Griffiths Model Research Articles

The phase diagrams of the [formula omitted] model on the Bethe lattice

The biquadratic exchange interaction is randomized in a bimodal form with probabilities (p) and (1−p) for the cases with K>0 (attractive case) and K<0 (repulsive case), respectively, and its effects on the phase diagrams of the spin-1 Blume–Emery–Griffiths model are studied on the Bethe lattice by using the recursion relations. It was found that the critical behaviors of the model change drastically.

Dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model in an oscillating field: the effective-field theory based on the Glauber-type stochastic dynamics

Using the effective-field theory based on the Glauber-type stochastic dynamics (DEFT), we investigate dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model under an oscillating magnetic field. We presented the dynamic phase diagrams in (T/J, h0/J), (D/J, T/J) and (K/J, T/J) planes, where T, h0, D, K and z are the temperature, magnetic field amplitude, crystal–field interaction, biquadratic interaction and the coordination number. The dynamic phase diagrams exhibit several ordered phases, coexistence phase regions and special critical points, as well as re-entrant behavior depending on interaction parameters. We also compare and discuss the results with the results of the same system within the mean-field theory based on the Glauber-type stochastic dynamics and find that some of the dynamic first-order phase lines and special dynamic critical points disappeared in the DEFT calculation.

Mixed spin thin films in the Blume–Emery–Griffiths model with biquadratic exchange interaction: A Monte Carlo study

In this paper, we investigate the Blume–Emery–Griffiths thin film model, for mixed spins σ=3/2 and S=2, with a biquadratic exchange interaction. More precisely, we study the effect of the coupling exchange interactions in the presence of both an external magnetic and crystal fields. We first elaborate analytically the ground state phase diagrams. Concretely, the stable phases have been discussed. Then, the magnetic properties, at non vanishing temperature, are illustrated using Monte Carlo calculations. To complete this study, we examine the hysteresis loops behavior of the system.

Dynamic phase transitions of the Blume–Emery–Griffiths model under an oscillating external magnetic field by the path probability method

Abstract By using the path probability method (PPM) with point distribution, we study the dynamic phase transitions (DPTs) in the Blume–Emery–Griffiths (BEG) model under an oscillating external magnetic field. The phases in the model are obtained by solving the dynamic equations for the average order parameters and a disordered phase, ordered phase and four mixed phases are found. We also investigate the thermal behavior of the dynamic order parameters to analyze the nature dynamic transitions as well as to obtain the DPT temperatures. The dynamic phase diagrams are presented in three different planes in which exhibit the dynamic tricritical point, double critical end point, critical end point, quadrupole point, triple point as well as the reentrant behavior, strongly depending on the values of the system parameters. We compare and discuss the dynamic phase diagrams with dynamic phase diagrams that were obtained within the Glauber-type stochastic dynamics based on the mean-field theory.

The sustainable organisation: blueprint for an integrated model

Purpose – The purpose of this article is to build on the original Dunphy, Griffiths and Benn (2007) model by proposing a new model of organisational sustainability which includes the economic dimension. There is a growing level of interest by senior executives in the role and potential impact that sustainability will have on their organisation’s future strategy and structure. Although management is keenly aware that sustainability is important in their future planning, there is much uncertainty about what level of involvement and commitment they should make towards sustainability endeavours. Design/methodology/approach – This article reviews the existing organisational sustainability change model while building a case to have the important economic strand added to the original change dimensions relating to the human and the environmental strands. Findings – This conceptual paper builds on previous work of sustainability organisational change theorists to produce an enhanced sustainability change model thus proposing a more comprehensive and integrated sustainability stage model that can guide managers in their quest to evolve effective and more sustainable organisations. Originality/value – This is a conceptual paper that builds on previous work of sustainability organisational change theorists to introduce an enhanced sustainability change model that includes the important economic dimension to the widely accepted social and environmental dimensions, thus proposing a comprehensive and integrated sustainability stage model that can guide managers in their quest to evolve effective and more sustainable organisations.

