multicritical phase diagrams of the ferromagnetic spin- blume–emery–griffiths model with...

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Journal of Magnetism and Magnetic Materials 320 (2008) 8–24

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Multicritical phase diagrams of the ferromagnetic spin-32

Blume–Emery–Griffiths model with repulsive biquadratic couplingincluding metastable phases: The cluster variation method and the path

probability method with the point distribution

Mustafa Keskin�, Osman Canko

Department of Physics, Erciyes University, 38039 Kayseri, Turkey

Received 19 February 2007; received in revised form 24 April 2007

Available online 18 May 2007

Abstract

We study the thermal variations of the ferromagnetic spin-32Blume–Emery–Griffiths (BEG) model with repulsive biquadratic coupling

by using the lowest approximation of the cluster variation method (LACVM) in the absence and presence of the external magnetic field.

We obtain metastable and unstable branches of the order parameters besides the stable branches and phase transitions of these branches

are investigated extensively. The classification of the stable, metastable and unstable states is made by comparing the free energy values of

these states. We also study the dynamics of the model by using the path probability method (PPM) with the point distribution in order to

make sure that we find and define the metastable and unstable branches of the order parameters completely and correctly. We present the

metastable phase diagrams in addition to the equilibrium phase diagrams in the (kT/J, K/J) and (kT/J, D/J) planes. It is found that the

metastable phase diagrams always exist at the low temperatures, which are consistent with experimental and theoretical works.

r 2007 Elsevier B.V. All rights reserved.

PACS: 05.70.Fh; 64.60.Cn; 64.60.Kw; 64.60.My; 75.10.Hk

Keywords: The spin-32Blume–Emery–Griffiths model; Cluster variation method; Path probability method; Metastable phase diagram

1. Introduction

The spin-32Blume–Emery–Griffiths (BEG) model has been paid much attention for many years because of their

simplicity and exhibits a variety of multicritical phenomena accompanied with the onset of first- and second-order phasetransitions. The spin-3

2BEG model is defined by the Hamiltonian

H ¼ �JX

ijh i

SiSj � KX

ijh i

S2i S2

j þDX

i

S2i

!, (1)

where each Si can take the values �32and �1

2and oij4 indicates summation over all pairs of nearest-neighbor sites. J, K

and D are the bilinear exchange, biquadratic exchange and crystal–field interactions, respectively.The phase diagrams of the ferromagnetic spin-3

2BEG model for K/J40 have been studied and its phase diagrams have

been presented by a variety of method such as renormalization-group (RG) methods [1], the effective field theory (EFT) [2],the Monte Carlo (MC) simulations and a density–matrix RG method [3]. An exact formulation of the model on a Bethelattice was studied by using the exact recursion equations [4]. The ground-state phase diagrams of the spin-3

2BEG model

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/j.jmmm.2007.05.005

onding author.

ddress: [email protected] (M. Keskin).

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ARTICLE IN PRESSM. Keskin, O. Canko / Journal of Magnetism and Magnetic Materials 320 (2008) 8–24 9

was worked out by Canko and Keskin [5]. On the other hand, the ferromagnetic spin-32BEG model with the repulsive

biquadratic coupling, i.e. K/Jo0 is now a subject of intense study. For example, an early attempt to study the spin-32BEG

model with K/Jo0 was made by Barretto and Bonfim [6], and Bakkali et al. [7] within the MFA and also the MCcalculation, and the EFT, respectively. Barreto and Bonfim calculated only the phase diagrams for the ferromagneticisotropic spin-3

2BEG model and Bakkali et al. [7] also presented two phase diagrams: One for the ferromagnetic spin-3

2BC

model in which is the spin-32Ising model with only J and D interactions, and the other for the ferromagnetic isotropic spin-3

2

BEG model that is the spin-32Ising model with only J and K interactions. Tucker [8] studied the ferromagnetic spin-3

2BEG

model with K/Jo0 by using the cluster variation method in pair approximation (CVMPA) and he only presented the phasediagrams of the spin-3

2BC model and isotropic spin-3

2BEG model for several values of the coordination number. Bakchich

and Bouziani [9] presented the phase diagram of the model in the (kT/J, D/J) plane for only the two different values of K/Jwithin an approximate RG approach of the Migdal–Kadanoff type. Ekiz et al. [10] investigated the ferromagnetic spin-3

2

BEG model on the Bethe lattice using the exact recursion equations and presented the phase diagrams in the (kT/J, K/J)plane for several values of D/J and in the (kT/J, D/J) plane for several values of K/J in the absence of an external magneticfield (H). Ekiz [11] extended the previous work, i.e. Ref. [10] for the presence of an external magnetic field. He presentedone phase diagram in the (kT/J, H/J) plane for K/J ¼ �0.5 and D/J ¼ 1.0 and the other phase diagram in the (kT/J, K/J)plane for H/J ¼ 2.0 and D/J ¼ 0.5. In both figures, he used the coordination number q ¼ 3, 4, 6 and 8. Recently, Pınaret al. [12] studied the ferromagnetic spin-3

2BEG model and found four new phase diagram topologies for H ¼ 0.0 and one

new phase diagram topology for H 6¼0.0 within the lowest approximation of the cluster variation method (LACVM).In spite of these studies, the critical behavior of the ferromagnetic spin-3

2BEG model with repulsive biquadratic

interaction (Ko0.0) has not been thoroughly explored. Especially, the metastable and unstable branches of the orderparameters and their phase transitions were not studied. Moreover, metastable phase diagrams of the model were also notcalculated. Whereas, the metastable phase diagrams of the spin-1 BEG model were investigated extensively for K40.0 [13]and also Ko0.0 [14]. Therefore, the purpose of this work is to investigate the behavior of the thermal variation of the orderparameters in depth and to obtain the metastable and unstable branches of the order parameters and to examine theirphase transitions for repulsive biquadratic interaction. We also study the dynamics of the model by using the pathprobability method (PPM) with the point distribution [15] in order to make sure that we find and classify the metastableand unstable branches of the order parameters completely and correctly. Finally, we present the metastable phase diagramsin addition to the equilibrium phase diagrams in the (kT/J, K/J) and (kT/J, D/J) planes. The LACVM, in spite of itslimitations, is an adequate starting point. Within this theoretical framework it is easy to determine the complete phasediagrams and find the some outstanding features in the temperature dependencies of the order parameters and as well asobtain metastable portion of the phase diagrams. It also offers a very practical and simple tool to solve most collectivephenomena.

The outline of the remaining part of this paper is as follows. In Section 2, we define the model briefly and obtain itssolutions at equilibrium within the LACVM. The thermal variations of the system are investigated in Section 3. Thedynamics of the model is studied by the PPM in Section 4. In Section 5, the transition temperatures are calculated precisely,and metastable phase diagrams are presented in addition to the equilibrium phase diagrams. Section 6 contains thesummary and conclusion.

2. Model and method

The spin-32BEG model is defined as a two-sublattice model with spin variables Si ¼ �

32; �1

2and Sj ¼ �

32; �1

2on sites of

sublattices A and B, respectively. The average value of each of the spin states will be denoted by X A1 ;X

A2 ;X

A3 and X A

4 on the

sites of sublattice A and X B1 ;X

B2 ;X

B3 and X B

4 on the sites of sublattice B, which are also called internal or the state or point

variables. X A1 ;X

B1 ; X A

2 ;XB2 ; X A

3 ;XB3 ; X A

4 ;XB4 are the fractions of the spins that have the values þ3

2; þ1

2; �1

2and � 3

2,

respectively. These variables obey the following two normalization relations for A and B sublattices:

X4i¼1

X Ai ¼ 1 and

X4j¼1

X Bj ¼ 1. (2)

In order to account for the possible two-sublattice structure, we need six long-range order parameters, which are

introduced as follows: MA � hSAi i, QA � hðS

Ai Þ

2i, RA � hðS

Ai Þ

3i, MB � hS

Bj i, QB � hðS

Bj Þ

2i, RB � hðS

Bj Þ

3i for A and B

lattices, respectively. /yS denotes the thermal average, MA and MB are the average magnetizations which is the excess ofone orientation over the other orientation, called magnetizations. QA and QB are the quadrupolar moments which are theaverage squared magnetizations, and RA and RB are the octupolar order parameters, that are the average cubedmagnetizations, for A and B sublattices, respectively.

