Number Of Sublattices Research Articles

Set based framework for Gibbs energy minimization

A new unified approach to Gibbs energy minimization is introduced. While it has only been tested on binary and ternary systems so far, it has a built in capability of handling arbitrary multicomponent multiphase systems with any number of sublattices, miscibility gaps, order–disorder transitions, and magnetic contributions. This new unified AMPL set-based Gibbs energy description optimizes the data representation and makes it possible to subject the task of phase diagram calculation to numerous existing general purpose optimization strategies as well as custom-made solvers. The approach is tested on a variety of systems, including Co–Mo, Al–Pt and Ca–Li–Na, all known to be computationally challenging for other approaches. In most of the tested systems, the AMPL code reproduces phase diagrams obtained via Thermo-Calc. In other systems, in-depth comparison of results suggests that in prior work a sub-optimal equilibrium might have been identified as a global one, and re-evaluation of previously published diagrams and databases might be necessary.

Optical activity of imperfect 1D superlattices with vacancies in impurity layers

The concentration dependence of the specific rotation angle of a gyrotropic photonic crystal (imperfect 1D superlattice with an arbitrary number of sublattices one of which (potassium dithionate K2S2O6) is optically active) is numerically simulated. Specific gyrotropic features of the photonic crystal related to the random substitution of layers of a model molecular crystal with vacancies for the layers of the K2S2O6 sublattice are revealed using microscopic analysis of the specific rotation angle of the optical polarization in the molecular crystal with a primitive lattice in the exciton spectral range.

Finite Size Effects in Magnetic and Optical Properties of Antiferromagnetic NiO Nanoparticles

We report systematic investigations on structural, magnetic and optical properties of NiO nanoparticles prepared by mechanical alloying. As-milled powders exhibit face centred cubic structure, but average particle size decreases and effective strain increases for the initial periods of milling. Lattice volume increases monotonically with a reduction in particle size. Antiferromagnetic NiO particles exhibit significant room temperature (RT) ferromagnetism with modest moment and coercivity. A maximum moment of 0.0147 μB/f.u at 12 kOe applied field and a coercivity of 160 Oe were obtained for 30 h milled NiO powder. Exchange bias decreases linearly with a decrease in NiO particle size. Thermo-magnetization data reveal the presence of mixed magnetic phases in milled powders and shifts magnetic phase transition towards high temperature with increasing milling. Annealing of milled NiO powder and photoluminescence studies show a large reduction in RT magnetic moment and blue-shifting of band edge emission peak. The observed properties are discussed on the basis of finite size effect, defect density, oxidation/reduction of Ni, increase in number of sublattices, uncompensated spins from surface to particle core, and interaction between uncompensated surfaces and particle core with lattice expansion.

On choosing a reference surface for a two-sublattice model

Hillert’s choice of a reference surface for the ( A , B ) a ( C , D ) c solution has vital advantages: the corresponding contribution to the Gibbs energy of a phase is straightforward to calculate; the surface is geometrically simple; it is continuously differentiable; the method can easily be generalized for multicomponent phases with any number of sublattices regardless of whether a component resides in one sublattice only or may inhabit several sublattices. Elegance, convenience, simplicity and naturalness of his construction may mask the fact that the choice of the reference frame is not unique. From the utilitarian angle, a discussion of the non-uniqueness undertaken here may be of little use, because Hillert’s method will forever be used for building the reference surfaces for phases with two or more sublattices. A sheer curiosity rather than pragmatism motivated revisiting a problem of defining the reference surface and considering Tikhonov regularization and the least-norm criterion as unconventional ways of selecting it. Mathematical particularities and peculiarities associated with these two approaches are discussed in the present work.

Crystals with sublattices and their symmetry in multidimensional crystal spaces

The symmetry of complex crystals composed of sublattices belonging to different Bravais typeshas been investigated. It is shown that description of this symmetry necessitates introduction of a multidimensional crystal space R 3k decomposing into the direct sum of k 3D orthogonal subspaces (S = 1, 2, …, k), where k is the number of sublattices. The symmetry of the subspaces R S 3 of the sublattices is a set of their crystallographic groups. The bases of subspaces R S 3 are related by the transformation of translation compatibility, at which the scale changes. A method for studying the origin of the spectra of elementary excitations in crystals is presented that is based on the analysis of the sublattice symmetry. Complex lattices composed of cubic sublattices are considered as an illustration.

