Equilateral Triangular Lattice Research Articles

General selection of lattice shape in ductile-reinforced brittle structures for increased stiffness

Ductile reinforcements are used in brittle structures to increase the overall stiffness and strength. However, the optimum lattice shape geometry according to which the ductile reinforcements must be laid down, has not been generally established. In this work, a ductile isotropic lattice structure is constructed from a unit cell and the brittle matrix is “filled” in the empty spaces thereafter. In homogeneous stress conditions, the brittle part will experience the same applied remote stress as is felt by the ductile lattice. The statistical theory of brittle failure is invoked to bring home the point that the unit cell area of the lattice is arbitrary (for a given total area) and hence the lattice shape geometry does not influence the probability of survival of the structure. Therefore, a suitable geometry shape (triangle or square) maybe chosen to further enhance the overall stiffness at a given probability of survival. It has been concluded that an (equilateral) triangular lattice is better suited compared to a square lattice if the ductile reinforcement lattice bars have the same stiffness. The technique and conclusion is generally true for any choice of brittle and ductile materials. Also, the conclusion holds true for both compressive and tensile loading.

Directional scaling symmetry of high-symmetry two-dimensional lattices.

Two-dimensional lattices provide the arena for many physics problems of essential importance, a scale symmetry, which rarely exists as noticed by Galileo, in such lattices can help reveal the underlying physics. Here we report the discovery and proof of directional scaling symmetry for high symmetry 2D lattices, i.e., the square lattice, the equilateral triangular lattice and thus the honeycomb lattice, with aid of the function y = arcsin(sin(2πxn)), where the parameter x is either the platinum number or the silver number , which are related to the 12-fold and 8-fold quasiperiodic structures, respectively. The directions and scale factors for the symmetric scaling transformation are determined. The revealed scale symmetry may have a bearing on various physical problems modeled on 2D lattices, and the function adopted here can be used to generate quasiperiodic lattices with enumeration of lattice points. Our result is expected to initiate the search of directional scaling symmetry in more complicated geometries.

A model for grid cells where spatially correlated place cells compete for the grid map nodes

Grid cells in the medial entorhinal cortex (mEC) encode space in a particular way: their firing rate intensity forms an equilateral triangular lattice as the animal moves in the environment [1]. Models addressing the generation of grid fields fall into two major classes, both with important limitations [2]. Most models based on interference of oscillations are not robust to noise and suffer from the prerequisite of two independent and stable network oscillations with similar frequency. Models based on recurrent networks, where the grid pattern emerges as a stable state of the network, suffer from the topographic assumption and the need for dense recurrent connections between mEC cells, which are both not supported by experimental data. Here we present a novel firing rate model for the generation of grid fields with two very important features: i) grid fields are built on the spatial information provided by place cells, and ii) plasticity on the synapses between place cells and grid cells depend on the place cell's activity, and therefore depend on space. First, grid cells are driven by the synaptic input provided by the population of place cells. We therefore assume that place cells are the main source of spatial information reaching grid cells, which is in agreement with their development order [3]. Secondly, synapses between place cells and grid cells are subject to a learning process which depends both on pre-synaptic activity, therefore depending on spatial location, and on post-synaptic activity. Plasticity is triggered when the grid cell's activity is above a threshold value. Synaptic efficacies are discrete (w- <w0<w+<w++) and their change depends on pre-synaptic place cell activity. Four firing rate domains are considered and the synaptic change for each domain, from low to high, is: static, w0; potentiated w+; depressed, w-; potentiated, w++. When a synapse w+ is potentiated again its efficacy is raised to w++.

Microscopic Phase Separation in Triangular-Lattice Quantum Spin Magnet κ-(BEDT-TTF)2Cu2(CN)3 Probed by Muon Spin Relaxation

The ground state of the quantum spin system kappa-(BEDT-TTF)2Cu2(CN)3 in which antiferromagnetically-interacting S=1/2 spins are located on a nearly equilateral triangular lattice attracts considerable interest both from experimental and theoretical aspects, because a simple antiferromagnetic order may be inhibited because of the geometrical frustration and hence an exotic ground state is expected. Furthermore, recent two reports on the ground state of this system have made it further intriguing by showing completely controversial results; one indicates the gapless state and the other gapped. By utilizing microscopic probe of muSR, we have investigated its spin dynamics below 0.1 K, unveiling its microscopically phase separated ground state at zero field.

