Configurations In Lattices Research Articles

Quantitative multiple mixing

We develop a method for providing quantitative estimates for higher order correlations of group actions. In particular, we establish effective mixing of all orders for actions of semisimple Lie groups as well as semisimple S -algebraic groups and semisimple adele groups. As an application, we deduce existence of approximate configurations in lattices of semisimple groups.

Modelling of Ring Geometry from von Neumann’s Point of View

The idea to consider different unions of points, lines, planes, etc., is rather old. Many important configurations of such kinds are geometric (or matroidal) lattices. In this work, we study Desargues, Pappus, and Pasch configurations in D-semimodular lattices.

The influence of individual lattice defects on the domain structure in magnetic antidot lattices

We numerically and experimentally investigate the influence of single defects consisting of a missing antidot on the spin configurations in rectangular permalloy antidot lattices. The introduction of such lattice defects leads to the nucleation of complex domain structures after the decay of a saturating magnetic field. Micromagnetic simulations yield four typical domain configurations around the defect having distinct energy densities. The existence of the four spin configurations is confirmed by magnetic force microscopy on antidot lattices containing individual defects.

Magnetic dipole configurations in honeycomb lattices: order and disorder

Dipolar spin ice has attracted much attention because of its intriguing ground state ordering and elementary excitation properties. We present experimental realizations of magnetic dipolar spin ice on periodic lattices with honeycomb symmetry. We have analyzed in particular the evolution and distribution of excitations with magnetic charges ±3 per vertex as a function of magnetic field and the distance b between the dipoles ranging from b = 0.4 to 1.7 μm. In all the dipole patterns investigated, we observe a surprisingly high abundance of ±3 magnetic charges at coercivity in the descending and ascending branches of the magnetic hysteresis. At the same time, these ±3 vertices form a charge ordered state with large domains, resembling an ionic crystal. Monte Carlo simulations of the magnetization reversal confirm in the framework of a macrospin model an enhanced abundance of ±3 magnetic charges at the coercive field. But much better agreement is achieved by taking into account the micromagnetic reversal mechanism, which proceeds via nucleation and domain wall propagation for dipoles aligned with the field and via coherent rotation for all others.

Existence and stability of multisite breathers in honeycomb and hexagonal lattices

We study the existence and stability of multisite discrete breathers in two prototypical non-square Klein–Gordon lattices, namely a honeycomb and a hexagonal one. In the honeycomb case we consider six-site configurations and find that for soft potential and positive coupling the out-of-phase breather configuration and the charge-two vortex breather are linearly stable, while the in-phase and charge-one vortex states are unstable. In the hexagonal lattice, we first consider three-site configurations. In the case of soft potential and positive coupling, the in-phase configuration is unstable and the charge-one vortex is linearly stable. The out-of-phase configuration here is found to always be linearly unstable. We then turn to six-site configurations in the hexagonal lattice. The stability results in this case are the same as in the six-site configurations in the honeycomb lattice. For all configurations in both lattices, the stability results are reversed in the setting of either hard potential or negative coupling. The study is complemented by numerical simulations which are in very good agreement with the theoretical predictions. Since neither the form of the on-site potential nor the sign of the coupling parameter involved have been prescribed, this description can accommodate inverse-dispersive systems (e.g. supporting backward waves) such as transverse dust-lattice oscillations in dusty plasma (Debye) crystals or analogous modes in molecular chains.

On the plasticity of short-range ordered and long-range ordered alloys

The plastic behaviour of short-range ordered and long-range ordered alloys shows remarkable similarities but also differences, e.g. in the temperature dependence of the critical shear stress, in the work hardening of single and polycrystals, in the slip characteristics and in the development of dislocation microstructures. These features are compared and illustrated by experimental data on concentrated fcc alloys showing tendencies to short-range ordering (Cu–Al) and on bcc intermetallic (long-range ordered) compounds (CuZn, Fe 3Al, Fe 3Si). In spite of strong differences of the developed dislocation microstructures and fine structures of slip distribution on the micro scale due to different dislocation configurations in fcc and bcc lattices, quite similar patterns of local shear occur on larger scales, e.g. slip band evolution and instabilities of slip in certain temperature regimes, indicating more general reasons. Even the frequently discussed ‘stress anomaly’ in the temperature dependence of the critical shear stress shows striking similarities in short- and long-range ordered alloys suggesting that at least part of its causes may lie in the counteracting processes of a kind of obstacle destruction mechanism by gliding dislocations and a restoration process of diffusional origin.

Long-lived localized field configurations in small lattices: application to oscillons

Long-lived localized field configurations such as breathers, oscillons, or more complex objects naturally arise in the context of a wide range of nonlinear models in different numbers of spatial dimensions. We present a numerical method, which we call the adiabatic damping method, designed to study such configurations in small lattices. Using three-dimensional oscillons in straight phi(4) models as an example, we show that the method accurately (to one part in 10(5) or better) reproduces results obtained with static or dynamically expanding lattices, dramatically cutting down in integration time. We further present results for two-dimensional oscillons, whose lifetimes would be prohibitively long to study with conventional methods.