Sublattices of product spaces: Hulls, representations and counting

The Cartesian product of lattices is a lattice, called a product space, with componentwise meet and join operations. A sublattice of a lattice L is a subset closed for the join and meet operations of L. The sublattice hull L Q of a subset Q of a lattice is the smallest sublattice containing Q. We consider two types of representations of sublattices and sublattice hulls in product spaces: representation by projections and representation with proper boundary epigraphs. We give sufficient conditions, on the dimension of the product space and/or on the sublattice hull of a subset Q, for L Q to be entirely defined by the sublattice hulls of the two-dimensional projections of Q. This extends results of Topkis (1978) and of Veinott [Representation of general and polyhedral subsemilattices and sublattices of product spaces, Linear Algebra Appl. 114/115 (1989) 681–704]. We give similar sufficient conditions for the sublattice hull L Q to be representable using the epigraphs of certain isotone (i.e., nondecreasing) functions defined on the one-dimensional projections of Q. This also extends results of Topkis and Veinott. Using this representation we show that L Q is convex when Q is a convex subset in a vector lattice (Riesz space), and is a polyhedron when Q is a polyhedron in R n . We consider in greater detail the case of a finite product of finite chains (i.e., totally ordered sets). We use the representation with proper boundary epigraphs and provide upper and lower bounds on the number of sublattices, giving a partial answer to a problem posed by Birkhoff in 1937. These bounds are close to each other in a logarithmic sense. We define a corner representation of isotone functions and use it in conjunction with the representation with proper boundary epigraphs to define an encoding of sublattices. We show that this encoding is optimal (up to a constant factor) in terms of memory space. We also consider the sublattice hull membership problem of deciding whether a given point is in the sublattice hull L Q of a given subset Q. We present a good characterization and a polynomial time algorithm for this sublattice hull membership problem. We construct in polynomial time a data structure for the representation with proper boundary epigraphs, such that sublattice hull membership queries may be answered in time logarithmic in the size | Q | of the given subset.

On the Toeplitz Determinant for Spin-Pair Correlation Function of 2-D Ising Model

In the present paper, we consider the correlation function of two spins i and f on a two-dimensional Ising model with non-crossing interactions. We assume that the two spins are separated by n lattice spacings. By extending a theory due to Vdovi­ chenko,1) giving the spontaneous magnetization of the Ising model on the square lattice, it was shown generally that the spin-pair correlation function is expressed by a 2n X 2n determinant. 2 ) If the lattice consists of a multiple number of sub lattices and t:he two spins i and f are on the same sublattice along an axis, the expression of the correlation function takes the form of an nlv X nlv block Toeplitz determinant with 2v X 2 v blocks, where v is the number of different sub lattices on the axis. If the lattice is monatomic on the axis along which the correlation is calculated, we have an n X n block Toeplitz determinant with 2 X 2 blocks. 3 ) It is known for simple lattices in which the lattice is monatomic on the axis and the interactions have a reflection symmetry with respect to the axis or some inversion symmetry, the 2 X 2 blocks are diagonal and the spin-pair correlation function is expressed by an n X n ordinary Toeplitz determinant. 4 )-7) In the present paper, we assume that the two spins i and f are on the same sublattice along an axis and that the interactions have either a reflection symmetry with respect to the axis or such an inversion symmetry that the midpoint between every pair of lattice sites on the axis is a centre of symmetry. We then show that the correlation function is expressed by an n X n determinant, instead of a 2n X 2n determinant. If there are v different sublattices on the axis, it takes the form of an nlv X nlv block Toeplitz determinant with v X v blocks. If the lattice is monatomic on the axis along which the correlation is calculated, we have an n X n ordinary Toeplitz determinant. In the diagram approach to the two-dimensional Ising model, the partition func­ tion and the pair correlation function are expressed in terms of diagrams of loops and chains, which consist of bonds, each connecting a pair of nearest-neighbour lattice sites. 8HO ) We put directions to the loops and chains, and then a bond with direction is called a step. We express the set of all the pairs of nearest-neighbour lattice sites with a direction by B, and then a step is an element in B_ The interaction between spins at both ends of a step labeled by /3 is denoted by fp. The number of elements

Free Energy for Layered Free Fermion Models

The exact expression of the free energy is obtained for the free fermion model on a lattice with a multiple number of sublattices. Compact exact expressions are presented, in particular, for two-sublattice systems. Vdovichenko's method is used in the derivation. Some remarks are given to justify the Vdovichenko's method for the free fermion model.

