Materials Science · Cross-listed from astro-ph.EP
Coupled Space Weathering: Nanophase Iron Formation by Micrometeoroid Impact and Solar Wind Sputtering
Ziyu Huang and Masatoshi Hirabayashi
Understanding the interplay between micrometeoroid impacts and
solar wind irradiation is crucial for interpreting lunar surface
evolution. Using reactive molecular dynamics simulations and surface binding energy (SBE) analyses, this study investigates the coupled effects of these two dominant space weathering processes on lunar regolith composed of Fe$_2$SiO$_4$. Our simulations reveal that micrometeoroid impacts significantly modify the lunar surface, creating structurally heterogeneous zones with varying SBEs across microcrater morphologies. Specifically, microcrater floors exhibit enhanced surface cohesion due to high-density compaction, whereas microcrater walls and ejecta show weakened structures. Applying Sigmund's sputtering theory with these SBEs indicates differential sputtering yields for Fe, Si, and O suggesting preferential retention of heavier elements like Fe. This selective sputtering mechanism supports the formation and growth of nanophase metallic iron (npFe$^0$) clusters, influencing the optical and compositional maturation of the lunar surface. These findings advance our understanding of lunar space weathering processes.
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Strongly Correlated Electrons
Study of the triangular-lattice Hubbard model with constrained-path quantum Monte Carlo
Shu Fay Ung, Ankit Mahajan and David R. Reichman
We
benchmark constrained-path Monte Carlo (CPMC) on the triangular-
lattice Hubbard model for several fillings and $U$ values and show that
symmetry-adapted trial wave functions substantially improve quantitative accuracy. Away from half-filling, simple free-electron-based trials that preserve the ground state
symmetry yield energy deviations $\lesssim 1\%$ from exact diagonalization and density
matrix renormalization group results. At half-filling, strong frustration in the intermediate to large $U$ regimes necessitates
symmetry-projected trials to reach comparable accuracy, where both free-electron and
symmetry-broken Hartree-Fock trials incur substantial constraint bias. Since the computational cost of CPMC with
symmetry projection scales polynomially with system size, our results motivate its use as a practical route for studying competing ground states in strongly correlated, frustrated systems.
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Materials Science
Predicting electron-phonon coupling and electronic transport at the moiré scale in twisted bilayer graphene
David J. Abramovitch and Marco Bernardi
First-principles calculations can accurately describe electron-
phonon (e-ph) interactions and electronic transport in a wide range of materials, but are currently limited to unit
cells with up to $\sim$100 atoms due to computational cost. Here, we develop an atomistic electronic potential with Holstein- and Peierls-like terms for modeling e-ph interactions and
phonon-limited electronic transport that enables the study of moiré systems with thousands of atoms per unit
cell. This method can accurately reproduce first-principles e-ph coupling and resistivity in
graphene and large-angle twisted
bilayer graphene (TBG). Using this approach, we study TBG over a range of twist angles down to 1.6$^\circ$ (5044-atom unit
cell), and report the
evolution of e-ph interactions and
phonon-limited resistivity with twist angle. The predicted resistivity increases by two orders of magnitude between 13.2$^\circ$ and 1.6$^\circ$, driven by the progressive reduction of the electronic energy scale. Our calculations can predict key experimental trends in 2.0$^\circ$ and 1.6$^\circ$ TBG, including the resistivity and its dependence on temperature and band filling. Our work establishes a scalable approach for quantitative studies of e-ph interactions and transport in moiré materials and other systems with previously inaccessible length scales.
