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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Resonance in radio frequency sheath admittance and enhanced impurity emission near the ion cyclotron frequency

Ion cyclotron resonance heating (ICRH) is of considerable interest among all auxiliary heating techniques, because it transfers power directly to ions, targets the high-density core, and involves the cheapest radio-frequency (RF) components. During ICRH operation, RF sheaths form on the ICRH antenna itself, nearby hardware, and far-field surfaces. These sheaths are associated with large hot-spot formation and impurity emissions. This work presents high-resolution numerical modelling of RF sheaths in nuclear fusion scenarios using hPIC2, a Debye-scale particle-in-cell code. The modelling reveals a new RF sheath phenomenon which occurs when the RF is resonant with a species ion cyclotron frequency or harmonic in an oblique magnetic field. This resonance of the RF sheath causes modifications of the energy-angle distributions of the ions impacting on the walls, with consequent increase in wall impurity emission. Due to the 1/ R scaling of the magnetic field in a tokamak, such cases are possible on divertor or vacuum chamber surfaces, depending on the geometry of the tokamak and the RF heating scenario. A simple physics interpretation using a driven damped harmonic oscillator is proposed, where the ion sheath transit time plays the role of damping the RF sheath admittance. Critically, the resonance leads to increased ion heat flux at the wall as well as increased physical sputtering, despite a lack of increase in RF rectified sheath potential.

ICRH antenna↗
DOE OSTI2022

Novel internal measurements of ion cyclotron frequency range fast-ion driven modes

Abstract Novel internal measurements and analysis of ion cyclotron frequency range fast-ion driven modes in DIII-D are presented. Observations, including internal density fluctuation ( n ~ ) measurements obtained via Doppler backscattering, are presented for modes at low harmonics of the ion cyclotron frequency localized in the edge. The measurements indicate that these waves, identified as coherent ion cyclotron emission (ICE), have high wave number, k ⊥ ρ fast ≳ 1, consistent with the cyclotron harmonic wave branch of the magnetoacoustic cyclotron instability, or electrostatic instability mechanisms. Measurements show extended spatial structure (at least ∼1/6 the minor radius). These edge ICE modes undergo amplitude modulation correlated with edge localized modes (ELM) that is qualitatively consistent with expectations for ELM-induced fast-ion transport.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Novel Strategy To Analyze Fourier Transform Ion Cyclotron Resonance Mass Spectrometry Data of Biomass Pyrolysis Oil for Oligomeric Structure Assignment

Among the most prominent challenges for the analysis of biomass pyrolysis products is the characterization of the abundant oligomer fraction of bio-oil. This fraction is principally made up of pyrolytic lignins and dehydrated, highly modified sugar oligomers, called humins, in liquid phase. An emerging technique for analysis of this oligomer fraction is high-resolution Fourier transform-ion cyclotron resonance mass spectrometry (FT-ICR MS), which allows for accurate determination of C x H y O z formulae for detected oligomers. Additionally, using simple dehydration and fragmentation reaction schemes, similar formulae can be developed from cellulose-, hemicellulose- and lignin-derived oligomers, which are primary depolymerization products. In this paper, FT-ICR MS analysis is coupled with combinatoric dehydration and fragmentation modeling in order to match experimentally detected bio-oil oligomers with hypothetical pathways for their formation during pyrolysis. In this way, we present a novel strategy by which oligomeric structures can be proposed for bio-oils. Using this approach, it becomes possible to advance the understanding of both the molecular structures comprising the bio-oil oligomer fraction and the pathways by which these structures form during biomass pyrolysis reactions.

09 BIOMASS FUELS↗
DOE OSTI2020

Vacuum Pyrolysis of Hybrid Poplar Milled Wood Lignin with Fourier Transform-Ion Cyclotron Resonance Mass Spectrometry Analysis of Feedstock and Products for the Elucidation of Reaction Mechanisms