ARTICLE IN PRESSM. Keskin, O. Canko / Journal of Magnetism and Magnetic Materials 320 (2008) 8–2410

The sublattice-order parameters can be expressed in terms of the internal variables and are given by

MA � SAi

� �¼ 3

2X A

1 � X A4

� �þ 1

2X A

2 � X A3

� �,

MB � SBj

D E¼ 3

2X B

1 � X B4

� �þ 1

2X B

2 � X B3

� �,

QA � SAi

� �2D E¼ 9

4ðX A

1 þ X A4 Þ þ

14ðX A

2 þ X A3 Þ,

QB � SBj

� �2� ¼ 9

4ðX B

1 þ X B4 Þ þ

14ðX B

2 þ X B3 Þ,

RA � SAi

� �3D E¼ 27

8 X A1 � X A

4

� �þ 1

8 X A2 � X A

3

� �,

RB � SBj

� �3� ¼ 27

8X B

1 � X B4

� �þ 1

8X B

2 � X B3

� �. ð3Þ

Using Eqs. (2) and (3), the internal variables can be expressed as a linear combinations of the order parameters,

X A1 ¼

14

QA �14

� �þ 1

6RA �

MA

4

�; X B

1 ¼14

QB �14

� �þ 1

6RB �

MB

4

�,

X A2 ¼

14

94�QA

� �þ 1

294MA � RA

� �; X B

2 ¼14

94�QB

� �þ 1

294MB � RB

� �,

X A3 ¼

14

94�QA

� �þ 1

2RA �

94MA

� �; X B

3 ¼14

94�QB

� �þ 1

2RB �

94MB

� �,

X A4 ¼

14

QA �14

� �þ 1

614MA � RA

� �; X B

4 ¼14

QB �14

� �þ 1

614MB � RB

� �. ð4Þ

The Hamiltonian of such a two-lattice ferromagnetic spin-32BEG model in an external magnetic field (H) is

H ¼ � JX

ijh i

SiSj � KX

ijh i

S2i S2

j

þDX

i

S2i þ

Xj

S2j

!�H

Xi

Si þX

j

Sj

!. ð5Þ

The equilibrium properties of the system are determined by the lowest approximation of the cluster-variation method(LACVM) which is identical to the mean-field approximation (MFA). The method consists of the following three steps: (i)consider a collection weakly interacting systems and define the internal variables, (ii) obtain the weight factor in terms ofthe internal variables, and (iii) find the free energy expression and minimize it. The LACVM, in spite of its limitations, is anadequate starting point. Within this theoretical framework it is easy to determine the complete phase diagrams and findsome outstanding features in the temperature dependencies of order parameters and as well as obtain the metastableportion of the phase diagrams. We should also mention that the bilinear and biquadratic interactions are restricted to the z

nearest-neighbor pair of spin. z is absorbed in J and K.The weight factors WA and WB can be expressed in terms of the internal variables for the A and B sublattices,

respectively, as

W A ¼NA!Q4

i¼1

X Ai NA

� �!

and W B ¼NB!Q4

j¼1

X Bj NB

� �!

, (6)

where NA and NB are the number of lattice points on the A and B sublattices, respectively. On the other hand, a simpleexpression for the internal energy of the system is found by working out Eq. (5) in the LACVM. This leads to

E

N¼ �JMAMB � KQAQB þDðQA þQBÞ �HðMA þMBÞ. (7)

Substituting Eq. (3) into Eq. (7) the internal energy per site can be written as

E

N¼ � J 3

2X A

1 � X A4

� �þ 1

2X A

2 � X A3

� �� 32

X B1 � X B

4

� �þ 1

2X B

2 � X B3

� �� K 9

4ðX A

1 þ X A4 Þ þ

14ðX A

2 þ X A3 Þ

� 94ðX B

1 þ X B4 Þ þ

14ðX B

2 þ X B3 Þ

� þD 9

4ðX A

1 þ X A4 Þ þ

14ðX A

2 þ X A3 Þ þ

94ðX B

1 þ X B4 Þ þ

14ðX B

2 þ X B3 Þ

� �H 3

2X A

1 � X A4

� �þ 1

2X A

2 � X A3

� �þ 3

2X B

1 � X B4

� �þ 1

2X B

2 � X B3

� �� , ð8Þ

where N (N ¼ NA+NB) is the number of total lattice points.


Using the LACVM, which is identical to the MFA, the free energy f (F ¼ E�TS) per site can be written as

f ¼F

N¼ �JMAMB � KQAQB þDðQA þQBÞ

�HðMA þMBÞ

þ1

b

X4i¼1

X Ai lnX A

i þX4j¼1

X Bj lnX B

j

( )

þ blA 1�X4i¼1

X Ai

( )þ blB 1�

X4j¼1

X Bj

( ), ð9Þ

where lA and lB are introduced to maintain the normalization conditions, b ¼ 1/kT, T is the absolute temperature, and k isthe Boltzmann factor. The minimization of Eq. (9) with respect to X A

i and X Bj and also using the Eq. (3), the self-consistent

equations are found to be

MA ¼3e

94bðKQB�DÞsinh 3

2b JMB þHð Þ�

þ e14bðKQB�DÞsinh 1

2b JMB þHð Þ�

2e94bðKQB�DÞcosh 3

2b JMB þHð Þ�

þ 2e14bðKQB�DÞcosh 1

2b JMB þHð Þ� ,

MB ¼3e

94bðKQA�DÞsinh 3

2b JMA þHð Þ�

þ e14bðKQA�DÞsinh 1

2b JMA þHð Þ�

2e94bðKQA�DÞcosh 3

2b JMA þHð Þ�

þ 2e14bðKQA�DÞcosh 1

2b JMA þHð Þ� ,

QA ¼9e

94bðKQB�DÞcosh 3

2b JMB þHð Þ�

þ e14bðKQB�DÞcosh 1

2b JMB þHð Þ�


2b JMB þHð Þ�



QB ¼9e

94bðKQA�DÞcosh 3

2b JMA þHð Þ�

þ e14bðKQA�DÞcosh 1

2b JMA þHð Þ�


2b JMA þHð Þ�


2b JMA þHð Þ� ,

RA ¼27e

94bðKQB�DÞsinh 3

2b JMB þHð Þ�

þ e14bðKQB�DÞsinh 1

2b JMB þHð Þ�


2b JMB þHð Þ�



RB ¼27e

94bðKQA�DÞsinh 3

2b JMA þHð Þ�

þ e14bðKQA�DÞsinh 1

2b JMA þHð Þ�


2b JMA þHð Þ�


2b JMA þHð Þ� . (10)

We are now able to examine the behavior of the order parameters of the ferromagnetic spin-32BEG model with the

repulsive biquadratic coupling in an external magnetic field by solving of the self-consistent equations, i.e., Eq. (10),numerically. In the following section, we shall examine the thermal variation of the systems.

