The Ising model with nearest- and next-nearest-neighbor interactions on the Bethe lattice: The exact recursion relations

The Ising model with nearest- and next-nearest-neighbor (NNN) bilinear interactions is examined on the Bethe lattice (BL) in terms of exact recursion relations (ERR) when the external magnetic [Formula: see text] is turned on. The thermal variation of the magnetization belonging to the central spin is investigated to calculate the possible phase diagrams of the model for given coordination numbers. Different phase regions, ferromagnetic (FM), antiferromagnetic (AFM) and paramagnetic (PM), are discovered and the phase lines in terms of first-order or second-order phase transitions are calculated. These lines are found to be order–disorder or order–order phase transition lines. It is also found that they combine at some special points or terminate at some end points for appropriate values of the model parameters.

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The magnetic phase diagrams of the ternary alloy ABpC1−p on the Bethe lattice

Modern Physics Letters B ◽

10.1142/s0217984918501774 ◽

2018 ◽

Vol 32 (16) ◽

pp. 1850177

Author(s):

Erhan Albayrak

Keyword(s):

Phase Diagrams ◽

Ternary Alloy ◽

Nearest Neighbor ◽

Magnetic Phase ◽

Bethe Lattice ◽

Recursion Relations ◽

Coordination Numbers ◽

First Order ◽

First Order Phase ◽

The Given

In this work, the ternary alloy (TA) of the form [Formula: see text] with spin-[Formula: see text], spin-2 and spin-[Formula: see text], respectively, is studied on the Bethe lattice in terms of exact recursion relations in the standard random approach. The bilinear interaction parameter [Formula: see text] is assumed to be ferromagnetic between the nearest-neighbor spins with spin-[Formula: see text] and spin-2, while [Formula: see text] is taken to be antiferromagnetic between spin-[Formula: see text] and spin-[Formula: see text]. The possible phase diagrams are obtained from the thermal analysis of the order parameters for the given coordination numbers z = 3,[Formula: see text]4,[Formula: see text]5 and 6. This analysis has also revealed that the model gives both second- and first-order phase transitions in addition to the compensation temperatures.

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The mixed spin-1/2 and spin-1 model with alternating coordination number

International Journal of Modern Physics B ◽

10.1142/s0217979219501029 ◽

2019 ◽

Vol 33 (11) ◽

pp. 1950102

Author(s):

Erhan Albayrak

Keyword(s):

Phase Transition ◽

Phase Transitions ◽

Coordination Number ◽

Bethe Lattice ◽

Recursion Relations ◽

Coordination Numbers ◽

First Order ◽

Mixed Spin ◽

Phase Transition Temperatures ◽

First Order Phase

The mixed spin-1/2 and spin-1 Blume–Capel model is studied with randomly alternated coordination numbers (CN) on the Bethe lattice (BL) by utilizing the exact recursion relations. Two different CNs are randomly distributed on the BL by using the standard–random (SR) approach. It is observed that this model presents first-order phase transitions and tricritical points for variations of CNs 3 and 4, even if these behaviors are not displayed for the regular mixed-spin on the BL. The phase diagrams are mapped by obtaining the phase transition temperatures of the first- and second-order on several planes.

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CRITICAL TEMPERATURES OF THE ISING MODEL ON THE BETHE LATTICE FOR ARBITRARY VALUES OF SPIN

International Journal of Modern Physics B ◽

10.1142/s0217979211102137 ◽

2012 ◽

Vol 26 (01) ◽

pp. 1250003 ◽

Cited By ~ 3

Author(s):

E. JURČIŠINOVÁ ◽

M. JURČIŠIN

Keyword(s):

Ising Model ◽

Spontaneous Magnetization ◽

Magnetic Moments ◽

Bethe Lattice ◽

Critical Temperatures ◽

Spin Variable ◽

Recursion Relations ◽

Coordination Numbers ◽

Full Structure ◽

Very High

Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.

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THE SPIN-1 BLUME–CAPEL MODEL WITH ALTERNATINGLY CHANGING BILINEAR EXCHANGE INTERACTIONS BETWEEN SUBLATTICES

International Journal of Modern Physics B ◽

10.1142/s0217979211102216 ◽

2012 ◽

Vol 26 (05) ◽

pp. 1250031 ◽

Cited By ~ 4

Author(s):

ERHAN ALBAYRAK

Keyword(s):

Crystal Field ◽

Nearest Neighbor ◽

Bethe Lattice ◽

Exchange Interactions ◽

Coordination Numbers ◽

First Order ◽

Crystal Fields ◽

Negative Crystal ◽

First Order Phase ◽

Second Order Phase Transition

The spin-1 Blume–Capel model is studied on a Bethe lattice which is divided into two sublattices A and B. Alternatingly changing bilinear exchange interactions, JAB and JBA, between the sublattices, i.e., between the nearest-neighbor shell spins, are assumed. The phase diagrams of the model are studied on the (JAB, T) planes for given values of JBA, crystal fields D and the coordination numbers q = 3, 4 and 6. It was found that the model either displays only second-order phase transition lines at higher crystal field values or second- and first-order phase transitions lines combined at tricritical points at lower negative crystal fields. It was also found that the tricritical points move to higher temperatures and to higher values of JAB as the crystal field becomes more negative.