The pyrolysis of lignocellulosic materials is a promising technique to produce fuels and chemicals. It is well known that the most abundant products of lignin pyrolysis are oligomeric molecules, known as pyrolytic lignin (PL). The chemical composition of PL has been extensively studied; however, there is still an important debate whether these oligomers are produced directly from the lignin or from the recombination of monomeric pyrolytic products. Existing theories are unable to describe the effect of vacuum on the distribution of pyrolysis products. Hybrid poplar milled wood lignin (MWL) was initially isolated and thoroughly characterized by Fourier transform-ion cyclotron resonance mass spectrometry (FT-ICR MS). Chemical formulas were assigned to each oligomeric compound detected. The MWL was also subjected to vacuum pyrolysis in a modified pyroprobe at 250, 750, and 1000 mbar (absolute pressure), and the resulting liquid products were analyzed by FT-ICR MS. A new strategy to assign structural representations to the oligomeric PL products is proposed, based on the plausible pyrolysis reaction mechanisms of depolymerization/fragmentation applied to original MWL oligomer formulas. Our results support the hypothesis that PL is formed from the removal of moieties from primary lignin pyrolysis products with between three and five aromatic rings. This depolymerization/fragmentation allows the oligomers to reduce their molecular weights to the point where they can be removed from the reaction zone by direct vaporization. Furthermore, this phenomenon highlights the importance of pressure on removal mechanisms and their impact on the molecular weight of the resulting products from lignin pyrolysis.

09 BIOMASS FUELS↗
DOE OSTI2026

Cyclotron breaking: a mechanism for parallel ion cyclotron waves to heat the fast solar wind

The Parker Solar Probe mission has observed near-continuous power in parallel ion cyclotron waves (PICWs) in the young, fast solar wind. These waves are unlikely to be directly produced by the turbulent cascade and are likely born of a local instability; yet, they are observed to both cool – and heat – the plasma. We propose that these observations can be self-consistently explained as the natural consequence of PICWs propagating in the inhomogeneous solar wind after they have been driven unstable. In this work, we argue that strong proton heating by a turbulent cascade of oblique ICWs will result in PICWs being driven unstable in a process known as quasi-linear focusing. Because the power in the turbulent cascade is concentrated at scales above the turbulent transition region, PICWs will be driven unstable within a range of wavenumbers parallel to the background magnetic field, 𝑘 ∥ , that is bounded from above by 𝑘$^{∗}_{∥P}$, corresponding to the start of the transition region. As unstable PICWs propagate away from the Sun to regions of lower proton density, their 𝑘 ∥ , multiplied by the proton inertial length 𝑑 p , increases. Eventually, 𝑘$^{∗}_{∥P}$ of the PICWs becomes larger than 𝑘$^{∗}_{∥P}$⁢𝑑 p and the waves damp, heating the solar wind. We call this effect ‘cyclotron breaking’, in analogy with ocean waves breaking on the shore. We then discuss the testable predictions of the theory, including a distinct heating signature in which PICWs cool fast protons and heat slow protons at any given heliocentric distance 𝑟. Finally, we conjecture that cyclotron breaking can lead to net heating by PICWs if the power emitted as PICWs decreases sufficiently rapidly with 𝑟 that local emission of PICWs is overwhelmed by the local damping of PICWs generated closer to the Sun.

plasma heating↗
DOE OSTI2025

Dependence of ion-cyclotron range of frequencies instabilities on species mix and fast-ion distribution: I. Stability

Ion cyclotron emission (ICE) driven by fast ions has the potential to be a reactor relevant diagnostic of confined and escaping fast ions; moreover, controlled experiments in magnetic fusion devices can shed light on similar magnetospheric instabilities below and above the ion cyclotron frequency f ci . Dedicated DIII-D experiments studied instabilities below f ci and at cyclotron harmonics l f ci by injecting thirteen different neutral beam populations into plasmas with different thermal compositions of H, D, and 3He at five different values of toroidal field. High-frequency magnetic loops diagnose mode properties. Each beam type is individually injected for an approximate slowing-down time (~100 ms), resulting in a database of 2529 conditions that are analyzed in frequency bands corresponding to ICE harmonics and < f ci global Alfvén eigenmodes. The sub-cyclotron modes are more unstable at low field and in plasmas with large H concentration. For D injection, first harmonic ICE at f cD is more unstable in hydrogen plasmas but, for H injection, the converse is true (ICE at f cH is more unstable in D), suggesting that thermal same-species ions damp fundamental ICE. ICE stability also depends upon the pitch-angle anisotropy gradient and upon the ‘bump-on-tail’ gradient ∂f/∂v. Although the frequency of unstable ICE in these plasmas always occurs near the central cyclotron frequency, the precise value also depends on the Alfvén speed and beam direction, consistent with the idea that instability occurs for modes that simultaneously satisfy ω ≃ 2πlf ci , the magnetosonic dispersion relation, and the Doppler-shifted resonance condition.

