Stable, metastable and unstable solutions of the Blume–Emery–Griffiths model

Physica A 267 (1999) 392–405www.elsevier.com/locate/physa

Stable, metastable and unstable solutions of theBlume–Emery–Gri�ths model

Mustafa Keskina;∗, Cesur Ekizb, Orhan Yal�c�nbaDepartment of Physics, Erciyes University, 38039 Kayseri, Turkey

bDepartment of Physics, Gaziosmanpa�sa University, 60110 Tokat, Turkey

Received 25 October 1998; received in revised form 3 December 1998

Abstract

The temperature dependence of the magnetization and quadrupole order parameters of theBlume–Emery–Gri�ths (BEG) model Hamiltonian with the nearest-neighbor ferromagnetic ex-change interactions [both bilinear (J ) and biquadratic (K)] and crystal �eld interaction (D) isstudied using the lowest approximation of the cluster variation method. Besides the stable so-lutions, metastable and unstable solutions of the order parameters are found for various valuesof the two di�erent coupling parameters, � = J=K and = D=K . These solutions are classi�edusing the free energy surfaces in the form of a contour map. The phase transitions of the stable,metastable and unstable branches of the order parameters are investigated extensively. The crit-ical temperatures in the case of a second-order phase transition are obtained for di�erent valuesof � and calculated by the Hessian determinant. The �rst-order phase transition temperaturesare found using the free energy values while increasing and decreasing the temperature. Thetemperature where both the free energies equal each other is the �rst-order phase transition tem-perature. Finally, the results are also discussed for the Blume–Capel model which is the specialcase of the BEG model. c© 1999 Elsevier Science B.V. All rights reserved.

PACS: 05.70.Fh; 64.60.My; 75.10.Hk; 75.50.Ee

Keywords: Spin-1 Ising system; Order parameters; Metastable and unstable states; Hessiandeterminant

1. Introduction

The spin-1 Ising model with arbitrary bilinear and biquadratic nearest-neighbor pairinteractions and a crystal �eld or a single-ion potential is known as the Blume–Emery–Gri�ths (BEG) model [1]. The model with vanishing biquadratic interaction is called

∗ Corresponding author. Fax: +90-352-437-4931; e-mail: [email protected].

0378-4371/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(98)00666 -9

M. Keskin et al. / Physica A 267 (1999) 392–405 393

the Blume–Capel (BC) model [2–6]. The BEG model was initially introduced to de-scribe the phase separation and super uid ordering He3–He4 mixtures [1]. Subsequently,the model was applied to study the thermodynamical behavior of certain cooperativephenomena such as multicomponent uids [7], microemulsions [8], ordering in semi-conductor alloys [9,10], electronic conduction models [11], the reentrant phenomenon inphase diagrams [12–19], critical behavior and multicritical phase diagrams [20–25,33–35]. The above calculations were done by the mean �eld approximation [1–7,26–28],renormalization-group techniques [20–25], the e�ective �eld theory [13–18], MonteCarlo Methods [29–31] and the cluster variation method [32] and its modi�ed version[33–35]. Besides these methods, the BEG model has been studied by other techniquessuch as the Monte Carlo renormalization technique [19], the constant coupling approx-imation [36,37] and transfer matrix method [38]. In this context, the exact solutionsof the BEG model on Bethe lattice, the honeycomb and square lattice in two dimen-sion [39–48] are worth mentioning. Due to its intrinsic complexity, the spin-1 BEGquantum model has also been studied [49]. On the other hand, the dynamics of thespin-1 BEG model is studied by the mean-�eld technique [50] and also the real-spacerenormalization-group approach [51]. Moreover, the non-equilibrium properties of theBEG model for zero crystal �eld interaction have been investigated by the path prob-ability method [52–56] and also by the mean-�eld technique based on the Glaubermodel [57,58].In spite of these studies, the temperature dependence of the magnetization and

