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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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At least 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Broadband hard X-ray attosecond pulses from extremely chirped electron beams

Attosecond pulses from free-electron lasers have opened the doors to atomic site-specific studies of bound electronic dynamics on their natural, sub-femtosecond timescales. Key to their success has been electron beam shaping techniques enabling the generation of sub-femtosecond current spikes with peak currents on the order of 10 kA. We demonstrate in an RF linac the generation of current spikes with extreme chirps on the order of 350 MeV/micron, directly competitive with the chirps expected from beam-driven plasma wakefield accelerators. Leveraging chirp-taper compensation, we use these highly chirped beams to generate hard X-ray attosecond pulses with bandwidths exceeding 30 eV, a factor of two beyond previous demonstrations. We simultaneously present the first explicit experimental evidence of chirp-taper compensation in an attosecond XFEL, finding that optimal tapering improves the bandwidth and pulse energy by factors of two and five, respectively, for our conditions. In addition to the immediate utility of such broadband hard X-ray pulses, electron beams with such extreme chirps can be utilized for unique new experimental modalities by performing further compression after the undulators. Such post-lasing compression can enable subsequent superradiant light emission at longer wavelengths, or direct excitation of quantum systems with the beam's intense space-charge field for unique attosecond pump-probe possibilities.

43 PARTICLE ACCELERATORS↗
DOE OSTI2021

Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics

Majorana bosons, that is, tight bosonic analogs of the Majorana fermionic quasiparticles of condensed-matter physics, are forbidden for gapped free bosonic matter within a standard Hamiltonian scenario. We show how the interplay between dynamical metastability and nontrivial bulk topology makes their emergence possible in noninteracting bosonic chains undergoing Markovian dissipation. This leads to a distinctive form of topological metastability, whereby a conserved Majorana boson localized on one edge is paired, in general, with a symmetry generator localized on the opposite edge. We argue that Majorana bosons are robust against disorder and identifiable by signatures in the zero-frequency steady-state power spectrum. Our results suggest that symmetry-protected topological phases for free bosons may arise in transient metastable regimes, which persist over practical timescales.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Hydrodynamically Controlled Self‐Organization in Mixtures of Active and Passive Colloids

Abstract Active particles are known to exhibit collective behavior and induce structure in a variety of soft‐matter systems. However, many naturally occurring complex fluids are mixtures of active and passive components. The authors examine how activity induces organization in such multi‐component systems. Mixtures of passive colloids and colloidal micromotors are investigated and it is observed that even a small fraction of active particles induces reorganization of the passive components in an intriguing series of phenomena. Experimental observations are combined with large‐scale simulations that explicitly resolve the near‐ and far‐field effects of the hydrodynamic flow and simultaneously accurately treat the fluid–colloid interfaces. It is demonstrated that neither conventional molecular dynamics simulations nor the reduction of hydrodynamic effects to phoretic attractions can explain the observed phenomena, which originate from the flow field that is generated by the active colloids and subsequently modified by the aggregating passive units. These findings not only offer insight into the organization of biological or synthetic active–passive mixtures, but also open avenues to controlling the behavior of passive building blocks by means of small amounts of active particles.

36 MATERIALS SCIENCE↗
DOE OSTI2025

A Polynomial-Time Classical Algorithm for Noisy Quantum Circuits

We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g., the computational basis). Our approach is based upon the intuition that noise exponentially damps nonlocal correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by keeping track of only the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasipolynomial time, so long as the distribution anticoncentrates. A number of implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: For constant noise rates, any quantum circuit for which error mitigation succeeds in polynomial-time on most input states can also be classically simulated in polynomial-time on most input states. Our algorithms scale exponentially in the inverse noise rate, which is fundamental and makes them impractical for current quantum devices.

decoherence↗
DOE OSTI2022

Grassland productivity responds unexpectedly to dynamic light and soil water environments induced by photovoltaic arrays

