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DOE OSTI2024

Investigation of Ring-heterogeneous Geometry Approximation for Efficient and Practical SFR Analysis

A novel geometry approximation scheme named ring-heterogeneous approximation for intermediate resolution multiphysics analysis of SFRs is presented. This approach particularly targets neutronics coupled with thermo-mechanics feedback, which dominate the passive safety of SFR designs. The ring-heterogeneous approximation casts an array of pins into volume-preserving bands of materials that constitute the pins. It allows to efficiently simulate SFRs with an unstructured mesh and also to explicitly consider the deformation of each material by retaining their heterogeneity. Verification results with single assembly and core problems of the ABTR benchmark demonstrated that the ring-heterogeneous approximation is a reasonable approximation for SFRs. For a 2D full core problem, the ring-heterogeneous approximation significantly reduced the number of elements by at least a factor of 70 from a fully-heterogeneous mesh while almost exactly preserving the reference fully-heterogeneous solutions with the RMS error of only 0.02% in the ring-wise fission reaction rate.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2020

Ring-opening and hydrodenitrogenation of indole under hydrothermal conditions over Ni, Pt, Ru, and Ni-Ru bimetallic catalysts

Here, the activity and selectivity of activated-carbon-supported Ni, Pt, Ru, and Ni-Ru bimetallic catalysts was examined for hydrothermal denitrogenation of indole. The molar yield of pyrrole ring-opening compounds, without an added hydrogen source, are in the order: Ni < Pt < Ni 90 Ru 10 < Ni 75 Ru 25 < Ni 50 Ru 50 ~ Ni 25 Ru 75 ~ Ru. Ru-containing catalysts facilitated production of hydrocarbons (hydrodenitrogenation (HDN) products) when used with added formic acid (hydrogen source). We elucidated catalytic hydrothermal HDN pathways for indole based on the product distributions and the variation of their yields with time. Hydrogenation of indole to indoline is the primary pathway and ring-opening of indoline to form alkyl anilines is faster than forming HDN products (alkyl benzenes). DFT calculations confirmed experimental activity trends, showing Ru is more active than Ni for indole ring opening and for o-toluidine deamination. If no hydrogen source is present, directly breaking the N-C bond in the pyrrole ring is more favorable than breaking the C-N bond with an aromatic carbon. If a H source is provided, the pyrrole ring hydrogenates first, forming indoline, followed by cleavage of the C-N bond.

42 ENGINEERING↗

DOE OSTI2025

Intrinsic activation energies for ring contraction of allylic cations in zeolites

Cyclic carbocations are important intermediates in zeolite-catalyzed chemistries such as methanol-to-hydrocarbons conversion, naphthenes ring opening, or coke formation. While information about their thermodynamic stability exists, little is known about the kinetics of formation and transformation of cyclic carbocations in zeolites. To fill this knowledge gap, ring contraction of the 1,3,5,5-tetramethylcyclohexenyl cation (C 10 H 17 + ), a representative of 6-membered ring allylic cations, was investigated by in situ UV–vis and IR spectroscopy. Protonic forms of zeolites served as catalysts, at temperatures from 80 °C to 135 °C. Significant oligomerization and hydride transfer in BEA and FAU hampered kinetics analysis, whereas ring contraction dominated in the channels of MOR. The reactant cation, characterized by an electronic absorption at 314 nm and an allylic stretch at 1549 cm −1 , contracted to both a 1,3-alkyl-substituted cyclopentenyl cation (287 nm and 1506 cm −1 ) and a 1,2,3-alkyl-substituted cyclopentenyl cation (297 nm and 1489 cm −1 ). Collection of time-resolved IR spectra and fitting of the intensities with various kinetic models revealed a third transformation, which is expected from thermodynamics: the 1,3-alkyl-substituted cyclopentenyl cation isomerizes to the 1,2,3-alkyl-substituted cyclopentenyl cation. Series of IR spectra recorded at different temperatures delivered intrinsic activation enthalpies (entropies) in MOR of 67 ± 2 kJ mol −1 (−130 ± 6 J mol −1 K −1 ) and 90 ± 3 kJ mol −1 (−70 ± 8 J mol −1 K −1 ) for the contraction to 1,3- and 1,2,3-substituted species, and of 88 ± 5 kJ mol −1 (−84 ± 12 J mol −1 K −1 ) for the isomerization. The findings characterize one path – via contraction of larger rings – to different cyclopentenyl species in zeolites; and the associated, moderate activation energies suggest such transformations contribute to many complex hydrocarbon reaction networks.