It is worthwhile to mention that the values of these sublattices order parameters define five different phases with thedifferent symmetry as follows:

(i)
Disordered phase (d): MA ¼MB ¼ 0, QA ¼ QBa0: (ii) Ferromagnetic-3
2phase (f3/2): MA ¼MB ¼ 3=2, QAaQBa0:

(iii)
Ferromagnetic-12phase (f1/2): MA ¼MB ¼ 1=2, QAaQBa0:
(iv)
Ferrimagnetic phase (i): MAaMBa0, QAaQBa0: (v) Antiquadrupolar phase (a): MA ¼MB ¼ 0, QAaQBa0:
In this point, we should mention that since the behavior of the stable branches of RA and RB are similar to the MA and MB,we have not used RA and RB to define these phases.

3. Thermal variations

In this section, we shall study the temperature dependency of the order parameters in the absence and presenceof an external magnetic field by solving the system of transcendental equations or the set of self-consistent equations,


i.e., Eq. (10), numerically. These equations are solved by using the Newton–Raphson method and the thermal variations oforder parameters for several values of D/J, K/J and H/J are plotted in Figs. 1 and 2. In the figures, subscript 1 denotes thestable states (solid lines), subscript 2 corresponds to metastable states (dashed-dotted lines) and three unstable states(dashed lines). This classification is done by matching the free energy values of these states and investigating solution of thedynamic or rate equations of the system, which is given in Section 4. TC and, Tt are the critical or the second-order phasetransition temperatures and the first-order phase transition temperatures for the stable branches of the order parameters,respectively. TCM and TCQ represent the critical or the second-order phase transition temperatures for only the stablebranches of the magnetizations and also the octupolar order parameters, and quadrupolar order parameters, respectively.Tfi and Tif are the second-order phase transition temperatures from the ferromagnetic-1

2phase (f1/2) to the ferrimagnetic

(i) phase and from the i phase to the f1/2 phase, respectively, for the stable branches of order parameters. TCM2 and TtQ2

represent the critical or the second-order phase transition temperatures for only the metastable branches of themagnetizations and also the octupolar order parameters, and the first-order phase transition temperature for themetastable branches of quadrupolar order parameters, respectively. TC2 and Tt2 are the second- and first-order phasetransition temperatures for the metastable branches of the order parameters.

First, we will investigate the temperature dependence of the order parameters in the absence of an external magneticfield, i.e., H ¼ 0.0. In this case, the behavior of the thermal variations temperature dependence of the order parametersdepends on D/J and K/J values and, by matching the free energy values of the solutions of the order parameters, thefollowing six main topological different types of behaviors, in which one of them exhibits only the stable branches, arefound and they are plotted in Fig. 1.

(a)

Fig.

lines

tran

tran

resp

(i) p

tran

phas

tran

(a)

(b)

(c)

(d)

(e)

(f)

Type 1: For K/J ¼ 1.25 and D/J ¼ 1.5, MA1 ¼MB1 ¼ 1.5, QA1 ¼ QB1 ¼ 2.25, and RA1 ¼ RB1 ¼ 3.5 at zerotemperature. On the other hand, MA2 ¼MB2 ¼M2 ¼ 0.5, QA2 ¼ QB2 ¼ Q2 ¼ 0.25, and RA2 ¼ RB2 ¼ R2 ¼ 0.125 atzero temperature. The stable branches of sublattice-order parameters decrease to zero discontinuously as the reducedtemperature (kT/J) increases, hence the system undergoes a first-order phase transition, seen in Fig. 1(a). The transitionis from the f3/2 phase to the d phase. Metastable branches of the magnetizations (M2) and octupolar moments (R2)decrease to zero continuously as the reduced temperature increases, therefore the M2 and R2 undergo a second-orderphase transition at TCM2, but metastable branches of the quadrupolar order parameters (Q2) undergoes a first-orderphase transition at TtQ2, because the discontinuity occurs, seen in the figure. Moreover, the values of M ¼ R ¼ 0.0below the Tt are unstable branches of the magnetizations (M3) and octupolar moments (R3), but Q3 is different thanzero and also occurs only below Tt.

(b)
Type 2: For K/J ¼ 0.25 and D/J ¼ 0.5, MA1 ¼MB1 ¼ 1.5, QA1 ¼ QB1 ¼ 2.25, and RA1 ¼ RB1 ¼ 3.5 at zerotemperature. On the other hand, MA2 ¼MB2 ¼M2 ¼ 0.5, QA2 ¼ QB2 ¼ Q2 ¼ 0.25, and RA2 ¼ RB2 ¼ R2 ¼ 0.125 at
1. The temperature dependences of the order parameters, subscript 1 indicates the stable state (solid lines), 2 the metastable state (dashed-dotted

), and 3 the unstable state (dashed lines). TC and Tt are the critical or the second-order phase transition temperatures and the first-order phase

sition temperatures for the stable branches of the order parameters, respectively. TCM and TCQ represent the critical or the second-order phase

sition temperatures for only the stable branches of the magnetizations and also the octupolar order parameters, and quadrupolar order parameters,

ectively. Tfi and Tif are the second-order phase transition temperatures from the ferromagnetic-12phase (f1/2) to the ferrimagnetic (i) phase and from the

hase to the f1/2 phase, respectively, for the stable branches of order parameters. TCM2 and TtQ2 represent the critical or the second-order phase

sition temperatures for only the metastable branches of the magnetizations and also the octupolar order parameters, and the first-order

e transition temperature for the metastable branches of quadrupolar order parameters, respectively. TC2 and Tt2 are the second- and first-order phase

sition temperatures for the metastable branches of the order parameters.

Exhibiting two first-order and a second-order phase transitions, one of the first-order phase transitions is for the stable branches of order parameters

(the transition is from the f3/2 phase to the d phase) and the other for the metasatable branches of only quadrupolar order parameters (the transition is

from the df3/2 phase to the f3/2 phase), and the second-order phase transition is from the df3/2 phase to the duf3/2 for the metastable branches of

magnetizations and also the octupolar order parameters. K/J ¼ 1.25 and D/J ¼ 1.5.

Exhibiting a second-order phase transition from the f3/2 phase to the d phase for the stable branches of order parameters and the first-order phase

transition from the df3/2 phase to the f3/2 phase for the metastable branches of branches of order parameters. K/J ¼ 0.25 and D/J ¼ 0.5.

Exhibiting two successive phase transitions for only the stable branches of order parameters in which the first one is a first-order phase transition from

the f3/2 phase to the f1/2 phase and the second one is a second-order phase transition from the f1/2 phase to the d phase. K/J ¼ �0.01 and D/J ¼ 0.5.

Exhibiting a second-order phase transition for only the stable branches of order parameters, the transition is from the f1/2 phase to the d phase.

K/J ¼ �0.5 and D/J ¼ 0.5.

Exhibiting two successive second-order phase transitions for the stable branches of order parameters in which the first one is from the i phase to the a

phase and the second one is from the a phase to the d phase. Also exhibiting the second-order phase transition for the metastable branches of order

parameters, the transition is from the da phase to the a phase. K/J ¼ �2.0 and D/J ¼ �2.5.

Exhibiting three successive second-order phase transitions for only the stable branches of order parameters. The first one is from the f1/2 phase to the i

phase at Tfi, the second one is from the i phase to the f1/2 phase at Tif and the third is from the f1/2 phase to the d phase. The first two second-order

transitions imply that the system exhibits a reentrant behavior. K/J ¼ �1.0 and D/J ¼ 0.05.


zero temperature. The stable branches of sublattice magnetizations and octupolar moments order parameters decreaseto zero continuously as the reduced temperature (kT/J) increases, hence the system undergoes a second-order phasetransition (TC), seen in Fig. 1(b). The transition is also from the f3/2 phase to the d phase. The stable branches ofquadrupolar order parameters, namely QA1 and QB1 make a cusp at TC. M2, Q2 and R2 undergo a first-order phasetransition at Tt2, because the discontinuity occurs, seen in the figure. Moreover, the values of M ¼ R ¼ 0.0 below theTC are unstable branches of the magnetizations (M3) and octupolar moments (R3), but Q3 6¼0.0 and it occurs below TC.