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Mesoscale and Nanoscale Physics
Universal tuning of quantum electrodynamic interactions from power laws to exponential screening and logarithmic antiscreening
Michael N. Leuenberger and Daniel Gunlycke
We introduce a material-agnostic platform for universal tuning of
quantum electrodynamic interactions from power laws to exponential screening and logarithmic antiscreening, realized in a dielectric spacer bounded by two gate-tunable two-dimensional conductors. The structured electromagnetic environment is completely specified by the transverse-
magnetic and transverse-electric reflection amplitudes \(r_{\mathrm{TM/TE}}(q_\perp,\omega)\) of the sheets. Starting from the QED action and a Green-function formulation, we resum the multiple-reflection series and show that the interactions are governed by a discrete set of transverse cavity harmonics. In the transparent limit \(r_{\rm TM}\to 0\), the interactions reduce to bulk power laws \(U(\rho)\propto \rho^{-\alpha}\). In the reflective limit \(|r_{\rm TM}|\to 1\), the phase/parity of \(r_{\rm TM}\) selects two qualitatively distinct branches: a Dirichlet/PEC (screening) branch \(r_{\rm TM}\to -1\) that removes the gapless transverse mode and yields an evanescent Bessel-\(K\) function \(U(\rho)\propto e^{-\pi\rho/d}/\sqrt{\rho/d}\) at \(\rho\gg d\), and an opposite Neumann/PMC-like (antiscreening) branch \(r_{\rm TM}\to +1\) that retains a gapless mode and can strongly enhance the long-range tail. Thus, the same
heterostructure provides in situ electrical control over both the range and the strength of mediated interactions.
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Strongly Correlated Electrons
Neural network backflow for ab-initio solid calculations
An-Jun Liu and Bryan K. Clark
Accurately simulating extended periodic systems is a central challenge in condensed matter physics. Neural
quantum states (NQS) offer expressive wavefunctions for this task but face issues with scalability. In this work, we successfully extend the neural network backflow (NNBF) approach to ab-initio solid-state materials. Building on our scalable optimization framework for molecules [Liu et al., PRB 112, 155162 (2025)], we introduce a two-stage
pruning strategy to manage the massive configuration space expansions: by utilizing a computationally cheap, physics-informed importance proxy, we devote exact NNBF amplitude evaluations solely to the most relevant determinants, significantly improving optimization efficiency, energy estimation, and convergence. Our framework achieves state-of-the-art accuracy across diverse solid-state
benchmarks. For 1D hydrogen chains, NNBF matches or surpasses DMRG and AFQMC, remains robust in strongly correlated bond-breaking regimes where coupled-cluster methods fail, and smoothly extrapolates to the thermodynamic limit. We further demonstrate its scalability by computing ground-state potential energy curves for 2D
graphene and 3D silicon. Finally, ablation studies validate the computational savings of our
pruning strategy and highlight the dependence of the NNBF energies on basis sets.
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Materials Science
Interfacial exchange and magnetostatic coupling in a CoFeB/Thulium Iron Garnet heterostructure
Walid Al Misba, Jenae E. Shoup, Miela J. Gross et al. (+5 more)
We investigate the exchange coupling between a ferrimagnetic
insulator (FI) thulium iron garnet (TmIG) deposited on a gadolinium gallium garnet (GGG) substrate, which shows perpendicular
magnetic anisotropy and a
ferromagnetic metal (FMM) stack with
oxide capping that consists of CoFeB(x)/W(0.4 nm)/CoFeB(0.8 nm)/MgO(1 nm)/W(5 nm). Vibrating sample magnetometry, magneto-optical Kerr microscopy and first-order reversal curve studies coupled with micromagnetic simulations are used to analyze the coupling between these layers. Strong interlayer exchange coupling and magnetostatic coupling are observed in the samples where the relative strength between these interactions can be controlled by varying the thickness of the CoFeB layer. Films with CoFeB thickness x <=1 nm are strongly exchange coupled, whereas the magnetostatic coupling dominates when the thickness is increased to 3 nm or above. These findings have important implications towards realizing fast and energy efficient spintronic devices using a FI, as its coupling to the FMM layer can be used for effective electrical read out of the
magnetic state of the FI.
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All authors Walid Al Misba, Jenae E. Shoup, Miela J. Gross, Dhritiman Bhattacharya, Kai Liu, Caroline A. Ross, Daniel B. Gopman, Jayasimha Atulasimha
Materials Science
Synchrotron-radiation X-ray topography and reticulography of bulk $\beta$-Ga$_2$O$_3$ crystals grown by the cold crucible method
Yongzhao Yao, Koki Mizuno, Kazuki Ohnishi et al. (+9 more)
The structural properties of a $\beta$-Ga$_2$O$_3$ single
crystal grown by the
oxide crystal growth from cold crucible (OCCC) method were investigated using synchrotron radiation X-ray topography and X-ray reticulography. The region grown beneath the seed exhibits high crystalline quality with a rocking curve full width at half maximum of about 26 arcsec. During diameter enlargement, a twist-type
lattice misorientation develops between the central and laterally expanded regions, originating near the shoulder and propagating along boundaries parallel to the $\langle010\rangle$ growth direction. Dislocation analysis reveals that $\langle010\rangle$-oriented screw dislocations dominate the defect structure with densities of ~$10^{5}$cm$^{-2}$, while higher densities (~$10^{6}$cm$^{-2}$) appear in the wing region. These results clarify defect formation in OCCC-grown $\beta$-Ga$_2$O$_3$ and provide insights for optimizing growth conditions.