electromagnetic ion cyclotron instabilities↗
DOE OSTI2025

Dependence of ion-cyclotron range of frequencies instabilities on species mix and fast-ion distribution: II. Time evolution

Dedicated DIII-D experiments studied instabilities below the ion cyclotron frequency f ci and at cyclotron harmonics by injecting thirteen different neutral beam populations into plasmas with different thermal compositions of H and D at five different values of toroidal field. The resulting database of 2529 conditions is described in Part 1: Stability. This paper focuses on the temporal evolution. For some beam geometries, the ion cyclotron emission (ICE) is much stronger at the beginning of injection, suggesting the importance of ‘bump-on-tail’ energy drive. Sawtooth crashes cause the emission to cease for some beam types and emission bands, but the opposite effect occurs for others. When unstable, for virtually all conditions and frequency bands, the emission bursts in a cycle reminiscent of predator-prey oscillations. Evidence for nonlinear interaction between ICE harmonics is observed. Strongly unstable modes typically appear within 1 ms from the start of beam injection and often decay within 0.1 ms when the beam turns off, but both the initial rise and final decay of the mode power exhibit a wide variety of behavior.

electromagnetic ion cyclotron instabilities↗
DOE OSTI2021

Simulating the effects of warm O + ions on the growth of electromagnetic ion cyclotron (EMIC) waves

Electromagnetic ion cyclotron (EMIC) waves are believed to play a crucial role in the dynamics of the Earth’s magnetosphere. It has been widely accepted that plasma compositions can influence the growth rate of EMIC waves in the inner magnetosphere, but how warm O + ions change the wave growth rate is not well known. In this study, we investigate the impact of ring current O + concentration on the EMIC growth rate during a specific storm when the O + ion flux significantly increases. We calculate the growth rate of EMIC waves by using physics-based ion distributions output from the Ring current–Atmosphere interactions Model with Self-Consistent magnetic (B) and Electric (E) fields (RAM-SCBE) model. The percentage of warm O + is parametrically varied to examine how the maximum EMIC growth rate changes over time and how the global distribution of the maximum EMIC growth rate is affected. We found that the maximum growth rate of H-band appears in the dusk-to-midnight sector near the plasmapause, while O-band is excited at a slightly outer region. The maximum growth rate of He-band is closely related to the cold plasma density. With the increase of warm O + density, the maximum growth rate of H-band and He-band EMIC wave is reduced, while that of O-band EMIC wave is increased, and the region with this wave excitation is widened. Furthermore, such variation in the maximum EMIC growth rate implies a potential impact on the associated wave–particle interactions and change of the decay rate in the ring current.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Ion cyclotron emission in Maxwellian plasmas

Destabilization of ion cyclotron waves—waves with frequencies close to ion cyclotron harmonics—in inhomogeneous plasmas with Maxwell velocity distribution is considered. A new mechanism of destabilization of these waves is found, whereas the known interaction between cyclotron waves and drift waves is shown to be hardly able to lead to instabilities in realistic tokamak plasmas. Our finding is that the resonance wave–particle interaction in the presence of temperature gradient can change the diamagnetic drift frequency in such a way that the destabilizing influence of the diamagnetic drift exceeds Landau damping. This occurs when the energy of resonance particles is sufficiently high, which is always the case due to the infinite tail of the Maxwell distribution. Particles with higher energies provide a larger ratio of drive to damping, but the number of resonance particles and, thus, instability growth rates are exponentially small when particle energy is very high. Therefore, only moderately suprathermal particles can lead to observable instabilities. A condition for instabilities driven by these particles is obtained. Destabilization of electrostatic cyclotron waves and ordinary cyclotron waves is studied.