Electromagnetically superconducting phase of the vacuum in a strong magnetic field: Structure of superconductor and superfluid vortex lattices in the ground state

Recently it was shown that vacuum in a background of a strong enough magnetic field becomes an electromagnetic superconductor due to interplay between strong and electromagnetic forces. The superconducting ground state of the vacuum is associated with a spontaneous emergence of quark-antiquark condensates which carry quantum numbers of charged $\ensuremath{\rho}$ mesons. The $\ensuremath{\rho}$-meson condensate is an inhomogeneous structure made of the so-called $\ensuremath{\rho}$ vortices, which are parallel to the magnetic field axis. The condensation of the charged $\ensuremath{\rho}$ mesons induces a (much weaker) superfluid-like condensate with quantum numbers of the neutral ${\ensuremath{\rho}}^{(0)}$ mesons. In this paper we show that the vortices in the superconducting condensate organize themselves in an equilateral triangular lattice similarly to an ordinary type-II superconductor. We show that each of these superconductor vortices is accompanied by three superfluid vortices and three superfluid antivortices made of the neutral $\ensuremath{\rho}$ meson condensate. The superconductor vortex overlaps with one of the superfluid vortices. The superposition of the superconductor and superfluid vortex lattices has a honeycomb pattern.

La morfología de la línea de esmalte en los molares de los roedores muroideos es explicada mediante una red hexagonal

La línea de esmalte observada en la corona de los roedores muroideos tiene formas geométricas que coinciden exactamente con una red plana triangular equilateral. Esta conclusión ayuda a explicar el modelo dental de los muroideos con una nueva perspectiva denominada odontohexasimetria, fundamentada en las formas hexagonales observadas en la línea de esmalte. &#x0D; La morfología de la corona del diente de los mamíferos es heredada de padres a hijos. Esto infiere la existencia en cada espécimen de un autocontrol inteligente codificado en el genoma, que coordina la morfología de los dientes durante el desarrollo embriológico y su oclusión. &#x0D; La hipótesis del desarrollo el bucle de las áreas de mitosis (Ruiz Bustos, 2004a, 2005, 2006 y 2007) propone, que en los momentos iniciales de la morfogénesis, es observada una masa de células en mitosis (área de mitosis de basal). Esta es repetida en un bucle, hasta constituir todas las restantes áreas de mitosis del diente. Al final de la morfogénesis, el proceso de biomineralización transforma las áreas de mitosis en los tubérculos de la corona del diente.

Electron Spin Resonance in the Quasi-Two-Dimensional Triangular-Lattice Antiferromagnet Rb4Mn(MoO4)3

We have performed electron spin resonance (ESR) measurements on single crystals of the newly synthesized Mn layered oxide Rb 4 Mn(MoO 4 ) 3 , in which Mn 2+ ions with S = 5/2 form an equilateral triangular-lattice. The magnetic properties of this compound are reported to be well reproduced by theoretical calculations in a two-dimensional Heisenberg triangular-lattice antiferromagnet (2D HTAFM) with Ising anisotropy. The compound exhibits two successive phase transitions ( T N1 , T N2 , T N1 < T N2 ) at zero-field. We have successfully explained the observed frequency dependence of the ESR resonance fields and the reported magnetization curves parallel and perpendicular to the easy-axis and the c -axis, at T < T N1 by mean-field approximation and a classical Monte Carlo simulation using a spin Hamiltonian of the 2D HTAFM with Ising anisotropy. Then, we have evaluated the anisotropy and exchange constants accurately. Consequently, we have confirmed that Rb 4 Mn(MoO 4 ) 3 is a quasi-2D HTAFM with Ising aniso...

Two-dimensional magnetism and spin-size effect in theS = 1 triangularantiferromagnet NiGa2S4

The triangular antiferromagnet is one of the most fundamental systems of geometrically frustrated magnets.NiGa2S4 is alayered chalcogenide compound with an equilateral triangular lattice, and it is a prime candidate foran S = 1 triangular antiferromagnet. Here we focus on low temperature magnetism inNiGa2S4,where quasi-static spins develop a spin-wave-like mode without forming any long-range ordering.We have studied low temperature magnetism of both polycrystalline samples and single crystals ofNi1 − xAxGa2S4 (A = Mn, Fe, Co, and Zn). A scaling law with a single energy scale of the Weisstemperature is found as an impurity effect and a hydrostatic pressure effect,providing evidence that it is in-plane interactions in the two-dimensionalNiS2 plane that drive the critical slowing down to the viscous spin liquid state atT* = 8.5 K and the spin-wave-likeexcitations of NiGa2S4 that emerge below T ∼ 3 K. Furthermore, we find spin-size dependent impurity effects in the temperature dependence of the specific heat ofNi1 − xAxGa2S4. Even with a highimpurity content, Zn2 + (S = 0) andFe2 + (S = 2) substitutedsystems with weak XY anisotropy and integral spins retain the quadratic temperature dependence of the magnetic specific heatlike pure NiGa2S4. A spin glass-like phase, on the other hand, emerges at low temperatureswith the substitution of magnetic impurities with half-odd integer spins: IsingCo2 + (S = 3⁄2) and weak XY Mn2 + S = 5⁄2) spins. This indicates that an integer size of spins is important for stabilizing thetwo-dimensional spin-wave-like behavior, and the unconventional spin state ofNiGa2S4 at low temperatures is distinct from a canonical spin glass.