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All authors Yongzhao Yao, Koki Mizuno, Kazuki Ohnishi, Yukari Ishikawa, Masanori Kitahara, Taketoshi Tomida, Rikito Murakami, Vladimir Kochurikhin, Liudmila Gushchina, Kei Kamada, Koichi Kakimoto, Akira Yoshikawa
Materials Science
Decoupling structural and bonding effects on ferroelectric switching in ScAlN via molecular dynamics under an applied electric field
Ryotaro Sahashi, Po-Yen Chen and Teruyasu Mizoguchi
ScxAl1-xN has emerged as a promising wurtzite-type ferroelectric material, where increasing the Sc composition reduces both the coercive field (Ec) and remanent polarization (Pr). This composition-dependent behavior is physically attributed to two simultaneous changes: the increase in the internal structural parameter u (structural effect) and the weakening of bond strength (bonding effect). Because these factors are strongly coupled in experiments, their individual contributions to ferroelectric switching remain unclear. In this study, we systematically decoupled these effects using machine-learning force field-based molecular dynamics (MD) simulations under an applied electric field. By artificially tuning u via in-plane strain at a fixed composition, we demonstrated that Pr is determined exclusively by the structural effect, exhibiting a universal linear dependence regardless of the composition. In contrast, Ec deviated from this structural trend, implying an additional compositional contribution. To isolate this, we evaluated configurations with identical u but varying Sc compositions; Pr remained constant, whereas Ec systematically decreased due to bond weakening. Furthermore, static nudged elastic band (NEB) calculations revealed that the static switching barrier depends solely on u, failing to explicitly capture the bonding effect on Ec. These results establish that while Pr is governed strictly by the structural effect, Ec is determined by a superposition of structural and bonding effects. Our findings highlight the necessity of dynamic MD simulations for fully understanding ferroelectric switching in compositionally tunable materials.
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Soft Condensed Matter
Molecular Origin of UV-Induced Irreversible Phase Changes in a Chromonic Liquid Crystal
Junghoon Lee, Seonghun Jeong, Jung-Min Kee et al. (+1 more)
Aqueous solutions of disodium cromoglycate (DSCG), a representative model system for chromonic
liquid crystals, exhibit temperature- and concentration-dependent phase behaviors spanning isotropic, nematic, and columnar phases, as well as their coexistence regions. Nastishin et al. (2018) reported that UV irradiation can alter the phase diagram, transforming a nematic phase into a nematic-isotropic biphasic state due to weakened molecular attractions, accompanied by a slow post-irradiation relaxation. Here, we revisit this phenomenon and elucidate the molecular origin of this phase diagram shift: the UV-induced photodegradation of DSCG into specific photodegradation products, which we identify using
liquid chromatography-mass spectrometry. Through an integrated approach combining in situ X-ray scattering and polarized optical microscopy, we demonstrate that these degradation products disrupt the self-assembly of DSCG aggregates, thereby expanding the isotropic and biphasic regions in the phase diagram. These findings demonstrate that chromonic assemblies and their phase behaviors are highly sensitive to minor chemical alterations, providing a potential route toward light-controlled self assembly of soft matter.
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All authors Junghoon Lee, Seonghun Jeong, Jung-Min Kee, Joonwoo Jeong
Strongly Correlated Electrons
Non-Abelian fractional Chern insulators from an exactly solvable two-body model
Joseph R. Cruise and Alexander Seidel
We construct a class of
lattice Hamiltonians whose single-particle spectrum consists of an arbitrary number of exactly degenerate flat bands that reproduce the analytic structure of the first $p$ Landau levels restricted to the
lattice. When combined with local bosonic contact interactions, these models become exactly solvable frustration-free parent
Hamiltonians for FCIs that realize both Abelian and non-Abelian parton
quantum Hall states. Using exact diagonalization, we confirm the expected zero-mode counting for variants of the model stabilizing the bosonic Jain-21 state as well as the non-Abelian 22- and 33-states, which are expected to support Ising- and Fibonacci-type anyons, respectively. Our construction provides an exactly solvable
lattice realization of multi Landau-level physics and offers a new framework for studying FCIs with Chern number $C > 1$. More broadly, it supplies a family of idealized
lattice models that capture the analytic structure of continuum Landau levels while remaining compatible with exponentially local hopping.
