Low-dimensional Behavior Research Articles

Dual Structure of a Vanadyl-Based Molecular Qubit Containing a Bis(β-diketonato) Ligand.

We designed [VO(bdhb)] (1') as a new electronic qubit containing an oxovanadium(IV) ion (S = 1/2) embraced by a single bis(β-diketonato) ligand [H2bdhb = 1,3-bis(3,5-dioxo-1-hexyl)benzene]. The synthesis afforded three different crystal phases, all of which unexpectedly contain dimers with formula [(VO)2(bdhb)2] (1). A trigonal form (1h) with a honeycomb structure and 46% of solvent-accessible voids quantitatively transforms over time into a monoclinic solvatomorph 1m and minor amounts of a triclinic solventless phase (1a). In a static magnetic field, 1h and 1m have detectably slow magnetic relaxation at low temperatures through quantum tunneling and Raman mechanisms. Angle-resolved electron paramagnetic resonance (EPR) spectra on single crystals revealed signatures of low-dimensional magnetic behavior, which is solvatomorph-dependent, being the closest interdimer V···V separations (6.7-7.5 Å) much shorter than intramolecular V···V distances (11.9-12.1 Å). According to 1H diffusion ordered spectroscopy (DOSY) and EPR experiments, the complex adopts the desired monomeric structure in organic solution and its geometry was inferred from density functional theory (DFT) calculations. Spin relaxation measurements in a frozen toluene-d8/CD2Cl2 matrix yielded Tm values reaching 13 μs at 10 K, and coherent spin manipulations were demonstrated by Rabi nutation experiments at 70 K. The neutral quasi-macrocyclic structure, featuring nuclear spin-free donors and additional possibilities for chemical functionalization, makes 1' a new convenient spin-coherent building block in quantum technologies.

Low-dimensional behavior of a Kuramoto model with inertia and Hebbian learning

We study low-dimensional dynamics in a Kuramoto model with inertia and Hebbian learning. In this model, the coupling strength between oscillators depends on the phase differences between the oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a five-dimensional dynamical system that decouples into a two-dimensional longitudinal system and a three-dimensional transverse system. We readily write an exact solution of the longitudinal system, and we then focus our attention on the transverse system. We classify the stability of the transverse system’s equilibrium points using linear stability analysis. We show that the transverse system is dissipative and that all of its trajectories are eventually confined to a bounded region. We compute Lyapunov exponents to infer the transverse system’s possible limiting behaviors, and we demarcate the parameter regions of three qualitatively different behaviors. Using insights from our analysis of the low-dimensional dynamics, we examine the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.

Minimal frustration underlies the usefulness of incomplete regulatory network models in biology

Regulatory networks as large and complex as those implicated in cell-fate choice are expected to exhibit intricate, very high-dimensional dynamics. Cell-fate choice, however, is a macroscopically simple process. Additionally, regulatory network models are almost always incomplete and/or inexact, and do not incorporate all the regulators and interactions that may be involved in cell-fate regulation. In spite of these issues, regulatory network models have proven to be incredibly effective tools for understanding cell-fate choice across contexts and for making useful predictions. Here, we show that minimal frustration-a feature of biological networks across contexts but not of random networks-can compel simple, low-dimensional steady-state behavior even in large and complex networks. Moreover, the steady-state behavior of minimally frustrated networks can be recapitulated by simpler networks such as those lacking many of the nodes and edges and those that treat multiple regulators as one. The present study provides a theoretical explanation for the success of network models in biology and for the challenges in network inference.