Acid catalysis↗

DOE OSTI2022

Facile polymerization in a bicellar template to produce polymer nano-rings

We report a well-defined discoidal bicelle composed of three lipids, specifically zwitterionic long-chain 1,2-dipalmitoyl phosphocholine (DPPC) and short-chain 1,2-dihexanoyl phosphocholine (DHPC) doped with anionic 1,2-dipalmitoyl phosphoglycerol (DPPG) provides a generalized template for the synthesis of hydrophobic polymer nano-rings. The lipid molar ratio of DPPC/DHPC/DPPG is 0.71/0.25/0.04. The detailed investigation and discussion were based on styrene but tested on three other vinyl monomers. The structure of nano-rings is identified through the detailed analysis of small angle X-ray/neutron scattering (SAXS and SANS) data and transmission electron micrographs (TEM), supported by the differential scanning calorimetric (DSC) data before and after polymerization. The investigation covers samples with a styrene-to-lipid ratio ranged varied from 1:50 to 1:10.The styrene monomers are initially located at both the discoidal planar (long-chain lipid rich) and rim (short-chain lipid rich) regions. During polymerization, they migrate to the more fluid rim region-section. The formation mechanism involves the interplay of hydrophobic interaction, mismatched miscibility of polystyrene between the ordered and disordered phases, and crystallinity of the long lipid acyl chains. This facile synthesis is proven applicable for several hydrophobic monomers. The well-defined nano-rings greatly enhance the interfacial area and have the potential to be the building blocks for functional materials, if monomers are incorporated with desirable functions, for future applications. Crosslinked polymer nano-rings are produced in well-defined discoidal bicelles. The enhanced immiscibility in the planar region drives the hydrophobic polymers to the fluidic rim. The platform can be generalized for synthesis of other hydrophobic polymer nano-rings.

36 MATERIALS SCIENCE↗

DOE OSTI2024

On the stability of a pair of vortex rings

The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

42 ENGINEERING↗

DOE OSTI2022

The Role of Methylaryl Radicals in the Growth of Polycyclic Aromatic Hydrocarbons: The Formation of Five-Membered Rings

The regions of the C 13 H 11 potential energy surface (PES) related to the unimolecular isomerization and decomposition of the 1-methylbiphenylyl radical and accessed by the 1-/2-methylnaphthyl + C 2 H 2 reactions have been explored by ab initio G3(MP2,CC)//B3LYP/6-311G(d,p) calculations. The kinetics of these reactions relevant to the growth of polycyclic aromatic hydrocarbons (PAH) under high-temperature conditions in circumstellar envelopes and in combustion flames has been studied employing the RRKM-Master Equation approach. The unimolecular reaction of 1-methylbiphenylyl proceeding via a five-membered ring closure followed by H elimination is predicted to be very fast, on a submicrosecond scale above 1000 K and to result in the formation of an embedded five-membered ring in the 9H-fluorene product. The 1-/2-methylnaphthyl + C 2 H 2 reaction mechanism involves acetylene addition to the radical on the methylene group followed by a six- or five-membered ring closure and aromatization via an H atom loss. Despite of the complexity of the C 13 H 11 PES, these straightforward pathways are dominant in the high-temperature regime (above ~1000 K), with the prevailing products being phenalene, with a significant contribution of 1H-cyclopenta(a)naphthalene, for 1-methylnaphthyl + C 2 H 2 , and 1H-cyclopenta(b)naphthalene and 3H-cyclopenta(a)naphthalene, for 2-methylnaphthyl + C 2 H 2 . The methylnaphthyl reactions with acetylene represent a clean source of the three-ring PAHs, but they are relatively slow owing to the high entrance barriers of ~10 kcal/mol, with the rate constants of about an order of magnitude lower as compared to those for naphthyl + allene and σ-aryl + C 2 H 2 . Here, the 1-methylnaphthyl + C 2 H 2 and 2-methylnaphthyl + C 2 H 2 reactions represent prototypes for PAH growth by an extra six- and five-membered ring on a zigzag edge or a corner of PAH and the generated modified Arrhenius expressions are recommended for kinetic modeling of PAH expansion by the mechanism of acetylene addition to methylaryl radicals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Microscopic Theory of Long-Time Center-of-Mass Self-Diffusion and Anomalous Transport in Ring Polymer Liquids