(c)
Type 3: For K/J ¼ �0.01 and D/J ¼ 0.5, MA1 ¼MB1 ¼ 1.5, QA1 ¼ QB1 ¼ 2.25, and RA1 ¼ RB1 ¼ 3.5 at zero
kT/J kT/J

0.5 1.51.0 2.0

0

1

2

3

M3, R3

0.0 0.5 1.0 1.5 2.0

0

1

2

3

MA1, MB1

QA1, QB1

RA1, RB1

MA1, MB1

QA1, QB1

RA1, RB1

MA1, MB1

MA1, MB1

MA1, MB1

QA1, QB1

QA1, QB1

Q A1, Q

B1

RA1, RB1

RA1, RB1

R A1, R B1

0.0 0.4 0.8

M, Q

, R

0

1

2

3

0.0 0.1 0.2 0.3

0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.2

0.4

0.6

0.8

1.0

R2 R2 Q3

Q2

Q2

Q2

Q3

Q3

Q3

M2

R2

R2

M2

M2

M2

Q2, Q3

M3, R3

M3, R3 M3, R3

M3, R3

0.0 0.1 0.2 0.3 0.4 0.5

0.00 0.15 0.30 0.45

M,

Q,

RM

, Q

, R

0

1

2

3

Tt

Tt

TCM2

Tt2TC

TC TC

TCM

TCQ

Tfi Tif

TC2 TC

QB1

QB1RB1

RB1

MB1

MB1MA1

MA1

M2

Q3

Q2

R2

RA1 RA1

QA1

QA1

Q2,Q3

0.1 0.2 0.3 0.4

0.4

0.3

0.2

0.5

R2

TtQ2

0.0


temperature, seen in Fig. 1(c). The stable branches of order parameters undergo two successive phase transitions, thefirst one is a first order from the f3/2 phase to the f1/2 phase and the second one is a second order, from the f1/2 phase tothe d phase, seen in Fig. 1(c). MA2 ¼MB2 ¼M2 and RA2 ¼ RB2 ¼ R2 are occur below Tt, but they do not undergo anyphase transitions. M3 ¼ R3 ¼ 0.0 but Q3 6¼0.0 that occur below TC.

(d)
Type 4: This type of behavior is presented for K/J ¼ �0.5 and D/J ¼ 0.5, seen in Fig. 1(d), and is similar to the type 2,except following differences: 1—MA1 ¼MB1 ¼ 0.5, QA1 ¼ QB1 ¼ 0.25, and RA1 ¼ RB1 ¼ 0.125 at zero temperature;hence the transition is from the f1/2 phase to the d phase. 2—The metastable branches of the order parameters do notexist. 3—The values of Q3 is close by QA1 ¼ QB1, not Q2 for low values of kT/J.
(e)
Type 5: For K/J ¼ �2.0 and D/J ¼ �2.5, MA1 ¼ 1.5, MB1 ¼ 0.5; QA1 ¼ 2.25, QB1 ¼ 1.25 and RA1 ¼ 3.5, RB1 ¼ 0.125at zero temperature. On the other hand, MA2 ¼MB2 ¼M2 ¼ 1.125, QA2 ¼ QB2 ¼ Q2 ¼ 2.5, and RA2 ¼ RB2 ¼
R2 ¼ 2.25 at zero temperature. The system undergoes two successive second-order phase transitions, the first one isfrom the ferrimagnetic (i) phase to the antiquadrupolar (a) phase and the second one is from the a phase to the d phase,seen in Fig. 1(d). The metastable branches of the order parameters undergo a second-order phase transition at TC2.M3 ¼ R3 ¼ 0.0 but Q3 6¼0.0 that occur below TCQ.

(f)
Type 6: For K/J ¼ �1.0 and D/J ¼ 0.05, MA1 ¼MB1 ¼ 0.5 and QA1 ¼ QB1 ¼ 0.25 and RA1 ¼ RB1 ¼ 0.1875 at zerotemperature. In this type, only the stable branches of order parameters exist and they undergo three successive second-order phase transitions, seen in Fig. 1(f). The first two second-order transitions; the first one is from the f1/2 phase to thei phase at Tfi and the second one is from the i phase to the f1/2 phase at Tif, imply that the system exhibits a reentrantbehavior, and the third one is from the f1/2 phase to the d phase at TC. This fact is seen the phase diagram of Fig. 7(b)explicitly, compare the figure with Fig. 7(b).
On the other hand, Fig. 2 illustrates the temperature dependence of the sublattice order parameters in the presence of anexternal magnetic field, namely H 6¼0.0, and the behavior depends on D/J, K/J and H/J. Following two fundamental typesof behavior are found.

(a)
Type 1: For K/J ¼ �3.0, D/J ¼ 0.0 and H/J ¼ 3.0, MA1 ¼ 1.5, MB1 ¼ 0.5; QA1 ¼ 2.25, QB1 ¼ 0.25 and RA1 ¼ 3.5,RB1 ¼ 0.125 at zero temperature. The stable branches of order parameters undergo a second-order phase transitionfrom the (i) phase to the (d) phase at Tif, illustrated in Fig. 2(a). Unstable branches of order parameters also occurbelow Tif, but they do not undergo any phase transition.
(b)
Type 2: For K/J ¼ �3.0, D/J ¼ 0.0 and H/J ¼ 12.75, MA1 ¼MB1 ¼ 1.5, QA1 ¼ QB1 ¼ 2.25 and RA1 ¼ RB1 ¼ 3.5 atzero temperature, seen in Fig. 1(c) at zero temperature. In this type, only the stable branches of order parameters existand they undergo two successive second-order phase transitions, seen in Fig. 2(b). The first one is from the f1/2 phase tothe i phase at Tfi and the second one is from i phase to the f1/2 phase at Tif. This implies that the system exhibits areentrant behavior.
Finally, we should mention that the metastable branches of order parameters and their phase transitions imply thatbesides the pure d, f3/2, f1/2, i and a phases in which only the stable branches of order parameters exist, we have followingphases:

(i)
Dense ferromagnetic-32phase (df3/2)
MA1 ¼MB1 ¼ 3=2; QA1aQB1a0 and MA2 ¼MB2a0; QA2 ¼ QB2a0.

(ii)
Dilute ferromagnetic-32 phase (duf3/2)
MA1 ¼MB1 ¼ 3=2; QA1aQB1a0 and MA2 ¼MB2 ¼ 0; QA2 ¼ QB2a0.

(iii)
Dense ferromagnetic-12phase (df1/2)
MA1 ¼MB1 ¼ 1=2; QA1aQB1a0 and MA2 ¼MB2a0; QA2 ¼ QB2a0.

(iv)
Dense ferrimagnetic phase (di)
MA1aMB1a0; QA1aQB1a0 and MA2 ¼MB2a0; QA2 ¼ QB2a0.

(v)
Dense antiquadrupolar phase (da)
MA1 ¼MB1 ¼ 0; QA1aQB1a0 and MBB2 ¼MB2a0; QA2 ¼ QB2a0.

ARTICLE IN PRESS

kT/J

0.0 1.0 2.0

M,

Q,

R

1.0

1.5

2.0

2.5

3.0

2.0

0

1

2

3

QA1

QB1

MA1

MB1M3

Q3

R3

RA1

RB1

QA

QB

MA

MB

RA

RB

M,

Q,

R

Tif

TifTif

0.0 0.5 1.0 1.5 2.5

1.50.5

Fig. 2. Same as Fig. 1, but (a) K/J ¼ �3.0, D/J ¼ 0.0 and H/J ¼ 3.0; (b) K/J ¼ �3.0, D/J ¼ 0.0 and H/J ¼ 12.75.