Data-driven discovery of intrinsic dynamics

Dynamical models underpin our ability to understand and predict the behaviour of natural systems. Whether dynamical models are developed from first-principles derivations or from observational data, they are predicated on our choice of state variables. The choice of state variables is driven by convenience and intuition, and, in data-driven cases, the observed variables are often chosen to be the state variables. The dimensionality of these variables (and consequently the dynamical models) can be arbitrarily large, obscuring the underlying behaviour of the system. In truth these variables are often highly redundant and the system is driven by a much smaller set of latent intrinsic variables. In this study we combine the mathematical theory of manifolds with the representational capacity of neural networks to develop a method that learns a system’s intrinsic state variables directly from time-series data, as well as predictive models for their dynamics. What distinguishes our method is its ability to reduce data to the intrinsic dimensionality of the nonlinear manifold they live on. This ability is enabled by the concepts of charts and atlases from the theory of manifolds, whereby a manifold is represented by a collection of patches that are sewn together—a necessary representation to attain intrinsic dimensionality. We demonstrate this approach on several high-dimensional systems with low-dimensional behaviour. The resulting framework provides the ability to develop dynamical models of the lowest possible dimension, capturing the essence of a system. Learning minimal representations of dynamical systems is essential for mathematical modelling and prediction in science and engineering. Floryan and Graham propose a deep learning framework able to estimate accurate global dynamical models by sewing together multiple local representations learnt from high-dimensional time-series data.

Comment on “Low-dimensional behavior of generalized Kuramoto model” by S. Ameli and K. A. Samani

Comment on “Low-dimensional behavior of generalized Kuramoto model” by S. Ameli and K. A. Samani

Impact of mixed anion ordered state on the magnetic ground states of S=1/2 square-lattice quantum spin antiferromagnets, Sr2NiO3Cl and Sr2NiO3F

The magnetic properties of the $S=1/2$ two-dimensional square-lattice antiferromagnets ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}X$ $(X=\mathrm{Cl}, \mathrm{F})$ with the trivalent nickel ions in a low-spin state were studied by magnetic susceptibility, heat capacity, neutron powder diffraction, high-field electron spin resonance (ESR), muon spin rotation and relaxation $({\ensuremath{\mu}}^{+}\mathrm{SR})$ measurements, and density functional theory (DFT) calculations. Both oxyhalides are isostructural to an ideal quantum square-lattice antiferromagnet ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}$, but the chlorine/fluorine anion exclusively occupies an apical site in an ordered/disordered manner with an oxygen anion, resulting in the formation of highly distorted ${\mathrm{NiO}}_{5}X$ octahedra with an off-center nickel ion. Magnetic susceptibility measurements revealed a remarkable difference between these two compounds: the magnetic susceptibility of ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{Cl}$ exhibited a broad maximum at approximately 35 K, which is typical of low-dimensional antiferromagnetic behavior. In contrast, the magnetic susceptibility of ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{F}$ exhibited spin-glass-like behavior below 12 K. No anomaly associated with long-range magnetic ordering was observed in the heat capacity, ESR, and neutron powder diffraction experiments. However, ${\ensuremath{\mu}}^{+}\mathrm{SR}$ measurements revealed the emergence of a static magnetic ordered state below ${T}_{\mathrm{N}}=28\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ in ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{Cl}$ and a short-range magnetic state below ${T}_{\mathrm{N}}=18\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ in ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{F}$. The DFT calculations suggested that the unpaired electron occupied a ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ orbital, and ferromagnetic couplings between the nearest-neighbor nickel spins were energetically favored. The mechanism of ferromagnetic superexchange interactions and the reason for the difference between the magnetic ground states in these nickel oxyhalides are discussed.

Relationship between morphological and physical properties in nanostructured CuMnO2