quadrupole order parameters of the spin-1 Ising BEG model has not been studiedextensively. Especially, the metastable and unstable branches of the order parametersand their phase transitions are not found. Therefore, the purpose of the present workis to investigate the behavior of the thermal variation of the order parameters and toobtain the metastable and unstable branches of the order parameters and their phasetransitions. It is worthwhile to mention that the stable, metastable and unstable solu-tions of the spin-1 BEG model Hamiltonian with zero crystal �eld interaction havebeen studied for zero magnetic �eld [59–62], an external magnetic �eld [63] and aswell as for magnetic �elds due to the dipole and quadrupole moments [64,65].The outline of the remaining part of this paper is as follows. In Section 2, we describe

the model brie y and present its solutions at equilibrium in the lowest approximationof the cluster variation method. The behavior of the thermal variation of the orderparameters is investigated and transition temperatures are found in Section 3. Thesummary and discussion are given in Section 4.

2. The spin-1 Ising BEG model and its solution at equilibrium

The spin-1 Ising BEG model is de�ned by the Hamiltonian

H =−J∑〈ij〉

SiSj − K∑〈ij〉

S2i S2j − D

∑Qi ; (1)

394 M. Keskin et al. / Physica A 267 (1999) 392–405

where each Si that can take the values 1; 0;−1 and 〈ij〉 indicates summation over allpairs of nearest-neighbor sites. J; K and D are the bilinear exchange, the biquadraticexchange and crystal-�eld interactions, respectively. It is a three-state and two-orderparameters system. The average fraction of each of the spin states will be indicatedby X1; X2 and X3, which are also called the state or point variables. X1 is the averagefraction of spins with value +1; X2 is the average fraction of spins that have the value0, and X3 is the average fraction of spins that have the value −1. These variables obeythe following normalization relation:

3∑i=1

Xi = 1 : (2)

Two long-range order parameters are introduced as follows: (1) the average mag-netization 〈S〉, which is the excess of one orientation over the other orientation, alsocalled dipole moment, and (2) the quadrupole moment Q, which is the average ofsquared magnetization 〈S2〉, written as

Q = 〈S2〉 : (3)

This de�nition of Q is di�erent from the de�nition Q=3〈S2〉− 2 used by Chen andLevy [66] and Keskin and co-workers [52–56,59–65]. The last de�nition ensures thatQ = 0 at in�nite temperature.The order parameters can be expressed in terms of the internal variables and are

given by

S ≡ 〈S〉= X1 − X3; Q ≡ 〈Q〉= X1 + X3 : (4)

Using Eqs. (2) and (4), the internal variables can be expressed as linear combinationsof the order parameters

X1 = 12 (Q + S); X2 = (1− Q) and X3 = 1

2 (Q − S) : (5)

The equilibrium properties of the system are determined by means of the lowestapproximation of the cluster variation method [67,68]. The method consists of thefollowing three steps: (1) consider a collection of weakly interacting systems and de�nethe internal variables; (2) obtain the weight factor W in terms of the internal variables;(3) �nd the free energy expression and minimize it.The weight factor W can be expressed in terms of the internal variables as

W =N∏3

i=1 (XiN ); (6)

where N is the number of lattice points. A simple expression for the internal energyof such a system is found by working out Eq. (1) in the lowest approximation of thecluster variation method. This leads to

E=N =−JS2 − KQ2 − DQ : (7)

Substituting Eq. (4) into Eq. (7), the internal energy per site can be written as

E=N =−J (X1 − X3)2 − K(X1 + X3)2 − D(X1 + X3) : (8)


Using the de�nition of the entropy Se (Se = k lnW ) with the Stirling approximation,the free energy F (F = E − TS) per site can be obtained

�=−FN= J (X1 − X3)2 + K(X1 + X3)2 + D(X1 + X3)

− 1�

3∑i=1

Xi ln Xi + �

(1−

3∑i=1

Xi

); (9)

where � is introduced to maintain the normalization condition, � = 1=kT; T is theabsolute temperature and k is the Boltzmann factor.For a system at equilibrium the free energy is minimum. In Eq. (9) the free energy

is given in terms of the internal variables. The minimization of Eq. (9) with respectto Xi gives

@�=@Xi = 0 (i = 1; 2; 3) : (10)

Using Eqs. (2), (9) and (10) the internal variables are found to be

Xi = ei=Z ; (11)

where

ei = exp(− �N@E@Xi

)and Z =

3∑i=1

ei (i = 1; 2; 3): (12)

