Triangular Order Research Articles

Discretization of singular systems and error estimation

Abstract This paper proposes a discretization technique for a descriptor differential system. The methodology used is both triangular first order hold discretization and zero order hold for the input function. Upper bounds for the error between the continuous and the discrete time solution are produced for both discretization methods and are shown to be better than any other existing method in the literature.

Mössbauer Study of Cubic Phase in the Mn-Sb System

One of the methods to produce novel materials is high temperature and high pressure treatment. In such a manner in the Mn Sb system the cubic modi cation of Mn3Sb could be produced [1, 2]. This compound has the L12-type of crystal structure (Cu3Au) with the lattice parameter a = 0.400 nm. The Mn atoms occupy 3c positions with the coordinates (0, 1/2, 1/2), the Sb atoms the positions 1a (0, 0, 0), Fig. 1. In normal conditions, the compound exists in a thermodynamically non-equilibrium state and decomposes at 420 K on the Mn2Sb + Mn phases. The magnetic measurements of the Mn3Sb were performed by the Faraday method in the eld 8.6 kOe and the temperature region from the liquid nitrogen to the temperature of phase decomposition. The derived value of speci c magnetization is very small, ≈ 1.0 A m/kg [2]. The neutron di raction analysis for the Mn3Sb was performed as well [3]. The data obtained were interpreted in the model of the ferrimagnetic ordering of Mn magnetic moments. The magnetic moments of Mn atoms are supposed to be antiparallel and have di erent values initially taking into account the similar local environment of Mn atoms in both tetragonal Mn2Sb and cubic Mn3Sb. However, it is possible to interpret the neutron di raction data in a di erent way, using the model of triangular magnetic ordering of Mn magnetic moments. In contrast to previous one, in this model the Mn atoms are considered to have equal magnetic moments ordered in the (111) plane and forming an equilateral triangle with each moment pointing to the center of the triangle. The similar model was used in [4] to describe the magnetic ordering of Mn in Mn3Pt with the same L12 type of structure below 400 K. Thus the model of the magnetic ordering of Mn atoms in the cubic Mn3Sb is ambiguous. Therefore the purpose of this study was to get Mossbauer information on the number of magnetically nonequivalent states of Mn atoms and to clarify a type of the magnetic ordering in Mn3Sb.

Excited states of quasi-one-dimensional hexagonal quantum antiferromagnets

We investigate the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the noncollinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compare our results for the longitudinal energy gaps at the magnetic wave vectors with the experimental results for several antiferromagnetic compounds with both integer and noninteger spin quantum numbers, and we find good agreement after the higher-order contributions are included in our calculations.

Rhombic preordering on a square substrate.

A competition of incommensurate symmetries occurs whenever a system is forced to conform to an ordering that is different from the intrinsically preferred structure of the system itself. As a model system of such a competition, we study the rivalry between the triangular ordering of hard disks and the square symmetry induced by a periodic square substrate. By using density functional theory as well as MonteCarlo computer simulations, we determine the full phase behavior for the case of one particle per minimum. We observe a rhombic preordering structure preceding the hexagonal solid as a direct consequence of the competing symmetries. Furthermore, the square-rhombic transition is reentrant with increasing substrate interaction. Our predictions can be verified in experiments of colloids in laser fields.

Effect of high pressure on the crystal structure, magnetic, and vibrational properties of multiferroic RbFe(MoO4)2

The structural, magnetic, and vibrational properties of RbFe(MoO${}_{4}$)${}_{2}$ have been studied by means of x-ray and neutron powder diffraction, magnetic susceptibility measurements, and Raman spectroscopy at pressures up to 10 GPa. The gradual structural phase transition from the initial trigonal $P\overline{3}m1$ phase to the monoclinic $C$2$/c$ phase via the intermediate $P\overline{3}$ phase was observed at $P\ensuremath{\sim}1$ GPa. The triangular antiferromagnetic order with a propagation vector $q$ $=$ (1/3, 1/3, ${q}_{\mathrm{z}}$) in the suppressed trigonal phase remains incommensurate, and the ${q}_{z}$ value increases from 0.45 to 0.48 in the pressure range 0--2 GPa at $T=2$ K. The pressure coefficient of the N\'eel temperature is equal to $(1/{T}_{N})d{T}_{N}/dP=0.09\phantom{\rule{0.16em}{0ex}}{\mathrm{GPa}}^{\ensuremath{-}1}$. No evidence of the formation of the long-range magnetic order in the monoclinic phase was found. The pressure dependencies of the vibrational modes for the trigonal and monoclinic phases were obtained.