Kolesnichenko, Ya. I. (ORCID:0000000323219681)↗
DOE OSTI2025

Physics of beam-driven ion cyclotron emission in the large plasma device

Abstract Ion cyclotron emission (ICE) is widely observed from spatially localised minority energetic ion populations in toroidal magnetically confined fusion (MCF) plasmas, both tokamaks and stellarators. Its spectral structure is typically regular with narrow suprathermal peaks, whose frequency separation matches a local energetic ion cyclotron frequency. Here we report the first computational (fully nonlinear self-consistent kinetic particle-in-cell code) and analytical (linear magnetoacoustic cyclotron instability (MCI)) studies of ICE observations from cylindrical plasmas contained in the Large Plasma Device (LAPD). Because LAPD is cylindrical, the plasma physics giving rise to the observed ICE spectrum necessarily excludes toroidal effects. Our approach, previously successful for toroidal plasma ICE, assumes slab geometry and hence is well adapted to LAPD. ICE from LAPD is strongly electrostatic, as distinct from electromagnetic, and is driven by 15 keV beam ions for which the ratio of perpendicular speed to the local Alfven speed, v ⊥ / v A , is 0.15, lower than in MCF plasmas from which beam-driven ICE has previously been observed. Our results are in good agreement with these observations. There is congruence between simulated ICE spectra, obtained in the saturated nonlinear regime of our computations, and observed ICE spectra; the underlying physics is essentially the same as in toroidal plasmas; and there is alignment with linear analytical theory where appropriate. The present work establishes a mapping from the cylindrical LAPD ICE observations to toroidal MCF ICE observations. The LAPD spectra are instances of beam-driven MCI-type ICE in its sub-Alfvenic, predominantly electrostatic manifestation, which has precedents in MCF stretching back to the 1990s. An interesting corollary is that, for many purposes, ICE in toroidal MCF plasmas ‘might as well’ be occurring in a cylinder.

Samant, O. (ORCID:0000000226055363)↗
DOE OSTI2025

Dependence of ion-cyclotron range of frequencies instabilities on species mix and fast-ion distribution: III. Novel sub-cyclotron modes

Dedicated DIII-D experiments studied instabilities above and below the ion cyclotron frequency by injecting thirteen different neutral beam populations into L-mode plasmas with different thermal compositions of H, D, and 3 He at five different values of toroidal field. Two companion papers describe the results of these experiments in detail. This paper focuses on data and analysis of two instabilities that occur at ${\sim}0.6$ of the deuterium and hydrogen cyclotron frequencies $f_\mathrm {cD}$ and $f_\mathrm {cH}$ in mixed hydrogen and deuterium plasmas. The modes below $f_\mathrm {cD}$ are often driven unstable by deuterium beams but are rarely destabilized by hydrogen beams. The modes below $f_\mathrm {CD}$ are reported here for the first time in a fusion plasma. These $\mathrm{sub-}f_\mathrm {cH}$ modes are only driven by deuterium beams but, unlike the $\mathrm{sub-}f_\mathrm {cD}$ modes, they are only unstable in plasmas with deuterium concentrations that are ${\lesssim}20\%$; they also have a smaller amplitude than the sub-deuterium modes. Addition of fully-ionized 3 He to the plasma has little effect on the sub-cyclotron deuterium modes but is stabilizing for the sub-cyclotron hydrogen modes. The modes are excited in the core and have toroidal mode numbers around 20. These trends are compared with calculations of the growth rate in a simplified model. The theory successfully predicts global Alfvén eigenmode instability with frequencies and mode numbers close to experimental values, the stronger amplitude of the sub-deuterium modes, and the species-mix dependence of the sub-hydrogen modes. Increased damping associated with the ion-ion hybrid, cutoff, and crossover frequencies plausibly accounts for the stabilization of the $\mathrm{sub-}f_\mathrm {cH}$ modes when 3 He is puffed. However, the relative stability of $\mathrm{sub-}f_\mathrm {cH}$ modes during hydrogen injection remains unexplained.

electromagnetic ion cyclotron instability↗
DOE OSTI2022

Mode structure measurements of ion cyclotron emission and sub-cyclotron modes on DIII-D