We construct a microscopic theory at the level of segment-scale correlated space–time intermolecular forces for the long-time center-of-mass (CM) diffusion constant and intermediate-time non-Fickian transport in dense solutions and melts of ring polymers. The approach combines ideas of polymer, colloid, and liquid-state statistical mechanics to quantify how the multifractal intra-ring conformational structure and inter-ring packing correlations determine dynamic caging constraints and time-dependent friction. Breakdown of Rouse theory is predicted to occur due to length scale-dependent temporal correlation of forces exerted on pairs of tagged ring segments from surrounding polymers. At large enough degrees of polymerization (N), a stronger scaling of the CM diffusion constant (D ∝ N –2 ) is predicted, with a crossover N D proportional to the product of the system-specific macromolecular volume fraction and dimensionless compressibility. In analogy with caging effects in glass-forming fluids, the theory does appear to begin to fail at sufficiently high N/N D for the center-of-mass diffusivity, likely due to another crossover to an even slower activated transport regime. However, use of N/N D with the theoretically predicted N D , in conjunction with dynamic blob scaling ideas for local Rouse friction, collapses semidilute and concentrated solution simulation data onto a master curve. Based on the same physical ideas employed to predict the diffusion constant, a generalized Langevin equation description is formulated for intermediate-time CM transport. It predicts two subdiffusive regimes that emerge due to the self-similar nature of internal ring structure. Analytic and numerical predictions for the apparent non-Fickian exponent as a function of time, N/N D and dimensionless compressibility, the evolution of the maximum degree of subdiffusive motion with N/N D , and the time scale for recovering Fickian diffusion are made, all of which are in good accordance with melt simulations. Furthermore, the present work sets the stage to address activated dynamics and glass formation on the macromolecular scale.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Ring Size Effects on the Structures of Sandwich Compounds with a Stoichiometry of C 12 H 12 M (M = Ti–Ni)

Ring size effects on geometries and electronic structures were investigated for the (C n H n )M(C m H m ) (n = 4, 5, or 6; m = 8, 7, or 6; m + n = 12; M = Ti–Ni) systems using density functional theory. The lowest-energy C 12 H 12 M structures for the early transition metals titanium, vanadium, and chromium are the experimentally known singlet (η 5 -C 5 H 5 )Ti(η 7 -C 7 H 7 ), doublet (η 5 -C 5 H 5 )V(η 7 -C 7 H 7 ), and singlet (η 6 -C 6 H 6 ) 2 Cr, respectively. The likewise experimentally known singlet (η 6 -C 6 H 6 ) 2 Ti, doublet (η 6 -C 6 H 6 ) 2 V, and singlet (η 5 -C 5 H 5 )Cr(η 7 -C 7 H 7 ) are the secondlowest- energy structures with only a small energy difference between the two vanadium structures. For the later transition metals, dibenzenemetal complexes are the lowest-energy C 12 H 12 M species with two fully bonded hexahapto benzene rings in the lowest-energy manganese and iron derivatives and one hexahapto and one dihapto benzene ring in the lowest-energy cobalt and nickel derivatives. The lowest-energy (C 5 H 5 )M(C 7 H 7 ) structures for the later transition metals iron, cobalt, and nickel have partially bonded nonplanar C 7 H 7 rings with one or two uncomplexed C=C bonds. The (C 4 H 4 )M(C 8 H 8 ) (M = Ti–Ni) structures with the metal sandwiched between four- and eight-membered rings were found to be much higher in energy than their (C 5 H 5 )M(C 7 H 7 ) and (C 6 H 6 ) 2 M isomers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Quantifying the Effect of Magnetic Field Line Curvature Scattering on the Loss of Ring Current Ions