In this study, crednerite CuMnO 2 nanostructures were prepared using a hydrothermal method at 100 °C with various amounts of NaOH mineralizator. Obtained nanostructured crednerite CuMnO 2 with monoclinic structure (space group C 2/ m ) exhibits two morphologies: nanobelts with lengths of 1–1.5 nm and thickness of 15–25 nm, and nanoplates with diameters of 50–70 nm. Comparative analyses of the prepared samples reveal a close relationship between morphological and physical properties of nanostructured CuMnO 2 . A low NaOH concentration promotes elongated crystal growth along the c-axis, resulting in nanobelt-shaped morphology. A strong base solution, on the other hand, promotes the formation of nanoplates. The different morphologies of nanostructured CuMnO 2 have different spectroscopic and magnetic properties. The Raman active A 1g mode at 637 cm −1 and a modified Curie-Weiss behavior characterize the nanobelt-shaped sample. This phase has two magnetic phase transitions: ferromagnetic at 9.2 K and antiferromagnetic at 42 K. The nanoplate-shaped sample, on the other hand, exhibits typical behavior as reported in the literature, namely the Raman active A 1g mode at 688 cm −1 and low-dimensional magnetism with antiferromagnetic ordering below 62 K. The variation in magnetic properties is presumably due to partial oxidation of Cu 1+ /Cu 2+ and Mn 3+ /Mn 4+ , as well as a change in Mn 3+ spin states from low-spin in nanobelt-shaped samples to high-spin in nanoplate-shaped samples. Both morphological and preferred crystal growth of nanostructured CuMnO2 can be tuned utilizing suitable amounts of NaOH mineralizator in the growth solution. A pure CuMnO2 phase of monoclinic structure has been prepared in the form of nanobelts with lengths up to 1.5 mm or nanoplates possessing 50–70 nm in diameter. The physical properties measurements imply distinct behaviour amongst these samples.The CuMnO2 nano-belts exhibit a Raman peak at a lower Raman number compared to that of the nanoplates. The nanobelt sample has been shown to be undergone two successive magnetic phase transitions: antiferromagnetically at 42 K and ferromagnetically at 9.2 K. The nanoplate samples have exhibited low-dimensional magnetic behaviour. The variation in the magnetic properties can be ascribed to the partial oxidation of Cu1+/Cu2+ and Mn3+/Mn4+. The variation of spin states of the Mn ions can be correlated with the change in the morphologies, i.e., from a low-spin state in the belt-shaped sample to a high-spin state in the plate-shaped samples. • Both morphological and preferred crystal growth of nanostructured CuMnO2 and then their physical properties can be tuned utilizing suitable amounts of NaOH mineralizator in the growth solution. Remarkably, a nanobelt CuMnO2 has been obtained for the first time. • Our experiments divulge that a pure CuMnO2 phase of monoclinic structure can been prepared in the form of nano-belts with lengths up to 1.5 mm or nanoplates possessing 50–70 nm in diameter. • The physical properties measurements imply distinct behaviour amongst these samples. The CuMnO2 nano-belts exhibits Raman peak at a lower Raman number compared to that of the nanoplates. • Regarding to the magnetic characteristics, the nanobelt samples have shown to be undergone into two successive magnetic phase transitions: antiferromagnetically at 42 K and ferromagnetically at 9.2 K. • The variation in the magnetic properties can be ascribed to the partial oxidation of Cu 1+ /Cu2+ and Mn 3+ /Mn4+. Remarkably, the variation of spin states of the Mn ions can be correlated with the change in the morphologies, i.e., from low-spin state in the belt-shaped sample to high-spin state in the plate-shaped samples.

Low-dimensional behavior of generalized Kuramoto model

Low-dimensional behavior of generalized Kuramoto model

Nonstoichiometric Ellenbergerite-Type Phosphates: Hydrothermal Synthesis, Crystal Chemistry, and Magnetic Behavior

We synthesized single crystals of Na0.55Ni6(OH)3(H0.61PO4)4 (I) and polycrystals of (Na, Ni)0.64Ni5.68(OH)3(H0.67PO4)4 (II) with ellenbergerite-like structures using the hydrothermal method. The phases crystallize in the hexagonal space group P63mc with the following unit cell parameters: a = 12.5342(1) Å, c = 4.9470(1) Å, and V = 673.08(2) Å3 for I; a = 12.4708(2) Å, c = 4.9435(2) Å, and V = 665.82(2) Å3 for II; and Z = 2. Their crystal structures are based on a 3D framework built from NiO6 octahedra and PO4 tetrahedra. The difference between I and II lies in the way the structural channels are filled along the [001] direction. These channels accommodate segments of Na- and (Na, Ni)-centered chains of face-sharing octahedra in the structures I and II, respectively. The magnetic susceptibility χ and the specific heat Cp evidence pronounced low-dimensional magnetic behavior at elevated temperatures and the formation of the weakly ferromagnetic long-range order at TNI = 61 K and TNII = 63 K. Analysis of the χ(T) data within both chain and dimer spin models allows the estimation of the leading exchange interaction parameters in the compounds under study.