M. Keskin, O. Canko / Journal of Magnetism and Magnetic Materials 320 (2008) 8–24 15

Again, we have not used RA and RB to define these phases due to the reason that the behavior of the metastable branches ofRA2 and RB2 are similar to the MA2 and MB2.

4. Dynamics of the system

In this section, the PPM with the point distribution of Kikuchi [15] used in order to derive a possible set of dynamic orrate equations, which describes the dynamical behavior of the system. The PPM is the natural extension into the timedomain of the cluster variation method (CVM) and provides a systematic derivation of the rate equations for successiveapproximation which are well known in the equilibrium statistical mechanics. It has been successfully applied to describethe nonequilibrium behavior of a number of homogeneous and inhomogeneous stationary systems, such as substitutionaldiffusion in ordered systems [16], diffusion and ionic conductivity in solid electrolytes [17], the kinetics of theorder–disorder transformation in body-centered-cubic (bcc) alloys [18] and a binary alloy [19]. It was also used toinvestigate the dynamics of following systems: The spin-1

2Ising model [20], spin-1 Ising systems [21,22], the isotropic spin-3

2

Ising model [23], phonon and atomic diffusion systems [24], a ternary system [25] and the microscopic mechanism of thecurrent-induced domain conversion phenomena on the Si (0 0 1) vicinal surface [26]. We should also mention that effortshave been made to show how the PPM can be used to evaluate atomistic parameters combining with experiments [27].Moreover, the PPM has been applied to study the influence of the interface disorder on the electronic properties of thesemiconductor heterostructures [28]. In addition, it has been employed to study the configurational kinetics of the disorder-L12 transition [29]. The kinetic evolution processes for a disorder-B2 transition has been also obtained by the PPM [30].The method was used to study the simple dynamics of voltage-gated ion channels [31].


In this method the rate of change of the state variables is written as

dX i

dt¼Xiaj

ðXji �XijÞ; (11)

where Xij is the path probability rate for the system to go from state i to j. The coefficients Xij are the product of threefactors: kij the rate constants with kij ¼ kji, a temperature-dependent factor which guarantees that the time-independentstate is the equilibrium state and a third factor which is the fraction of the system that is in the state i, e.g., Xi. Detailedbalancing requires that

Xij ¼ Xji. (12)

The following two options were introduced by Kikuchi [15]:

ðAÞ Xij ¼ kijZ�1X i exp �

b2

qE

qX i

�qE

qX j

� � �, (13a)

ðBÞ Xij ¼ kijZ�1X iexp �b

qE

qX i

�, (13b)

which both fulfill the necessary requirements expressed by Eq. (12), and again Z is the partition function, and E is theinternal energy which is given in Eq. (8). Assumption A is called recipe I and B is called recipe II by Kikuchi [15]. The rateconstants kij can be function of the temperature. The simplest assumption for this temperature dependence is to use anArrhenius factor: k0 e

�u=kBT , where u is the activation energy, which ensures that the rate will go to zero at T ¼ 0 asrequired. The rate constants are chosen as follows: The first rate constant is k12 ¼ k34 ¼ k1 which corresponds to thetranslation of the particles through the lattice. The second rate constant k14 ¼ k23 ¼ k2 associated with rotation of aparticle on a given site and the third one is k13 ¼ k24 ¼ k3 that is associated with the simultaneous translation and rotationof particles, hence we will take k3 ¼ k1 k2. It is assumed that double processes, the simultaneous translation or rotation oftwo particles do not take place. The occurrence of the rate constants is given in Table 1.

We will use the recipe II in order derive the dynamic equations because the general behavior of the solution of thedynamic equations, namely relaxation curves and also flow diagrams, is not drastically changed, i.e., the lines flow moreand less the same pattern using either recipe I or II [21,32]. Using Eqs. (3), (8), (11) and (13b), the set of dynamic equationsfor the order parameters are obtained:

ZA;B dMA;B

dt¼ k1

98eA;B1 þ

124eA;B2 þ

124eA;B3 þ

98eA;B4

� �MA;B þ �

14eA;B1 �

14eA;B2 þ

14eA;B3 þ

14eA;B4

� �QA;B

�þ �1

2eA;B1 �

16eA;B2 �

16eA;B3 �

12eA;B4

� �RA;B þ

916eA;B1 þ

116eA;B2 �

116eA;B3 �

916

eA;B4

� � þ k2

18eA;B1 �

98eA;B2 �

98eA;B3 þ

18eA;B4

� �MA;B þ

34eA;B1 �

14eA;B2 þ

14eA;B3 �

34eA;B4

� �QA;B

�þ �1

2eA;B1 þ

12eA;B2 þ

12eA;B3 �

12eA;B4

� �RA;B þ �

316eA;B1 þ

916eA;B2 �

916eA;B3 þ

316eA;B4

� � þ k3 �

94eA;B1 þ

112

eA;B2 þ

112eA;B3 �

94eA;B4

� �MA;B þ �

12eA;B1 þ

12eA;B2 �

12eA;B3 þ

12eA;B4

� �QA;B

�þ eA;B

1 �13eA;B2 �

13eA;B3 þ eA;B

4

� �RA;B þ

98eA;B1 �

18eA;B2 þ

18eA;B3 �

98eA;B4

� � ,

ZA;B dQA;B

dt¼ 2k1

98eA;B1 þ

124eA;B2 �

124eA;B3 �

98eA;B4

� �MA;B þ �

14eA;B1 �

14eA;B2 �

14eA;B3 �

14eA;B4

� �QA;B

�þ �1

2eA;B1 �

16eA;B2 þ

16eA;B3 þ

12eA;B4

� �RA;B þ

916eA;B1 þ

116eA;B2 þ

116eA;B3 þ

116eA;B4

� � þ 2k3 �

98e

A;B1 �

124e

A;B2 þ

124e

A;B3 þ

98e

A;B4

� �MA;B þ �

14e

A;B1 �

14e

A;B2 �

14e

A;B3 �

14e

A;B4

� �QA;B

�þ 1

2eA;B1 þ

16eA;B2 �

16eA;B3 �

12eA;B4

� �RA;B þ

916eA;B1 þ

116eA;B2 þ

116eA;B3 þ

916eA;B4

� � ,

Table 1

The description of the rate constants, k3 is taken as k1k2

X1 X2 X3 X4

X1 k1 k3 k2X2 k1 k2 k3X3 k3 k2 k1X4 k2 k3 k1


ZA;B dRA;B

dt¼ k1

11732eA;B1 þ

1396eA;B2 þ

1396eA;B3 þ

11732eA;B4

� �MA;B þ �

1316eA;B1 �

1316eA;B2 þ

1316eA;B3 þ

1316eA;B4

� �QA;B

�þ �13

8eA;B1 �

1324eA;B2 �

1324eA;B3 �

1396eA;B4

� �RA;B þ

11764eA;B1 þ

1364eA;B2 �

1364eA;B3 �

11764eA;B4

� � þ k2

932eA;B1 �

932eA;B2 �

932eA;B3 þ

932eA;B4

� �MA þ

2716eA;B1 �

116eA;B2 þ

116eA;B3 �

2716eA;B4

� �QA;B

�þ �9

8eA;B1 þ

18eA;B2 þ

18eA;B3 �

98eA;B4

� �RA;B þ �

2764eA;B1 þ

964eA;B2 �

964eA;B3 þ

2764eA;B4

� � þ k3 �

6316eA;B1 þ

748eA;B2 þ

748eA;B3 �

6316eA;B4

� �MA;B þ �

78eA;B1 þ

78eA;B2 �

78eA;B3 þ

78eA;B4

� �QA;B

�þ 7

4eA;B1 �

712eA;B2 �

712eA;B3 þ

74eA;B4

� �RA;B þ

6332eA;B1 �

732eA;B2 þ

732eA;B3 �

6332eA;B4

� � ,

where

eAi ¼ exp �

bN

qE

qX Ai

!; eB

j ¼ exp �bN

qE

qX Bj

!; ZA ¼

X3i¼1

eAi and ZB ¼

X3j¼1

eBj :

Subscripts A and B indicate the set of dynamic equations for the A and B sublattices, respectively, hence we have sixcoupled differential equations.

