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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

Search for contact interactions and large extra dimensions in the dilepton channel using proton–proton collisions at √s = 8 TeV with the ATLAS detector

Research is conducted for non-resonant new phenomena in dielectron and dimuon final states, originating from either contact interactions or large extra spatial dimensions. The LHC 2012 proton–proton collision dataset recorded by the ATLAS detector is used, corresponding to 20 fb –1 at √s = 8 TeV. The dilepton invariant mass spectrum is a discriminating variable in both searches, with the contact interaction search additionally utilizing the dilepton forward-backward asymmetry. No significant deviations from the Standard Model expectation are observed. Lower limits are set on the ℓℓqq contact interaction scale Λ between 15.4 TeV and 26.3 TeV, at the 95% credibility level. For large extra spatial dimensions, lower limits are set on the string scale MS between 3.2 TeV to 5.0 TeV.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2014

Linear growth of structure in massive bigravity

The ghost-free theory of massive gravity with two dynamical metrics has been shown to produce viable cosmological expansion, where the late-time acceleration of the Universe is due to the finite range of the gravitational interaction rather than a nonzero cosmological constant. Here the cosmological perturbations are studied in this theory. The full perturbation equations are presented in a general gauge and analyzed, focusing on subhorizon scales in the quasistatic limit during the matter-dominated era. An evolution equation for the matter inhomogeneities and the parameters quantifying the deviations from general relativistic structure formation are expressed in terms of five functions whose forms are determined directly by the coupling parameters in the theory. The evolution equation has a similar structure to Horndeski-type scalar-tensor theories, exhibiting a modified growth rate and scale-dependence at intermediate wavenumbers. Predictions of the theory are confronted with observational data on both background expansion and large-scale structure, although care must be taken to ensure a model is stable. It is found that while the stable models fit the data well, they feature deviations from the standard cosmology that could be detected or ruled out by near-future experiments.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

Variance reduction for Fokker–Planck based particle Monte Carlo schemes

Recently, Fokker–Planck based particle Monte Carlo schemes have been proposed and evaluated for simulations of rarefied gas flows [1–3]. In this paper, the variance reduction for particle Monte Carlo simulations based on the Fokker–Planck model is considered. First, deviational based schemes were derived and reviewed, and it is shown that these deviational methods are not appropriate for practical Fokker–Planck based rarefied gas flow simulations. This is due to the fact that the deviational schemes considered in this study lead either to instabilities in the case of two-weight methods or to large statistical errors if the direct sampling method is applied. Motivated by this conclusion, we developed a novel scheme based on correlated stochastic processes. The main idea here is to synthesize an additional stochastic process with a known solution, which is simultaneously solved together with the main one. By correlating the two processes, the statistical errors can dramatically be reduced; especially for low Mach numbers. To assess the methods, homogeneous relaxation, planar Couette and lid-driven cavity flows were considered. For these test cases, it could be demonstrated that variance reduction based on parallel processes is very robust and effective.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

PolyPole-1: An accurate numerical algorithm for intra-granular fission gas release

This paper describes the development of a new numerical algorithm (called PolyPole-1) to efficiently solve the equation for intra-granular fission gas release in nuclear fuel. The work was carried out in collaboration with Politecnico di Milano and Institute for Transuranium Elements. The PolyPole-1 algorithms is being implemented in INL's fuels code BISON code as part of BISON's fission gas release model. The transport of fission gas from within the fuel grains to the grain boundaries (intra-granular fission gas release) is a fundamental controlling mechanism of fission gas release and gaseous swelling in nuclear fuel. Hence, accurate numerical solution of the corresponding mathematical problem needs to be included in fission gas behaviour models used in fuel performance codes. Under the assumption of equilibrium between trapping and resolution, the process can be described mathematically by a single diffusion equation for the gas atom concentration in a grain. In this work, we propose a new numerical algorithm (PolyPole-1) to efficiently solve the fission gas diffusion equation in time-varying conditions. The PolyPole-1 algorithm is based on the analytic modal solution of the diffusion equation for constant conditions, with the addition of polynomial corrective terms that embody the information on the deviation from constant conditions. The new algorithm is verified by comparing the results to a finite difference solution over a large number of randomly generated operation histories. Furthermore, comparison to state-of-the-art algorithms used in fuel performance codes demonstrates that the accuracy of the PolyPole-1 solution is superior to other algorithms, with similar computational effort. Finally, the concept of PolyPole-1 may be extended to the solution of the general problem of intra-granular fission gas diffusion during non-equilibrium trapping and resolution, which will be the subject of future work.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2015

