Lattice Statistics Research Articles

The inversion relation method for some two-dimensional exactly solved models in lattice statistics

The partition-functions-per-siteκ of several two-dimensional models (notably the eight-vertex, self-dual Potts and hard-hexagon models) can be easily obtained by using an inversion relation for local transfer matrices, together with symmetry and analyticity properties. This technique is discussed, the analyticity properties compared, and some equivalences (and nonequivalences) pointed out. In particular, the critical hard-hexagon model is found to have the sameκ as the self-dualq-state Potts model, withq=(3 + √5)/2 = 2.618 .... The Temperley-Lieb equivalence between the Potts and six-vertex models is found to fail in certain nonphysical antiferromagnetic cases.

Lattice statistics for highly heterogeneous populations of rigid rectangular particles

DiMarzio’s lattice model is used to calculate the configurational entropy of a population of rigid rectangular parallelepipeds which is arbitrarily heterogeneous in both lengths and widths. An analytical expression is obtained which is a product of a small number of terms and is readily applicable to statistical thermodynamic calculations. The limit of infinitesimal lattice cell size is considered as an approximation of a continuum description. In this continuum limit, the alignment transition for monodisperse nematogen is found to occur at a lower density, and with a smaller density discontinuity, than is predicted by the lattice model in its original form. Although, at a given density, the alignment and the pressure are generally greater in the continuum description than in the discrete description, the alignment and pressure are less at the transition in the continuum case than they are at the transition in the discrete case. Whereas the original lattice model predicts that no transition occurs for particles of axial ratio less than 3.65, in the continuum limit the transition disappears only when the axial ratio is less than 3.06. Alternative approaches for treating solvents are discussed.

Exact Algorithm ford-Dimensional Walks on Finite and Infinite Lattices with Traps

An exact algorithm is formulated to calculate the expected walk length $〈n〉$ for a walker (atom, molecule) undergoing random displacements on a finite or infinite (periodic) $d$-dimensional lattice with traps (reactive sites). The method is illustrated for the case of a single deep trap surrounded by shallow traps and the calculated value of $〈n〉$ agrees to within 0.3% of the Monte Carlo result for all lattices considered. The theory introduced is capable of generalization to many new classes of problems in lattice statistics.

Generating functions for enumerating self-avoiding rings on the square lattice

It is shown that generating function techniques provide an efficient means of enumerating the number of self-avoiding rings (polygons) on the square lattice. The techniques can be applied to a number of related problems in lattice statistics and statistical mechanics. The enumeration has been extended to polygons of up to 38 steps.

Finite lattice series expansions for the triangular lattice

It is shown that, on the triangular lattice, series expansions for the Ising model and related problems in lattice statistics can be obtained by combining the contributions of finite hexagonal regions. This generalises earlier results for square lattice systems.

Calculation of relaxation volumes, dipole tensors and Kanzaki forces for point defects

The relaxation volumes and dipole tensors associated with point defects in metals are commonly deduced from the results of lattice statistics and computer simulation calculations. Often an approximation introduced by Hardy for weakly disturbing defects is applied in such deductions. However, for strongly disturbing defects such as interstitials, and even in some models vacancies, this approximation breaks down. A more rigorous treatment of the problem which avoids these difficulties is presented.

Hard hexagons: exact solution

The hard-hexagon model in lattice statistics (i.e. the triangular lattice gas with nearest-neighbour exclusion) has been solved exactly. It has a critical point when the activity z has the value 1/2(11+5 square root 5)=11.09017..., with exponents alpha =1/3, beta =1/9. More generally, a restricted class of square-lattice models with nearest-neighbour exclusion and non-zero diagonal interactions can be solved.

Lattice statistical aspects of electron correlation problems

A technique for the lattice statistics, the Kikuchi's cluster variation method, is applied to improve the Gutzwiller's method for the ground state problem of the many-electron system on the assumption that small number of macroscopic variables are enough to describe the feature of a whole configuration. The momentum distribution in the present approximation is no longer a step function consisting of two constant values, though some people might have been thinking that too simple distribution is an intrinsic weak point of the Gutzwiller's method. A metal-insulator transition of the Brinkman-Rice type takes place in the half-filled case also in the higher order approximations. The merit of the present scheme is in the realization of the correlated wavefunction rather than in the lowering of the ground state energy. Some spatial correlation functions are calculated. The role of the interaction range is studied.