We report mode structure measurements of coherent ion cyclotron emission (ICE) and sub-cyclotron modes on DIII-D. Through a dedicated experiment, we aimed to characterize a variety of modes via the upgraded ICE diagnostic in both L- and H-mode plasmas. In the L-mode plasmas, autopower spectrum peaks at harmonics of the ion cyclotron frequency fci were observed and are localized in the core of the plasma. Sub-cyclotron modes (f ~ 0.5f ci ) were also observed in L-mode plasmas when the toroidal magnetic field strength was lowered from B T = 2.17 T. In H-mode plasmas, many ICE harmonics localized to the plasma edge were observed, with some exceeding the nominal ICE diagnostic bandwidth of f ϵ [0, 100] MHz. Polarization estimates made using an orthogonal pair of vertical and horizontal magnetic pickup loops on the outer wall of the machine indicate that ICE and sub-cyclotron modes have compressional polarization at the plasma edge, the latter being consistent with simulation efforts for comparable DIII-D plasmas. For all modes, the same harmonics are observed on both the centerpost and outer wall loops, indicating that ICE is poloidally extended. Finally, toroidal mode numbers were calculated using three outer wall loops for both core ICE and sub-cyclotron modes in L-mode plasmas. Furthermore, the sub-cyclotron case served as a benchmark for our calculation method, with measured numbers of n ϵ [–24, –18] roughly agreeing with heuristic estimates of n ϵ [–20, –13]. Core ICE mode numbers were measured to be n ϵ [–10, 5], which is comparable to measurements made on other machines.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

New interpretation of ion cyclotron emission from a tokamak

The tempting interpretation of ion cyclotron emission in terms of compressional Alfvén eigenmodes involving energetic ions is inconsistent with recent TCV experimental observations in some important aspects, such as (i) the perturbed poloidal field is exceeding the parallel perturbed magnetic field significantly, and (ii) the modes are near cyclotron harmonic and exhibit Alfvèn scaling of their frequency. We show that these characteristics can be explained by considering finite Larmor radius effects of thermal ions in shear Alfvén waves that allow such waves to exist well above the ion cyclotron frequency in the form of wave-packets bouncing within the plasma volume.

Plasma waves↗
DOE OSTI2024

Simulating energetic ions and enhanced fusion rates from ion-cyclotron resonance heating with a full-wave/Fokker–Planck model

Reproducing fast-ion enhanced fusion rates from ion-cyclotron resonance heating (ICRH) in tokamaks requires the self-consistent coupling of a full-wave solver and a Fokker–Planck solver, which evolves multiple simultaneously resonant ion species. We introduce a new self-consistent model that iterates the TORIC full-wave solver with the CQL3D Fokker–Planck solver using the integrated plasma simulator (IPS). This model evolves the bounce-averaged ion distribution functions in both parallel and perpendicular velocity-space with a quasilinear radio frequency (RF) diffusion operator valid in the ion finite Larmor radius (FLR) limit and the RF electric fields with the resultant non-Maxwellian FLR dielectric tensor. This produces non-Maxwellian ICRH simulations that are fully self-consistent, fast, and interoperable with integrated modeling frameworks, such as TRANSP/GACODE/IPS-FASTRAN. We demonstrate our model's capabilities by validating it against experimental data in Alcator C-Mod. We then perform the first RF heating simulations of SPARC using self-consistent non-Maxwellian ion distributions to investigate the potential to enhance fusion rates using ion cyclotron resonance heating generated fast ions.