During geomagnetic storms, the ring current ions sometimes exhibit rapid loss as suggested by the fast recovery of the Dst index on a time scale of a few hours. Here, the effects of magnetic field line curvature (FLC) scattering on the loss of ring current ions, which have not been well quantified, are studied here by test particle simulations under the T89c magnetic field model. Our simulation results show that the prediction of ion loss based on a single–value cutoff of the κ parameter or maximum of δμ/μ of a single FLC scattering is not accurate. Instead, the e–folding lifetime (τ) for the loss of ring current ions due to cumulative FLC scattering has been calculated for different initial ion energies, equatorial pitch angles, and L shells under different geomagnetic conditions. The results show that in general the FLC scattering loss is faster for ions of higher energy, higher mass, smaller pitch angle, higher L, and at high Kp level. Specifically, we find that at Kp = 6 the lifetime can be <10 h at L > 5 for 100 s keV protons and at L > 4 for 100 s keV O + , which demonstrates that FLC scattering can be an important mechanism for the observed fast loss (τ < 10 h) of ring current ions during geomagnetic storms. Furthermore, we formulate an empirical formula for τ as a function of ion energy, pitch angle, position, species, and Kp. The empirical formula can be directly included in ring current models to account for the FLC scattering effects.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Comparison of the Distribution of the In Situ Ring Current Density Based on Magnetic Field Observations From THEMIS, MMS and Cluster

Abstract We review and extend previous work on the ring current density distribution by applying the curlometer technique to multi‐spacecraft magnetic field data on different spatial scales, directly comparing the results of three space missions: Time History of Events and Macroscale Interactions during Substorms (THEMIS), The Magnetospheric Multiscale (MMS) and Cluster, that have so far accrued decades of data on a range of spatial separation scales. We suggest that the statistical results of Cluster data over two decades can illustrate well the overall form of the ring current (in the ring plane), but can underestimate the current density, while THEMIS (partially) and MMS are able to observe the detailed structure of the ring current and indeed show different behavior on some (smaller) spatial scales. These results suggest that, in addition to the local time and radial form, reported previously, the morphology of the ring current may also depend on magnetic latitude, as influenced by the tilt angle between the inner and outer magnetic fields.

Tan, X. [School of Space and Earth Sciences Beihan↗

DOE OSTI2022

The Milky Way’s middle-aged inner ring

Aims. We investigate the metallicity, age, and orbital anatomy of the inner Milky Way, specifically focussing on the outer bar region. Methods. We integrated a sample of APOGEE DR16 inner Galaxy stars in a state of the art bar-bulge potential with a slow pattern speed and investigated the link between the resulting orbits and their [Fe/H] and ages. By superimposing the orbits, we built density, [Fe/H], and age maps of the inner Milky Way, which we divided further using the orbital parameters eccentricity, |Xmax|, and |Zmax|. Results. We find that at low heights from the Galactic plane, the Galactic bar gradually transitions into a radially thick, vertically thin, elongated inner ring with average solar [Fe/H]. This inner ring is mainly composed of stars with AstroNN ages between 4 and 9 Gyr with a peak in age between 6 and 8 Gyr, making the average age of the ring ~6 Gyr. The vertical thickness of the ring decreases markedly towards younger ages. We also find very large L4 Lagrange orbits that have average solar to super-solar metallicities and intermediate ages. Lastly, we confirm a clear X-shape in the [Fe/H] and density distributions at large Galactic heights. Conclusions. The orbital structure obtained for the APOGEE stars reveals that the Milky Way hosts an inner ring-like structure between the planar bar and corotation. This structure is on average metal rich, intermediately aged, and enhances the horizontal metallicity gradient along the bar’s major axis.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Unified understanding of the impact of semiflexibility, concentration, and molecular weight on macromolecular-scale ring diffusion