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Phase diagrams of the random nearest-neighbor mixed spin-1/2 and spin-3/2 Blume–Capel model

Modern Physics Letters B ◽

10.1142/s0217984918503256 ◽

2018 ◽

Vol 32 (27) ◽

pp. 1850325 ◽

Cited By ~ 1

Author(s):

Erhan Albayrak

Keyword(s):

Phase Diagrams ◽

Nearest Neighbor ◽

Bethe Lattice ◽

Recursion Relations ◽

Mixed Case ◽

Coordination Numbers ◽

System Parameters ◽

Mixed Spin ◽

Critical Lines ◽

Random Approach

The mixed spin-1/2 and spin-3/2 Blume–Capel (BC) model is considered on the Bethe lattice (BL) with randomly changing coordination numbers (CN) and examined in terms of exact recursion relations. A couple of two different CNs are changed randomly on the shells of the BL in terms of a standard–random approach to obtain the phase diagrams on possible planes of the system parameters. It is found from the thermal analysis of the order-parameters that the model only gives the second-order phase transitions as in the regular mixed case. As the probability of having larger CN increases, the temperatures of the critical lines also increase as expected.

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The thermal properties of the mixed spin-1/2, 1, 3/2 Ising model on the Bethe lattice

Modern Physics Letters B ◽

10.1142/s0217984921500792 ◽

2020 ◽

pp. 2150079

Author(s):

J. Kple ◽

E. Albayrak ◽

F. Hontinfinde

Keyword(s):

Thermal Properties ◽

Magnetic System ◽

Bethe Lattice ◽

Thermal Fluctuations ◽

Model Parameters ◽

Phase Boundaries ◽

Recursion Relations ◽

First Order ◽

Mixed Spin ◽

Number Formula

A triple mixed-spin Ising system defined on the Bethe lattice is numerically investigated by means of exact recursion relations (ERRs) calculations. The lattice is constituted by three types of magnetic atoms A, B, C with spins [Formula: see text], [Formula: see text], [Formula: see text] respectively arranged in the form ABCABC. The effects of bilinear exchange and crystal-field interactions as well as those of thermal fluctuations on the order parameters and phase diagrams are thoroughly studied and specified. First-order transitions and tricritical points are present for the coordination number [Formula: see text] whereas at [Formula: see text] they are absent. Global compensation phenomena are absent for the magnetic system. Instead, it is shown that it can only occur between the sublattice magnetizations B and C of the system. Several novel kinds of reentrance of the phase boundaries while varying the values of model parameters have been reported.

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First-order transition features of the triangular Ising model with nearest- and next-nearest-neighbor antiferromagnetic interactions

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2007.04.051 ◽

2007 ◽

Vol 383 (2) ◽

pp. 351-371 ◽

Cited By ~ 14

Author(s):

A. Malakis ◽

N.G. Fytas ◽

P. Kalozoumis

Keyword(s):

Ising Model ◽

Nearest Neighbor ◽

Order Transition ◽

First Order ◽

First Order Transition

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Developing Practical Models of Complex Salts for Molten Salt Reactors

Thermo ◽

10.3390/thermo1020012 ◽

2021 ◽

Vol 1 (2) ◽

pp. 168-178

Author(s):

Theodore M. Besmann ◽

Juliano Schorne-Pinto

Keyword(s):

Molten Salt ◽

Nearest Neighbor ◽

System Modeling ◽

Model Parameters ◽

Calphad Approach ◽

Coordination Numbers ◽

Regression Methods ◽

Complex Salts ◽

Salt Systems ◽

Fitting Model

Molten salt reactors (MSRs) utilize salts as coolant or as the fuel and coolant together with fissile isotopes dissolved in the salt. It is necessary to therefore understand the behavior of the salts to effectively design, operate, and regulate such reactors, and thus there is a need for thermodynamic models for the salt systems. Molten salts, however, are difficult to represent as they exhibit short-range order that is dependent on both composition and temperature. A widely useful approach is the modified quasichemical model in the quadruplet approximation that provides for consideration of first- and second-nearest-neighbor coordination and interactions. Its use in the CALPHAD approach to system modeling requires fitting parameters using standard thermodynamic data such as phase equilibria, heat capacity, and others. A shortcoming of the model is its inability to directly vary coordination numbers with composition or temperature. Another issue is the difficulty in fitting model parameters using regression methods without already having very good initial values. The proposed paper will discuss these issues and note some practical methods for the effective generation of useful models.

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Random crystal field effects on antiferromagnetic spin-1 Blume–Capel model

Modern Physics Letters B ◽

10.1142/s0217984921502869 ◽

2021 ◽

pp. 2150286

Author(s):

Erhan Albayrak

Keyword(s):

Crystal Field ◽

Critical Points ◽

Bethe Lattice ◽

Honeycomb Lattice ◽

Recursion Relations ◽

First Order ◽

Field Effects ◽

Number Formula ◽

First Order Phase ◽

Antiferromagnetic Spin

The outcome of the random crystal field effects on the antiferromagnetic spin-1 Blume–Capel model and external magnetic field are examined on the Bethe Lattice in terms of exact recursion relations. It is assumed that the crystal field is either turned on or off randomly with probability [Formula: see text] and [Formula: see text], respectively. The phase diagrams are constructed from the thermal analysis of the order parameters with the coordination number [Formula: see text] which corresponds to honeycomb lattice. It is explored that the system goes both second- and first-order phase transitions, along with the reentrant behavior and a few critical points. The reentrant behavior is stronger for lower values of [Formula: see text] and disappears as [Formula: see text] gets closer to 1.0. The first-order lines are observed to be either linked to the tricritical points or decomposed. The critical end points and double critical points are also observed.

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