A Posteriori Error Analysis for the Postprocessed MITC Plate Elements

We present an a posteriori error analysis for the postprocessed mixed interpolated tensorial components (MITC) finite element methods approximating the solution of the Reissner--Mindlin plate bending problem. We first prove a new a priori error estimate by utilizing the results for the postprocessing method. Then we introduce a residual-based a posteriori error estimator and prove its reliability and efficiency with techniques similar to those recently used for the original MITC elements. The basic features of the error estimator in adaptive computations are illustrated by a series of numerical tests for a triangular low order element.

Epitaxially Constrained Hexagonal Ferroelectricity and Canted Triangular Spin Order in LuFeO3 Thin Films

Epitaxially Constrained Hexagonal Ferroelectricity and Canted Triangular Spin Order in LuFeO₃ Thin Films

Kinetic control of structural and magnetic states in LuBaCo4O7

The $R$BaCo${}_{4}$O${}_{7}$ (R $=$ Ca, Y, Tb, Ho, Tm, Yb, Lu) compounds provide a novel topology for studying the competition between triangular geometry and magnetic order. Here, we report the structural and magnetic behavior of the Lu member of this series via neutron and synchrotron x-ray diffraction, magnetization, and resistivity measurements. We determined sequential phase transitions and a strong competition between a stable and a metastable low-temperature state that critically depends on controlled cooling rates and the associated heat removal kinetics. No evidence for long-range ordered magnetism was detected by neutron diffraction at any temperature. However, very slow spin dynamics are evidenced by time-dependent neutron diffraction measurements and can be explained by several competing magnetic phases with incommensurate short-range correlations coexisting in this material.

“Volume-point” heat conduction constructal optimization based on entransy dissipation rate minimization with three-dimensional cylindrical element and rectangular and triangular elements on microscale and nanoscale

Based on constructal theory, the constructs of three “volume-point” heat conduction models with three-dimensional cylindrical element and rectangular and triangular elements on microscale and nanoscale are optimized by taking minimum entransy dissipation rate as optimization objective. The optimal constructs of the three “volume-point” heat conduction models with minimum dimensionless equivalent thermal resistance are obtained. The results show that the optimal constructs of the three-dimensional cylindrical assembly based on the minimizations of dimensionless equivalent thermal resistance and dimensionless maximum thermal resistance are different, which is obviously different from the comparison between those of the corresponding two-dimensional rectangular assembly based on the minimizations of these two objectives. The optimal constructs based on rectangular and triangular elements on microscale and nanoscale when the size effect takes effect are obviously different from those when the size effect does not take effect. Because the thermal current density in the high conductivity channel of the rectangular and triangular second order assemblies are not linear with the length, the optimal constructs of these assemblies based on the minimization of entransy dissipation rate are different from those based on the minimization of maximum temperature difference. The dimensionless equivalent thermal resistance defined based on entransy dissipation rate reflects the average heat transfer performance of the construct. The studies on “volume-point” heat conduction constructal problems at three-dimensional conditions and microscale and nanoscale by taking minimum entransy dissipation rate as optimization objective extend the application range of the entransy dissipation extremum principle.

Vortex states and the phase diagram of a multiple-component Ginzburg-Landau theory with competing repulsive and attractive vortex interactions

We investigate the behavior of vortices of multicomponent superconductivity with intercomponent coupling, realized in ${\mathrm{MgB}}_{2}$ and Fe-based superconductors, within the framework of Ginzburg-Landau (GL) theory in terms of numerical calculations of the time-dependent GL equations and the variational method. It is revealed that close to the critical point of the composite system the intercomponent coupling makes the system behave as a single component superconductivity. However, when the bare mean-field critical points of the two components coincide with each other, and furthermore the interband coupling disappears at the same temperature, interesting phenomena occur as follows. Vortices interact attractively at large separation and repulsively at short distance in certain parameter space. Because of the nonmonotonic interaction profile, phase separations between vortex clusters of triangular order and the Meissner state take place, which indicates a first-order phase transition associated with the penetration of the magnetic field into a superconductor sample. Phase diagrams of vortex states are then constructed with the associated magnetization curve. It is found that the system shares some features of both type-I and -II superconductors.