Structure of β-AgGaO{sub 2}; ternary I–III–VI{sub 2} oxide semiconductor with a wurtzite-derived structure

The structure of the wurtzite-derived β-AgGaO{sub 2} was refined by Rietveld analysis of high-resolution powder diffraction data obtained using synchrotron X-ray radiation. The space group of the crystal is Pna2{sub 1} with lattice parameters of a{sub 0}=5.56175 Å, b{sub 0}=7.14749 Å, and c{sub 0}=5.46875 Å. The deviation of O–Ag–O and M–O–M bond angles from the regular tetrahedral angle of 109.5° was very large at ∼8° and ∼11°, respectively. The electronic structure of β-AgGaO{sub 2} is discussed based on its structure, and the indirect band gap of β-AgGaO{sub 2} was related to significant tetrahedral distortion. Although β-AgGaO{sub 2} decomposes into metallic silver and Ga{sub 2}O{sub 3} at a high temperature in any atmosphere, β-AgGaO{sub 2} is stable up to 690 °C under an O{sub 2} atmosphere. No direct transformation from the wurtzite-derived phase to a delafossite phase occurs in β-AgGaO{sub 2}. - Graphical abstract: Crystal structure of β-AgGaO{sub 2} was refined by Rietveld analysis. AgO{sub 4} and O(Ag,Ga){sub 4} tetrahedra are significantly distorted from ideal tetrahedron. - Highlights: • Orthorhombic β-AgGaO{sub 2} with a wurtzite-derived β-NaFeO{sub 2} structure was synthesized. • Its structure was refined by Rietveld analysis of high-resolution XRD data. • Silver and oxygen tetrahedra are significantly distorted from an ideal tetrahedron. • The extent of this tetrahedral distortion is related to the band gap nature. • β-AgGaO{sub 2} is a metastable phase but is stable up to 690 °C in an O{sub 2} atmosphere.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Measurement of top-quark pair production in association with charm quarks in proton–proton collisions at √s = 13 TeV with the ATLAS detector

Inclusive cross-sections or top-quark pair production in association with charm quarks are measured with proton-proton collision data at a center-of-mass energy of 13 TeV corresponding to an integrated luminosity of 140 fb -1 , collected with the ATLAS experiment at LHC between 2015 and 2018. The measurements are performed by requiring one or two charged leptons (electrons and muons), two b-tagged jets, and at least one additional jet in the final state. A custom flavor-tagging algorithm is employed for the simultaneous identification of b-jets and c-jets. In a fiducial phase space that replicates the acceptance of the ATLAS detector, the cross-sections for $t\bar{t}$ + ≥ 2c and $t\bar{t}$ + 1c production are measured to be $1.28^{+0.27}_{-0.24}$ pb and $6.4^{+1.0}_{-0.9}$ pb, respectively. The measurements are primarily limited by uncertainties in the modeling of inclusive $t\bar{t}$ and $t\bar{t}$ + $b\bar{b}$ production, in the calibration of the flavor-tagging algorithm, and by data statistics. Cross-section predictions from various $t\bar{t}$ simulations are largely consistent with the measured cross-section values, though all underpredict the observed values by 0.5 to 2.0 standard deviations. In a phase-space volume without requirements on the $t\bar{t}$ decay products and the jet multiplicity, the cross-section ratios of $t\bar{t}$ + ≥ 2c and $t\bar{t}$ + 1c to total $t\bar{t}$ + jets production are determined to be (1.23 ± 0.25)% and (8.8 ± 1.3)%.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2016