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Mesoscale and Nanoscale Physics
Spatiotemporal Magnonic Vortex Beams with Alternating Transverse Orbital Angular Momentum
Muyang Xie, Chenchen Liu, Jian Huang et al. (+3 more)
Recent theoretical and experimental advances have demonstrated spatiotemporal photonic and acoustic
vortex beams in free space. Such spatiotemporal
vortex beams possess orbital angular momentum oriented perpendicular to the wave propagation direction. Herein, we report the discovery of spatiotemporal magnonic
vortex beams in a confined
ferromagnetic nanostrip geometry. The spatiotemporal magnonic
vortex beam features stationary phase dislocations and exhibits wave propagation along a zigzag-like trajectory. Notably, the transverse orbital angular momentum carried by these phase dislocations displays spatial alternation. Our results differ distinctly from their photonic and acoustic counterparts, offering new insights into the fundamental research of spatiotemporal
vortex beams.
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All authors Muyang Xie, Chenchen Liu, Jian Huang, Zhenyu Wang, Xinwei Dong, Ruifang Wang
Mesoscale and Nanoscale Physics · Cross-listed from physics.bio-ph
Digital unzipping of DNA through a solid-state nanopore: A theoretical study for base-by-base ratcheting
Tomoki Ohkubo
Solid-state nanopore DNA sequencers present mechanical and chemical stability, reusability, and large-scale integrability. However, their development is hindered by the absence of a
protein-free mechanism for controlling DNA translocation, which is accomplished by motor
proteins in their biological counterparts. Here, we propose and theoretically analyze a
protein-independent ratchet mechanism based on the unzipping of double-stranded DNA at the nanopore rim. When the transmembrane bias exceeds a certain threshold, the base pairs mechanically dissociate, allowing one strand to translocate while the other remains upstream. This unzipping process is known to slow DNA motion, suggesting that voltage pulses can trigger individual unzipping events at externally defined times, a concept referred to as digital unzipping. However, the intrinsic unzipping barrier is insufficient to provide the dwell times required for a reliable ionic-current readout; therefore, an additional mechanism is needed to hold the DNA in place between voltage pulses. To overcome this limitation, we introduce a reversible hold mechanism implemented via electrostatic attraction between DNA and a charged nanopore wall, which temporarily immobilizes the strand once the unzipping fork catches on the nanopore rim. Using a statistical-mechanical model, we track the
evolution of the mean and variance of DNA position through each ratchet cycle. Analytical expressions for the corresponding error probabilities show that submicrosecond switching of the hold mechanism enables base-by-base stepping with an error rate <5%. These results theoretically demonstrate that digital unzipping combined with a reversible hold mechanism can yield deterministic single-base motion, thus opening a viable route toward all-solid-state nanopore sequencing.
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Statistical Mechanics
Scaling Autoregressive Models for Lattice Thermodynamics
Xiaochen Du, Juno Nam, Sulin Liu et al. (+1 more)
Predicting how materials behave under realistic conditions requires understanding the statistical distribution of atomic configurations on
crystal lattices, a problem central to
alloy design, catalysis, and the study of phase transitions. Traditional Markov-chain Monte Carlo sampling suffers from slow convergence and critical slowing down near phase transitions, motivating the use of generative models that directly learn the thermodynamic distribution. Existing autoregressive models (ARMs), however, generate configurations in a fixed sequential order and incur high
memory and training costs, limiting their applicability to realistic systems. Here, we develop a framework combining any-order ARMs, which generate configurations flexibly by conditioning on any known subset of
lattice sites, with marginalization models (MAMs), which approximate the probability of any partial configuration in a single forward pass and substantially reduce
memory requirements. This combination enables models trained on smaller
lattices to be reused for sampling larger systems, while supporting expressive
Transformer architectures with
lattice-aware positional encodings at manageable computational cost. We demonstrate that
Transformer-based any-order MAMs achieve more accurate free energies than multilayer perceptron-based ARMs on both the two-dimensional Ising model and CuAu
alloys, faithfully capturing phase transitions and critical behavior. Overall, our framework scales from $10 \times 10$ to $20 \times 20$ Ising systems and from $2 \times 2 \times 4$ to $4 \times 4 \times 8$ CuAu supercells at reduced computational cost compared to conventional sampling methods.