Icosahedral density waves

Abstract A theoretical model is proposed for the formation of quasicrystalline alloys. The icosahedral phase is shown to result from a reconstructive transition of the displacive type from a parent crystalline structure. Two order parameters are involved in the transition mechanism: a displacement field which transforms a number of sublattices into icosahedral clusters, and an icosahedral density wave which breaks incommensurately the initial translational order. The main features of the model are introduced through the illustrative example of AlMnSi.

EXCHANGE SPIN WAVES AND THEIR MANIFESTATION IN TWO-MAGNON ABSORPTION AND RAMAN SCATTERING

EXCHANGE SPIN WAVES AND THEIR MANIFESTATION IN TWO-MAGNON ABSORPTION AND RAMAN SCATTERING

Ion-exchange equilibria in zeolites and clay minerals. Different concentration scales and derived thermodynamic functions

Treatments have been made of the thermodynamics of ion exchange which use mole fractions and equivalent cation fractions as concentration scales. Recently, substantial criticisms have been made of the widely used equivalent-fraction scale in ion exchange. These criticisms require an examination of the treatments based on the two scales and of the criticisms of one of these. It is shown that the criticisms are not valid and that the two approaches are completely equivalent and compatible. Reasons are given for preferring equivalent cation fractions as the concentration scale for zeolite exchangers where cations are often distributed among a number of sub-lattices.

Orientational disordering in crystals—I: A model for the evaluation of thermodynamic properties of melting transitions in atomic and plastic crystals

A generalisation of two-sublattice models of melting transitions in atomic and molecular crystals (Lennard-Jones and Devonshire; Pople and Karasz; Amzel and Becka) has been attempted. The intention was to develop a scheme by which parameters, such as the number of sublattices and distinguishable molecular orientations can be selected. It is found that the choice of these parameters made by Lennard Jones and Devonshire, and by Pople and Karasz, protects the theory to some extent from some of the more difficult physical issues. A single-lattice model also emerges as another possibility in the series of models that may be called upon. Thus a single-lattice model, based on concepts from Eyring's Significant Structure Theory of Liquids for the melt, and standard thermodynamic functions for the solid, has been developed to evaluate thermodynamic properties of melting transitions in atomic and molecular crystals. Despite the remarkable simplicity of the model predicted values of entropy of fusion for both argon and methane are in good agreement with experimental data. Mean positional disordering energies for the same substances have also been computed.

The quantum theory of the spontaneous magnetization of ferrimagnetics, anti-ferromagnetics and thin films with arbitrary spin at high temperatures

Using Van Vleck's modification of the Heisenberg theory of ferro-magnetism for substances with arbitrary spin per atom, a generalization of the theory is given for the case of ferrimagnetics, anti-ferromagnetics and thin films with an arbitrary number of sub-lattices and with arbitrary spin per atom. The exchange part of the energy operator is formulated by means of “rotated” spin operators. The theory contains the quantum-mechanical generalization of non-quantum theories given by Neel and later generalized by Yafet and Kittel. It also contains the theory of thin films proposed recently by the author. The magnetization laws are completely analogous to the relations derived from the theory of molecular fields. The constants of the molecular fields are interpreted by means of exchange integrals. For the special case of spin S=1/2the theory agrees with the results of the work of Vlasov and Ismuchametov for ferrimagnetics and anti-ferromagnetics and with the author's for thin films.

Equation of State for Vapor-Liquid-Solid System

Generalizing the idea of order-disorder in the alloy theory, we can calculate the unified equation of state for a lattice gas with many interstitial sublattices in the degree of approximation similar to that developed by Bragg-Williams. For n =9 ( n is the number of sublattices), the concrete equation of state is estimated and the ratio of the number of molecules belonging to each sublattice to their total number is given as a function of molecular density. Then the critical temperature and density for vapor-liquid transition are estimated for n =4, 9, 16. Furthermore, it is shown that the density region in the stable liquid phase is very narrow for n =4, but that it becomes wider for n =9, 16.