainties of the measurements, with the results obtained by the Cluster/PEACE electron spectrometers. The data analyzed was selected because it represented a potential reconnection event as currently published.

Figueroa-Vinas, Adolfo↗

NASA NTRS2011

High Frequency Acoustic Response Characterization and Analysis of the Deep Throttling Common Extensible Cryogenic Engine

The Common Extensive Cryogenic Engine program demonstrated the operation of a deep throttling engine design. The program, spanning five years from August 2005 to July 2010, funded testing through four separate engine demonstration test series. Along with successful completion of multiple objectives, a discrete response of approximately 4000 Hz was discovered and explored throughout the program. The typical low-amplitude acoustic response was evident in the chamber measurement through almost every operating condition; however, at certain off-nominal operating conditions, the response became discrete with higher amplitude. This paper summarizes the data reduction, characterization, and analysis of the 4,000 Hz response for the entire program duration, using the large amount of data collected. Upon first encountering the response, new objectives and instrumentation were incorporated in future test series to specifically collect 4,000 Hz data. The 4,000 Hz response was identified as being related to the first tangential acoustic mode by means of frequency estimation and spatial decomposition. The latter approach showed that the effective node line of the mode was aligned with the manifold propellant inlets with standing waves and quasi-standing waves present at various times. Contour maps that contain instantaneous frequency and amplitude trackings of the response were generated as a significant improvement to historical manual approaches of data reduction presentation. Signal analysis and dynamic data reduction also uncovered several other features of the response including a stable limit cycle, the progressive engagement of subsequent harmonics, the U-shaped time history, an intermittent response near the test-based neutral stability region, other acoustic modes, and indications of modulation with a separate subsynchronous response. Although no engine damage related to the acoustic mode was noted, the peak-to-peak fluctuating pressure amplitude achieved 12.1% of the mean chamber pressure at its highest. The identification of this response in terms of an instability is also discussed.

Casiano, M. J.↗

NASA NTRS2009

Exploring Divisibility and Summability of 'Photon' Wave Packets in Nonlinear Optical Phenomena

Formulations for second and higher harmonic frequency up and down conversions, as well as multi photon processes directly assume summability and divisibility of photons. Quantum mechanical (QM) interpretations are completely congruent with these assumptions. However, for linear optical phenomena (interference, diffraction, refraction, material dispersion, spectral dispersion, etc.), we have a profound dichotomy. Most optical engineers innovate and analyze all optical instruments by propagating pure classical electromagnetic (EM) fields using Maxwell s equations and gives only lip-service to the concept "indivisible light quanta". Further, irrespective of linearity or nonlinearity of the phenomena, the final results are always registered through some photo-electric or photo-chemical effects. This is mathematically well modeled by a quadratic action (energy absorption) relation. Since QM does not preclude divisibility or summability of photons in nonlinear & multi-photon effects, it cannot have any foundational reason against these same possibilities in linear optical phenomena. It implies that we must carefully revisit the fundamental roots behind all light-matter interaction processes and understand the common origin of "graininess" and "discreteness" of light energy.