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All authors Xiaochen Du, Juno Nam, Sulin Liu, Rafael Gómez-Bombarelli
Soft Condensed Matter · Cross-listed from physics.bio-ph
A Unified Variational Principle for Branching Transport Networks: Wave Impedance, Viscous Flow, and Tissue Metabolism
Riccardo Marchesi
The branching geometry of biological transport networks is characterized by a diameter scaling exponent $\alpha$. Two structural attractors compete: impedance matching ($\alpha \sim 2$) for pulsatile flow and viscous-metabolic minimization ($\alpha = 3$) for steady flow. Neither predicts the empirically observed $\alpha_{\mathrm{exp}} = 2.70 \pm 0.20$ in mammalian arterial trees. Incorporating sub-linear vessel-wall scaling $h(r) \propto r^p$ ($p = 0.77$) into a three-term metabolic cost rigorously breaks Murray's cubic law – via Cauchy's functional equation – bounding the static optimum to $\alpha_t \in [2.90, 2.94]$. We formulate a unified network-level Lagrangian balancing wave-reflection penalties against transport-metabolic costs. Because the operational duty cycle $\eta$ is uncertain over developmental timescales, we cast the optimization as a zero-sum game between network architecture and environment. Von Neumann's minimax
theorem – proved via strict monotonicity of the cost curves – yields a unique saddle point $(\alpha^, \eta^)$ satisfying an exact equal-cost condition. We further prove $N = 2$ uniquely maximizes the network stiffness ratio $\kappa_{\mathrm{eff}}(N)$, deriving binary branching as a structural consequence of the framework. For the porcine coronary tree ($G = 11$ generations), $\alpha^* = 2.72$, within $0.1\sigma$ of morphometric data. Sensitivity analysis confirms $|\Delta\alpha^*| < 0.01$ across physiological metabolic ranges; the prediction depends critically only on the histological exponent $p$ – a zero-parameter derivation from fundamental scaling principles that simultaneously recovers a cumulative wave dissipation of 6.3%, consistent with independent clinical estimates.
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Statistical Mechanics
Observation-Time-Induced Crossover in Driven Anomalous Transport
Masahiro Shirataki and Takuma Akimoto
We investigate how a weak constant force becomes detectable through fluctuations in anomalous transport in strongly heterogeneous media. Rather than focusing on the mean drift, we show that the key signature of the force appears in the variance of the particle displacement. As representative models, we study a biased continuous-time random walk (CTRW) with nearest-neighbor jumps and a biased quenched trap model (QTM) with a power-law waiting-time tail. By analysing the force dependence of the displacement variance, we quantify how fluctuations respond to weak driving. We find that for $\alpha<2$, the response exhibits an observation-time-induced crossover: at fixed bias, the variance initially follows its unbiased scaling and only at later times crosses over to a force-dominated nonequilibrium regime. Equivalently, at fixed observation time $t$, there exists a threshold bias $\varepsilon_c(t)$ separating an apparently
equilibrium-like regime from a detectable nonequilibrium response. This threshold decreases with increasing $t$, implying that weaker forces become observable over longer measurement windows. Quenched disorder further lowers the detection threshold compared with CTRW, and the crossover reflects a competition between the finite observation time and the intrinsic relaxation time of the driven heterogeneous system.
