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Physica A 364 (2006) 263–270

www.elsevier.com/locate/physa

Multicritical behavior of the antiferromagnetic spin-32

Blume–Emery–Griffiths model

Mustafa Keskina,�, M. Ali Pınarb, Ahmet Erdinc-a, Osman Cankoa

aDepartment of Physics, Erciyes University, 38039 Kayseri, TurkeybInstitute of Science, Erciyes University, 38039 Kayseri, Turkey

Received 9 June 2005; received in revised form 22 August 2005

Available online 17 October 2005

Abstract

The multicritical behavior of the antiferromagnetic spin-32Blume–Emery–Griffiths model in an external magnetic field is

studied within the lowest approximation of the cluster variation method. We have investigated the thermal variations of

order parameters for different values of interaction parameters and external magnetic field and constructed the resulting

phase diagrams. The model exhibits distinct critical regions, including the first-order, second-order tricritical point, double

critical point and zero critical point. Comparison of the phase diagrams with closely related systems is made.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Spin-32Blume–Emery–Griffiths model; Cluster variation method; Phase diagrams

1. Introduction

The spin-32Blume–Emery–Griffiths (BEG) model is described by the Hamiltonian in an external magnetic

field

H ¼ �JXhi;ji

SiSj � KXhi;ji

S2

i S2

j þDX

i

S2

i þX

j

S2

j

!�H

Xi

Si þX

j

Sj

!, (1)

where the spins Si located at site i on a discrete lattice can take the values �32and �1

2, and the first two

summations run over all nearest-neighbor pair of sites. J, K, D and H describe the bilinear interaction,biquadratic interaction, single-ion anisotropy and effect of an external magnetic field, respectively. TheHamiltonian and phase diagrams are invariant under the transformations (H-�H and S-�S). The spin-3

2

BEG model is the most general spin-32Ising model. The spin-3

2Ising model Hamiltonian with only J and D

interaction is known as the spin-32Blume–Capel (BC) model, and the spin-3

2Ising model Hamiltonian with only

J and K interaction is known as the isotropic spin-32BEG model.

The critical properties of the spin-32BEG model for K=JX0 have been studied and its phase diagrams have

been calculated by renormalization-group (RG) techniques [1], the effective field theory (EFT) [2], the Monte

e front matter r 2005 Elsevier B.V. All rights reserved.

ysa.2005.08.077

ing author. Tel.: +90352 4374901x33105; fax: +90352 4374933.

ess: keskin@erciyes.edu.tr (M. Keskin).

ARTICLE IN PRESSM. Keskin et al. / Physica A 364 (2006) 263–270264

Carlo (MC) and a density-matrix-RG method [3]. An exact formulation of the spin-32BEG model on a Bethe

lattice was investigated by using the exact recursion equations [4]. The ferromagnetic spin-32BEG model with

repulsive biquadratic coupling, i.e., K=Jo0, has been also investigated. An early attempt to study theferromagnetic spin-3

2BEG model was made by Barreto and Bonfim [5] and Bakkali et al. [6] within the mean-

field approximation (MFA) and MC calculation, and the EFT, respectively. Barreto and Bonfim calculatedonly the phase diagrams for the isotropic spin-3

2BEG model. Bakkali et al. also presented two phase diagrams,

one for the spin-32BC model and one for the isotropic spin-3

2BEG model. Tucker [7] studied the spin-3

2BEG

model with K=Jo0 by using the cluster variation method in pair approximation (CVMPA) and onlypresented the phase diagrams of the spin-3

2BC and isotropic spin-3

2BEG model for several values of the

coordination number. Bakchich and Bouziani [8] calculate the phase diagram of the model only in (T/J,D/J)for the two different values of K/J within an approximate RG approach of the Migdal–Kadanoff type.Recently, Ekiz et al. [9] investigated the model on Bethe lattice using the exact recursion equations andpresented the phase diagrams in (kT/J,K/J) plane for several values of D/J and in (kT/J,D/J) plane for severalvalues of K/J. Ekiz [10] extended the previous work, i.e. Ref. [9], for the presence of an external magnetic field.He considered only the ferromagnetic case.

