Planar Triangular Lattice Research Articles

Ground states of the Triangular Ising model with two-and three-spin interactions

The possible ground state spin configurations of an Ising model on a plane triangular lattice are investigated. The model incorporates competing interactions between spins at nearest and next-nearest neighbour sites as well as a coupling between three spins at the vertices of a nearest-neighbour triangle, and an external magnetic field. Models of this type are frequently used to describe the structures of adsorbate layers on hexagonal substrates. The analysis is based on linear inequalities involving the magnetization, two- and three-spin correlations, and on simple convexity arguments. Part of the inequalities needed are proved with the aid of a computer. For vanishing three-spin coupling the results of earlier studies are confirmed; in addition, the resulting seven topologically distinct structures are shown to be unique. Two of these structures are energetically degenerate; the degeneracy cannot be lifted by any further two-spin interaction. For nonzero three-spin coupling only an “almost complete” solution is given, involving four additional spin configurations. The “ground state phase diagrams” are discussed.

Thermodynamics and molecular freedom of dimers on plane triangular lattices

This is an extension of three previous papers dealing with dimers on rectangular lattices (one, two, and three dimensions). The technique presented in the first paper in this series continues to be fruitful for dimers on plane triangular lattices. Entropy, isothermal compressibility, constant pressure heat capacity, and molecular freedom per dimer at close packing are obtained exactly for lattices infinite in one direction and finite in the other. Observations made in the third paper of the series concerning molecular freedom per dimer at close packing on rectangular lattices are used to extrapolate our results to infinite plane triangular lattices. At close packing, the molecular freedom per dimer on an infinite plane triangular lattice is calculated to be 2.356 527... in agreement with the value obtained by Nagle. Based on our earlier findings, the value of 2.356 527... was used to obtain the analytic fit for the thermodynamic quantities in terms of the normalized number density.

On the statistics of adsorption: Active clusters in alloy catalysts

The dependence of the sticking coefficient S on the surface coverage θ like S ∼ (1 − θ) n is well known. A similar formula holds for the dependence of the sticking coefficient on the surface concentration of an active species of atoms, x s (diluted by some inactive species) S ∼ x n s. Here n denotes the number of active atoms per adsorption site ( n = 2 for bridge sites, n = 4 for fourfold hollow sites and so on). This formula is especially useful in showing the strong influence of small amounts of inactive atoms on the catalytic activity involving the adsorption of large molecules, e.g. C x H y . Its validity however is limited to the case of a purely statistical mixture of active and inactive atoms. If there is an interaction between the surface atoms, e.g. a tendency towards clustering of inactive atoms, the local probabilities of active clusters are changed forcing a corrective factor in the above equation. This factor is calculated for the simple quadratic and the plane triangular lattices using the method of Kikuchi for the cases: (a) pair. square (= fourfold hollow adsorption site), two squares, square of four squares, on the sq lattice; (b) triangle (= threefold hollow site), adjacent pair of triangles, hexagon (six triangles), on the pt lattice. It can be shown that especially an attraction between active and inactive atoms leads to strong deviations from the simple formula even at moderate interaction enthalpies.

On the statistics of adsorption: Active clusters in alloy catalysts

Abstract The dependence of the sticking coefficient S on the surface coverage θ like S ∼ (1 − θ ) n is well known. A similar formula holds for the dependence of the sticking coefficient on the surface concentration of an active species of atoms, x s (diluted by some inactive species) S ∼ x n s . Here n denotes the number of active atoms per adsorption site ( n = 2 for bridge sites, n = 4 for fourfold hollow sites and so on). This formula is especially useful in showing the strong influence of small amounts of inactive atoms on the catalytic activity involving the adsorption of large molecules, e.g. C x H y . Its validity however is limited to the case of a purely statistical mixture of active and inactive atoms. If there is an interaction between the surface atoms, e.g. a tendency towards clustering of inactive atoms, the local probabilities of active clusters are changed forcing a corrective factor in the above equation. This factor is calculated for the simple quadratic and the plane triangular lattices using the method of Kikuchi for the cases: (a) pair. square (= fourfold hollow adsorption site), two squares, square of four squares, on the sq lattice; (b) triangle (= threefold hollow site), adjacent pair of triangles, hexagon (six triangles), on the pt lattice. It can be shown that especially an attraction between active and inactive atoms leads to strong deviations from the simple formula even at moderate interaction enthalpies.

A molecular dynamics simulation study of octopoles in the field of a planar surface

The molecular dynamics simulation method is used to investigate a model of tetrahedral molecules, such as methane, adsorbed in a monolayer at a flat surface, such as graphite. The model consists of 64 octopoles arranged in a planar triangular lattice, with nearest neighbour interactions, and with an external (surface) field applied. The equations of motion are framed entirely in terms of quaternion parameters. Orientational order parameters, and thermodynamic properties, are monitored, and a single phase transition, from a rotationally ordered to a disordered state, with a broad anomaly in the heat capacity, is observed.