These dynamic equations are solved by two different methods: The first one is to express the solution of the equations bymeans of a flow diagram [33], which shows the solution of these equations in a three-dimensional phase space of M, Q andR, starting with initial values very close to the boundaries. As time progresses by given small steps, the values of M, Q andR are computed and the point representing them moves in the plane. A set of solution curve is obtained by considering alldifferent initial values. The results are presented in Fig. 3 for K/J ¼ �0.01, D/J ¼ 0.5 and kT/J ¼ 0.18 that corresponds toFig. 1(c) at kT/J ¼ 0.18. In the figure, the open circle is the stable equilibrium solution which corresponds to the lowestvalues of the free energy or the deepest minimum, the filled square is the metastable state because the system relaxes into itand it does not correspond to the deepest minimum but corresponds to the secondary minimum, and the filled circle is theunstable solution or state which corresponds to local maxima (the peaks) or saddle point. Solid curves representk1 ¼ k2 ¼ 1 and the dashed curves represent k1 ¼ 1 and k2 ¼ 10. If one studies this figure, one can see that the systemrelaxes into only two different states. One is the stable state (MA1 ¼MB1 ¼ 1.346, QA1 ¼ QB1 ¼ 1.943, andRA1 ¼ RB1 ¼ 2.877) which corresponds to the lowest values of the free energy or the deepest minimum, and the other isthe metastable state (MA2 ¼MB2 ¼ 0.494, QA2 ¼ QB2 ¼ 0.353, and RA2 ¼ RB2 ¼ 0.278) which does not correspond to thedeepest minimum but corresponds to the secondary minimum. Moreover, the unstable solution (MA3 ¼MB3 ¼ 0.0,QA3 ¼ QB3 ¼ 0.257, and RA3 ¼ RB3 ¼ 0.0) marked with a filled circle can be seen explicitly, because it is seen as a saddle

Fig. 3. The flow diagram of the system in a three-dimensional phase space for K/J ¼ �0.01, D/J ¼ 0.5, kT/J ¼ 0.18 and two different sets of rate

constants: (solid) k1 ¼ k2 ¼ 1 and (dashed) k1 ¼ 1, k2 ¼ 10. Subscript 1 denotes the stable state, 2 metastable state, and 3 unstable state. The open circle

corresponds to the stable solution, the closed square to the metastable solution and closed circles is the unstable solution.


point. If one compare this figure with Fig. 1(c) at kT/J ¼ 0.18, one can see that the stable and metastable states are foundby the dynamic study are exactly the same with the equilibrium study, but to obtain the unstable state by the dynamic studyis a tedious procedure. The second one is the Runge–Kutta method. We use this method to study relaxation curves of theorder parameters and to see the flatness property of the metastable state and as well as overshooting phenomena.Relaxation curves of order parameters are seen in Fig. 4 for K/J ¼ �0.01, D/J ¼ 0.5 and kT/J ¼ 0.18. Thick solid curvesrepresent k1 ¼ k2 ¼ 1 and the thin solid curves represent k1 ¼ 1 and k2 ¼ 10. If one compares this figure with Figs. 3 and1(c) at kT/J ¼ 0.18, one can see that the stable and the metastable solutions coincide each other exactly. However, onecannot obtain the unstable solutions from this method. Moreover, one can obtain following three important results fromFigs. 3 and 4. (1) If the initial values close to the metastable solution or state, the system relaxes into it, otherwise into thestable state. (2) If k24k1 and initial values are not very close to the stable state, the system relaxes into the metastable statemore than k1 ¼ k2. These two facts or results have been also observed experimentally [34]. For example, in order to obtainan amorphous metallic alloy or a metallic glass that corresponds to a metastable phase or state, one needs following twofundamental conditions: (i) to use certain compositions for certain alloys. Hence, the initial conditions are very importantto obtain a metastable phase or state and (ii) to cool these well-prepared liquid alloys very rapidly, therefore the values ofrate constants are very important. (3) The flatness properties of metastable states and overshooting phenomena [18,35]have also seen in Fig. 4. In this stage, it is worthwhile to mention that if the systems stay in their metastable state or phase,their properties change drastically. For example, the rapid cooling alloys leads to amorphous structures, namely metastablephase and it is known that the properties of alloys improve significantly [34]. On the other hand, the metastability isbecoming a serious problem in high-performance very large scale integration (VLSI) in complimentary metal–oxidesemiconductor dynamic D-latch [36]. Recently, Bouabci and Carneiro [37], and Rachadi and Benyoussef [38] havepresented the Monte Carlo cluster algorithm to eliminate the metastability in the first-order phase transitions of the spinmodels. We should also note that these figures, especially Fig. 3, suggest that how one can avoid that the system relaxesinto the metastable state.

Finally, in order to explore the flow diagrams in more detail we have obtained the free energy surface in the terms M, Q

and R for K/J ¼ �0.01, D/J ¼ 0.5 and kT/J ¼ 0.18 that corresponds Figs. 3 and 4, and Fig. 1(c) for given values of kT/J¼ 0.18, seen in Fig. 5. In this figure, the open circle is the stable equilibrium solution which corresponds to the lowestvalues of the free energy or the deepest minimum, the filled square is the metastable state because the system relaxes into itand it does not correspond to the deepest minimum but corresponds to the secondary minimum, and the filled circle is theunstable solution or state which corresponds to local maxima (the peaks) or saddle point. Hence, the stable, metastable andunstable states are seen very clearly on the free energy surface in terms of order parameters, i.e. Fig. 5. If one can compares

Fig. 4. Relaxation curves of the order parameters M, Q and R for K/J ¼ �0.01, D/J ¼ 0.5, kT/J ¼ 0.18 and different set of values the rate constants:

k1 ¼ k2 ¼ 1 (thick solid lines), k1 ¼ 1 and k2 ¼ 10 (thin solid lines). Subscript 1 and 2 represent to the stable state and metastable state, respectively.

ARTICLE IN PRESS

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.0

0.5

1.0

0.00.5

1.01.5

2.0

f

M

Q

Fig. 5. The free energy surface in terms of M and Q for K/J ¼ �0.01, D/J ¼ 0.5, kT/J ¼ 0.18. The open circle corresponds to the stable solution, the

closed square to the metastable solution and closed circle is the unstable solution.


this figure with Figs. 1(c) at kT/J ¼ 0.18 and Figs. 3 and 4, it is easily seen the stable, metastable and unstable solutionsexactly coincide with each other.

5. The metastable phase diagrams in addition to the equilibrium phase diagrams

In this section, we present the metastable phase diagrams in addition to the equilibrium phase diagrams of the spin-32

Ising BEG model for K/Jp0 since we make sure that the metastable and unstable branches of the order parametersobtained completely and correctly in Section 4. The critical or second-order phase transition temperatures for the orderparameters in the case of a second-order phase transition are calculated numerically by the investigation of the behavior ofthe order parameters as a function of the temperature as follow. If the order parameters become equal as reduced thetemperature is lowered and the temperature where the order parameters become equal is the critical temperature or theorder parameters decrease to zero continuously as the reduced temperature increases, the temperature whereMA1 ¼MB1 ¼ 0.0 and RA1 ¼ RB1 ¼ 0.0 (also MA2 ¼MB2 ¼ 0.0 and RA2 ¼ RB2 ¼ 0.0) is the second-order phasetransition temperature. QA1 and QB1 (also QA2 and QB2) make a cusp at this temperature. On the other hand, the first-order phase transition temperatures for the stable branches of order parameters are found by matching the values of thetwo branches of the free energy followed while increasing and decreasing the temperature. The temperature at which thefree energy values equal is the first-order phase transition temperature (Tt) for the stable branches order parameters.Furthermore, the first-order phase transition temperature (Tt2) for the metastable branches of the order parameters is thetemperature where the discontinuity occurs first for SA2, SB2; QA2, QB2 and RA2, RB2.