On the other hand, as far as we know, multicritical behavior of the antiferromagnetic spin-32BEG model has

not been investigated whereas that of the antiferromagnetic spin-32BC model was studied by Bakchich et al.

[11] within the MFA, Bekhechi and Benyoussef [12] within the transfer matrix finite-size-scaling (TMFSS)calculations and MC simulations, Ekiz [13,14] by means of the exact recursion relations on Bethe lattice andby Keskin at al. [15] within the LACVM. These works [11–15] show that the antiferromagnetic spin-3

2BC

model exhibits a rich variety of behavior: second-order, first-order and critical points of different order.Therefore, the purpose of this work is to study multicritical behavior of the antiferromagnetic spin-3

2BEG and

to calculate the phase diagrams by using the LACVM which is identical to the MFA. Our recent works [15,16]display that the LACVM, in spite of its limitations such as the correlations of spin fluctuations not beingconsidered, is an adequate starting point in which within this theoretical framework, it is easy to determine thecomplete phase diagrams. It also predicts the existence of the multicritical points. It is worthwhile to mentionthat the spin-3

2BEG model with the J and K nearest-neighbor interactions was introduced to explain phase

transition in DyVO4 [17] qualitatively within the MFA by Sivardiere and Blume [18]. Later, this model wasused in a study of tricritical properties of a ternary fluid mixture and compared with the result of theexperimental observations on the system of ethanol–water–carbon dioxide [19]. Moreover, the study of theantiferromagnetic phase is an important and actual area in magnetism.

The remainder of this work is organized as follows. In Section 2, we define the model briefly and obtain itssolutions at equilibrium within the LACVM. Thermal variations of the order parameters are investigated inSection 3. In Section 4, transition temperatures are calculated precisely and the phase diagrams are presentedin (H/|J|,kT/|J|) plane. Section 5 contains the summary 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

2

on sites of sublattices A and B, respectively. The average value of each of the spin states will be denoted by XA1 ,

XA2 , XA

3 and XA4 on the sites of sublattice A and XB

1 , XB2 , XB

3 and XB4 on sublattice B, which are also called the

state or point variables. XA1 ,X

B1 ; XA

2 ,XB2 ; XA

3 ,XB3 and XA

4 ,XB4 are the fractions of the spin values þ3

2, þ1

2, �1

2and

�32on A and B sublattices, respectively. In order to account for the possible two-sublattice structure we need

six long-range order parameters which are introduced as follows: MA � hSA

i i, QA � hðSA

i Þ2i � 5

4,

RA �53hðSA

i Þ3i � 41

12hSA

i i, MB � hSB

j i, QB � hðSB

j Þ2i � 5

4, RB �

53hðSB

j Þ3i � 41

12hSB

j i for A and B sublattices,respectively. MA and MB are the average magnetizations which is the excess of one orientation over theother, called magnetizations; QA and QB are the quadrupolar moments which are the average squaredmagnetizations; and RA and RB are the octupolar-order parameters for A and B sublattices, respectively. Thesublattice magnetizations define four different phases of the antiferromagnetic spin-3

2BEG model: (i) The

disordered (D) phase with MA ¼MB40, (ii) the antiferromagnetic (AF) phase with MA ¼ �MBa0, (iii) theantiferrimagnetic (AI) phase with MAa�MBa0 and (iv) the ferrimagnetic (I) phase with MAaMBa0.