Neutron Scattering Study of Magnetic Excitations in Quasi-One-Dimensional Triangular Lattice Antiferromagnet CsVCl3

CsVCl 3 is composed of one-dimensional antiferromagnetic chains which are coupled weakly antiferromagnetically in the basal hexagonal plane. In order to clarify the anomalous properties hitherto observed, an inelastic neutron scattering has been studied. We observed good one-dimensionality of J '/ J =2.7×10 -4 . But, though S =3/2, the spin wave dispersion along the chain disagrees with the classical picture. The excitation spectra along the triangular lattice plane are unusual. The excitation is dominated by a dispersionless mode of strong intensity. Compared to this mode, the conventional acoustic spin wave branch seems to be of minor excitation at lower temperatures than the Neel point.

Anomalous spin correlation in a linear-chain triangular lattice antiferromagnet CsVCl 3

We have demonstrated that CsVCl 3 is a very good realization of spin 3 2 , 1D Heisenberg antiferromagnetic system with J'/ J = 2.7 x 10 4 through the inelastic scattering of neutrons. In this substance, the antiferromagnetic coupling in the basal triangular lattice plane causes a frustration effect. We have observed certain “mysterious” properties. For instance, the excitation spectra in the plane are difficult to explain by the simple spin wave theory.

Adsorption ensembles and catalysis: a mathematical method

Adsorption and subsequent catalytic reactions of large molecules (e.g. higher hydrocarbons) usually require an ensemble of adsorption sites, i.e. a number of surface atoms being of the active species. The probabilities of certain ensembles (bridge sites and triangles) on a plane triangular lattice are calculated for fixed surface concentration of one atomic species depending on the interactions between different kinds of atoms. In a similar way the surface coverage of interacting adsorbed atoms on a perfect triangular lattice of adsorption sites is investigated with respect to its dependence on the partial pressure and the interaction energies. In both cases ordered surface structures can be found; the conditions of their formation and their consequences are pointed out.

The effect of anisotropic higher order dispersive interactions on the conformational stabilities of argon clusters

The point–dipole interaction model of polarizability developed by Applequist is applied to a theory of long-range intermolecular dispersive forces to explore the effect of anisotropic many body interactions on the properties of various van der Waals complexes. Our anisotropic point–dipole interaction formalism appears to be significantly more accurate than isotropic pairwise formalisms at medium to large internuclear separations. At short internuclear distances, however, our procedures overestimate the dispersion energy due to the inherent neglect of electron exchange effects. Nevertheless, for large clusters of atoms or molecules, the present formalism provides a dependable and relatively rapid calculational approach to the analysis of many body dispersion forces. The only-time consuming calculation of the entire procedure is the inversion of a 3N×3N matrix, where N is the number of atoms in the van der Waals complex. Potential energy surfaces for three noble gas atoms are predicted to have minima for both the triangular and linear configurations. Isotropic pairwise calculations predict a minimum only for the triangular configuration. Three other types of clusters are also investigated: a linear array, a planar triangular lattice, and a hexagonal closest packed cluster. The typical effect of the anisotropic many body interactions is to lower the calculated dispersion energy by 5%–20%. The magnitude of the energy lowering is approximately proportional to the optical anisotropy of the system. Our results imply that anisotropic effects are only important when the optical anisotropy is larger than 0.001.

Forces between plates with surface dipole layers

Two parallel plates in an aqueous electrolyte medium each carry surface dipoles which are immersed in the adjacent electrolyte. The object of this paper is to calculate the potential field between the plates and hence obtain the interplate force due to the dipole layers. The discreteness of the dipoles at each interface is represented by regarding them as situated at the sites of a plane triangular lattice. The potential then has the same periodicity in planes parallel to the plates and is expressed as a Fourier series. Series expressions for the free energy and force are deduced in the linear approximation and the effect of lateral plate shift is examined. However, except at very small plate separations, only the zero wave number or “smeared dipole” term proves to be appreciable and this is calculated using the nonlinear Poisson-Boltzmann equation. A comparison is made between expressions for the force derived, respectively, from point-dipole and finite-dipole models. Finally the effect on the force of specific adsorption of ions into the dipole layers is considered.

Magnetic phase diagrams of mixed metamagnetic dihalides by magnetically induced light scattering

The magnetic ordering in a dilute metamagnet (FepMg1−pCl2) and a mixed metamagnet with competing spin anisotropies (FepCo1−pCl2) has been studied by direct optical methods. The mixed phase region is mapped in (H, T, p) space. The tricritical temperatures fall to zero at p ≈ 0.6, the percolation limit for the planar triangular lattice. In this regime Fe(II) orders independently of the Co(II) spins.