Conformationally fluctuating, globally compact macromolecules such as polymeric rings, single-chain nanoparticles, microgels, and many-arm stars display complex dynamic behaviors due to their rich topological structure and intermolecular organization. Synthetic rings are hybrid objects with conformations that display both ideal random walk and compact globular features, which can serve as models of genomic DNA. To date, emphasis has been placed on the effect of ring molecular weight on their unusual behaviors. Here, we combine simulations and a microscopic force-level theory to build a unified understanding for how key aspects of ring dynamics depend on different tunable molecular properties including backbone rigidity, monomer concentration, degree of traditional entanglement, and molecular weight. Our large-scale molecular dynamics simulations of ring melts with very different backbone stiffnesses reveal unanticipated behaviors which agree well with our generalized theory. This includes a universal master curve for center-of-mass diffusion constants as a function of molecular weight scaled by a chemistry and thermodynamic state-dependent critical molecular weight that generalizes the concept of an entanglement cross-over for linear chains. The key physics is how backbone rigidity and monomer concentration induced changes of the entanglement length, interring packing, degree of interpenetration, and liquid compressibility slow down space-time dynamic-force correlations on macromolecular scales. A power law decay of the center-of-mass diffusion constant with inverse molecular weight squared is the first consequence, followed by an ultraslow activated hopping transport regime. Our results set the stage to address slow dynamics and kinetic arrest in different families of compact synthetic and biological polymeric systems.

Science & Technology - Other Topics↗

DOE OSTI2023

The frequency and sizes of inner bars and nuclear rings in barred galaxies and their dependence on galaxy properties

ABSTRACT Using a volume- and mass-limited (D < 30 Mpc, $\log \, (M_{\star }/M_{\odot })\ge 9.75$) sample of 155 barred S0–Sd galaxies, I determine the fraction with secondary structures within their bars. Some 20 ± 3 per cent have a separate inner bar, making them double-barred; an identical fraction have nuclear rings, with $11^{+3}_{-2}$ per cent hosting both. The inner-bar frequency is a strong, monotonic function of stellar mass: only $4^{+3}_{-2}$ per cent of barred galaxies with $\log \, (M_{\star }/M_{\odot })= 9.75$–10.25 are double-barred, while 47 ± 8 per cent of those with $\log \, (M_{\star }/M_{\odot })\gt 10.5$ are. The nuclear-ring frequency is a strong function of absolute bar size: only $1^{+2}_{-1}$ per cent of bars with semimajor axes <2 kpc have nuclear rings, while $39^{+6}_{-5}$ per cent of larger bars do. Both inner bars and nuclear rings are absent in very late-type (Scd–Sd) galaxies. Inner bar size correlates with galaxy stellar mass, but is clearly offset to smaller sizes from the main population of bars. This makes it possible to define ‘nuclear bars’ in a consistent fashion, based on stellar mass. There are eight single-barred galaxies where the bars are nuclear-bar-sized; some of these may be systems where an outer bar failed to form, or previously double-barred galaxies where the outer bar has dissolved. Inner bar size is even more tightly correlated with host bar size, which is likely the primary driver. In contrast, nuclear ring size is only weakly correlated with galaxy mass or bar size, with more scatter in size than is true of inner bars.