Local-field excitations in two-dimensional lattices of resonant atoms

We study excitations of the local field (locsitons) in nanoscale two-dimensional (2D) lattices of strongly interacting resonant atoms and various unusual effects associated with them. Locsitons in low-dimensional systems and the resulting spatial strata and more complex patterns on a scale of just a few atoms were predicted by us earlier [A. E. Kaplan and S. N. Volkov, Phys. Rev. Lett. 101, 133902 (2008)]. These effects present a radical departure from the classical Lorentz-Lorenz theory of the local field (LF), which assumes that the LF is virtually uniform on this scale. We demonstrate that the strata and patterns in the 2D lattices may be described as an interference of plane-wave locsitons, build an analytic model for such unbounded locsitons, and derive and analyze dispersion relations for the locsitons in an equilateral triangular lattice. We draw useful analogies between 1D and 2D locsitons but also show that the 2D case enables locsitons with the most diverse and unusual properties. Using the nearest-neighbor approximation, we find the locsiton frequency band for different mutual orientations of the lattice and the incident field. We demonstrate a formation of distinct vector locsiton patterns consisting of multiple vortices in the LF distribution and suggest a way to design finite 2D lattices that exhibit such patterns at certain frequencies. We illustrate the role of lattice defects in supporting localized locsitons and also demonstrate the existence of ``magic shapes,'' for which the LF suppression at the exact atomic resonance is canceled.

Electron spin resonance in a new triangular-lattice Mn layered oxide

We have performed electron spin resonance (ESR) measurements on single crystals of a new Mn layered oxide Rb4Mn(MoO4)3. The Mn2+ ions with S = 5/2 form an equilateral triangular-lattice. The magnetization process of this compound is well reproduced by a theoretical calculation in terms of an Ising-like Heisenberg antiferromagnet on a triangular-lattice. The frequency dependence of the ESR resonance fields along the easy-axis in the low-temperature phase is well explained by the calculations expected from the Ising-like Heisenberg model. We have evaluated the anisotropy and exchange interaction constants, and confirmed that this compound is a good example of the Ising-like Heisenberg antiferromagnet on the regular triangular-lattice.

Adsorption on Nanotubes Having Equilateral Triangular Geometry with First- and Second-Neighbor Interactions: Attractive First Neighbors

The recently published general method of studying adsorption on terraces and nanotubes is applied to adsorption on nanotubes in hollows or on-tops, which form a zigzag equilateral triangular adsorbate lattice. We consider adsorbate-adsorbate first- and second-neighbor interactions, with attractive first neighbors. In addition to empty and full coverage, there are three phases of occupational characteristics which are independent of the diameter of the nanotube. The low temperature energy phase diagram is the same for nanotubes of increasing diameters and, thus, is also valid in the infinite-diameter limit. As expected, this diagram is the same as that of the infinite-width limit of an equilateral triangular terrace, on which we have recently reported. The current study also includes temperature effects on the phases and the transition between phases.

A geometric approach to deploying robot swarms

We discuss the fundamental problems and practical issues underlying the deployment of a swarm of autonomous mobile robots that can potentially be used to build mobile robotic sensor networks. For the purpose, a geometric approach is proposed that allows robots to configure themselves into a two-dimensional plane with uniform spatial density. Particular emphasis is paid to the hole repair capability for dynamic network reconfiguration. Specifically, each robot interacts selectively with two neighboring robots so that three robots can converge onto each vertex of the equilateral triangle configuration. Based on the local interaction, the self-configuration algorithm is presented to enable a swarm of robots to form a communication network arranged in equilateral triangular lattices by shuffling the neighbors. Convergence of the algorithms is mathematically proved using Lyapunov theory. Moreover, it is verified that the self-reparation algorithm enables robot swarms to reconfigure themselves when holes exist in the network or new robots are added to the network. Through extensive simulations, we validate the feasibility of applying the proposed algorithms to self-configuring a network of mobile robotic sensors. We describe in detail the features of the algorithm, including self-organization, self-stabilization, and robustness, with the results of the simulation.