We can now obtain the equilibrium and metastable phase diagrams of the spin-32 Ising BEG model and the calculatedphase diagrams are presented in Figs. 6 and 7. In these phase diagrams, thin dashed and thick solid lines represent the first-and second-order phase transitions for the stable branches of the order parameters; thin and thick dashed-dotted linesindicate the first- and second-order phase transitions of the metastable branches of the order parameters, respectively. Thedotted line separates the different phases. A, T, D and Z are the special points for the equilibrium phase diagrams in whichdenote the multicritical, tricritical, double critical and zero-temperature critical points, respectively. Am, Dm and Em are themulticritical, double critical and critical end points for the metastable phase diagrams, respectively.

Fig. 6 shows the phase diagram of the model in the absence of an external magnetic field in the (kT/J, K/J) plane forvarious values of D/J. Four different phase diagrams topologies, in which three of them contain the metastable phases,have been found in this plane, the topology depending on D/J values.

(a)
For D/J ¼ �0.5; besides the disordered phase (d) and the ferromagnetic-32(f3/2), the ferromagnetic-1
2(f1/2), ferrimagnetic

(i), antiquadrupolar phase (a) also exits in the equilibrium phase diagram, seen in Fig. 6(a). The phase diagram alsocontains the dense ferrimagnetic (di), where the metastable and unstable branches of order parameters occur besides

ARTICLE IN PRESS

012

0.0

0.5

1.0

1.5

2.0

-0.50.00.5

kT

/J

0.0

0.5

1.0

1.5

d

-4-20

kT

/J

0.0

0.5

1.0

1.5

a

i

d

A

Z Z

D

EmT

A

K/J

df3/2

di

K/J

-3-2-10

0.0

0.5

1.0

1.5

d

i

da

Z Z

df1/2

f1/2

f3/2

Amf3/2

f3/2

f1/2

f1/2

f3/2

f1/2

T

Dm

df3/2

duf3/2

d

Fig. 6. The equilibrium phase diagrams (thick solid and thin dashed lines) and metastable phase diagrams (thin and thick dashed-dotted lines) of the spin-32

BEG model with the repulsive biquadratic interaction in the (kT/J, K/J) plane. The dashed and thin dashed-dotted lines represent the first-order phase

transition for the stable and metastable branches of the order parameters, respectively. Thick solid and thick dashed-dotted lines represent the second-

order phase transition for the stable and metastable branches of the order parameters, respectively. The ferromagnetic-32(f3/2) and ferromagnetic-1

2(f1/2),

dense ferromagnetic-32and -1

2(df3/2 and df1/2), dilute ferromagnetic-3

2(duf3/2), ferrimagnetic (i), dense ferromagnetic (di), antiquadrupolar (a), dense

antiquadrupolar (da) and disordered (d) phases are found. The dotted line separates the different phases. A, T, D and Z are the special points in which

denote the multicritical, tricritical, double critical and zero-temperature critical points, respectively. Am, Dm and Em are the multicritical, double critical

and critical end points for the metastable phase diagram, respectively. (a) D/J ¼ �0.5; (b) D/J ¼ 0.0; (c) D/J ¼ 0.5; (d) D/J ¼ 1.0.

M. Keskin, O. Canko / Journal of Magnetism and Magnetic Materials 320 (2008) 8–2420

the stable branches, and the dense antiquadrupolar phase (da) where the metastable and unstable branches of orderparameters occur besides the stable branches. All the phase boundaries among these phases are second-order lines. Thephase diagrams also exhibits the following special points: The two multicritical (A), and two zero-temperature critical(Z) points for equilibrium phase diagram and one multicritical (Am) for the metastable phase diagram. We should alsomention that the equilibrium phase diagram exhibits a reentrant behavior, e.g., as the reduced temperature is lowered,

ARTICLE IN PRESS

-4 -2 0

kT

/J

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

f3/2

i

f1/2

da

D/J

0

kT

/J

0.0

0.5

1.0

1.5

2.0

d

d

Z Z

T

TEm

D

A

Ad

a

-2 -1 0

0.0

0.5

1.0

1.5

i

d

A

-1.5 -1.0

kT

/J

0.0

0.5

1.0

1.5

i

d

ZZ

ZZ

di

df3/2

df 1/2

f1/2

f3/2

f1/2df3/2

f3/2

Am

f 3/2

f 1/2

f3/2

f1/2

Dm

duf3/2

-0.5 0.0 0.5 0.0 0.5 1.0

2 4

Fig. 7. Same as Fig. 6 but in the (kT/J, D/J) plane. (a) K/J ¼ �2.0; (b) K/J ¼ �1.0; (c) K/J ¼ �0.68; (d) K/J ¼ 0.0; (d) K/J ¼ 1.25.


there are transitions from the d phase to the a phase, from the a phase to the i phase and from the i phase to the f1/2phase. Moreover, the phase boundaries between the a and da phases, and between the i and di phases are the second-order phase lines for the metastable branches of order parameters. The boundaries between the i phase and a phase,and between the da and di phases are the second-order phase lines for the stable branches of order parameters.


This equilibrium phase diagram has only been obtained by the MFA [12], which is either absent from previousapproaches, such as the MC calculation, RG techniques, the EFT, etc., or has gone unnoticed.

(b)
For D/J ¼ 0.0, the phase diagram presents the d, f3/2, f1/2 and i phases and it exhibits only two special Z points,illustrated in Fig. 6(b); hence the system does not exhibit a metasatable phase diagram. The i phase lies at lowtemperatures and the phase boundaries among these phases are all second-order lines. One should also notice apronounced reentrance occurring in this diagram. The similar equilibrium phase diagram has been obtained by usingthe MC calculation [6], EFT [7], and CVMPA for qo6 (q is the coordination number) [8] as well as in the exactformulation of the model on Bethe lattice [10,11] for qo6, but only differs in that the reentrant behavior does notfind in these works. However, the exactly similar equilibrium phase diagram was obtained by the MFA [6], and also inRefs. [8,11] for q46.
(c)
For D/J ¼ 0.5, the equilibrium phase diagram is similar to Fig. 6(b), except the i phase and as well as two zero-temperature critical points (Z) disappear, seen in Fig. 1(c). The phase boundary between the f3/2 and the f1/2 phase is afirst-order line that starts from the zero temperature and terminates a double critical end (D) point, where two differentcritical systems coexist for the equilibrium phase diagram. On the other hand, besides the d, f3/2 and f1/2 phases we havethe dense ferromagnetic-3
2(df3/2) and dense ferromagnetic-1

2(df1/2) phases, where the metastable and unsatable branches

of order parameters occur besides the stable branches, occurs for low values of reduced temperature. The first-orderphase line for the metastable branches of order parameters separates the f3/2 phase from the df3/2 phase and the first-order phase line for the stable branches of order parameters separates the f1/2 phase from the df1/2. Moreover, thindotted line separates the df3/2 phase from the df1/2 phase. We should also mention that the similar equilibrium phasediagram, except the first-order phase line, has been obtained in ferromagnetic spin-32 BEG model within the CVMPA [8]and the exact formulation of the model on the Bethe lattice by using the exact recursion equations [10].