ARTICLE IN PRESSM. Keskin et al. / Physica A 364 (2006) 263–270 265

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

MA � hSA

i i ¼32ðXA

1 � XA4 Þ þ

12ðXA

2 � XA3 Þ,

MB � hSB

j i ¼32ðXB

1 � XB4 Þ þ

12ðXB

2 � XB3 Þ,

QA � hðSA

i Þ2i � 5

4¼ XA

1 � XA2 � XA

3 þ XA4 ,

QB � hðSB

j Þ2i � 5

4¼ XB

1 � XB2 � XB

3 þ XB4 ,

RA �53hðSA

i Þ3i � 41

12hSA

i i ¼12ðXA

1 � XA4 Þ þ

32ðXA

3 � XA2 Þ,

RB �53hðSB

j Þ3i � 41

12hSB

j i ¼12ðXB

1 � XB4 Þ þ

32ðXB

3 � XB2 Þ. ð2Þ

Using Eq. (2) with the normalization relations for XAi and XB

i , namelyP4

i¼1XAi ¼ 1 and

P4

j¼1XBj ¼ 1, the

internal variables can be expresses as linear combinations of the order parameters:

XA1 ¼

14ð1þQAÞ þ

110ðRA þ 3SAÞ ; XB

1 ¼14ð1þQBÞ þ

110ðRB þ 3SBÞ;

XA2 ¼

14ð1�QAÞ þ

110ðSA � 3RAÞ ; XB

2 ¼14ð1�QBÞ þ

110ðSB � 3RBÞ

XA3 ¼

14ð1�QAÞ þ

110ð3RA � SAÞ ; XB

3 ¼14ð1�QBÞ þ

110ð3RB � SBÞ;

XA4 ¼

14ð1þQAÞ �

110ðRA þ 3SAÞ ; XB

4 ¼14ð1þQBÞ �

110ðRB þ 3SBÞ:

(3)

Finally, the Hamiltonian of such a two-sublattice spin-32BEG model in an external magnetic field is

H ¼ �JXhi;ji

SiSj � KXhi;ji

S2

i S2

j þDX

i

S2

i þX

j

S2

j

!�H

Xi

Si þX

j

Sj

!. (4)

The equilibrium properties of the system are determined by the LACVM. The method consists of thefollowing three steps: (i) consider a collection of weakly interacting systems and define the internal variables,(ii) obtain the weight factor in terms of the internal variables and (iii) find the free energy expression andminimize it.

The weight factors WA and WB can be expressed in terms of the internal variables for the A and Bsublattice, respectively, as

WA¼

NA!Q4i¼1

ðXAi NAÞ!

and WB¼

NB!Q4j¼1

ðXBi NBÞ!

, (5)

where NA and NB are the number of lattice points on the A and B sublattices, respectively. On the other hand,a simple expression for internal energy of the system is found by working out Eq. (4) in the LACVM. Thisleads to

E

N¼ �JMAMB � KQAQB þDðQA þQBÞ �HðMA þMBÞ. (6)

Substituting Eq. (2) into (6), the internal energy per site can be written as

E

N¼ � J 3

2ðXA

1 � XA4 Þ þ

12ðXA

2 � XA3 Þ

� �32ðXB

1 � XB4 Þ þ

12ðXB

2 � XB3 Þ

� �� KðXA

1 � XA2 � XA

3 þ XA4 ÞðX

B1 � XB

2 � XB3 þ XB

4 Þ

þD ðXA1 � XA

2 � XA3 þ XA

4 Þ�

þ ðXB1 � XB

2 � XB3 þ XB

4 Þ�

�H 32ðXA

1 � XA4 Þ þ

12ðXA

2 � XA3 Þ

�þ3

2ðXB

1 � XB4 Þ þ

12ðXB

2 � XB3 Þ�, ð7Þ

where N ¼ NAþNB is the total lattice points.