Quantum mechanics capacitance molecular mechanics modeling of core-electron binding energies of methanol and methyl nitrite on Ag(111) surface

We study a newly devised quantum mechanics capacitance molecular mechanics (QMCMM) method for the calculation of core-electron binding energies in the case of molecules adsorbed on metal surfaces. This yet untested methodology is applied to systems with monolayer of methanol/methyl nitrite on an Ag(111) surface at 100 K temperature. It was found out that the studied C, N, and O 1s core-hole energies converge very slowly as a function of the radius of the metallic cluster, which was ascribed to build up of positive charge on the edge of the Ag slab. Further analysis revealed that an extrapolation process can be used to obtain binding energies that deviated less than 0.5 eV against experiments, except in the case of methanol O 1s where the difference was as large as 1.8 eV. Additional QM-cluster calculations suggest that the latter error can be connected to the lack of charge transfer over the QM–CMM boundary. Thus, the results indicate that the QMCMM and QM-cluster methods can complement each other in a holistic picture of molecule-adsorbate core-ionization studies, where all types of intermolecular interactions are considered.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

TURBULENT GENERAL MAGNETIC RECONNECTION

Plasma flows with a magnetohydrodynamic (MHD)-like turbulent inertial range, such as the solar wind, require a generalization of general magnetic reconnection (GMR) theory. We introduce the slip velocity source vector per unit arclength of field line, the ratio of the curl of the non-ideal electric field in the generalized Ohm’s Law and magnetic field strength. It diverges at magnetic nulls, unifying GMR with null-point reconnection. Only under restrictive assumptions is the slip velocity related to the gradient of quasi-potential (which is the integral of parallel electric field along magnetic field lines). In a turbulent inertial range, the non-ideal field becomes tiny while its curl is large, so that line slippage occurs even while ideal MHD becomes accurate. The resolution is that ideal MHD is valid for a turbulent inertial range only in a weak sense that does not imply magnetic line freezing. The notion of weak solution is explained in terms of renormalization group (RG) type theory. The weak validity of the ideal Ohm’s law in the inertial range is shown via rigorous estimates of the terms in the generalized Ohm’s Law. All non-ideal terms are irrelevant in the RG sense and large-scale reconnection is thus governed solely by ideal dynamics. We discuss the implications for heliospheric reconnection, in particular for deviations from the Parker spiral model. Solar wind observations show that reconnection in a turbulence-broadened heliospheric current sheet, which is consistent with Lazarian–Vishniac theory, leads to slip velocities that cause field lines to lag relative to the spiral model.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Multivariate analysis for the estimation of target localization errors in fiducial marker-based radiotherapy