Phenomenological Model of Nonlinear Dynamics and Deterministic Chaotic Gas Migration in Bentonite: Experimental Evidence and Diagnostic Parameters

Understanding gas migration in compacted clay materials, e.g., bentonite and claystone, is important for the design and performance assessment of an engineered barrier system of a radioactive waste repository system, as well as many practical applications. Existing field and laboratory data on gas migration processes in low-permeability clay materials demonstrate the complexity of flow and transport processes, including various types of instabilities, caused by nonlinear dynamics of coupled processes of liquid–gas exchange, dilation, fracturing, fracture healing, etc., which cannot be described by classical models of fluid dynamics in porous media. We here show that the complexity of gas migration processes can be explained using a phenomenological concept of nonlinear dynamics and deterministic chaos theory. To do so, we analyzed gas pressure and gas influx (i.e., input) and outflux (i.e., output), recorded during the gas injection experiment in the compact Mx80-D bentonite sample, and calculated a set of the diagnostic parameters of nonlinear dynamics and chaos, such a global embedding dimension, a correlation dimension, an information dimension, and a spectrum of Lyapunov exponents, as well as plotted 2D and 3D pseudo-phase-space strange attractors, based on the univariate influx and outflux time series data. These results indicate the presence of phenomena of low-dimensional deterministic chaotic behavior of gas migration in bentonite. In particular, during the onset of gas influx in the bentonite core, before the breakthrough, the development of gas flow pathways is characterized by the process of chaotic gas diffusion. After the breakthrough, with inlet-to-outlet movement of gas, the prevailing process is chaotic advection. During the final phase of the experiment, with no influx to the sample, the relaxation pattern of gas outflux is resumed back to a process of chaotic diffusion. The types of data analysis and a proposed phenomenological model can be used to establish the basic principles of experimental data-gathering, modeling predictions, and a research design.

Diaphite-structured nanodiamonds with six- and twelve-fold symmetries

Nanodiamonds (ND) with 1-5 nm dimensions found in meteorites or produced by chemical vapour deposition (CVD) and detonation synthesis are typically described in terms of an sp3-bonded carbon network. However, ultra-high-resolution transmission electron microscopy (uHRTEM) combined with density functional theory (DFT) modelling leads to a different structural interpretation. uHRTEM imaging and nanodiffraction studies of many NDs show six-fold symmetry features whose identity has long been controversial. We also observe diffraction patterns with twelve equally-spaced and symmetrically but unequally arranged reflections, indicating structures with crystallographically-forbidden ideal and distorted twelve-fold symmetry. Structural models based on our DFT calculations lead to an interpretation of these unusual features found throughout the meteoritic and CVD samples in terms of sp3 domains arranged around and coherently bonded to graphitic domains embedded within the diamond matrix. The bonding at the sp2-sp3 interface can explain the unusual features observed in electron energy-loss spectra (EELS) below the onset of the main diamond C1s core-loss edge leading to predictions of low-dimensional conductivity behaviour. The presence of sp2- as well as sp3-bonded regions allows us to interpret previously unexplained features of the Raman spectra and EELS data of ND materials.

Formula omitted]-dominant switched linear systems

This paper studies the asymptotic behavior of switched linear systems, beyond classical stability. We focus on systems having a low-dimensional asymptotic behavior, that is, systems whose trajectories converge to a common time-varying low-dimensional subspace. We introduce the concept of path-complete p-dominance for switched linear systems, which generalizes the approach of quadratic Lyapunov theory by replacing the contracting ellipsoids by families of quadratic cones whose contraction properties are dictated by an automaton. We show that path-complete p-dominant switched linear systems are exactly the ones that have a p-dimensional asymptotic behavior. Then, we describe an algorithm for the computation of the cones involved in the property of p-dominance. This allows us to provide an algorithmic framework for the analysis of switched linear systems with a low-dimensional asymptotic behavior.