A Monte Carlo study of the spin-1 Blume–Emery–Griffiths phase diagrams within biquadratic exchange anisotropy

The effect of the bi-quadratic exchange coupling anisotropy on the phase diagram of the spin-1 Blume–Emery–Griffiths model on simple-cubic lattice is investigated using mean field theory (MFT) and Monte Carlo simulation (MC). It is found that the anisotropy of the biquadratic coupling favors the stability of the ferromagnetic phase. By decreasing the parallel and/or perpendicular bi-quadratic coupling, the ferrimagnetic and the antiquadrupolar phases broaden in contrast, the ferromagnetic and the disordered phases become narrow. The behavior of magnetization and quadrupolar moment as a function of temperature is also computed, especially in the ferrimagnetic phase.

Martensitic and austenitic transformations in core-surface cubic nanoparticles

As a continuation of our recently published work, we have used the pair approximation in Kikuchi version to investigate martensitic and austenitic transformations in homogeneous (HM) and composite (CM) cubic nanoparticles (CNPs) based on the Blume–Emery–Griffiths model. A single cubic nanoparticle made of a core surrounded by a surface is considered as shaped in two dimensional (2D) square arrays instead of hexagonal array. From the phase diagrams of HM and CM-CNPs it has been observed that the martensitic–austenitic transformations (MT–AT) occurred. The influence of the exchange coupling and single-ion anisotropy parameters in the model Hamiltonian on the MT–AT is studied and analyzed in comparison with the results for hexagonal nanoparticles. Significant changes of the phase transition points and hysteresis behaviours depending upon the particle structure have been discussed. • This paper realized cubic nanoparticles. • This nanoparticles are investigated with single-ion anisotropy. • The hysteretic splitting are plotted.

Rates of Convergence in the Blume–Emery–Griffiths Model

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature \(\beta \) and the interaction strength \(K\). The rates of convergence results are obtained as \((\beta ,K)\) converges along appropriate sequences \((\beta _n,K_n)\) to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.

Phase boundaries of a spin-3/2 Blume–Emery–Griffiths model on a honeycomb lattice

The spin-3/2 Blume–Emery–Griffiths model on a honeycomb lattice is studied by Monte Carlo simulations with the goal to determine phase diagrams for a range of the model parameters and to investigate the nature of the phase transitions between the respective phases. For positive values of the biquadratic to bilinear interaction ratio α, we find two ferromagnetically ordered phases, F1 and F2, with the sublattice magnetizations (1/2,1/2) and (3/2,3/2), respectively, and our results confirm the discontinuous character of the order-disorder critical line as a function of the single-ion anisotropy strength, predicted by the effective-field theory (EFT). For negative values of α, there is another ferrimagnetic phase of the type (1/2,3/2), located between F1 and F2. However, the step-like variation of the order-disorder critical frontier obtained from EFT for large negative α is not reproduced and thus deemed artifact of the EFT approximation. Finite-size scaling analysis performed at various points between the respective identified phases gave the ratio of critical exponents γ/ν consistent with the 2D Ising universality class, except in the vicinity of the boundary intersection, where the results deviated from the standard values beyond the measurement errors.

Potts models with invisible states on general Bethe lattices

The number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order has attracted recent attention. In the q = 2 case, a Bragg–Williams (mean-field) approach necessitates four such invisible states while a 3-regular random graph formalism requires seventeen. In both of these cases, the changeover from second- to first-order behaviour induced by the invisible states is identified through the tricritical point of an equivalent Blume–Emery–Griffiths model. Here we investigate the generalized Potts model on a Bethe lattice with z neighbours. We show that, in the q = 2 case, invisible states are required to manifest the equivalent Blume–Emery–Griffiths tricriticality. When z = 3, the 3-regular random graph result is recovered, while z → ∞ delivers the Bragg–Williams (mean-field) result.

Quaternary Ni–Mn–In–Y Heusler alloys: a way to achieve materials with better magnetocaloric properties?