Purpose: To assess the target localization error (TLE) in terms of the distance between the target and the localization point estimated from the surrogates (|TMD|), the average of respiratory motion for the surrogates and the target (|aRM|), and the number of fiducial markers used for estimating the target (n). Methods: This study enrolled 17 lung cancer patients who subsequently underwent four fractions of real-time tumor tracking irradiation. Four or five fiducial markers were implanted around the lung tumor. The three-dimensional (3D) distance between the tumor and markers was at maximum 58.7 mm. One of the markers was used as the target (P{sub t}), and those markers with a 3D |TMD{sub n}| ≤ 58.7 mm at end-exhalation were then selected. The estimated target position (P{sub e}) was calculated from a localization point consisting of one to three markers except P{sub t}. Respiratory motion for P{sub t} and P{sub e} was defined as the root mean square of each displacement, and |aRM| was calculated from the mean value. TLE was defined as the root mean square of each difference between P{sub t} and P{sub e} during the monitoring of each fraction. These procedures were performed repeatedly using the remaining markers. To provide the best guidance on the answer with n and |TMD|, fiducial markers with a 3D |aRM ≥ 10 mm were selected. Finally, a total of 205, 282, and 76 TLEs that fulfilled the 3D |TMD| and 3D |aRM| criteria were obtained for n = 1, 2, and 3, respectively. Multiple regression analysis (MRA) was used to evaluate TLE as a function of |TMD| and |aRM| in each n. Results: |TMD| for n = 1 was larger than that for n = 3. Moreover, |aRM| was almost constant for all n, indicating a similar scale for the marker’s motion near the lung tumor. MRA showed that |aRM| in the left–right direction was the major cause of TLE; however, the contribution made little difference to the 3D TLE because of the small amount of motion in the left–right direction. The TLE calculated from the MRA ({sup MRA}TLE) increased as |TMD| and |aRM| increased and adversely decreased with each increment of n. The median 3D {sup MRA}TLE was 2.0 mm (range, 0.6–4.3 mm) for n = 1, 1.8 mm (range, 0.4–4.0 mm) for n = 2, and 1.6 mm (range, 0.3–3.7 mm) for n = 3. Although statistical significance between n = 1 and n = 3 was observed in all directions, the absolute average difference and the standard deviation of the {sup MRA}TLE between n = 1 and n = 3 were 0.5 and 0.2 mm, respectively. Conclusions: A large |TMD| and |aRM| increased the differences in TLE between each n; however, the difference in 3D {sup MRA}TLEs was, at most, 0.6 mm. Thus, the authors conclude that it is acceptable to continue fiducial marker-based radiotherapy as long as |TMD| is maintained at ≤58.7 mm for a 3D |aRM| ≥ 10 mm.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2024

Foundations of automatic feature extraction at LHC–point clouds and graphs

Abstract Deep learning algorithms will play a key role in the upcoming runs of the Large Hadron Collider (LHC), helping bolster various fronts ranging from fast and accurate detector simulations to physics analysis probing possible deviations from the Standard Model. The game-changing feature of these new algorithms is the ability to extract relevant information from high-dimensional input spaces, often regarded as “replacing the expert” in designing physics-intuitive variables. While this may seem true at first glance, it is far from reality. Existing research shows that physics-inspired feature extractors have many advantages beyond improving the qualitative understanding of the extracted features. In this review, we systematically explore automatic feature extraction from a phenomenological viewpoint and the motivation for physics-inspired architectures. We also discuss how prior knowledge from physics results in the naturalness of the point cloud representation and discuss graph-based applications to LHC phenomenology.

Bhardwaj, Akanksha↗
DOE OSTI2014

Raman microscopy of lithium-manganese-rich transition metal oxide cathodes

Lithium-rich and manganese-rich (LMR) layered transition metal (TM) oxide composites with general formula xLi 2 MnO 3 ·(1-x)LiMO 2 (M = Ni, Co, Mn) are promising cathode candidates for high energy density lithium ion batteries. Lithium-manganese-rich TM oxides crystallize as a nanocomposite layered phase whose structure further evolves with electrochemical cycling. Raman spectroscopy is a powerful tool to monitor the crystal chemistry and correlate phase changes with electrochemical behavior. While several groups have reported Raman spectra of lithium rich TM oxides, the data show considerable variability in terms of both the vibrational features observed and their interpretation. In this paper, Raman microscopy is used to investigate lithium-rich and manganese-rich TM cathodes as a function of voltage and electrochemical cycling at various temperatures. No growth of a spinel phase is observed within the cycling conditions. However, analysis of the Raman spectra does indicate the structure of LMR-NMC deviates significantly from an ideal layered phase. Finally, the results also highlight the importance of using low laser power and large sample sizes to obtain consistent data sets.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Applying Information Theory to Design Optimal Filters for Photometric Redshifts

In this paper we apply ideas from information theory to create a method for the design of optimal filters for photometric redshift estimation. We show the method applied to a series of simple example filters in order to motivate an intuition for how photometric redshift estimators respond to the properties of photometric passbands. We then design a realistic set of six filters covering optical wavelengths that optimize photometric redshifts for z <= 2.3 and i < 25.3. We create a simulated catalog for these optimal filters and use our filters with a photometric redshift estimation code to show that we can improve the standard deviation of the photometric redshift error by 7.1% overall and improve outliers 9.9% over the standard filters proposed for the Large Synoptic Survey Telescope (LSST). We compare features of our optimal filters to the LSST and find that the LSST filters incorporate key features for optimal photometric redshift estimation. Finally, we describe how information theory can be applied to a range of optimization problems in astronomy.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Salt Sample Statistical Study