Abstract Agrivoltaic (AV) systems are designed to coproduce photovoltaic (PV) energy on lands simultaneously supporting agriculture (food/forage production). PV infrastructure in agroecosystems alters resources critical for plant growth, and water‐limited agroecosystems such as grasslands are likely to be particularly sensitive to the unique spatial and temporal patterns of incident sunlight and soil water inherent within AV systems. However, the impact of resource alteration on forage production, the primary ecosystem service from managed grasslands, is poorly resolved. Here, we evaluated seasonal patterns of soil moisture (SM) and diurnal variation in incident sunlight (photosynthetic photon flux density [PPFD]) in a single‐axis‐tracking AV system established in a formerly managed semiarid C 3 grassland in Colorado. Our goals were to (1) quantify dynamic patterns of PPFD and SM within a 1.2 MW PV array in a perennial grassland, and (2) determine how aboveground net primary production (ANPP) and photosynthetic parameters responded to the resource patterns created by the PV array. We hypothesized that spatial variability in ANPP would be strongly related to SM patterns, typical of most grasslands. We measured significant reductions in ANPP directly beneath PV panels, where SM and PPFD were both low. However, in locations with significantly increased SM from the shedding and redistribution of precipitation by PV panels, ANPP was not increased. Instead, ANPP was greatest in locations where plants were shaded in the afternoon but received high levels of PPFD in the morning hours, when air temperatures and vapor pressure deficits were relatively low. Thus, contrary to expectations, we found relatively weak relationships between SM and ANPP despite significant spatial variability in both. Further, there was little evidence that light‐saturated photosynthesis ( A sat ) and quantum yield of CO 2 assimilation (ϕCO 2 ) differed for plants growing directly beneath (lowest PPFD) versus between (highest PPFD) PV panels. Overall, the AV system established in this semiarid managed grassland did not alter patterns of ANPP in ways predictable from past studies of controls of ANPP in open grasslands. However, our results suggest that the diurnal timing of low versus high periods of PPFD incident on plants is an important determinant of productivity patterns in grasslands.

14 SOLAR ENERGY↗
DOE OSTI2024

Directional Superradiance in a Driven Ultracold Atomic Gas in Free Space

Ultracold atomic systems are among the most promising platforms that have the potential to shed light on the complex behavior of many-body quantum systems. One prominent example is the case of a dense ensemble illuminated by a strong coherent drive while interacting via dipole-dipole interactions. Despite being subjected to intense investigations, this system retains many open questions. A recent experiment carried out in a pencil-shaped geometry [Ferioli Nat. Phys. 19, 1345 (2023)] has reported measurements that have seemed consistent with the emergence of strong collective effects in the form of a “superradiant” phase transition in free space, when looking at the light-emission properties in the forward direction. Motivated by the experimental observations, we carry out a systematic theoretical analysis of the steady-state properties of the system as a function of the driving strength and atom number N . We observe signatures of collective effects in the weak-driving regime, which disappear with increasing drive strength as the system evolves into a single-particle-like mixed state comprised of randomly aligned dipoles. Although the steady state features some similarities to the reported superradiant-to-normal nonequilibrium transition, also known as cooperative resonance fluorescence, we observe significant qualitative and quantitative differences, including a different scaling of the critical drive parameter (from N to N ). We validate the applicability of a mean-field treatment to capture the steady-state dynamics under currently accessible conditions. Furthermore, we develop a simple theoretical model that explains the scaling properties by accounting for interaction-induced inhomogeneous effects and spontaneous emission, which are intrinsic features of interacting disordered arrays in free space. Published by the American Physical Society 2024