ARTICLE IN PRESSM. Keskin et al. / Physica A 364 (2006) 263–270266

Using the definition of the entropy SeðSe ¼ k lnW Þ with the Stirling approximation, the free energy F ðF ¼

E � TSÞ per site can now be found as

f ¼F

N¼ �JMAMB � KQAQB þDðQA þQBÞ �HðMA þMBÞ

þ1

b

X3i¼1

XAi ln XA

i þX3j¼1

XBj ln XB

j

( )þ b lA 1�

X3i¼1

XAi

( )þ b lB 1�

X3j¼1

XBj

( ), ð8Þ

where lAand lB are introduced to maintain the normalization condition, b ¼ 1=kT , T is the absolutetemperature and k is the Boltzmann factor. The minimization of Eq. (8) with respect to XA

i and XBj and using

the Eq. (2), the self-consistent equations are found to be

MA ¼3ebðKQB�DÞ sinh

32bðJMBþHÞ

� �þe�bðKQB�DÞ sinh

12bðJMBþHÞ

� �2ebðKQB�DÞ cosh

32bðJMBþHÞ

� �þ2e�bðKQB�DÞ cosh

12bðJMBþHÞ

� � ;MB ¼

3ebðKQA�DÞ sinh32bðJMAþHÞ

� �þe�bðKQA�DÞ sinh

12bðJMAþHÞ

� �2ebðKQA�DÞ cosh

32bðJMAþHÞ

� �þ2e�bðKQA�DÞ cosh

12bðJMAþHÞ

� � ;QA ¼

2ebðKQB�DÞ cosh32bðJMBþHÞ

� ��2e�bðKQB�DÞ cosh

12bðJMBþHÞ

� �2ebðKQB�DÞ cosh

32bðJMBþHÞ

� �þ2e�bðKQB�DÞ cosh

12bðJMBþHÞ

� � ;QB ¼

2ebðKQA�DÞ cosh32bðJMAþHÞ

� ��2e�bðKQA�DÞ cosh

12bðJMAþHÞ

� �2ebðKQA�DÞ cosh

32bðJMAþHÞ

� �þ2e�bðKQA�DÞ cosh

12bðJMAþHÞ

� � :

(9)

At this point, we should mention that since the behaviors of RA and RB are similar to that of SA and SB, wehave not obtained RA and RB and investigated their behaviors as many researchers have made. We are nowable to examine the behavior of the order parameters of the antiferromagnetic spin-3

2BEG model in an

external magnetic field by solving the self-consistent equations, i.e., Eq. (9), numerically. In the followingsection, we shall examine the thermal variation of the systems.

3. Thermal variations

In this section, we shall study the temperature dependencies of the order parameters by solving these fournonlinear algebraic equations: namely, the set of self-consistent equations (9), numerically. These equationsare solved by using the Newton–Raphson method and the thermal variations of MA, MB, QA and QB forseveral of K, D and H for Jo0 (we fixed J ¼ �1) are plotted in Fig. 1. In the figures, Tc, Tc

0 and Tt are thecritical or the second-order phase transition (also called the Ne’el temperature) and the first-order phasetransition temperatures, respectively. The behavior of the temperature dependence of the order parametersdepends on H/|J|, K/|J| and D/|J| values and the following five main topological different types are found byinvestigating these behaviors.

(a)

Type 1: For H=jJj ¼ 0:5, K=jJj ¼ 2:25 and D=jJj ¼ �1:0, MA ¼ þ32, MB ¼ �

32, QA ¼ QB ¼ 1:0 at

zero temperature; this type corresponds to the AF phase (MA;B ¼ �32). MA decreases and MB

increases continuously and they meet at kT=jJj ¼ 2:085 which is the Tc as the temperature increases;therefore, a second-order phase transition occurs, seen in Fig. 1(a). The transition is from the AF tothe D phase.

(b)

Type 2: For H=jJj ¼ 0:01, K=jJj ¼ 2:25 and D=jJj ¼ 3:0, behavior of this type is very similar to type 1except MA ¼

12, MB ¼ �

12and QA ¼ QB ¼ �1:0. Hence, the transition is from the AI phase with MA;B ¼ �

12

to the D phase.