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations — the bivariate Coxian representation — for generalized Erlang distributions.

Higher Class Groups of Locally Triangular Orders over Number Fields

In this paper, we study the K-theory of triangular rings. As an application, we show that for a locally triangular order Λ, the p-torsion in the higher class group Cl2n(Λ) can only occur for primes p which lie under the prime ideals ℘ of [Formula: see text], at which Λ is not maximal.

Al nanocluster arrays onSi(111)−7×7surfaces: Formation process and interactions among clusters

By means of variable-temperature scanning tunneling microscopy, we have investigated systematically the formation process of Al magic cluster arrays on the $\mathrm{Si}(111)\text{\ensuremath{-}}7\ifmmode\times\else\texttimes\fi{}7$ surface at high temperature in situ. It was found that the magic clusters form only when the Al coverage is over a critical value, $\ensuremath{\sim}0.08\ifmmode\pm\else\textpm\fi{}0.015$ ML. These clusters occupy preferentially on the faulted-half-unit cells of the $\mathrm{Si}(111)\text{\ensuremath{-}}7\ifmmode\times\else\texttimes\fi{}7$ surface, but this preference is coverage dependent. By analyzing systematically the spatial distribution of magic clusters below the saturation coverage, an attractive interaction between clusters was found, which prevents a triangular ordering of the nanocluster array.

Synthesis and characterization of copper fiber reinforced Zr 41.2Ti 13.8Cu 12.5Ni 10.0Be 22.5 bulk metallic glass

Highly conductive Cu fibers were introduced into Zr 41.2Ti 13.8Cu 12.5Ni 10Be 22.5 (Vitreloy1) Bulk Metallic Glass (BMG) of poor conductivity to produce a composite with a thermal anisotropy. The composites with circular and square cross-sections with the volume fraction ranging from 7 to 60% were processed successfully by pressure-gravity infiltration. Differential Scanning Calorimetry (DSC) showed that the BMG matrix remained predominantly amorphous after adding up to 60 vol.% of Cu fibers. The limited crystallization at the Cu/Vitreloy1 interface was observed under optical microscope. This crystallization is attributed to heterogeneous nucleation starting from the Cu/Vitreloy1 interface followed by the growth in the region with higher Cu concentration around the copper fibers. A combination of two-dimensional triangular and square ordering of the fibers was observed in the composite samples. X-ray diffraction patterns of the composites showed the peaks from the reinforced fibers and crystals around the Cu/Vitreloy1 interface superimposed on the broad diffuse maxima from the amorphous phase. Best processing parameters in terms of processing time and temperature were determined for the composite reinforced with 60 vol.% fibers.

Polarized neutron scattering studies of chiral criticality, and new universality classes of phase transitions

Using a novel polarised neutron scattering technique, the critical exponents for the spin chirality and chiral susceptibility are determined for the triangular lattice antiferromagnet (TLA) CsMnBr 3 in the ranges of reduced temperature τ>10 −3 and τ>7×10 −3, respectively. Their values, β C=0.44(2) and γ C=0.85(3), together with the scaling relation α + 2 β C + γ C = 2.13 ( 9 ) , including the critical exponent where α for the specific heat, prove that the spin-ordering transition belongs to the XY chiral universality class. In the case of helimagnet Ho, it is found that β C - 2 β = 0.14 ( 4 ) , where β is the staggered magnetisation exponent. The scaling relation α + 2 β + γ = 2 could be fulfilled with a reasonable α=0.23(4), although for the chiral critical exponents β C =0.90(2) and γ C =0.69(5) one needs α=−0.49(5) in contradiction with any experimental data. As the scaling relation always holds, we assume that the spin-ordering transition in Ho is of the first order. In the quantum antiferromagnet CsCuCl 3, a triangular spin order coexists with a long-period Dzyaloshinskii helix. The Dzyaloshinskii axial vector should remove the helix chiral degeneracy, which has not been observed in reality. The critical exponent β=0.22(2) is found to be in agreement with the XY chiral scenario for a TLA. Chiral scattering above T N is very weak, probably being masked by zero-point quantum fluctuations. A modulation of the crystal structure with the periodicity of the helix is observed, indicating strong coupling of the Dzyaloshinskii–Moriya interaction with the lattice.