Agarwal, Sanaa (ORCID:0000000336863804)↗
DOE OSTI2024

Dynamical Aspects of the Quark Gluon Plasma

Shortly after the Big Bang the entire universe was filled with a nearly perfect fluid known as the Quark Gluon Plasma. Relativistic heavy ion collisions can now reproduce this fluid in the laboratory where the Quark Gluon Plasma exhibits a rapid but smooth cross-over phase transition into hadrons at vanishing net-baryon densities. Recent experiments plan to explore finite baryon densities where a critical point is expected. If found, this would mark the first discovery of a critical point in a relativistic system described by a fundamental theory of nature, which would have far-reaching consequences for high-energy nuclear physics and nuclear astrophysics (such as in neutron star mergers). Characteristic temperatures of equilibrium (e.g. the inflection point of the entropy density) and transport coefficients (e.g. minimum of the shear viscosity over entropy density) vary widely at a cross-over phase transition but converge at a critical point, and extracting the behavior of these characteristic temperatures is a major focal point of this research. Specifically, the interplay between strange and light hadrons is exploited to study the flavor hierarchy in the cross-over region. To investigate this, a viscous relativistic hydrodynamics framework with two conserved charges is being developed into a new open-source code along with initial conditions that contain baryon number and strangeness. Flow observables sensitive to the equation of state and transport coefficients are calculated across beam energies. New techniques are being developed to study this flavor hierarchy from first principles and to extract the characteristic temperatures from experimental data. Through this new dynamical framework, this project provides essential guidance to the Beam Energy Scan II runs at Relativistic Heavy-Ion Collider and the future Facility for Antiproton and Ion Research facility in the search for the Quantum Chromodynamic critical point and subsequent investigation of the baryon-rich Quark Gluon Plasma.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Dynamically controlled random lasing with colloidal titanium carbide MXene

Control of lasing properties through tailorable and dynamically tunable materials and reconfigurable compositions can augment the performance of random lasers for a wide range of applications. Here, a colloid of randomly dispersed weakly scattering single-layer titanium carbide (Ti 3 C 2 T x ) MXene flakes embedded within rhodamine 101 gain medium is experimentally shown to provide feedback for random lasing. Additionally, in contrast to previously reported random laser systems where the optical properties of scatterers are static, the relative permittivity of Ti 3 C 2 T x MXene flakes can be varied under optical pumping due to the saturable absorption properties. Numerical simulations indicate that the observed nonlinear response of Ti 3 C 2 T x MXene flakes enables dynamically tunable random lasing. Thus, pumping the Ti 3 C 2 T x MXene flakes with a second optical source decreases the gain threshold required to obtain random lasing. Also, using numerical simulations, it is shown that the control over the intensity of the second pump enables tuning the field distribution of the random lasing modes. Considering the diversity of the MXenes family, the proposed MXene colloidal metamaterial design opens up a new avenue to advanced control of lasing properties for photonic applications.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Data from: "Towards CONUS-Wide ML-Augmented Conceptually-Interpretable Modeling of Catchment-Scale Precipitation-Storage-Runoff Dynamics"

This data package was generated to support the manuscript “Towards CONUS-Wide Machine Learning-Augmented Conceptually Interpretable Modeling of Catchment-Scale Precipitation-Storage-Runoff Dynamics.” It provides input files, model outputs, plotting data, scripts, notebooks, and documentation used to develop, evaluate, and reproduce Mass-Conserving Perceptron (MCP)-based hydrologic modeling experiments across 513 selected Catchment Attributes and Meteorology for Large-sample Studies in the United States (CAMELS-US) basins. The files are organized by modeling component and analysis purpose, including rainfall–runoff experiments, snow module experiments, coupled hydrologic-snow experiments, Long Short-Term Memory (LSTM) benchmark results, model skill metrics, initialization and epoch records, cell-state normalization files, Akaike Information Criterion (AIC)-based model comparison files, and data used to generate manuscript figures. Tabular files can be opened using standard spreadsheet software or Python/R data-analysis tools. Python scripts, Jupyter notebooks, and selected MATLAB scripts are included for model execution, postprocessing, plotting, and statistical analysis. Quality assurance and quality control were conducted through the source-data selection and modeling workflow. Meteorological forcing, streamflow, and static catchment attributes were derived from the CAMELS-US dataset, and snow water equivalent data were derived from the University of Arizona (UA) Snow Water Equivalent dataset. Selected basins and time periods were screened during the associated research workflow to avoid missing observations or poor-quality cases. Static geospatial features were processed primarily using Quantum Geographic Information System (QGIS) and Geospatial Data Abstraction Library (GDAL) workflows. Additional details are provided in the associated manuscript and documentation.