(c) Type 3: For H=jJj ¼ 4:25, K=jJj ¼ �0:68 and D=jJj ¼ 0:55, MA ¼MB ¼

32and QA ¼ QB ¼ 1:0 at

zero temperature. As the temperature increase, the sublattice order parameters undergo successivetwo second-order phase transitions at two different temperatures, i.e., Tc and Tc

0, seen in Fig. 1(c). This

ARTICLE IN PRESS

0.0 0.5 1.0 1.5 2.0

MA,

MB,

QA,

QB

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

MB

MA

QA, QB

Tc

(a) 0.00 0.05 0.10 0.15 0.20 0.25

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Tc

(b)

MB

MA

QA, QB

0.00 0.15 0.30 0.45

MA,

MB,

QA,

QB

MA,

MB,

QA,

QB

0.50

0.75

1.00

1.25

1.50

Tc'

(c)

Tc

MB

MA

QA

QB

0.00 0.25 0.50 0.75 1.00

Tc

(d)Tt

MB

MA

QA

QB

MB

MA

QA

QB

MB

MA

QA

QB

kT/|J|

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

1.5

Tc

(e)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Fig. 1. Thermal variations of the sublattice order parameters, MA, MB, QA and QB. Tc, Tc0 and Tt are the second- and first-order phase

transition temperatures for the sublattice order parameters, J ¼ �1. (a) Exhibiting a second-order phase transition for only the sublattice

magnetizations, H ¼ 0:5, K ¼ 2:25 and D ¼ �1:0. (b) Same as (a), but H ¼ 0:01, K ¼ 2:25 and D ¼ 3:0. (c) Exhibiting the two successive

second-order phase transitions for sublattice order parameters, H ¼ 4:25, K ¼ �0:68 and D ¼ 0:55. (d) Exhibiting two successive phase

transitions in which the first one is a first-order and the second one is a second-order phase transition for the order parameters, H ¼ 0:3,K ¼ �0:68, D ¼ 0:55. (e) Exhibiting a second-order phase transition for the sublattice order parameters, H ¼ 3:0, K ¼ �0:68 and

D ¼ 0:55.

M. Keskin et al. / Physica A 364 (2006) 263–270 267

implies that the system exhibits a reentrant behavior. This fact is seen explicitly in the phase diagram,namely Fig. 2(d).

(d)

Type 4: For H=jJj ¼ 0:3, K=jJj ¼ �0:68 and D=jJj ¼ 0:55, MA ¼12, MB ¼ �

32and QA ¼ �1:0, QB ¼ 1:0 at

zero temperature. In this type, the system undergoes two successive phase transitions in which the first oneis a first-order phase transition from the AI with MA ¼

12, MB ¼ �

32to the AF phase with MA ¼

32,

MB ¼ �12and the second one is a second-order phase transition from the AF phase with MA ¼

32, MB ¼ �

12

to the D phase, seen in Fig. 1(d).

(e) Type 5: For H=jJj ¼ 3:0, K=jJj ¼ �0:68 and D=jJj ¼ 0:55, MA ¼

32, MB ¼

12and QA ¼ 1:0, QB ¼ �1:0 at

zero temperature, this type corresponds to the I phase (MA ¼32, MB ¼

12). As the temperature increases,

MA decreases and MB increases continuously and they meet at kT=jJj ¼ 0:986 which is Tc, hence a second-order phase transition occurs, seen in Fig. 1(e). The transition is from the I phase with MA ¼

32, MB ¼

12to

the D phase. These facts are seen explicitly in the phase diagram of Fig. 2(d).

ARTICLE IN PRESS

0.00 0.05 0.10 0.15 0.20 0.25

H/IJ

I

0.0

0.2

0.4

0.6

(a)

Z

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

(b)

Z

0.0 0.5 1.0 1.5

H/IJ

I

0

1

2

3

4

5

(c)

Z

(d)

ZC

0.00 0.25 0.50 0.75 1.00

0.0

1.5

3.0

Hc

kT/IJI

0.0 0.4 0.8 1.2

H/IJ

I

0.0

0.5

1.0

1.5

(e)

T

Fig. 2. Phase diagrams of the antiferromagnetic spin-32BEG model in the (H/|J|, kT/|J|) plane. The D, AF and AI phases are found.