Frequency-Dependent Shape Changes of Colloidal Clusters under Transverse Electric Field

We have studied clustering of colloidal particles under the influence of an ac electric field as a function of frequency. The field was applied in a direction perpendicular to the confining walls. Two regimes are observed, a low frequency regime where the clusters are isotropic with a local triangular order, as reported earlier in the literature, and a new high-frequency regime where the clusters are highly elongated (anisotropic) with no local order. The crossover from one regime to the other occurs at a critical frequency, f(c). The threshold field for the cluster formation, E(th), increases with frequency in both the regimes. An increase in the particle size leads to a reduction in both E(th) and f(c). We present evidence to show that the elongated structures seen at high frequency are related to the field inhomogeneities at imperfections on the conducting surface. We also propose a possible mechanism based on hydrodynamic flow considerations to explain the formation of these clusters.

A quantum spin liquid in a two-layer triangular antiferromagnet

The possibility of implementing a quantum-spin-liquid-type state in a two-layer triangular spin-1/2 antiferromagnet at T = 0 is investigated. The ratio of intra-to interlayer exchange constants (j) is found under which a transition from the classical state with 120° triangular order to a quantum state with zero magnetization per site occurs; in this case, the spins of adjacent layers form singlets that are separated from triplet excitations by an energy gap. Compared with an analogous system with the square lattice, the range of j in which the classical ordered state is realized turns out to be an order of magnitude smaller due to the effects of frustration; in this case, the behavior of thermodynamic quantities is analogous, on the whole, to that in two-layer square lattices; a difference manifests itself in the behavior of the gap in the spectrum of quasiparticles in an external magnetic field h. For small fields h, a j-h phase diagram is constructed that determines the domains in which the 120° and the singlet phases exist. It is established that, in the neighborhood of the second-order phase transition, the contribution, to the thermodynamic quantities, of longitudinal spin fluctuations, which are disregarded in the spin-wave description, is comparable to the contribution of transverse fluctuations.

Non-classical nucleation in supercooled nickel

The dynamics of homogeneous nucleation and growth of crystalline nickel from the supercooled melt is examined during rapid quenching using molecular dynamics and a modified embedded atom method potential. The character of the critical nuclei of the crystallization transition is examined using common neighbour analysis and visualization. At nucleation, the saddle point droplet consists of randomly stacked planar structures with an in-plane triangular order. These results support previous theories predicting that in the presence of long-range forces the critical nucleus is non-classical.

Statistical mechanics of nucleation in solids: a kinetics driven morphological transition

We examine in detail, through extensive molecular dynamics simulations, the process of nucleation of a triangular phase in a model two-dimensional square crystal consisting of particles interacting via a pair and three-body potentials. We find that at high temperatures, the critical nucleus of the triangular phase is isotropic with local triangular order. At low temperatures, the critical nucleus consists of a twinned bicrystal which is anisotropic and breaks the symmetry of the underlying square phase. We show that this morphological transition is caused by the dynamics of non-elastic degrees of freedom, viz. interfacial defect fields. This transition is similar to that between ‘ferrite’ and ‘martensite’ as observed in real materials when a solid such as iron is cooled across a structural transition. Based on our simulation results, we arrive at a single theoretical formalism which qualitatively describes the transformation to both a ferrite and a martensite.

The structure of frozen phases in slit nanopores: A grand canonical Monte Carlo study

Freezing of soft spheres in slit nanopores is investigated using Grand canonical Monte Carlo simulations. The pores are in equilibrium with a liquid located close to the liquid–solid coexistence region in the bulk Lennard-Jones phase diagram. In addition to layering, the confined fluid is found to possess in-plane order, leading to the formation of frozen phases which give rise to a sequence of solid–solid transformations as the pore width is varied. Transformations between n layered triangular to n+1 layered square lattices and between n layered square to triangular lattices, are observed for n=1, 2, 3, and 4. The transition from triangular to square lattices occurs via an intermediate buckled phase which is characterized by increased out-of-plane motion, while maintaining in-plane triangular order. Buckling was found to decrease with increasing number of layers. The transition between square to triangular lattices at a fixed number of layers is accompanied by a lowering of the solvation force, resulting in a doublet in the solvation force maxima. Influence of fluid–wall interactions on the nature of the frozen phases are studied by comparing the structures formed with a 10-4-3 and 10-4 fluid–wall potential. The solid structures are classified based on their closest 3D counterparts.