ESS-DIVE CSV File Formatting Guidelines Reporting ↗
DOE OSTI2021

CTRAMER: An open-source software package for correlating interfacial charge transfer rate constants with donor/acceptor geometries in organic photovoltaic materials

In this paper, we present CTRAMER (Charge-Transfer RAtes from Molecular dynamics, Electronic structure, and Rate theory)—an opensource software package for calculating interfacial charge-transfer (CT) rate constants in organic photovoltaic (OPV) materials based on ab initio calculations and molecular dynamics simulations. The software is based on identifying representative donor/acceptor geometries within interfacial structures obtained from molecular dynamics simulation of donor/acceptor blends and calculating the corresponding Fermi’s golden rule CT rate constants within the framework of the linearized-semiclassical approximation. While the methods used are well established, the integration of these state-of-the-art tools originating from different disciplines to study photoinduced CT processes with explicit treatment of the environment, in our opinion, makes this package unique and innovative. The software also provides tools for investigating other observables of interest. After outlining the features and implementation details, the usage and performance of the software are demonstrated with results from an example OPV system.

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

Problem-tailored Simulation of Energy Transport on Noisy Quantum Computers

The transport of conserved quantities like spin and charge is fundamental to characterizing the behavior of quantum many-body systems. Numerically simulating such dynamics is generically challenging, which motivates the consideration of quantum computing strategies. However, the relatively high gate errors and limited coherence times of today's quantum computers pose their own challenge, highlighting the need to be frugal with quantum resources. In this work we report simulations on quantum hardware of infinite-temperature energy transport in the mixed-field Ising chain, a paradigmatic many-body system that can exhibit a range of transport behaviors at intermediate times. We consider a chain with L = 12 sites and find results broadly consistent with those from ideal circuit simulators over 90 Trotter steps, containing up to 990 entangling gates. To obtain these results, we use two key problem-tailored insights. First, we identify a convenient basis – the Pauli Y basis – in which to sample the infinite-temperature trace and provide theoretical and numerical justifications for its efficiency relative to, e.g., the computational basis. Second, in addition to a variety of problem-agnostic error mitigation strategies, we employ a renormalization strategy that compensates for global nonconservation of energy due to device noise. We discuss the applicability of the proposed sampling approach beyond the mixed-field Ising chain and formulate a variational method to search for a sampling basis with small sample-to-sample fluctuations for an arbitrary Hamiltonian. This opens the door to applying these techniques in more general models.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Zinc oxide (ZnO) hybrid metasurfaces exhibiting broadly tunable topological properties

Extreme light confinement observed in periodic photonic structures, such as the vortex singularities in momentum (k) space, has been associated with their topological nature. Consequently, by exploiting and tuning their topological properties, optical metasurfaces have been demonstrated as an attractive platform for active photonics. However, given the fact that most active media under external excitations can only provide limited refractive index change, the potential advancements offered by the topological character of active metasurfaces have remained mostly unexplored. Zinc oxide (ZnO), which has recently exhibited optically-induced extraordinarily large permittivity modulations at visible and near-infrared frequencies, is an excellent active material for dynamic metasurfaces exhibiting strong tuning. This work demonstrates that a hybrid metasurface consisting of an array of ZnO nanodisks on a silver backplane displays broadly tunable topological properties. In particular, by performing k-space scattering simulations using measured pump-fluence-dependent material properties of ZnO, we study in detail the light reflection from the hybrid metasurface. Our results validate that the large k-space topology tuning of the metasurface can result in enormously strong polarization manipulation of near-infrared light in the vicinity of the topological features. The observed polarization switching effect is highly sensitive to the polarization and wavelength of an incident wave, owing to the symmetry and dispersion characteristics of the proposed system. Our study indicates that leveraging a combination of the extraordinary material properties and the k-space topology, hybrid metasurfaces based on ZnO may open new avenues for creating all-optical switchable metadevices.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Excited-state electronic structure of molecules using many-body Green’s functions: Quasiparticles and electron–hole excitations with VOTCA-XTP