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Strongly Correlated Electrons
Giant anomalous Hall conductivity in frustrated magnet EuCo2Al9
Sheng Xu, Jian-Feng Zhang, Shu-Xiang Li et al. (+16 more)
The interaction between conduction electrons and localized
magnetic moments profoundly influences the electrical and
magnetic properties of materials, giving rise to a variety of fascinating physical phenomena and
quantum effects. Here, we discover a
giant anomalous
Hall effect (AHE) in a frustrated Eu-based magnet, exhibiting a
giant anomalous
Hall conductivity (AHC) of 31000 {\Omega}-1cm-1 and a remarkable anomalous
Hall angle (AHA, tan{\theta}H) of 12 %–surpassing conventional mechanisms (either intrinsic or extrinsic) by two orders of magnitude. Combining magnetotransport,
quantum oscillations,
neutron diffraction and ab initio calculations, we establish that the
giant AHC originates from fluctuating
spin chirality skew scattering, generated by indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions of Eu-4f moments. Simultaneously, Hund's coupling of itinerant electrons and localized Eu-4f
spins triggers
giant exchange splitting, evidenced by temperature-dependent
Fermi surface reconstruction. This work establishes a frustrated
magnetic platform for engineering the AHE and elucidates the governing role of exchange interactions and
spin textures in
quantum transport, while also providing a framework for designing unconventional spintronic systems that harness emergent
spin-texture dynamics.
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All authors Sheng Xu, Jian-Feng Zhang, Shu-Xiang Li, Junfa Lin, Xiaobai Ma, Wenyun Yang, Jun-Jian Mi, Zheng Li, Tian-Hao Li, Yue-Yang Wu, Jiang Ma, Qian Tao, Wen-He Jiao, Xiaofeng Xu, Zengwei Zhu, Yuanfeng Xu, Hanjie Guo, Tian-Long Xia, Zhu-An Xu
Mesoscale and Nanoscale Physics
Composite boson theory of Hall crystals and their transitions to Wigner crystals
Julian May-Mann, Sayak Bhattacharjee and Srinivas Raghu
We consider the crystallization of a two-dimensional electron system in a perpendicular
magnetic field using composite
boson theory. There are three possible states to consider: the
Hall liquid, the Wigner
crystal, and the
Hall crystal (a state with both broken translation
symmetry and a quantized
Hall response). Within composite
boson theory, these states map onto a
superconductor, a Mott
insulator, and a supersolid of composite
bosons respectively. We show that when a $\nu = 1$
Hall liquid has a sufficiently soft roton, there is a first order transition to a triangular
lattice Hall crystal. If we continue to decrease the roton mass, there is a continuous transition from the
Hall crystal to a Wigner
crystal. {When the
Hall crystal exhibits the integer
quantum Hall effect,} this transition {is} described by a free Dirac
fermion and, at the critical point, the coupling to the
phonons of the
crystal is irrelevant, {in the {renormalization group} sense}. We extend this analysis to fractional $\nu = 1/m$
Hall liquids. There, due to kinetic frustration arising from flux attachment, honeycomb
lattice Hall crystals are preferred over triangular ones at intermediate interaction strength.
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Materials Science · Cross-listed from physics.chem-ph
Stochastic Collision Theory of Magnetism in Radical Fluids
Yoshiaki Uchida and Ryohei Kishi
How
stochastic, microscopic events generate deterministic, macroscopic properties is a fundamental question in physics. We address this question by developing a
quantum master equation model for concentrated radical solutions, where random molecular collisions govern the
magnetic properties of the system. Our theory reveals a simple mechanism: the first-order exchange contribution averages to zero over collisions, while the second-order term survives as an effective
ferromagnetic coupling that enhances magnetization. The model captures the experimentally observed trends in
magnetic behavior that deviate from conventional theories. Because the mechanism arises from statistical averaging, it may apply to a broader class of soft matter phenomena, including
liquid crystals.
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Materials Science
Acrylamide Conformers: A Revision of Published Density Functional Theory Studies
William Scott and Estela Blaisten-Barojas
Acrylamide, with PubChem identifier CID=6579 is broadcasted to have four stable conformers contrasting with several journal publications characterizing only two or three. In this revision summary the discrepancy is clarified. Through very high precision density functional theory (DFT) calculations, three stable conformers and the three transition state barriers existing between them are verified to exist and validated with our own DFT calculations The most stable conformer is a planar molecular structure termed "sys" or "trans" in the literature. Meanwhile, a less stable structure termed "skew" pertains to two 3-dimensional structures that are energy-degenerate, but differ in their structure for being mirrored images of each other. Vibrational spectra, partial atomic charges, Cartesian coordinates, and Intrinsic Reaction Coordinate paths are summarized and recalculated with DFT at the $\omega$B97XD/Def2TZVPP level for the three stable acrylamide isomers: the sys/trans lowest in energy structure, and the two skew mirrored structures.
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