Calculation of binodals and spinodals of binary solid solution with plane triangular and fcc lattices

The cluster-variation method was used to calculate the binodal and spinodal of a binary solid solution with an fcc and plane triangular lattice. The position of the binodal on the cA-T diagram for the plane triangular lattice is compared with the exact Ising solution.

Glass-like phase on the close packed lattices

Abstract The random mixture of the ferro- and antiferromagnetic exchange integrals (bonds) on a triangular cactus tree, which represents properties of the close packed lattices (and also of the Kagome lattice), such as the plane triangular lattice and the fcc lattice, is discussed. It is shown that the glass-like phase (GLP) exists, in addition to the ferromagnetic and paramagnetic phases. The phase boundaries between them are determined.

Low-Temperature Approach to the Renormalization-Group Study of Critical Phenomena

A new method of exploring the contents of the renormalization-group equations for discrete spins is introduced. The equations are expanded in low-temperature series and the truncated series are used to obtain the critical exponents and critical temperature of a system. The method is demonstrated on the planar triangular Ising lattice and the critical parameters are found to be within a few percent of the exactly known values in third nonvanishing order of approximation.

Magneto-optical study of the magnetic phase diagram of the dilute metamagnet FepMg1-pCl2

The mixed phase region of the magnetic phase diagram of FepMg1-pCl2 (p=1.00-0.71) has been mapped by a direct optical method. The tricritical temperature falls with increasing dilution by magnesium and is extrapolated to zero at p approximately 0.6, close to the percolation limit for a plane triangular lattice rather than the three-dimensional analogue, despite the essential three-dimensional nature of the metamagnetic transition. In contrast, the tricritical field tends to a finite value as p decreases.

Derivation of low-temperature series expansions for the Ising model with triplet interactions on the plane triangular lattice

Derivation of low-temperature series expansions for the Ising model with triplet interactions on the plane triangular lattice

Derivation of low-temperature series expansions for the Ising model with triplet interactions on the plane triangular lattice

The derivation of low-temperature (high-field) series expansions for the Ising model with pure triplet interactions on the plane triangular lattice is described. Euler's law of the edges is used to transform the linkage rule into a form convenient for the derivation of ferromagnetic polynomials. Explicit results are given for the ferromagnetic polynomials corresponding to the first twelve powers of the temperature variable (u3) both as functions of the field variable ( mu ) and, in zero field, as functions of the temperature variable (u2) of the simple Ising model.

High temperature free energy of the Schrodinger model of ferromagnetism for arbitrary single-ion contributions

A system is considered described by the hamiltonian H=-J Sigma (i,j) Pij- Sigma i=1N Qi( zeta ) where the first term corresponds to the Schrodinger spin S exchange model and the second term represents an arbitrary single-ion contribution (characterized by the set of parameters zeta ). Terms in the free energy through (J/kBT)7 have been obtained for the face-centered cubic, body-centred cubic, simple cubic, plane triangular and plane square lattices, explicit results for the face-centred cubic lattice are given.

On randomly dilute ferromagnetic models

The paper considers the I(1/2), H(1/2), and H( infinity ) ferromagnetic models in which there are nonmagnetic impurities distributed at random on the sites of the underlying physical lattice. High temperature power series expansions, which are presented, are analysed by conventional methods to find the Curie temperature Tc(p) and susceptibility exponent gamma (p), where p is the concentration of magnetic elements. The series are analysed for p>1, as well as p<1, thus making contact with the nonselfintersecting chain problem. At p=1, gradients of TC(p)/TC(1) against p, are found, for the face centred cubic lattice, to be 1.05+or-0.02 for I(1/2), 1.15+or-0.01 for H( infinity ) and 1.36+or-0.03 for H(1/2). For the plane triangular lattice, the initial gradient for the I(1/2) model is 1.45+or-0.05. The behaviour found for gamma (p) is rejected as physically implausible, and the implications of its anomalous behaviour are discussed.

Crystal structure and melting in a cell model. I. Solvable lattices and isomorphisms

The free energy and crystal structure of some particular cases of a cell model of melting are obtained. These include (i) a class of one-dimensional cases with interactions extending to many neighbouring cells, (ii) the plane hexagonal, bcc and triangular lattices with nearest-neighbour interactions, (iii) a case of the plane, square lattice with first- and second-neighbour (diagonal) interactions, and others. The solutions for the simple cubic lattices in all dimensions were obtained in a previous paper (abstr. A46569 of 1971). It is shown that some of the 2, 3, etc dimensional cases are closely related ('isomorphic') to one-dimensional cases.