Monomer Adsorption on Equilateral Triangular Lattices with Attractive First-neighbor Interactions

We have recently studied a model of monomer adsorption on infinitely long equilateral triangular lattices with terraces of finite width M and nonperiodic boundaries. This study was restricted to the case of repulsive adsorbate-adsorbate first-neighbor interactions but included attractive, repulsive, and negligible second-neighbor interactions. The present work extends this study to the case of attractive first-neighbors, and the phases are determined, as before, with a confidence exceeding 10 significant figures. Phase diagrams are included for terrace widths M < or =11. Most of the occupational characteristics of the phases fit exact analytic expressions in M. The infinite-M limit of these expressions, combined with other analyses, provide the complete phase diagram for the infinite two-dimensional lattice. In addition to the empty and full coverage phases, there are three phases exhibiting stripe and cluster features that were not observed in the case of repulsive first-neighbors.

Monomer Adsorption on Equilateral Triangular Lattices with Repulsive First-Neighbor Interactions

A model of monomer adsorption on infinitely long, finite-width M equilateral triangular lattices with nonperiodic boundaries is presented. The study includes adsorbate-adsorbate first- and second-neighbor interactions with results obtained for repulsive first neighbors. The matrix method and numerical algorithms presented here allow determination of the occupational characteristics of the adsorption crystallization phases, which fit exact analytic expressions in the width M of the lattice. The limit as M approaches infinity provides the complete energy phase diagram for the infinite two-dimensional surface and recovers the results obtained by different methods that were often applied only in restricted energy regions of the phase diagram. The ordered phases are (2x1), (2x2), (3x1), (square root of 3xsquare root of 3) R30 degrees, and the complementary phases of (2x2) and (square root of 3xsquare root of 3) R30 degrees. Comparison is made with other theoretical studies and with experimental observations on adsorption systems consistent with the limitations of the model. In some cases, comparison with experimental data yields bounds on the interaction energies between adsorbates. On the basis of the model, suggestions are made on the manner in which to conduct relatively low temperature experiments to allow determination of most, if not all, of the interaction energies from the knowledge of the sequences of phases and the conditions prevailing at the transitions between phases.

Geometric frustration inherent to the trillium lattice, a sublattice of the B20 structure

We study the classical Heisenberg model on a recently identified three dimensional corner-shared equilateral triangular lattice, a magnetic sublattice to a large class of systems with the symmetry group P2$_1$3. Since the degree of geometric frustration of the nearest neighbor antiferromagnetic model on this lattice lies on the border between the pyrochlore (not ordered) and hexagonal (ordered) lattices, it is non-trivial to predict its ground state. Using a classical rotor model, we find an ordered ground state with wavevector $(\frac{2\pi}{3a_0},0,0)$ featuring 120$^o$ rotated spins on each triangle. However, a mean field approximation on this lattice fails to find an ordered ground state, finding instead a non-trivially degenerate ground state. As the mean field approach is known to agree with Monte Carlo on the pyrochlore lattice, the reasons for this discrepancy are discussed. We also discuss the possible relevance of our results to MnSi.

Superconductivity in Organic Compounds with Pseudo-Triangular Lattice

We study spin fluctuation (SF) mediated superconductivity (SC) in a half-filled square lattice Hubbard model with the transfer matrices -t between nearest neighbor sites and -t' between a half of next nearest neighbor sites neighboring along only one of the <1,1> directions, considering application of this model to organic kappa-(BEDT-TTF)2X compounds. Varying the t'/t value from 0 to 1, one can interpolate between a square and an equilateral triangular lattice, the latter giving frustration to antiferromagnetically (AF) coupled spin systems. Within the fluctuation exchange (FLEX) approximation, we calculate chi(q,omega), Tc and the SC order parameter for various model parameter values and find that both AF and SC are suppressed as one approaches the frustration geometry or |(t'/t)-1| \to 0. The SC phase, however, extends beyond the AF phase boundary fairly close to t'/t=1 for realistic U/t values. The order parameter is of x2-y2-type for t'/t<1 and of xy-type for t'/t>1.

