A Double Chain Approximation to the Ising Model

Abstract

J. A. PLASCAK and K. P. SILVA: Double Chain Approximation to the Ising Model 669 phys. stat. sol. (b) 110, 669 (1982)To calcnlate thermodynamic properties of the anisotropic Ising model, a double chain approximation (DCA), based on Bogolyubov's variational procedure, is presented. The critical temperatures are compared t o those obtained by the usual mean field approximation (MFA), the linear chain approximation (LCA), and the exact or series results, when existing. A fitting of the spontaneous polarization and the dielectric constant data for the quasi-one-dimensional ferroelectric crystal PMDP (PbDPOJ is also made.Zur Berechnung der thermodynamischen Eigenschoften eines anisotropen Ising-Modells wird eine auf dem Bogolyubov-Vsriationsverfahren beruhende DoppelkettennHherung (DCA) angcgeben.Die kritischen Temperaturcn werden mit denen verglichen, die mit der ublichen Naherung dcs mittleren Feldes (JIFA) und der linearen Kettenniiherung (LCA) erhalten werden und wo vorhanden, mit den exakten oder Serienergebniasen. Eine Anpassung der spontanen Polarisierung und der Dielektrizitltskonstante fur den quasi-eindimensionalen ferroelektrischen Kristall PNDP (PbDPO,) wird ebenfalls durchgefuhrt.

Critical Temperature of a Quasi‐One‐Dimensional Ising Model. The Bethe‐Peierls Approximation

“…The so-called linear chain approximation (LCA) (see, e.g., [5]) is one of the simplest ways of critical temperature calculation. I n this approach, the one-dimensional chains are selected in the system.…”

Re-entrant phase transitions of the Blume–Emery–Griffiths model for a simple cubic lattice on the cellular automaton

“…The parameters of J and K are the bilinear and biquadratic interaction energies, respectively and D is the single-ion anisotropy constant. The three-dimensional BEG model has been extensively studied by different techniques, using the mean-field approximation (MFA) (1,5−7) , effective-field theory (8−11) , two-particle cluster approximation (TPCA) 12 , Bethe approximation 13 , high-temperature series expansion 14 , renormalization group theory 15 , Monte Carlo simulations (13,16−17) , linear chain approximation (18,19) and cellular automaton (20,21) In this paper we studied the three-dimensional BEG model using an improved heating algorithm from the Creutz Cellular Automaton (CCA) for simple cubic lattice. The CCA algorithm is a microcanonical algorithm interpolating between the canonical Monte Carlo and molecular dynamics techniques on a cellular automaton, and it was first introduced by Creutz 22 .…”

Gaussian Fluctuations of Molecular Field in Quasi-One-Dimensional Ising Model

“…The results are presented graphically. Figure 1 shows the relative Curie temperature tc = kBTc/J 1 versus the anisotropy J2/J 1 , as obtained within the molecular field approximation (MFA), linear chain approximation (LCA) [5] (for Ñ = oo), and the approximation assumed in the present paper (MGFA) as well as the exact results for a square lattice (sq). For J2/J 1 = 1 we obtained in MGFA for a square lattice tc = 0.584532, while the exact value is tc = 0.567296 [7].…”