(d)
For D/J ¼ 1.0, the equilibrium phase diagram contains first-order and second-order phase transition lines, seen inFig. 6(d). Besides two tricritical points (T), the double critical (Dm) and critical end (Em) points for the metastable phasediagram also exist. The phase boundary between the f3/2 and d phase for very high values of the reduced temperatures(kT/J) and the boundary between the f1/2 and d phases for very low values kT/J are second-order phase lines. Betweenthese very high and low values of kT/J, the first-order phase line occurs and it separates the f3/2 phase from the d phase.Therefore, two tricritical points (T) exist in the equilibrium phase diagram. On the other hand, the df3/2 and duf3/2phases occur for low values of kT/J. The phase boundaries between the df3/2 phase and duf3/2 phase for low values ofK/J, and between the df3/2 phase and f3/2 phase for high values of K/J are the second-order phase lines for themetastable branches of magnetization and also octupolar order parameters. The boundaries between the f3/2 phase anddf3/2 phase is the first-order phase line for the metastable branches magnetizations and also the octupolar orderparameters below Em, and between the f3/2 phase and duf3/2 phase is also a first-order line for the metastable branchesof quadrupolar order parameters above Em. The dotted line separates the f3/2 phase from the duf3/2 phase. Thisequilibrium phase diagram is also only found by the MFA, more precisely the LACVM [12], which is either absentfrom the other approximations or the exact calculation on the Bethe lattice or has gone unnoticed.
Fig. 7 illustrates the phase diagram of the model in the (kT/J, D/J) plane for various values of K/J for the absence of anexternal magnetic field. Study of the phase diagram in the (kT/J, D/J) plane yields five fundamental phase diagramsdepending on the value of K/J. (i) Fig. 7(a) represents the metastable phase diagram in addition to the equilibrium phasediagram for K/J ¼ �2.0. In the equilibrium phase diagram, besides the ordered phase (d) and the ferromagnetic-3

2(f3/2), the

ferromagnetic-12(f1/2), ferrimagnetic (i) and antiquadrupolar (a) phases also exit and all the phase boundaries among these

phases are second-order lines. The equilibrium phase diagram also exhibits two multicritical (A) and two zero-temperaturecritical points (Z). We should also mention that the reentrance also occurs for the second-order phase transition line whichseparates i phase (actually di phase, because the metastable phase is considered) from the f3/2 phase. The similarequilibrium phase diagram topology has been obtained in ferromagnetic spin-3

2BEG model on the Bethe lattice by using

the exact recursion equations by Ekiz et al. [10], except the following differences: (1) the reentrant behavior has not beenobserved for the second-order phase transition line in which separates i phase from the f3/2 phase, (2) the phase boundarybetween the a and d phases is a first-order phase line. On the other hand, the behavior of the metastable phase diagram isvery similar to the Fig. 6(a); hence the system exhibits the di and da phases. (ii) Fig. 7(b) shows only the equilibrium phasediagram for K/J ¼ �1.0; hence the system does not exhibit the metastable phase diagram. The phase diagram is similar tothe equilibrium phase diagram of Fig. 7(a), except that the a phase disappears, hence only one multicritical point exist. Thisequilibrium phase diagram is in a very good agreement with the work of Ekiz et al. [10]. (iii) For K/J ¼ �0.68, this phasediagram is illustrated in Fig. 7(c) that only the equilibrium phase diagram exists. The similar equilibrium phase diagramwas found by using the RG approach of the Migdal–Kadanoff type [9] and the exact recursion equations [10] but onlydiffers from these works in that the reentrant behavior does not occur for the second-order phase transition line in whichseparates i phase from the f3/2 and f1/2 phases. (iv) Fig. 7(d) illustrates the metastable phase diagram in addition tothe equilibrium phase diagram for K/J ¼ 0.0, the topology of this phase diagram is very similar to the phase diagram of


Fig. 6(c), except obtained in the (kT/J, D/J) plane. The equilibrium phase diagram is in very good agreement with theequilibrium phase diagram found by the MFA and MC Calculation [6], the EFT [7] and the exact recursion equations [10].(v) For K/J ¼ 1.25, the topology of the equilibrium and metastable phase diagrams are also very similar to the phasediagram of Fig. 6(d), except obtained in the (kT/J, D/J) plane, seen in Fig. 7(e). This equilibrium phase diagram is also onlyobtained by the MFA [12], which is either absent from the other approximations or the exact calculation on the Bethelattice or has gone unnoticed.

Finally, we should also mention that the system does not exhibit the metastable phase diagrams in the presence of anexternal magnetic field in the (kT/J, H/J) and (kT/J, K/J) planes. Since the equilibrium phase diagrams in these planes werepresented and discussed in Ref. [12] in detail, we will not give the equilibrium phase diagram again in this work.

6. Summary and conclusion

In this paper, we have investigated the thermal variations of the order parameters of the spin-32 Ising BEG model withrepulsive biquadratic interaction by using the LACVM, which is identical the MFA, in the absence and the presence of theexternal magnetic field. Besides the stable branches of the order parameters, we obtain the metastable and unstable parts ofthese curves and phase transitions of the metastable branches of the order parameters are also investigated. Unstablebranches of order parameters do not undergo any phase transitions. The classification of the stable, metastable andunstable states is made by matching the free energy values of these solutions. We also studied the dynamics of the model bythe path probability method with the point distribution [15] in order to make sure that we have found and defined themetastable and unstable branches of the order parameters completely and correctly. Then, we have constructed themulticritical phase diagrams of the spin-32 BEG model with repulsive biquadratic interaction including the phase transitionsof metastable branches of the order parameters. Therefore, we presented the metastable phase diagrams in addition to theequilibrium phase diagrams for the model with in the (kT/J, K/J) and (kT/J, D/J) planes. In the (kT/J, K/J) plane, wefound that the behavior of the system strongly depends on the values of the D/J and four different phase diagramtopologies have been found, seen in Fig. 6. Only, one of them, namely Fig. 6(b), does not contain the metastable phasediagram. In the (kT/J, D/J) plane, we found that the behavior of the system strongly depends on the values of the K/J andfive fundamental phase diagrams have been found, illustrated in Fig. 7. In this case, two of them, namely Figs. 7(b) and (c),do not contain the metastable phase diagram. We have not presented the phase diagram of the model in the presence of anexternal magnetic field in the (kT/J, H/J) and (kT/J, K/J) planes. The reason is that the system does not exhibit themetastable phase diagrams and the equilibrium phase diagrams were presented and discussed in Ref. [12], extensively.

Finally, it should be also mention that we have found that the metastable phase diagrams of the spin-32Ising BEG model

for K/Jp0, which has served as a paradigm for a large number of physically important phenomena, always exists at the lowtemperatures that are consistent with experimental and theoretical works on some alloys [39–41], semiconductors [42–45],polymers [46], water [47] and the ternary system [47]. Moreover, the spin-1 BEG model have been used to calculate themetastable phase diagrams of some real systems such as Cu–Al–Mn shape-memory alloys [41], semiconductor alloys[44,45] and the ternary system [48] and the unstable continuation of the second-order phase transition line in(AIIIBV)1�xC

IV2 x-semiconductor alloys [45]. We hope that our investigation stimulates further theoretical and experimental

works, especially, to obtain the metastable phase diagrams within more accurate techniques, such as the PPM with pairdistribution, dynamical Monte Carlo simulations, etc. and also to use the spin-3

2Ising systems, or its variations, to calculate

the metasable phase diagrams of the some real systems. Therefore, the system studied in the paper may be simple, butfruitful from the theoretical and material science points of view.

Acknowledgments

This work was supported by the Research Fund of Erciyes University, Grant numbers: FBT-05-03 and FBA-06-01. Thepart of this work was also supported by the Scientific and Technological Research Council of Turkey (TUBITAK) Grantno: 105T114. We would also like to thank M. Ali Pınar and Muharrem Kirak for some of the numerical calculations.

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