Historically there has, at times, been large variability in measured uranium concentration amongst salt samples taken at the same time from the electrorefiner with relative standard deviations ranging from 0.59% up to a high of 114%. This has led to uncertainty in the actual uranium content of the salt. A series of surrogate experiments were performed with gadolinium to study the effects of Li2O, Gd2O3, and Gd metal additions to an LiCl-KCl-GdCl3 salt on the gadolinium content of the salt. These experiments showed no increase in salt sample variability following the additions of the mentioned species, indicating that the carryover of these species from the oxide reduction process may not lead to the historic variability issues.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2022

Automated exploitation of the big configuration space of large adsorbates on transition metals reveals chemistry feasibility

Mechanistic understanding of large molecule conversion and the discovery of suitable heterogeneous catalysts have been lagging due to the combinatorial inventory of intermediates and the inability of humans to enumerate all structures. Here, we introduce an automated framework to predict stable configurations on transition metal surfaces and demonstrate its validity for adsorbates with up to 6 carbon and oxygen atoms on 11 metals, enabling the exploration of ~10 8 potential configurations. It combines a graph enumeration platform, force field, multi-fidelity DFT calculations, and first-principles trained machine learning. Clusters in the data reveal groups of catalysts stabilizing different structures and expose selective catalysts for showcase transformations, such as the ethylene epoxidation on Ag and Cu and the lack of C-C scission chemistry on Au. Deviations from the commonly assumed atom valency rule of small adsorbates are also manifested. This library can be leveraged to identify catalysts for converting large molecules computationally.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

TH-AB-201-11: Small-Field Beam Quality Corrections and Their Clinical Impact for Flattened and Unflattened Fields of the TrueBeam

Purpose: To determine small-field beam quality correction factors for several detectors used with flattened and unflattened beams of the Varian TrueBeam accelerator, and to quantify the clinical impact of these corrections through a treatment planning study. Methods: Small-field beam quality correction factors were measured and simulated in EGSnrc for several microchambers and diodes. The Exradin W1 plastic scintillation detector was used as a reference for the measured results, and its response was validated using alanine measurements traceable to a primary standard. Symmetric, jaw-defined field sizes ranging from 6 mm to 10 cm were used. Twelve hypofractionated clinical cases were then recalculated and/or reoptimized using beam models that were modified based on the resulting output factors. Three-dimensional dose distributions and cumulative dose-volume-histogram plots were compared between the models. The impact of these measurements on the flattening-filter-scatter-source (FFSS) model, and its subsequent effect on treatment plans, was also investigated. Results: The diodes exhibited over-response in small fields, whereas the microchambers tended to under-respond as field size decreased. Field-size-dependent effects were less significant for the flattening-filter-free beams (i.e., corrections were closer to unity) than for flattened fields of the same size. Recalculated and reoptimized plans displayed varying levels of deviation from the original dose distributions, with larger discrepancies at effective jaw settings of 3×3 cm{sup 2} or smaller. Dose distributions calculated using uncorrected microchamber output factors and/or modified FFSS showed large discrepancies in the low-dose regions, which manifested as DVH changes for the organs at risk (OAR). Conclusion: Field-size-dependent beam quality corrections were measured and simulated for ionization chambers and diodes typically used in small fields. Plans calculated with models based on these measured data showed dosimetric deviations mainly out of field, which affected OAR dose metrics. Accordingly, these beam quality corrections should be accounted for during commissioning to ensure greater treatment planning accuracy.