Here, we present the open-source VOTCA-XTP software for the calculation of the excited-state electronic structure of molecules using many-body Green’s function theory in the GW approximation with the Bethe–Salpeter equation (BSE). This work provides a summary of the underlying theory and discusses the details of its implementation based on Gaussian orbitals, including resolution-of-identity techniques and different approaches to the frequency integration of the self-energy or acceleration by offloading compute-intensive matrix operations using graphics processing units in a hybrid OpenMP/Cuda scheme. A distinctive feature of VOTCA–XTP is the capability to couple the calculation of electronic excitations to a classical polarizable environment on an atomistic level in a coupled quantum- and molecular-mechanics (QM/MM) scheme, where a complex morphology can be imported from Molecular Dynamics simulations. The capabilities and limitations of the GW–BSE implementation are illustrated with two examples. First, we study the dependence of optically active electron–hole excitations in a series of diketopyrrolopyrrole-based oligomers on molecular-architecture modifications and the number of repeat units. Second, we use the GW–BSE/MM setup to investigate the effect of polarization on localized and intermolecular charge-transfer excited states in morphologies of low-donor content rubrene–fullerene mixtures. These showcases demonstrate that our implementation currently allows us to treat systems with up to 2500 basis functions on regular shared-memory workstations, providing accurate descriptions of quasiparticle and coupled electron–hole excited states of various characters on an equal footing.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Spin excitations and dynamics in 2D magnets: An overview of magnons and magnetic skyrmions

van der Waals magnetic materials open up exciting possibilities to investigate fundamental spin properties in low-dimensional systems and to build compact functional spintronic structures. This review focuses on the recent progress in two-dimensional (2D) magnets that explore beyond the homogenous magnetically ordered state, including magnons (spin waves), magnetic skyrmions, and complex magnetic domains. Properties of these spin and topology excitations in 2D magnets provide insights into spin-orbit interactions and other forms of coupling between electrons, phonons, and spin-dependent excitations. Such spin-based quasiparticles can also serve as information carriers for next-generation high-speed computing elements. Furthermore, we will first lay out the general theoretical basis of dynamical responses in magnetic systems, followed by detailed descriptions of experimental progress in magnons and spin textures (including magnetic domains and skyrmions). Discussion on the experimental techniques and future perspectives are also included.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Ultrafast nonlinear absorption of TMDC quantum dots

Here, we study theoretically absorption properties of MoS 2 quantum dots placed in an ultrashort and strong optical pulse with the duration of a few femtoseconds and the amplitude up to 1.5 V/Å. While the monolayer of MoS 2 has a bandgap of ≈ 1.8 eV, the quantum dots of MoS 2 have in-gap edge states, which open additional absorption channels. The absorbance of the system strongly depends on the amplitude of the pulse. At small amplitude, less than 0.5 V/Å, the absorbance varies by more than 20% with the frequency of the pulse. The system shows saturable absorption if the frequency of the pulse is larger than the bulk bandgap and reverse saturable absorption if the frequency of the pulse is in the bandgap. At large amplitudes, > 0.5 V/Å, of the pulse, the ultrafast absorbance of a quantum dot has a weak dependence on the pulse frequency. If the frequency of the pulse is less than the corresponding bulk bandgap, the absorbance has a nonmonotonic dependence on the pulse amplitude with the maximum at ≈ 0.5 V/Å. We also present the results for QDs of different TMDC materials, which show that at large field amplitude, > 0.5 V/Å, the absorbance is universal and has weak dependence on the type of TMDC.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Environmental Contributions to Proton Sharing in Protein Low-Barrier Hydrogen Bonds