Lattice model studies of the effect of chain flexibility on the nematic–isotropic transition. III. Completely flexible chain solutes in rigid rod solvents

A steric, mean field, simple cubic lattice model using DiMarzio lattice statistics for mixtures of completely flexible chain solutes in rigid rod solvents is presented. The chain solutes depress the nematic–isotropic transition temperature of the pure solvent. The slopes of the two-phase coexistence curves in the reduced temperature/mole fraction plane, solute and solvent order parameters, and solute activity coefficients for mixtures of rods and chains of various lengths are discussed and compared with relevant experimental data. The results indicate that although completly flexible chains in this model are too flexible compared with n-alkane solutes, the chain flexibility of the solute is important in determining the ability of the solute to disrupt the nematic order of the solvent.

On the normalization of one-dimensional lattice statistics

It is shown that the normalization of the statistics associated with the distribution of λ-bell particles [λ=1 for simple particles; λ=2 for dumbbell particles, etc.] on a one-dimensional lattice space, is given by a simple, two-term recursion relation. This normalization, for a lattice consisting of N sites, is shown to be (1/x+)N, where x+ is the real positive root of the equation xλ +x−1=0. A general recursion expression is also developed that describes the higher order moments of the statistics.

Odd operators and spinor algebras in lattice statistics:n-point functions for the rectangular Ising model

An extension is given of the development by Schultz et al. [3] of the pioneering work of Kaufman and Onsager [2] on the planar Ising ferromagnet. This involves a novel form of Wick's theorem [19] for fermions. In subsequent papers the method will be applied to determine then-point functions and to evaluate rigorously critical indices.

Lattice model studies of the effect of chain flexibility on the nematic–isotropic transition. II. Rigid cores having semiflexible tails

A steric, mean field, three-dimensional, simple cubic lattice model (using DiMarzio lattice statistics), with and without attractive forces, for molecules composed of rigid cores and semiflexible pendant tails is presented. The calculated trends and absolute values of the following properties at the nematic–isotropic transition for cores of length-to-breadth ratios of 4 and 5 with tails of varying length are compared with relevant experimental data: temperature, core order parameter, intramolecular and intermolecular tail order parameters, densities, relative density change, and entropy change. With results intermediate between those for the completely flexible tail model and those for the rigid rod models, this semiflexible tail model reasonably describes the general features and trends observed experimentally in homologous series of liquid crystals at the transition. Tail chain flexilibity (important in the transition) at a given temperature is rather adequately treated as an entirely internal property in this model. Steric repulsions are sufficient to generate the transition; attractive interactions, while not essential, do allow some adjustments in trends and some improvements in energetics. Preliminary calculations indicate that a softened repulsive potential leads to somewhat improved values for the core order parameter and the relative density change.

Lattice model studies of the effect of chain flexibility on the nematic–isotropic transition. I. Rigid cores having completely flexible tails

A steric, mean field, three-dimensional, simple cubic lattice model using DiMarzio lattice statistics for molecules composed of rigid cores and completely flexible pendant tails is presented. The calculated trends and absolute values of the following properties at the nematic–isotropic transition for cores of length-to-breadth ratios of 5 and 10 with tails of varying length are discussed and compared with relevant experimental data: temperature, core order parameter, intermolecular tail order parameter, densities, relative density change, and entropy change. The results indicate that although completely flexible tails in this model are too flexible compared with hydrocarbon tails in real liquid crystals, tail flexibility of some degree should be important in the nematic–isotropic transition.

Adsorption, reactivity and surface structural chemistry of hydrogen cyanide on Pt(110)

The adsorption, desorption, surface structural chemistry, and reactivity of HCN adsorption layers on Pt(110) have been studied by LEED and thermal desorption mass spectrometry. HCN adsorbs at 300 K via a mobile precursor state, and induces a (1 × 2) → (1 × 1) change in the periodicity of the clean reconstructed Pt(110) surface. Desorption occurs yielding H 2, HCN and C 2N 2 in comparable quantities, and the observed behavior is accounted for in terms of associative desorption from a fully dissociated layer of H and CN species. A simple model is developed using lattice statistics which gives a fairly good account of the observed desorption spectra and their coverage dependence, both as regards peak temperature shifts and product distribution. Comparison of the observed and calculated spectra for C 2N 2 leads to the following conclusions: (i) the initial sticking probability of HCN is of the order of unity; (ii) H and CN occupy different kinds of adsorption site; (iii) CN itself is a “two-site” adsorbate at 300 K. c(2 × 4) and (1 × 1) LEED patterns due to adsorbed CN are observed which are consistent with this view; under certain conditions a more compressed, poorly ordered ¦ 1 2 3 , 1 2 3 ¦ structure is also observed.

Validity of Using Lattice Statics at All Temperatures

It is shown that zero-point and thermal vibrations change the vacancy formation volume calculated from classical lattice statistics by usually no more than ${10}^{\ensuremath{-}3}$ at all temperatures. In view of the recent interest in relating the temperature variation of the vacancy formation volume to Arrhenius plot curvature, the validity of lattice statics, as a tool for the calculation of temperature-dependent defect properties, is established.