Quasi-one-dimensional magnetic properties of NiNb2−xVxO6 compounds synthesized at high pressure in a nonstandard columbite-type structure

We report on the low-dimensional magnetic behavior of the series of compounds ${\mathrm{NiNb}}_{2\ensuremath{-}x}{\mathrm{V}}_{x}{\mathrm{O}}_{6}$ ($0\ensuremath{\le}x\ensuremath{\le}2$), with a columbite-type crystal structure stabilized at high pressure and temperature. Based on susceptibility, magnetization, and specific-heat measurements, the system is characterized as presenting quasi-one-dimensional magnetism, with ${\mathrm{Ni}}^{2+}$ magnetic moments that can be modeled as Ising spins, placed along zigzag chains in the crystal structure. The low-temperature phase is found to consist of an antiferromagnetic arrangement of ferromagnetic chains, and a metamagnetic transition to uniform ferromagnetic order is observed under magnetic fields slightly above ${\ensuremath{\mu}}_{0}H=1\phantom{\rule{4pt}{0ex}}\mathrm{T}$. We discuss the effects of substituting vanadium for niobium, maintaining the same crystal structure along the whole series of samples. In particular, the long-range magnetic order, most clearly seen for $x=0$, tends to be suppressed as the vanadium content is increased. The exchange interactions are quantified, revealing that the ferromagnetic intrachain interactions vary from about $7\phantom{\rule{4pt}{0ex}}\mathrm{K}$ for ${\mathrm{NiNb}}_{2}{\mathrm{O}}_{6}$ to $2\phantom{\rule{4pt}{0ex}}\mathrm{K}$ for ${\mathrm{NiV}}_{2}{\mathrm{O}}_{6}$, remaining one order of magnitude larger than the mean antiferromagnetic interchain coupling.

Low-dimensional magnetism in calcium nitridonickelate(II) Ca2NiN2.

Calcium nitridonickelate(II) Ca2NiN2 has been prepared through a high-temperature and high-pressure azide-mediated redox reaction, demonstrating that this method can stabilise nitrides of late transition metals in relatively high oxidation states. Ca2NiN2 crystallizes in the Na2HgO2 structure type and displays low-dimensional antiferromagnetic ordering of Ni2+ spins.

Thermoelectric Properties of InA Nanowires from Full-Band Atomistic Simulations.

In this work we theoretically explore the effect of dimensionality on the thermoelectric power factor of indium arsenide (InA) nanowires by coupling atomistic tight-binding calculations to the Linearized Boltzmann transport formalism. We consider nanowires with diameters from 40 nm (bulk-like) down to 3 nm close to one-dimensional (1D), which allows for the proper exploration of the power factor within a unified large-scale atomistic description across a large diameter range. We find that as the diameter of the nanowires is reduced below d < 10 nm, the Seebeck coefficient increases substantially, as a consequence of strong subband quantization. Under phonon-limited scattering conditions, a considerable improvement of ~6× in the power factor is observed around d = 10 nm. The introduction of surface roughness scattering in the calculation reduces this power factor improvement to ~2×. As the diameter is decreased to d = 3 nm, the power factor is diminished. Our results show that, although low effective mass materials such as InAs can reach low-dimensional behavior at larger diameters and demonstrate significant thermoelectric power factor improvements, surface roughness is also stronger at larger diameters, which takes most of the anticipated power factor advantages away. However, the power factor improvement that can be observed around d = 10 nm could prove to be beneficial as both the Lorenz number and the phonon thermal conductivity are reduced at that diameter. Thus, this work, by using large-scale full-band simulations that span the corresponding length scales, clarifies properly the reasons behind power factor improvements (or degradations) in low-dimensional materials. The elaborate computational method presented can serve as a platform to develop similar schemes for two-dimensional (2D) and three-dimensional (3D) material electronic structures.

