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

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.

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

2012 ◽  
Vol 26 (05) ◽  
pp. 1250031 ◽  
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.

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

2018 ◽  
Vol 32 (27) ◽  
pp. 1850325 ◽  
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.

Trimodal-random field Blume-Capel model

2021 ◽  
pp. 2150270
Author(s):  
Erhan Albayrak
Keyword(s):  
Magnetic Field ◽  
Phase Transitions ◽  
Random Field ◽  
Phase Diagrams ◽  
Critical Points ◽  
Bethe Lattice ◽  
Recursion Relations ◽  
First Order ◽  
Random Magnetic Field ◽  
First Order Phase

The external random magnetic field [Formula: see text] with three nodes, i.e. acting up and down along the [Formula: see text]-axis and zero, effective on the spins in the Blume-Capel model is analyzed on the Bethe lattice in terms of the exact recursion relations. All the nodes are assumed to have the same probability, [Formula: see text], so that the model could give various kinds of phase transitions. As a mapping of the phase transitions, the phase diagrams are constructed on two different planes which present very rich and interesting phase diagrams. In addition to the second- and first-order phase transitions, a few critical points, reentrant and double reentrant behaviors are also observed.

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

2020 ◽  
Vol 34 (10) ◽  
pp. 2050087
Author(s):  
Erhan Albayrak
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Nearest Neighbor ◽  
Bethe Lattice ◽  
Model Parameters ◽  
Thermal Variation ◽  
Recursion Relations ◽  
Coordination Numbers ◽  
First Order ◽  
Special Points

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.

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

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.

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

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.

SPIN-2 ISING MODEL ON THE BILAYER BETHE LATTICE

2008 ◽  
Vol 22 (24) ◽  
pp. 4189-4203 ◽  
Author(s):  
ERHAN ALBAYRAK ◽  
SEYMA AKKAYA ◽  
SABAN YILMAZ
Keyword(s):  
Phase Diagrams ◽  
Ground State ◽  
Nearest Neighbor ◽  
Lower Layer ◽  
Bethe Lattice ◽  
Branching Ratio ◽  
Interlayer Coupling ◽  
Coupling Constants ◽  
First Order Phase ◽  
Over The Top

A spin-2 system consisting of two layers of Bethe lattices each with a branching ratio of q Ising spins was analyzed by the use of the exact recursion relations in a pairwise approach. The upper layer interacting with nearest-neighbor (NN) bilinear interaction J1 is laid over the top of the lower layer interacting with bilinear NN interaction J2, and the two layers are tied together via the bilinear interaction between the vertically aligned adjacent NN spins denoted as J3. The study of the ground state phase diagrams on the (J2/|J3|, J1/|J3|) plane with J3>0 and J3<0 and on the (J2/J1, J3/q J1) plane with J1>0 has yielded five distinct ground state configurations. The temperature dependent phase diagrams are obtained for the case with intralayer coupling constants of the two layers with ferromagnetic type J1 and J2>0, and the interlayer coupling constant of the layers with either ferromagnetic J3>0 or antiferromagnetic type J3<0 on the (kT/J1, J3/J1) planes for given values of the J2/J1 for various values of the coordination numbers. As a result, we have found that the model presents both second- and first-order phase transitions, therefore, tricritical points.

The quaternary alloy on the Bethe lattice

2018 ◽  
Vol 32 (21) ◽  
pp. 1850226 ◽  
Author(s):  
Erhan Albayrak
Keyword(s):  
Phase Diagrams ◽  
Coordination Number ◽  
Ternary Alloy ◽  
Interaction Parameter ◽  
Bethe Lattice ◽  
Quaternary Alloy ◽  
Spin Models ◽  
Recursion Relations ◽  
Mixed Spin ◽  
Random Approach

The quaternary alloy (QA) is simulated on the Bethe lattice (BL) in the form of ABpCqDr and its phase diagrams are calculated by using the exact recursion relations (ERR) for the coordination number z = 3. The QA is designed on the BL by placing A atoms (spin-1/2) on the odd shells and randomly placing B (spin-3/2), C (spin-5/2) or D (spin-1) atoms with probabilities p, q and r, respectively, on the even shells. A compact form of formulation for the QA is obtained in the standard-random approach which can easily be reduced to ternary alloy (TA) and mixed-spin models by the appropriate values of the random variables p, q and r. The phase diagrams are calculated on the temperature and ratio of bilinear interaction parameter planes for given values of probabilities.

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

2012 ◽  
Vol 26 (01) ◽  
pp. 1250003 ◽  
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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