The influence of nonmagnetic impurities (Y) on the magnetocaloric effect (MCE) in the Heusler alloy Ni50Mn34In16Y in which ferromagnetic and antiferromagnetic coupling coexists in the martensitic state is investigated by means of ab initio and Monte Carlo simulations. Different impurity configurations are considered. Either Mn on the Mn sublattice or Ni on the Ni sublattice is substituted randomly by Y or substitution may occur simultaneously on both sublattices. The magnetic and magnetocaloric properties are obtained as a function of temperature and magnetic field using a mixed type of Potts and Blume–Emery–Griffiths model where the model parameters are obtained from ab initio calculations. When varying the impurity concentration, the shape of the magnetization curves is drastically changed at the martensitic transformation temperature due to the complex competing magnetic interactions. We show that the inverse MCE effect increases if Mn is substituted by Y compared with Y substituting for Ni. For example, the alloy system Ni50Mn32.75Cu1.25In16 where 5 at% Mn is replaced by Cu exhibits an inverse MCE which is twice as large compared with Ni50Mn34In16. Such predictive simulations may be used to design Heusler composites which display larger MCEs.

Potts models with (17) invisible states on thin graphs

The order of a phase transition is usually determined by the nature of the symmetry breaking at the phase transition point and the dimension of the model under consideration. For instance, q-state Potts models in two dimensions display a second order, continuous transition for q = 2, 3, 4 and first order for higher q. Tamura et al recently introduced Potts models with ‘invisible’ states which contribute to the entropy but not the internal energy and noted that adding such invisible states could transmute continuous transitions into first order transitions (Tamura et al 2010 Prog. Theor. Phys. 124 381; 2011 J. Phys: Conf. Ser. 297 012022; 2012 Interface Between Quantum Information and Statistical Physics; Tanaka and Tamura 2011 J. Phys.: Conf. Ser. 320 012025). This was observed both in a Bragg–Williams type mean-field calculation and 2D Monte-Carlo simulations. It was suggested that the invisible state mechanism for transmuting the order of a transition might play a role where transition orders inconsistent with the usual scheme had been observed. In this paper we note that an alternative mean-field approach employing 3-regular random (‘thin’) graphs also displays this change in the order of the transition as the number of invisible states is varied, although the number of states required to effect the transmutation, 17 invisible states when there are 2 visible states, is much higher than in the Bragg–Williams case. The calculation proceeds by using the equivalence of the Potts model with two visible and r invisible states to the Blume–Emery–Griffiths (BEG) model, so a by-product is the solution of the BEG model on thin random graphs.

Ground states of the frustrated Blume–Emery–Griffiths model in a field

Ground-state properties of the Blume–Emery–Griffiths model with antiferromagnetic nearest-neighbor interactions on a triangular lattice are investigated in the presence of an external magnetic field. In particular, we explore the modelʼs parameter space and identify regions with different degenerate ground states that may give rise to different magnetic phases also at finite temperatures. We demonstrate the presence of such phases by Monte Carlo simulations of magnetization processes for selected values of parameters.

Monte Carlo simulations of the Blume–Emery–Griffiths model with S=3/2 and S=5/2

The magnetic properties of the Blume–Emery–Griffiths(BEG) model [1] with fractional large spin(S=3/2 and S=5/2) on two-dimensional square lattice are discussed by making use of the Monte Carlo (MC) simulation for the magnetization 〈Sz〉, sublattice magnetizations 〈Sz(A)〉 and 〈Sz(B)〉, the magnetic specific heat CM and spin structures. The ground state phase diagram of these models has been obtained for exchange parameter J2/J1 and anisotropy parameter D/J1. The temperature dependence of sublattice magnetizations 〈Sz(A)〉 and 〈Sz(B)〉 is investigated for various values of the anisotropy D. The relations between the Curie temperature Tc and the critical temperature Ts at which sublattice magnetizations begin taking different values are also studied.

Phase transitions in presence of surfactants: From discrete to continuum

We study by \Gamma -convergence the discrete-to-continuum limit of the Blume–Emery–Griffiths model describing the phase transition of a binary mixture in presence of a third surfactant phase. In the case of low surfactant concentration we study the dependence of the surface tension on the density of the surfactant and we describe the microstructure of the ground states. We then consider more general (n-dimensional) energies modeling phase transitions in presence of different species of surfactants and, in the spirit of homogenization theory, we provide an integral representation result for their \Gamma -limit. As an application we study the ground states of these systems for prescribed volume fractions of the phases.