EPR probing of bonding and spin localization of the doublet-quartet states in a spin-frustrated equilateral triangular lattice: Cu 3(O 2C 16H 23) 6·1.2 C 6H 12

We present high-frequency (34 and 95 GHz) EPR spectroscopic measurements of the magnetic parameters of both the ground state (spin-doublet) and the excited state (spin-quartet) of the model frustrated-spin triangular lattice of Cu 3(O 2C 16H 23) 6·1.2 C 6H 12, containing the Cu 3 6+ core. From 295 down to about 100 K, the EPR spectra from single crystals consist of a well-resolved triplet, but with the central component being overlapped by a single peak. At 4 K, the triplet is replaced by a singlet. The triplet is shown to arise from the quartet state, located at 324 K above the ground state. Its magnetic parameters are: D = –535 G, E = 0, g // = 2.209, g ⊥ = 2.057 with the parallel direction being the three-fold axis of the Cu 3 6+ core. The singlet is assigned to the S = 1/2 ground state, with g xx = 2.005, g yy = 2.050, and g zz = 2.282. Its hyperfine structure was that from a single Cu nucleus, with A zz = 157 G, and A xx = A yy < 60 G, demonstrating that in the doublet state the unpaired electron is localized on only one of the three Cu 2+ ions. We ascribe this localization to an antiferromagnetic exchange interaction between Cu 3 6+ cores, with zJ′ = –0.15 K. These results serve as a basis for detailed theoretical calculations of spin dynamics and electronic bonding in a frustrated triangular magnetic lattice. To cite this article: B. Cage et al., C. R. Chimie 6 (2003) 000–000.

Efficient Interleaving Schemes of Volume Holographic memory

Like the conventional digital storage systems, volume holographic memory can be deteriorated by burst errors due to its high-density storage characteristics. These burst errors are used byoptical defects such as scratches, dust particles, etc. and are two-dimensional in a data page. To deal with these errors, we introduce some concepts for describing them and propose efficient two- dimensional interleaving schemes. The schemes are two-dimensional lattices of an error-correction code word and have equilateral triangular and square structures. Using these structures, we can minimize the number of code words that are interleaved and improve the efficiency of the system. For large size burst errors, the efficient interleaving structure is an equilateral triangular lattice. However, for some small size burst errors, it is reduced to a square lattice.

Hyperfine Coupling and Magnetic Interaction of Layer-Structured Compound Ni(OD) 2 Studied by Solid-State High-Resolution Deuterium NMR

Magnetic interaction and hyperfine coupling were investigated for Ni(OD) 2 as a reference of our series of studies of layer-structured Cu(II) assembled complexes. In crystal of Ni(OD) 2 , Ni(II) ions with S = 1 are bridged by hydroxyl ions to form an equilateral planer triangular lattice. Hyperfine coupling constant of deuterium of hydroxyl ions was determined to be −0.17 MHz from temperature dependence of isotropic shift of a magic angle spinning deuterium NMR spectrum. Ferromagnetic exchange interaction J the layer was estimated to be +3 K from the deuterium NMR shift by assuming Heisenberg model for planer triangular lattice. Present results indicated that magic angle spinning deuterium NMR is also useful to determine magnetic local structure and magnetic interaction of transition metal complexes with electron spins greater than S = 1/2.

Circle Covering With a Margin

The problem of the thinnest (i.e., minimum density) covering of the plane with equal circles, solved by Kershner in 1939, can be interpreted as searching for the most economical distribution of transmission towers over a very large plane region, all towers having equal circular range, and collectively to provide reception at each point of the region. The problem of the thinnest 2-fold circle covering can be interpreted similarly, with the stronger requirement that the region should remain covered even if one of the towers stops functioning. Here we consider the intermediate variation, in which the region's coverage is to be maintained even if one of the towers experiences partial loss of power resulting in a certain decrease of its range radius. Specifically, we say that a covering of the plane with unit circular disks has margin µ (where 0 ≤ µ ≤ 1) if any one arbitrarily chosen disk can be replaced with a concentric disk of radius r = 1 - µ and the plane still remains covered. This concept provides a continuous transition from simple covering (margin 0) to 2-fold covering (margin 1). In this paper we determine the minimum density \(\vartheta _L^{1 + \mu } \) among all lattice circle coverings that have margin µ, and the lattice λµ that produces the minimum density. Somewhat surprisingly, the optimal lattice λµ behaves discontinuously and is non-unique at a pair of values of µ. As µ varies from 0 to 1, at one point λµ changes suddenly from an equilateral triangular lattice into a square lattice and then, at another point, into a certain rectangular lattice.