Prasad, Narasimha↗

DOE OSTI2020

Dimension-free path-integral molecular dynamics without preconditioning

Convergence with respect to imaginary-time discretization (i.e., the number of ring-polymer beads) is an essential part of any path-integral-based molecular dynamics (MD) calculation. However, an unfortunate property of existing non-preconditioned numerical integration schemes for path-integral molecular dynamics—including essentially all existing ring-polymer molecular dynamics (RPMD) and thermostatted RPMD (T-RPMD) methods—is that for a given MD time step, the overlap between the exact ring-polymer Boltzmann–Gibbs distribution and that sampled using MD becomes zero in the infinite-bead limit. This has clear implications for hybrid Metropolis Monte Carlo/MD sampling schemes, and it also causes the divergence with bead number of the primitive path-integral kinetic-energy expectation value when using standard RPMD or T-RPMD. We show that these and other problems can be avoided through the introduction of “dimension-free” numerical integration schemes for which the sampled ring-polymer position distribution has non-zero overlap with the exact distribution in the infinite-bead limit for the case of a harmonic potential. Most notably, we introduce the BCOCB integration scheme, which achieves dimension freedom via a particular symmetric splitting of the integration time step and a novel implementation of the Cayley modification [R. Korol et al., J. Chem. Phys. 151, 124103 (2019)] for the free ring-polymer half-steps. More generally, we show that dimension freedom can be achieved via mollification of the forces from the external physical potential. The dimension-free path-integral numerical integration schemes introduced here yield finite error bounds for a given MD time step, even as the number of beads is taken to infinity; these conclusions are proven for the case of a harmonic potential and borne out numerically for anharmonic systems that include liquid water. The numerical results for BCOCB are particularly striking, allowing for nearly three-fold increases in the stable time step for liquid water with respect to the Bussi–Parrinello (OBABO) and Leimkuhler (BAOAB) integrators, while introducing negligible errors in the calculated statistical properties and absorption spectrum. Importantly, the dimension-free, non-preconditioned integration schemes introduced here preserve ergodicity and global second-order accuracy, and they remain simple, black-box methods that avoid additional computational costs, tunable parameters, or system-specific implementations.

Korol, Roman (ORCID:0000000193076351)↗

DOE OSTI2023

A Refinement-by-Superposition -Method for (curl)- and (div)-Conforming Discretizations

Here, we present refinement-by-superposition (RBS) hp-refinement infrastructure for computational electromagnetics (CEMs), which permits exponential rates of convergence. In contrast to dominant approaches to hp-refinement for continuous Galerkin methods, which rely on explicit constraint equations, the multilevel strategy presented drastically reduces the implementation complexity. Through the RBS methodology, enforcement of continuity occurs by construction, enabling arbitrary levels of refinement with ease, and without the practical (but not theoretical) limitations of constrained-node refinement. We outline the construction of the RBS hp-method for refinement with H (curl)- and H (div)-conforming finite cells. Numerical simulations for the 2-D finite element method (FEM) solution of the Maxwell eigenvalue problem demonstrate the effectiveness of RBS hp-refinement. As an additional goal of this work, we aim to promote the use of mixed-order (low- and high-order) elements in practical CEM applications.

42 ENGINEERING↗

DOE OSTI2020

Radiation shielding analysis of the concrete hutch of a long beamline of the Advanced Photon Source Upgrade project

The radiation shielding analysis of the external concrete enclosure of a feature long beamline of the Advanced Photon Source Upgrade project is described in this work. The beamline has 2 undulators that can operate in two different configurations and the source terms are estimated for the worst configuration using the SRW code in the OASYS package as well as the STAC8 code. The shielding analysis is carried out using the STAC8 code, and for the case of pink beam, with the FLUKA code also. Effective dose rates on contact with the outside surface of the concrete walls for pink, reflected full spectrum and monochromatic beams are calculated. For the monochromatic beams, 2 sets of discrete bandwidths (BW) from the Double Crystal Monochromator (DCM) and the full reflectance of the Double Multilayer Monochromator (DMM) are used. Further, due to the size of the direct beam and the energy domain of interest in the direct and scattered fields, the importance of the equivalent dose to the lens of the eye is discussed with respect to the shielding calculations of such facilities. The dose rates outside the concrete hutch in the lateral directions calculated using STAC8 is about 50% higher than the FLUKA results for the case of pink beam but may be considered as a reasonable agreement due to the differences in the calculational methodologies. Those dose rates in the lateral and forward directions are higher for the pink beam to allow full occupancy from the radiation protection point of view. For the full spectrum after 3 reflections, the effective dose rates are less than 0.5 Sv/h in the lateral direction but is higher in the forward region below about 22° with respect to the beam direction. For the DCM beams, except for the scenario with one mirror and 0.1%BW, all dose rates are less than 0.5 Sv/h outside the concrete hutch in the lateral directions. Here, the higher harmonics are what contributes to the dose outside the shield and the bandwidths assumed to derive the photon flux thus becomes important. The dose rates with the DMM are generally lower compared to the 0.1% BW DCM beam but are higher than the results obtained with XOP calculated bandwidths.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