Z represents the partition function, e1; e2 and e3 are calculated using Eq. (8) as follows:

e1 = exp�(2JS + 2KQ + D) ;

e2 = 1 ;

e3 = exp�(−2JS + 2KQ + D) : (13)

One can easily �nd the following set of self-consistent equations by using Eqs. (4)and (11)–(13):

S =2e�(k +2KQ) sinh 2�K�S

1 + 2e�(k +2KQ) cosh 2�K�S;

Q =2e�(k +2KQ) cosh 2�K�S

1 + 2e�(k +2KQ) cosh 2�K�S; (14)

where � = J=K and = D=K are called the ratio of the coupling constants. Thesetwo non-linear algebraic equations are solved by using the Newton–Raphson method.Thermal variations of the S and Q for several values of � and are plotted inFigs. 1–5. The discussion of these solutions will be given in the last section. In the �g-ures, subscript 1 indicates the stable solutions (solid lines), subscript 2 corresponds tothe metastable solutions (dashed-dotted lines) and 3 unstable solutions (dashed lines).This classi�cation is done by comparing the free energy values of these solutions.Especially, we display the free energy surfaces by means of a contour map in twodimensional phase space of S and Q. For example, Fig. 6 illustrates the free energy


Fig. 1. Thermal variations of the quadrupolar order parameter Q for � = 0. Subscript 1 indicates the stablestate (solid lines), 2 the metastable state (dashed-dotted lines) and 3 the unstable state (dashed lines). Thehorizontal dotted line corresponds to Q = 2=3. The thick, thin and dotted arrows represent the �rst-orderphase transition temperature for the stable, metastable and unstable branches of Q, respectively. Tu is theupper limit of stability temperature. The stable branch of Q becomes the metastable after Tt . (a) Thinlines correspond to = −1:5 and exhibiting a �rst-order phase transition for the metastable branch of thequadrupolar order parameter. The thick line corresponds to = −3:0. (b) Very thin lines correspond to =−1:30, thick lines with =−4=3 and they both illustrate a �rst-order phase transition for Q1. Thin linesmatch with = −1:34 and exhibit a �rst-order phase transition for Q1 and Q2. (c) Thick lines correspondto =−0:9 and illustrate a �rst-order phase transition for Q3. A thin line matches with = 1:0.

Fig. 2. The order parameters S and Q as a function of temperature, exhibiting a second-order phase transitionfor S1. Subscript 1 indicates the stable state (solid lines) and 3 the unstable state (dashed line). The horizontaldotted line corresponds to Q = 2=3. (a) � = 0:4 and = 0:5. (b) � = 1:5 and = 0:5.

surfaces in the form of a contour map for �=0:8; =−1:6; kT=K=0:8 and N=10000.In this �gure, the open circle corresponds to the stable solution which is the lowestvalue of the free energy, or the deepest minimum, the �lled square corresponds tothe metastable solution which is the second lowest value of the free energy or thesecondary minimum and the �lled circle is the unstable solution which correspondsto a higher value of the free energy, the peak or the saddle point in the free energysurfaces. If one compares Fig. 4(a) with Fig. 6 at kT =0:8; it is easily seen that stable,


Fig. 3. The order parameters S and Q as a function of temperature, exhibiting a second-order phase transitionfor S1 and S3. Subscript 1 indicates the stable state (solid lines), 2 the metastable state (dashed-dotted lines)and 3 the unstable state (dashed lines). The horizontal dotted line corresponds to Q = 2=3. Tc and Tc3 arethe critical temperatures for S1 and S3, respectively. Tt3 is the �rst-order phase transition for Q3, indicatedby a dotted arrow. (a) �=1:2 and =−1:6. (b) A �rst-order phase transition for Q3; Tt3 = Tc; �=0:7 and =−4=3. (c) A �rst-order phase transition for Q2, namely Tt2, indicated by a thin arrow and also for Q3(� = 0:7 and =−1:34).