07 ISOTOPE AND RADIATION SOURCES↗
DOE OSTI2018

Microfocus diffraction from different regions of a protein crystal: structural variations and unit-cell polymorphism

Real macromolecular crystals can be non-ideal in a myriad of ways. This often creates challenges for structure determination, while also offering opportunities for greater insight into the crystalline state and the dynamic behavior of macromolecules. To evaluate whether different parts of a single crystal of a dynamic protein, EutL, might be informative about crystal and protein polymorphism, a microfocus X-ray synchrotron beam was used to collect a series of 18 separate data sets from non-overlapping regions of the same crystal specimen. A principal component analysis (PCA) approach was employed to compare the structure factors and unit cells across the data sets, and it was found that the 18 data sets separated into two distinct groups, with large R values (in the 40% range) and significant unit-cell variations between the members of the two groups. This categorization mapped the different data-set types to distinct regions of the crystal specimen. Atomic models of EutL were then refined against two different data sets obtained by separately merging data from the two distinct groups. A comparison of the two resulting models revealed minor but discernable differences in certain segments of the protein structure, and regions of higher deviation were found to correlate with regions where larger dynamic motions were predicted to occur by normal-mode molecular-dynamics simulations. The findings emphasize that large spatially dependent variations may be present across individual macromolecular crystals. This information can be uncovered by simultaneous analysis of multiple partial data sets and can be exploited to reveal new insights about protein dynamics, while also improving the accuracy of the structure-factor data ultimately obtained in X-ray diffraction experiments.

Thompson, Michael C. (ORCID:0000000260992027)↗
DOE OSTI2017

Pseudospectral Maxwell solvers for an accurate modeling of Doppler harmonic generation on plasma mirrors with particle-in-cell codes

With the advent of petawatt class lasers, the very large laser intensities attainable on target should enable the production of intense high-order Doppler harmonics from relativistic laser-plasma mirror interactions. At present, the modeling of these harmonics with particle-in-cell (PIC) codes is extremely challenging as it implies an accurate description of tens to hundreds of harmonic orders on a broad range of angles. In particular, we show here that due to the numerical dispersion of waves they induce in vacuum, standard finite difference time domain (FDTD) Maxwell solvers employed in most PIC codes can induce a spurious angular deviation of harmonic beams potentially degrading simulation results. This effect was extensively studied and a simple toy model based on the Snell-Descartes law was developed that allows us to finely predict the angular deviation of harmonics depending on the spatiotemporal resolution and the Maxwell solver used in the simulations. Our model demonstrates that the mitigation of this numerical artifact with FDTD solvers mandates very high spatiotemporal resolution preventing realistic three-dimensional (3D) simulations even on the largest computers available at the time of writing. Finally, we finally show that nondispersive pseudospectral analytical time domain solvers can considerably reduce the spatiotemporal resolution required to mitigate this spurious deviation and should enable in the near future 3D accurate modeling on supercomputers in a realistic time to solution.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma

In order to enable kinetic simulation of non-thermal edge plasmas at a reduced computational cost, a new hybrid-Lagrangian δf scheme has been developed that utilizes the phase space grid in addition to the usual marker particles, taking advantage of the computational strengths from both sides. The new scheme splits the particle distribution function of a kinetic equation into two parts. Marker particles contain the fast space-time varying, δf, part of the distribution function and the coarse-grained phase-space grid contains the slow space-time varying part. The coarse-grained phase-space grid reduces the memory-requirement and the computing cost, while the marker particles provide scalable computing ability for the fine-grained physics. Weights of the marker particles are determined by a direct weight evolution equation instead of the differential form weight evolution equations that the conventional delta-f schemes use. The particle weight can be slowly transferred to the phase space grid, thereby reducing the growth of the particle weights. The non-Lagrangian part of the kinetic equation – e.g., collision operation, ionization, charge exchange, heat-source, radiative cooling, and others – can be operated directly on the phase space grid. Deviation of the particle distribution function on the velocity grid from a Maxwellian distribution function – driven by ionization, charge exchange and wall loss – is allowed to be arbitrarily large. The numerical scheme is implemented in the gyrokinetic particle code XGC1, which specializes in simulating the tokamak edge plasma that crosses the magnetic separatrix and is in contact with the material wall.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