A transformation including the weak-graph theorem and the duality transformation

A transformation in classical lattice statistics which generalizes the weak-graph theorem and includes the duality transformation is described. The energy of the configurations is expressed by a function of “quantum numbers” which are subject to certain constraints. These constraints generate the variables of the transformed system. Examples of transformations in vertex models, and two- and many-component Ising models are given. This method simplifies the derivation of a number of transformations. As an example, the transformation of the Ising model on a triangular lattice into a dual on a triangular lattice can be accomplished without an intermediate step via the honeycomb lattice.

Effects of bundling in a lattice gas model of liquid crystals

Phase transition properties in a lattice gas model of liquid crystals, consisting of rigid rods with attractive, anisotropic, nearest-neighbor interaction, are calculated with the method of Guggenheim-Flory-DiMarzio lattice statistics. It is shown that in this model the properties of the isotropic-anisotropic phase transition are altered by a bundling phenomenon, in which parallel molecules are grouped into energetically favorable clusters of various sizes. The bundling of molecules increases the density at which the isotropic-anisotropic phase transition occurs and decreases the values of the accompanying density and order parameter discontinuities at the transition. The cases where the molecular length to breadth ratio l = 5, 10 are calculated and illustrated.

Small‐angle x‐ray scattering of semicrystalline polymers. II. Analysis of experimental scattering curves

AbstractThe small‐angle x‐ray scattering (SAXS) patterns of a number of linear polyethylene (PE) and polyoxymethylene (POM) samples have been measured and compared to the intensity functions of one‐dimensional paracrystalline lattices. It was found that the ratio of the angular positions of the second and first scattering maxima (θ2/θ1) is generally less than or equal to 2.0, implying that the paracrystalline lattice statistics are symmetric or moderately skewed to larger periods. The Bragg spacing (“long period”) of such samples is within 3% of the identity period of the macrolattice. With quenched POM the ratio θ2/θ1 is substantially larger than 2.0, which indicates either extremely asymmetric lattice statistics or coexisting structures within the material. From consideration of the reduced widths of the first scattering maxima, it was found that some broadening is present in addition to that from the paracrystallinity. This excess broadening could result from a finite lattice length of ∼1000 Å. The need for careful experimental technique for obtaining the actual position of the scattering maximum is emphasized. In addition, it is demonstrated that the scattering curve and the correlation function of the system yield essentially the same apparent structural periods.

Mössbauer and magnetic investigation of point defects in iron-doped cadmium fluoride

Iron impurity point defects in cadmium fluoride are investigated by 57Fe Mössbauer effect spectroscopy, which offers an extremely sensitive probe into the environment of the iron atoms. This study on dilute iron in cadmium fluoride has not revealed any static Jahn-Teller distortion or static off-center stabilization of Fe2+ for temperatures above 2.08°K. The system can be characterized by the presence of nearest neighbor point defects and random strains from distant defects distributed statistically throughout the lattice. Lattice statistics for codoped systems are developed and used to explain the results for iron-ytterbium and iron-sodium codoped cadmium fluoride. Magnetic susceptibility measurements coorrborate the presence of divalent iron and indicate the onset of antiferromagnetic ordering below 5°K (θ=−5.7°K).

Small‐angle x‐ray scattering of semicrystalline polymers. I. Review of existing models

AbstractThe effect of previously proposed distributions of particle size and interparticle “gap” lengths on the small‐angle x‐ray scattering of a paracrystalline one‐dimensional macrolattice has been examined. It was concluded that the general paracrystalline model, in which the fluctuations of crystalline and amorphous thickness both contribute to the destruction of long‐range order, best describes the structure of lamellar aggregates in semicrystalline polymers. By using this model, the influence of symmetric and asymmetric lattice statistics on the positions of the scattering maxima were investigated. It was found that positively skewed thickness distributions result in the second‐order maximum occurring at an angle greater than twice that of the first‐order maximum (sx̄2/sx̄1 > 2.0); the position of the first‐order maximum is generally greater than the Bragg angle of the structure. With negatively skewed distributions, the ratio of the scattering angles, sx̄2/sx̄1, is less than 2.0, and the first maximum is displaced below the Bragg angle. Qualitatively similar behavior is found with lattices characterized by symmetric lattice statistics, though these deviations from the Bragg conditions are smaller than in the case of negatively skewed distributions. The ratio of the scattering angles of the second and first maxima best reflects the general shape of the lattice statistics in a paracrystalline lattice. The effect of a transition zone, having properties intermediate between those of the crystalline and amorphous regions, was also considered. While the intensity of the higher‐order maxima is decreased, no significant shift of the scattering angles results from the incorporation of such a transition zone.