Astronomy & Astrophysics↗

DOE OSTI2022

Autoresonant excitation of nonlinear beam motion in storage rings

Understanding and controlling of nonlinear beam dynamics are essential to the performance of storage ring light sources. In order to measure and correct limitations due to nonlinear beam dynamics, an effective method is needed to excite the beam oscillation to large amplitudes. Autoresonance is a method that enables the control of the amplitude of a driven nonlinear oscillator by sweeping the frequency of the driver. In this study, autoresonant excitation of beam transverse oscillations in storage rings is studied. The threshold of the drive amplitude for autoresonance is theoretically obtained, for cases with or without damping. The results are in good agreement with simulations for a simple storage ring model, as well as for models of actual storage rings. Application of the theory to experimental data on the SPEAR3 storage ring also validates the results

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

Ring patterns generated by an expanding colloidal meniscus

The drop-and-dry is a common technique allowing for creation of periodic nanoparticle (NP) structures for sensing, photonics, catalysis, etc. However, the reproducibility and scalability of this approach for fabrication of NP-based structures faces serious challenges due to the complexity of the simple, at first glance, evaporation process. In this work we study the effect of the spatial confinement on the NP self-assembly under slow solvent evaporation, when the air-liquid-substrate contact line (CL) expands from the center towards the walls of a cylindrical cell, forming a toroid. Using in situ video monitoring of the stick-slip CL motion, we find regular hydrodynamic perturbations in the meniscus, and reveal fine details of the formation of quasiperiodic rings of close packed NP layers. We report that drying of the toroidal NP droplet has a number of important differences from drying of the classical hemispherical colloidal drops. In toroidal drops we observe linear-in-time average meniscus motion, in contrast to the hemispherical drops where the meniscus moves as a square root of time. While both droplet geometries produce NP ring patterns, the ring width for the toroidal drop decreases with increasing ring radius, while it decreases with decreasing the radius of the hemispherical drop. We suggest that free ligands are the main cause of the Marangoni instabilities driving the periodic vorticity in the meniscus. Finally, we show that the usually ignored contact line tension may yield a considerable contribution to the CL pinning causing the CL slip-stick motion and the ring formation.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Vortex rings drive entrainment and cooling in flow induced by a spark discharge

Spark plasma discharges induce vortex rings and a hot gas kernel. We develop a model to describe the late stage of the spark-induced flow and the role of the vortex rings in the entrainment of cold ambient gas and the cooling of the hot gas kernel. The model is tested in a plasma-induced flow using density and velocity measurements obtained from simultaneous stereoscopic particle image velocimetry and background oriented schlieren. We show that the spatial distribution of the hot kernel follows the motion of the vortex rings, whose radial expansion increases with the electrical energy deposited during the spark discharge. Here, the vortex ring cooling model establishes that entrainment in the convective cooling regime is induced by the vortex rings and governs the cooling of the hot gas kernel, and the rate of cooling increases with the electrical energy deposited during the spark discharge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Saturation of Vortex Rings Ejected from Shock-Accelerated Interfaces

Structures evoking vortex rings can be discerned in shock-accelerated flows ranging from astrophysics to inertial confinement fusion. By constructing an analogy between vortex rings produced in conventional propulsion systems and rings generated by a shock impinging upon a high-aspect-ratio protrusion along a material interface, we extend classical, constant-density vortex-ring theory to compressible multifluid flows. Here, we further demonstrate saturation of such vortex rings as the protrusion aspect ratio is increased, thus explaining morphological differences observed in practice.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Measuring Topological Constraint Relaxation in Ring-Linear Polymer Blends

Polymers are an effective test bed for studying topological constraints in condensed matter due to a wide array of synthetically available chain topologies. When linear and ring polymers are blended together, emergent rheological properties are observed as the blend can be more viscous than either of the individual components. This emergent behavior arises since ring-linear blends can form long-lived topological constraints as the linear polymers thread the ring polymers. Here, in this work, we demonstrate how the Gauss linking integral can be used to efficiently evaluate the relaxation of topological constraints in ring-linear polymer blends. For majority-linear blends, the relaxation rate of topological constraints depends primarily on reptation of the linear polymers, resulting in the diffusive time τ d,R for rings of length N R blended with linear chains of length N l to scale as τ d,R ~ N$^2_R$N$^{3.4}_L$.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