Dashed and solid lines indicate, respectively, first- and second-order phase transitions. The specials points are the tricritical (T), critical (C)

and zero-temperature critical points (Z), J ¼ �1. (a) K ¼ 2:25 and D ¼ 3:0; (b) K ¼ 2:25 and D ¼ �1:0; (c) K ¼ �1:25 and D ¼ 0:5; (d)K ¼ �0:68 and D ¼ 0:55; (e) K ¼ 2:5 and D ¼ 1:0.

M. Keskin et al. / Physica A 364 (2006) 263–270268

4. Phase diagrams

In this section, we present the phase diagram of the antiferromagnetic spin-32BEG model in an external

magnetic field. The critical or second-order phase transition temperatures for the sublattice order parameters inthe case of a second-order phase transition are calculated numerically, i.e., the investigation of the behavior ofthe order parameters as a function of temperature in which the sublattice order parameters become equal as thetemperature is lowered and the temperature where the sublattice order parameters become equal is the critical orsecond-order phase transition temperature. On the other hand, the first-order phase transition temperatures forthe sublattice order parameters are found by matching the values of the two branches of the free energy followedwhile increasing and decreasing the temperature. The temperature at which the free energy values equal eachother is the first-order phase transition temperature (Tt) for the sublattice order parameters.

We can now obtain the phase diagrams of the antiferromagnetic spin-32BEG model in the (kT/|J|, H/|J|) plane

for different values of K, D, and Jo0 (we fixed J ¼ �1) shown in Fig. 2. In the phase diagrams, the solid line

ARTICLE IN PRESSM. Keskin et al. / Physica A 364 (2006) 263–270 269

represents the second-order phase transition line and the dashed line is the first-order phase transition line. T, C

and Z are the special points which denote the tricritical, critical and zero-temperature critical points, respectively.The arrows on H/|J| axis represent the ground state. For H=jJj40, there are two ordered ground states,antiferromagnetic (MA ¼ �MB) and antiferrimagnetic (MAa�MB). Fig. 2(a) shows the phase diagram forK ¼ 2:25 and D ¼ 3:0. The AF phase (1

2,�1

2) is separated from the D phase by the second-order phase transition

line that terminates at the zero-temperature critical point (Z). The similar phase diagram has been obtainedrecently for the antiferromagnetic Ising model on two-fold cayley tree for qo6, q is the coordination number [20].Fig. 2(b) illustrates the phase diagram for K=jJj ¼ 2:25 and D=jJj ¼ �1:0. This phase diagram is similar to thatgiven in Fig. 2(a), except the system exhibits a reentrant behavior. As the temperature is lowered, the system passesfrom the D phase to the AF phase, and back to the D phase. The similar reentrant phenomenon has been found byWang and Rauchwarger [21] in the antiferromagnetic Blume–Capel (BC) model in an external magnetic fieldwithin the MFA, by Hui [22] in the ‘‘in-plane antiferromagnet’’ model, which is an Ising cubic lattice with nearest-neighbor antiferromagnetic coupling in the x and y directions and nearest-neighbor ferromagnetic couplings in thez direction within the extended MFA and by Erdinc- et al. [16] in the antiferromagnetic Blume–Emery–Griffithsmodel with LACVM. In spin systems, the reentrant behavior can be understood as follows. At high temperatures,the entropy is the most important factor and uncorrelated fluctuations determine the thermodynamics. The systemis then in the D phase bias due to the applied field. As the temperature is lowered, the energy and entropy are bothimportant, and the correlated fluctuations affect the dominance of either phase significantly. The system enters theordered phase. At the low temperatures, the energy is important, not the entropy and the system reenters the Dphase again [22]. However, in this work since we are using the LACVM which is identical to the MFA, thecorrelation of spin fluctuations have not been considered. Therefore, the origin of the reentrant phenomenon is dueto the frustration generated by the competition between the external magnetic field and the exchange interactions[20]. Fig. 2(c) represents the phase diagram for K=jJj ¼ �1:25 and D=jJj ¼ 0:5. This phase diagram looks likethat of Fig. 2(b), except the following two differences: (1) the second-order phase transition line separates the AIphase from the D phase and (2) for 0:946pkT=jJjp1:646, H/|J| becomes double valued and beyondkT=jJj ¼ 1:646, no solution for H/|J| is obtained. Therefore, the second-order phase line has a bulge for thisregion, suggesting the occurrence of the some sort of a reentrant phenomenon, because as the H/|J| values areincreased, the system passed from the D to the AI phase, and back to the D phase again. Fig. 2(d) illustrates thephase diagram for K=jJj ¼ �0:68, D=jJj ¼ 0:55. This phase diagram is similar to that given in Fig. 2(c) except atlow values of kT/|J| and H/|J|, a first-order phase transition line exists that separates the AI phase with MA ¼