Peculiarity of a magnetic structure in a quasi-one-dimensional columbite Co0.4Ni0.6Nb2O6

Quasi-one-dimensional (Q1D) spin chain systems have great potential applications in high-density information storage devices, quantum information and computers, because of their quantum magnetism properties. The low-dimensional magnetic behavior has been investigated in ANb2O6, (A = Mn, Fe, Co or Ni) compounds, the structural and magnetic properties are very interesting because the system presents weakly interacting Ising chains, which leads to this quasi-one-dimensional magnetic order. Our investigation combines specific heat and magnetic measurements; x-ray and neutron diffraction (ND). In this work, we present a Co/Ni orthorhombic structure, called columbite, which crystallizes with Pbcn space group, whose formula is Co0.4Ni0.6Nb2O6. Co for Ni substitution induces a continuous lattice volume decrease, preserving the orthorhombic crystal structure. Magnetic susceptibility and specific heat measurements reveal that antiferromagnetic order occurs at 3.4 K, as a consequence of weak interchain interactions. Partial substitution of the magnetic ion tends to change the magnetic ordering observed in the CoNb2O6 and NiNb2O6. Lastly, we present this magnetic structure changes with the Ni-Co substitution.

Modeling Microbial Interactions across Nutritional Environments using Maximum Entropy

Microbial communities have exponential capacity for complexity. (In principle, among k species, each of (2∧k - 1) subsets (every pair, triplet, or larger group up to size k) could modulate the consortium's macroscopic dynamics.) Indeed, recent studies on synthetic communities have argued that high-dimensional (many-species) interactions dominate communal function. Here we explore a contrasting view. Despite the combinatorial potential for multiscale interactions, two features of microbial social life—localized diffusion of intercellular signals and prevalent evolutionary incentives for trophic specialization—provide some physical grounds for relatively low-dimensional behavior. We show how a commonly-used null model used to interpret consortial function can misidentify low-order dynamics as high-dimensional. As an alternative, we develop a mechanism-free model of microbial interactions based on the principle of maximum entropy. The model encodes the effects of potential species interactions via correlated random variables governed by a probability distribution informed by measurements. Our model is statistically-coherent, mathematically-flexible, and compatible with (though not limited to) plausible dynamics of bacterial growth. We test the model using data on microbial growth across carbon sources from a recent droplet microfluidics technology that enables wide sampling over the space of microbial compositions. Over hundreds of synthetic small-scale synthetic communities, our model better generalizes over carbon sources in capturing more nonlinear structure of interactions relative to a set of alternative models. The model's failures identify intriguing ecological behavior. These results highlight the potential predictability of some microbial ecosystems from limited measurements. These ideas also show a framework to identify genuinely emergent interactions in a statistically-principled fashion.

Data-driven discovery of coordinates and governing equations.

The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam's razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom deep autoencoder network to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. This method places the discovery of coordinates and models on an equal footing.

Rb2CaCu6(PO4)4O2, a novel oxophosphate with a shchurovskyite-type topology: synthesis, structure, magnetic properties and crystal chemistry of rubidium copper phosphates.

Single crystals of Rb2CaCu6(PO4)4O2 were synthesized by a hydrothermal method in the multicomponent system CuCl2-Ca(OH)2-RbCl-B2O3-Rb3PO4. The synthesis was carried out in the temperature range from 690 to 700 K and at the general pressure of 480-500 atm [1 atm = 101.325 kPa] from the mixture in the molar ratio 2CuO:CaO:Rb2O:B2O3:P2O5. The crystals studied by single-crystal X-ray analysis were found to be monoclinic, space group C2, a = 16.8913 (4), b = 5.6406 (1), c = 8.3591 (3) Å, β = 93.919 (3)°, V = 794.57 (4) Å3. The crystal structure of Rb2CaCu6(PO4)4O2 is similar to that of shchurovskyite and dmisokolovite and is based upon a heteropolyhedral open framework formed by polar layers of copper polyhedra linked via isolated PO4 tetrahedra. The presence of well-isolated 2D heteropolyhedral layers in the title compound suggests low-dimensional magnetic behavior which is masked, however, by the fierce competition between multiple ferromagnetic and antiferromagnetic exchange interactions. At TC = 25 K, Rb2CaCu6(PO4)4O2 reaches a magnetically ordered state with large residual magnetization.

One-Dimensional Behavior of Imidazolium Lead Iodide

Organic–inorganic hybrid crystals with lead halides as the inorganic moiety are remarkable, in that they show low-dimensional behavior despite their three-dimensional structures. Here, we report on...