Hydrogen bonds (H-bonds) are central to biomolecular structure and dynamics. Although H-bonds are typically characterized by well-defined proton positions, proton delocalization has been proposed to play a role in facilitating enzyme catalysis and allostery in some systems. Experimentally locating protons is difficult, hampering the study of proton mobility in H-bonds. We used neutron crystallography, atomic resolution X-ray bond length analysis, and large quantum mechanics/molecular mechanics-Born–Oppenheimer molecular dynamics (QM/MM-BOMD) simulations to comprehensively characterize the shared proton/deuteron in a Glu–Asp low-barrier hydrogen bond (LBHB) in the bacterial protein YajL that is a conventional H-bond in the homologous disease-associated human protein DJ-1. X-ray bond length analysis of protiated and perdeuterated DJ-1 and YajL shows no significant effect of deuteron substitution on these carboxylic acid-carboxylate H-bonds but does reveal an effect at the active site glutamic acid near a cysteine thiolate. Residues in an H-bonded network that might favor LBHB formation in YajL were interrogated by the mutation of homologous residues in DJ-1. A distal DJ-1 substitution increases proton delocalization in the Glu–Asp H-bond, demonstrating that mutations within extended H-bond networks can modulate proton transfer barriers in carboxylic acid-carboxylate H-bonds. In addition, proton mobility in the H-bond is correlated with dimer-spanning motions in the QM/MM-BOMD simulations of YajL and DJ-1. Our results show that proton delocalization can be tuned using combined bioinformatic, structural, and computational information, opening the possibility of using engineered proton delocalization as a probe of H-bonding environments and as a tool to test hypotheses about LBHB function.

Lin, Jiusheng [University of Nebraska, Lincoln, NE↗
DOE OSTI2025

Lines of Bound States in the Continuum in Monodisperse Microsphere Structures

Incorporating epitaxially grown active semiconductor media into self-assembled microsphere crystals has remained a major challenge as the curved geometry of the spheres is incompatible with conventional planar growth and patterning techniques. As a result, active photonic devices based on such self-assembled structures, which have other desirable properties such as enabling the rapid assembly of large area systems, have been elusive. Here, we numerically demonstrate an alternative route to creating high-Q resonances in microsphere-based nanophotonic assemblies compatible with standard optical gain materials. We show that close-packed layers of monodisperse microspheres can exhibit symmetry bandgaps, regardless of their layer stacking order. Thus, this microsphere environment can be used to create lines of bound states in the continuum (BICs) of an embedded slab made of active material. Using a realistic ZnS@SiO 2 (core-shell) architecture and sphere diameters compatible with scalable colloidal assembly, we predict sharp spectral features in angle-resolved transmission that trace these lines of BICs and persist in the presence of some fabrication disorder. Altogether, in enabling the integration of unpatterned epitaxial gain slabs with large-area colloidal microsphere assemblies, this approach opens a practical path toward active, dynamically reconfigurable photonic devices that leverage BIC physics for narrow-linewidth lasing and enhanced light–matter interactions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Structural magnetic glassiness in the spin ice Dy 2 Ti 2 O 7

The origin and nature of glassy dynamics presents one of the central enigmas of condensed-matter physics across a broad range of systems ranging from window glass to spin glasses. The spin-ice compound Dy 2 Ti 2 O 7 , which is perhaps best known as hosting a three-dimensional Coulomb spin liquid with magnetically charged monopole excitations, also falls out of equilibrium at low temperature. How and why it does so remains an open question. Based on an analysis of low-temperature diffuse neutron-scattering experiments employing different cooling protocols alongside recent magnetic noise studies, combined with extensive numerical modeling, we argue that upon cooling, the spins freeze into what may be termed a “structural magnetic glass,” without an a priori need for chemical or structural disorder. Specifically, our model indicates the presence of frustration on two levels, first producing a near-degenerate constrained manifold inside which phase ordering kinetics is in turn frustrated. A remarkable feature is that monopoles act as sole annealers of the spin network and their pathways and history encode the development of glass dynamics, allowing the glass formation to be visualized. Our results suggest that spin ice Dy 2 Ti 2 O 7 provides one prototype of magnetic glass formation specifically and a setting for the study of kinetically constrained systems more generally.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