Phase transition properties of the spin-1 Blume–Emery–Griffiths model with random transverse crystal field

Phase transition properties of a spin-1 Blume–Emery–Griffiths model (BEGM) with random transverse crystal field is studied by the effective field theory for a simple cubic lattice. In T − D x space, we obtain the phase diagrams with the ratio α between the biquadratic interaction and the exchange interaction as well as a tunable parameter l of the transverse crystal field. The tricritical point (TCP) appears at α < 0 , which undergoes a crossover from positive to negative direction of the transverse crystal field when l < 0 . The TCP cannot be observed for α > 0 . The maximum critical temperature increases with the increase of α . The position of the peak value tends to the drift of negative or positive direction for a different magnitude or an imperfect ( ± ) transverse crystal field distribution. In T − α space, the range of ordered phase is magnified when the ratio is changed from α < 0 to α > 0 . The random transverse crystal field obviously affects the TCP.

Kinetics and Structural Evolution During Aging of Coker Vapor Deposits

Unwanted deposition of carbonaceous material from the vapor phase is a chronic problem in piping and equipment downstream of coking units and may be a limitation to achieving longer run length. The laboratory deposits were found to exhibit major differences from those in industry. An aging study was conducted to rationalize these differences. After days to weeks of aging at elevated temperature, the laboratory deposits become very similar to those of the graphitic industrial deposits. The differences in morphology that remain after aging were attributed to differences in hydrodynamic conditions during the deposit laydown. During aging, the composition and structure of deposits change with the time. This process includes complex chemical reactions and finally leads to more condensed polyaromatic and graphitic structures. Deposit aging kinetics was also studied through thermogravimetric analysis. Deposit characterization and kinetic studies yielded a consistent picture of the evolution from the heavy fluid phase components, which initially deposit from the vapor due to physical condensation, to the final massive, graphitic, coke-like solids found in industrial units. This structural evolution is described following the Marsh–Griffiths model.

Blume–Emery–Griffiths dynamics in social networks

We introduce the Blume–Emery–Griffiths (BEG) model in a social networks to describe the three-state dynamics of opinion formation. It shows that the probability distribution function of the time series of opinion is a Gaussian-like distribution. We also study the response of BEG model to the external periodic perturbation. One can observe that both the interior thermo-noise and the external field result in phase transition, which is a split phenomena of the opinion distributions. It is opposite between the effect acted on the opinion systems of the amplitude of the external field and of the thermo-noise.

Generalized partition function zeros of 1D spin models and their critical behavior at edge singularities

Here we study the partition function zeros of the one-dimensional Blume–Emery–Griffiths model close to their edge singularities. The model contains four couplings (H, J, Δ, K) including the magnetic field H and the Ising coupling J. We assume that only one of the three couplings (J, Δ, K) is complex and the magnetic field is real. The generalized zeros zi tend to form continuous curves on the complex z-plane in the thermodynamic limit. The linear density at the edges zE diverges usually with ρ(z) ∼ |z − zE|σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang–Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either Δ or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings (H, Δ, K) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.

Dynamic phase transition in the kinetic spin-5/2 Blume–Emery–Griffiths model in an oscillating external magnetic field

Employing a mean-field approach, we study the stationary states of the kinetic spin-5/2 Blume–Emery–Griffiths (BEG) model under the presence of a time-varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. We employ the Glauber transition rates to construct the set of dynamic mean-field equations. We investigate the time variation in average order parameters to find the phases in the system, and the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the dynamic phase transtions and to the dynamic phase transition temperature. The dynamic phase diagrams are presented in three different planes. The phase diagrams contain the ferromagnetic-5/2, the ferromagnetic-3/2, the ferromagnetic-1/2, the ferroquadrupolar, and disordered fundamental phases. They also include the nine coexisting or mixed phases composed of binary and ternary combinations of fundamental phases that strongly depend on the interaction parameters. The phase diagrams display the critical end point, double critical end point, triple point, quadruple point, and one, two, or three special points and the dynamic tricritical point that depends on the interaction parameters.