Magnetic multi-valued recording by new magnetic configurations in 2D and 3D artificial lattices

New magnetic configurations in 2D and 3D artificial lattices are studied by solving the non-linear equation based on the classical Heisenberg models for 2D and 3D artificial lattices. The results indicate a strong possibility of use as an ultra-high density magnetic recording media achieved by magnetic multi-valued (MMV) recording. The MMV states are based on the magnetic anisotropic configurations in 2D and 3D artificial lattices. In this paper, the magnetic properties of these artificial lattices and the density of MMV recording are discussed.

Spin-Reorientation Transition in Superlattices by Large Scale Dynamic Simulation

New magnetic configurations in 2D and 3D artificial lattices are studied by solving the non-linear equation based on the classical two- or three- dimensional Heisenberg models with full long range dipolar interaction. The results indicate strong possibility to be used as an ultra-high density magnetic recording media achieved by magnatic multi-valued(MMV) recoding. The magnetic properties and the density of MMV recoding are discussed based on the magnetic anisotropic configurations in 2D and 3D artificial lattices.

Application of nested CPA theory to hydrogenated amorphous silicon

The electronic structure of hydrogenated amorphous Silicon is studied as a function of hydrogen concentration, and of local configurations. The theory used for the calculations is based on a Cluster-Bethe lattice method, using a tight-binding Hamiltonian. The alloy disorder is treated with a theory, based on the Coherent Potential Approximation (CPA), which is able to introduce effects of different local configurations in the Bethe lattices through nested self-consistent solutions.

Vacancy-interstitial recombination coefficients in radiation-induced growth models

The jump method for the calculation of the intrinsic recombination coefficient α∗ for vacancies and interstitials, as used in the analysis of radiation-induced growth in metals, is re-examined. A straightforward procedure is presented which can accommodate simple or complex defect configurations in isotropic or anisotropic lattices. The method is applied to two recent analyses of radiation growth in HCP zirconium. It is shown that differences in the values of α∗ are not, as previously suggested, associated with the interstitial configuration, but with an inconsistent definition of the atomic jump frequencies. The removal of the inconsistency is such that it would improve the agreement between calculated and measured growth rates in one of the analyses.

Configurations in Crystals, Rod Lattices, and Lamellar Systems

For three-dimensional ordered media of periodic and partially periodic structure, homotopy theory allows the existence of nonsingular stable distortion fields, which are finite in size and thus resemble extended particles. It is shown that because of integrability conditions homotopy arguments are not adequate for these systems. By methods of differential topology it is proved that in three dimensions topological particles of this type are unstable, but that there are analogous fields in higher dimensions that cannot decay to a uniform ground state.

Microscopic formulation of a lattice-defect model

A microscopic construction method of lattice defects which can be represented by dislocation configurations in simple-cubic lattices and for interaction by central forces is given. In contrast to standard theories, where dislocations are introduced via topological operations (Burgers circuit), the author starts from a microscopic ad hoc Hamiltonian which is suitable only for central-force interacting systems. The various sectors of this microscopic Hamiltonian are associated with dislocation configurations. A detailed discussion of the stability and symmetry properties of the microscopic Hamiltonian is given. Possible extensions of the theory to non-central-force interacting systems are pointed out.

Configurations of close-packed clusters of substitutional point defects in crystals

Abstract A theoretical study is described of the possible crystallographic configurations of small groups of either vacancies or solutes in which each point defect has at least one other point defect in a nearest-neighbour position. The number of cluster configurations that may arise for a given number of point defects depends critically on the crystal structure and in some cases a large number of types is possible each with a group of crystallographically equivalent variants. General procedures for describing and classifying these configurations are discussed. Clusters of three and four defects in all possible single lattice structures and in some important double lattice structures are enumerated. The relationships which exist between configurations in different lattices are emphasized and possible extensions and applications of the analysis are discussed. In particular the results provide a basis for a study of the nucleation of extended crystal defects using computer simulation techniques.

Magnetic ground state spin configurations in face-centred cubic and Cu3Au-type crystals

Assuming various interaction pairs, the generalizedLuttinger-Tisza method (GLTM) is used to determine the ground state spin configurations in face-centred cubic and Cu3Au-type lattices. The configurations assigned as obtained in terms of magnetic interactions include, with a few exceptions, every magnetic structure observed in the above types of crystal lattices and permit for systems with various ground state spin configurations to establish unambiguously the predominant interactions and to estimate their sign as well as their relative magnitudes. Finally some experimental observations are discussed in the light of the results obtained by the GLT method.

Excluded Volume Effect in Polymer Chains. I

The number of allowed configurations of a polymer chain is considered on the basis of a random walk on an arbitrary ``regular'' lattice. Upper and lower bounds for the number of nonoverlapping configurations in various lattices have been derived by means of a recursion formula method. The probability density function and its moments for the ``head-to-tail'' distance for short-range nonoverlapping chains are shown to the calculable by the use of a generating function. The order of the logarithm of the number of nonsuperposable ring polymer chains has been shown to be directly proportional to the number of segments composing the ring.