12,

MB ¼ �32from the AI phase with MA ¼

32, MB ¼ �

12for H=jJjoHc. Therefore, for H=jJjoHc and below the

first-order line, the phase is the AF phase with MA ¼12, MB ¼ �

32and for H=jJjoHc and above the first-order

phase line, the phase is again the AF phase with MA ¼32, MB ¼ �

12. Moreover, for H=jJj4Hc, the phase is always

the I phase with MA ¼32, MB ¼

12, hence, the second-order phase line separates the I phase from the D phase.

Finally, Fig. 2(e) represents the phase diagram for K=jJj ¼ 2:25, D=jJj ¼ 1:0. The system undergoes a second-order phase transition for high values of H/|J| and a first-order phase transition for low values of H/|J|; hence, thetricritical behavior appears in the phase diagram. Moreover, the system also exhibits a reentrant behavior andsecond- and first-order phase transition lines separate the AF from the D phase.

5. Summary and conclusions

In this work, first we have investigated the thermal variations of the antiferromagnetic spin-32BEG model in

an external magnetic field using the LACVM. The behaviors of the temperature dependence of the sublatticemagnetizations depend on K, D, H and Jo0 (we fixed J ¼ �1:0) values and five main topologies of differenttypes are found by investigating these behaviors. Then, we have presented the phase diagrams of the system inthe (kT/|J|, H/|J|) plane. We found that the behavior of the system strongly depends on the values of K and D

nearest-neighbor interactions and five different phase diagram topologies were found. In general, the AF or AIphases are separated from the D phase by the second-order phase line that terminates at the zero-temperaturecritical point (Z), see Figs. 2(a)–(c). For small values of repulsive K, and D values, at low values of kT/|J|,there is a range of H/|J| where one more first-order phase line also occurs that terminates at the critical point(C), besides the second-order phase line, see Fig. 2(d). In this figure, the first-order phase line separates the twodistinct AI phases: namely, the AI phase with MA ¼ þ

12, MB ¼ �

32and the AI phase with MA ¼ þ

32, MB ¼ �

12.

ARTICLE IN PRESSM. Keskin et al. / Physica A 364 (2006) 263–270270

Moreover, for H=jJj4Hc the phase is the I phase with MA ¼32, MB ¼

12. Hence, the second-order phase line

separates the I phase from the D phase. However, for H=jJ=joHc, the second-order phase line separates theAI phase with MA ¼

32, MB ¼ �

12from the D phase. The special point Z disappears in only one phase diagram,

namely Fig. 2(e), where the system exhibits the tricritical behavior. The phase diagram presented in Fig. 2(a) issimilar to that obtained by Ekiz [20] and the phase diagram of Fig. 2(b) is similar to that presented by Wangand Rauchwarger [21], Hui [22] and recently by Erdinc- et al. [16].

