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Oin the method of Yvon in crystal statistics

C. iDOMB, Bove Riley

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

Closed-form approximations in crystal statistics have suffered from the defect that no steady series of approximations was available from which the rate of convergence could be assessed. The method of Yvon, based on a cluster-integral type of development, can furnish such a series of approximations. It is shown how to construct the partition functions for such approximations, the key being the use of the Mayer theory for multi-component assemblies. The method is applied to various properties of the Ising model of a ferromagnet and antiferromagnet, and results are obtained which are consistent with those based on series expansions. Previous investigations of Fournet which gave different results are shown to have made use of an insufficient number of terms in the approximation. The statistical mechanics of interacting systems on crystal lattices has attracted a good deal of attention in recent years. Exact results have been confined to a restricted class of problems in two dimensions, but for more general problems two independent approaches have been pursued. Closed-form approximations have been developed which ignore certain features of the statistical problem; and limited numbers of terms of exact series expansions for the partition function at high and low temperatures have been calculated. (See Domb (i 960) for a detailed review.) Series expansions are most useful when the terms are all consistent in sign. Convergence is then assured in the critical region, and they can yield reliable information on critical behaviour. An example of this kind is provided by the magnetic susceptibility of the Ising model above the Curie point (Domb & Sykes 1957; Fisher i959). When the terms are not consistent in sign some rearrangement must be sought, and the position is less satisfactory. For some problems, such as antiferromagnetism in a magnetic field, little information about critical behaviour can be derived from series expansions. Closed-form approximations could also be used for estimating critical behaviour if a series of successive approximations were available, so that the general trend of a particular property could be assessed. However, for most of the approximations discussed in the literature, the labour increases rapidly with the order of the approximation, and the process has not been followed for more than two or three stages. A method which does offer the possibility of a steady series of approximations,

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Closed-form approximations in crystal statistics have suffered from the defect that no steady series of approximations was available from which the rate of convergence could be assessed. The method of Yvon, based on a cluster-integral type of development, can furnish such a series of approximations. It is shown how to construct the partition functions for such approximations, the key being the use of the Mayer theory for multi-component assemblies. The method is applied to various properties of the Ising model of a ferromagnet and antiferromagnet, and results are obtained which are consistent with those based on series expansions. Previous investigations of Fournet which gave different results are shown to have made use of an insufficient number of terms in the approximation. The statistical mechanics of interacting systems on crystal lattices has attracted a good deal of attention in recent years. Exact results have been confined to a restricted class of problems in two dimensions, but for more general problems two independent approaches have been pursued. Closed-form approximations have been developed which ignore certain features of the statistical problem; and limited numbers of terms of exact series expansions for the partition function at high and low temperatures have been calculated. (See Domb (i 960) for a detailed review.) Series expansions are most useful when the terms are all consistent in sign. Convergence is then assured in the critical region, and they can yield reliable information on critical behaviour. An example of this kind is provided by the magnetic susceptibility of the Ising model above the Curie point (Domb & Sykes 1957; Fisher i959). When the terms are not consistent in sign some rearrangement must be sought, and the position is less satisfactory. For some problems, such as antiferromagnetism in a magnetic field, little information about critical behaviour can be derived from series expansions. Closed-form approximations could also be used for estimating critical behaviour if a series of successive approximations were available, so that the general trend of a particular property could be assessed. However, for most of the approximations discussed in the literature, the labour increases rapidly with the order of the approximation, and the process has not been followed for more than two or three stages. A method which does offer the possibility of a steady series of approximations,
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
Abstract This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.
This work has been concentrated on a detailed study of low temperature (and high field) expansions for the simple Ising model. Methods were considered for their derivation and derived long series; these have been analyzed to provide evidence on critical indices. Also studied were the derivation of series expansions for the Ising model with spin S > 1/2 and the corresponding low temperature behavior. Also reported is a study of the configurational problem presented by the Potts model and the planar Potts model. The former model in particular has attracted particular attention recently in relation to the properties of liquid crystals. There is evidence that its behavior differs significantly from that of the Ising model and may in certain circumstances lead to a first order transition. Work has continued with a general program of tabulation of lattice constants for all standard two and three dimensional lattices. (Modified author abstract)
The aim of this series of papers is to help bridge the gap in our knowledge of the properties of the Ising and Heisenberg models between very short-range forces corresponding to nearest-neighbour interactions, and very long-range forces corresponding to the mean-field approximation. The present paper is concerned with the equivalent neighbour model in which the interactions between a spin and a certain finite number of its neighbours are equal, and the remaining interactions are all zero. Methods derived previously for obtaining series expansions at high temperatures are generalized to include second- and third-neighbour shells for standard two- and three-dimensional lattices. It is found that as the co-ordination number q increases the properties of the model approach an asymptotic value which depends on dimension, and on q in a given dimension, but not significantly on the type of lattice structure. An estimate is made of this asymptotic behaviour as a function of q .
Abstract The exact solution of the three-dimensional Ising model of a ferromagnetic presents diffi­culties of a very fundamental nature. It therefore seems that the most reliable information on the behaviour of the model is provided by exact series expansions of the partition func­tion at low and high temperatures. However, the usual low -temperature expansion fails to converge in the neighbourhood of the critical point. By rearranging the terms of the series on the basis of physical considerations, it is possible to obtain a systematic set of successive approximations, each approximation taking exact account of clusters of a given size or less (metastable approximations). By extrapolation accurate estimates can be derived of the Curie point and critical values of the energy and entropy. It is found that there is a marked difference in behaviour between two- and three-dimensional lattices, a far larger proportion of the entropy change taking place in the temperature region below the Curie point in the latter case. The corre­sponding specific heat curves are therefore much closer to those observed experimentally. Finally, a brief discussion is given of the dependence of the specific heat curve on lattice structure.