Physics↗
DOE OSTI2023

Interaction of Inner Heliosheath Ions with Electromagnetic Ion Cyclotron Waves

Abstract Inner heliosheath (IHS) ions are expected to be exposed to various waves, shocks, and turbulence, which can affect ion distributions and thus their charge-exchange rates with interstellar neutral atoms. This work addresses the potential significance of electromagnetic ion cyclotron (EMIC) waves under expected IHS conditions. From a kinetic dispersion relation, we find the possibility of frequent triggering of EMIC instability in the IHS. The threshold anisotropy of proton temperatures required for the instability is small, ( T ⊥ – T ∣∣ )/ T ∣∣ ≈ 0.1 or less, mainly due to high plasma β (ratio of the plasma to magnetic pressures). Numerical calculations on the scattering of ions (protons, He + , and He 2+ with energy of 0.01–50 keV) based on two models for EMIC waves with a moderate intensity indicate significant scattering in the pitch angle (mostly a few tens of degrees) and energy (mostly a few tens of percent) although details depend on the energy and pitch angle of each species and adopted EMIC wave models. This occurs on a short timescale (<100 times the gyro-period of each ion species). Resonant scattering in a few to a few tens of keV (corresponding to the expected pickup ion energy in the IHS) is easily expected unless the wavenumber is too large. The scattering effect is distinguished among different species such that for lower gyrofrequency ions (He + versus He 2+ and He 2+ versus protons), the main scattering effect moves toward a lower energy domain. All these results imply continuous disturbance of ion distributions by possibly prevailing EMIC waves in the IHS.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Ion cyclotron heating at high plasma density in Proto-MPEX

The physics of ion cyclotron heating (ICH) relevant to the steady-state linear machine MPEX (Material Plasma Exposure eXperiment) has been explored in its predecessor, short-pulse device: Proto-MPEX. MPEX will utilize fundamental ICH to increase heat flux at the target and produce ion temperatures and velocity distributions with improved fidelity to those found in a tokamak divertor region, in comparison to those produced by substrate biasing. In the experiments on Proto-MPEX described here, bulk ion temperatures up to ~15 eV have been achieved with 20 kW net ICH power at 6.5 MHz, using ICH heating of a deuterium plasma produced by a helicon plasma source. The heat flux at the target has been observed to increase throughout the plasma cross section, including in the core region. Core $T_i$ and target heat flux are observed to scale linearly with injected ICH power. Further, measurements of plasma loading and target heat flux as a function of the magnetic field strength at the antenna, together with modeling of the wave propagation from the antenna to the ion cyclotron resonance using the ANTENA and COMSOL codes with a warm plasma dielectric tensor, indicate that power is coupled to the core plasma via fast wave excitation of a kinetic Alfvén wave.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

WEST actively cooled load resilient ion cyclotron resonance heating system results

Three identical new WEST ion cyclotron resonance heating (ICRH) antennas have been designed, assembled then commissioned on plasma from 2013 to 2019. The WEST ICRH system is both load-resilient and compatible with long-pulse operations. The three antennas have been successfully operated together on plasma in 2019 and 2020, with up to 5.8 MW of coupled power. In this work, the load resilience capability has been demonstrated and the antenna feedback controls for phase and matching have been developed. The breakdown detection systems have been validated and successfully protected the antennas. The use of ICRH in combination with lower hybrid has triggered the first high confinement mode transitions identified on WEST.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

The role of RF-induced E×B flows in the mitigation of scrape-off-layer convective transport during ion cyclotron resonance heating

Abstract While multiple experiments have reported a decrease in intermittent fluctuations in the far Scrape-Off-Layer (SOL) during ion cyclotron resonance heating (Antar et al 2010 Phys. Rev. Lett. 105 165001, Li et al 2022 Nucl. Eng. Technol. 54 207–19, Antar et al 2012 Nucl. Fusion 52 103005), the physical mechanism behind this observation has not been fully established yet. In this work, we demonstrate, for the first time, a direct correlation between the amplitude of RF-induced E×B flows and turbulence suppression in the far SOL. Using the Gas Puff Imaging (GPI) diagnostic on Alcator C-Mod, we show again that Ion Cyclotron Range of Frequencies can significantly alter the flow in the SOL and introduce a shear layer in regions magnetically connected close to the antenna (Cziegler et al 2012 Plasma Phys. Control. Fusion 54 105019). With the 4-strap field-aligned antenna operated in dipole phasing, the ratio of the power coupled by the central two straps to the power coupled by the outer two straps was varied. The resulting RF-induced radial electric field magnitude thus varied, and we show that the impact on the far SOL turbulence correlates with the modified E×B velocity. We then apply a newly-developed blob tracking algorithm (Han et al 2022 Sci. Rep. 12 18142) to higher-resolution GPI videos in order to directly observe the process of blob shearing by RF-induced E×B flows. We show that the radially sheared poloidal flows act as a transport barrier by stretching, stopping, and destroying filaments, which is consistent with the observed difference in turbulence statistics.

Diab, R. (ORCID:0000000342065868)↗