Finally, it is worthwhile mentioning that there are similarities between our theoretical study andexperimental results on successive phase transitions in some rare-earth compounds such as antiferromagneticDyVO4 [17]. In relation with experimental results on phase transitions in antiferromagnetic DyVO4, we havestudied the statistical mechanical properties of the Ising-like Hamiltonian. This Hamiltonian is not directlyapplicable to antiferromagnetic DyVO4 because the energy-level scheme is not that of an effective spin-3

2nor

are the interactions in that substance strictly Ising-like. We obtained the qualitative picture of the model.Moreover, we should also mention that the tricritical point, which appears in Fig. 2(e), has been observedexperimentally in various systems, such as multicomponent mixtures, orientational order–disorder transitionsin NH4Cl, and certain magnetic substances, e.g., the metamagnetic transitions dysprosium aluminum garnet(DAG), FeCl2, FeBr2, Ni(NO3)2 � 2H2O, etc. [23].

Acknowledgment

This research was supported by the Research Fund of Erciyes University, Grant no. FBT-05-03.

References

[1] A. Bakchich, A. Bassir, A. Benyoussef, Physica A 195 (1993) 188;

N. Tsushima, T. Horiguchi, J. Phys. Soc. Japan 67 (1998) 1574.

[2] T. Kaneyoshi, M. Jascur, Phys. Lett. A 177 (1993) 172.

[3] N. Tsushima, Y. Honda, T. Horiguchi, J. Phys. Soc. Japan 66 (1997) 3053.

[4] E. Albayrak, K. Keskin, J. Magn. Magn. Mater. 241 (2002) 249.

[5] F.C. Sa Barreto, O.F. De Alcantara Bonfim, Physica A 172 (1991) 378.

[6] A. Bakkali, M. Kerouad, M. Saber, Physica A 229 (1996) 563.

[7] J.W. Tucker, J. Magn. Magn. Mater. 214 (2000) 121.

[8] A. Bakchich, M. El Bouziani, J. Phys.: Condens. Matter 13 (2001) 91.

[9] C. Ekiz, E. Albayrak, M. Keskin, J. Magn. Magn. Mater. 256 (2003) 311.

[10] C. Ekiz, Phys. Stat. Sol. (b) 241 (2004) 1324.

[11] A. Bakchich, S. Bekhechi, A. Benyoussef, Physica A 210 (1994) 415.

[12] S. Bekhechi, A. Benyoussef, Phys. Rev. B 56 (1997) 13954.

[13] C. Ekiz, J. Magn. Magn. Mater. 284 (2004) 409.

[14] C. Ekiz, Phys. Lett. A 325 (2004) 99.

[15] M. Keskin, M.A. Pınar, A. Erdinc- , O. Canko, Phys. Lett. A, submitted.

[16] A. Erdinc- , O. Canko, M. Keskin, J. Magn. Magn. Mater., in press.

[17] A.H. Cooke, D.M. Martin, M.R. Wells, Solid State Commun. 9 (1971) 519;

A.H. Cooke, D.M. Martin, M.R. Wells, J. Phys. 32 (1971) C1 488.

[18] J. Sivardiere, M. Blume, Phys. Rev. B 5 (1972) 1126.

[19] S. Krinsky, D. Mukamel, Phys. Rev. B 11 (1975) 399.

[20] C. Ekiz, Phys. Lett. A 327 (2004) 374.

[21] Y.L. Wang, K. Rauchwarger, Phys. Lett. A 59 (1976) 73.

[22] K. Hui, Phys. Rev. B 38 (1988) 802.

[23] J.M. Kincaid, E.G.D. Cohen, Phys. Rep. 22 (1975) 57;

K. Barat, B.K. Chakrabarti, Phys. Rep. 258 (1995) 377;

T.S. Chang, D.D. Vvedensky, J.F. Nicoll, Phys. Rep. 217 (1992) 279;

C. Rottman, M. Wortis, Phys. Rep. 103 (1984) 59.