Fig. 4. The order parameters S and Q as a function of temperature, exhibiting a �rst-order phase transitionfor S1 and Q1. Tu is the upper limit of the stability temperature. A thick arrow, a thin arrow and a dottedarrow represent a �rst-order phase transition temperature for the stable, metastable and unstable branchesof the order parameters, respectively. The stable branches of the order parameters become the metastableafter Tt . The horizontal dotted line corresponds to Q= 2=3. (a) �= 0:8 and =−1:6. (b) Also exhibiting a�rst-order phase transition for Q2 and Q3, and a second-order phase transition for S3, �=0:4 and =−1:34.(c) Also exhibiting a �rst-order phase transition for Q3, and a second-order phase transition for S3, �= 0:4and =−4=3. (d) Also exhibiting a �rst-order phase transition for Q2 and Q3, � = 0:36 and =−1:34.


Fig. 5. The order parameters S and Q as a function of temperature subscript 1 indicates the stable state (solidlines), 2 the metastable state (dashed dotted lines) and 3 the unstable state (dashed lines). A thick arrowand a thin arrow indicate the �rst-order phase transition temperature for the stable and metastable branchesof the order parameters, respectively. Tc2 and Tc3 are the critical temperatures for S2 and S3, respectively.The stable branch of Q becomes the metastable after Tt . The horizontal dotted line corresponds to Q= 2=3.(a) Exhibiting a �rst-order phase transition for S2 and Q2, �=0:4 and =−1:6. (b) Exhibiting a �rst-orderphase transition for Q1 and a second-order phase transition for S2 and S3, � = 0:2 and = −1:30. (c)Exhibiting a �rst-order phase transition for Q1 and a second-order phase transition for S2 and S3, � = 0:2and =−4=3. (d) Exhibiting a �rst-order phase transition for Q1 and Q2 and a second-order phase transitionfor S2 and S3, � = 0:2 and =−1:34.

Fig. 6. The contour mapping of the free energy for � = 0:8; = −1:6; kT=K = 0:8 and N = 10000. Theopen circle corresponds to the stable solution, the �lled square to the metastable solution and the �lled circlerepresents the unstable solution.

metastable and unstable solutions coincide exactly with each other. On the other hand,Tc; Tc2 and Tc3 are the critical temperatures in the case of a second-order phase transi-tion for the stable, metastable and unstable branches of the magnetization, respectively.Tt ; Tt2 and Tt3 are the �rst-order phase transition temperatures for the stable, metastableand unstable branches of the order parameters and they are indicated by the thick, thinand dotted arrows, respectively, in the �gures. The stable branches of order parameters


become the metastable after Tt . Finally, Tu is the temperature in which the disconti-nuity occurs �rst for the stable branches of the order parameters. We will give moreinformation about these transition temperatures in the next section.It is worthwhile to mention that if the biquadratic interaction, namely K , is zero in

Eq. (14), we will obtain the set of self-consistent equations for the Blume–Capel (BC)model which is the special case of the BEG model [6].

3. Transition temperatures

In order to determine the critical temperatures for the stable, metastable and unstablebranches of the magnetization in the case of a second-order phase transition easily andprecisely, we introduce the matrix of the second derivative of the free energy withrespect to internal variables which is called the Hessian determinant. Therefore, theHessian determinant can be calculated using the equation

@2�@Xi@Xj

= 0 (i; j = 1; 2; 3) : (15)

Substituting Eq. (9) into Eq. (15), the determinant can be obtained explicitly as

A=

∣∣∣∣∣∣∣∣∣∣∣

2�J + 2�K − 1X1

0 −2�J + 2�K

0 − 1X2

0

−2�J + 2�K 0 2�J + 2�K − 1X3

∣∣∣∣∣∣∣∣∣∣∣: (16)

In the absence of the biquadratic interaction in Eq. (16) reduces to the Hessiandeterminant of the BC model.The values of the determinant are calculated and plotted as a function of the tem-

perature for various values of � and , e.g., �=1:2 and =−1:6, seen in Fig. 7 whichcorresponds to the second-order phase transition for S1 and S3. The change in sign ofthe determinant corresponds to the critical temperature. In the �gure the determinantchanges sign at two di�erent temperatures, the �rst one corresponds to the critical tem-perature for the unstable branch of S, namely Tc3 and the second one is the criticaltemperature for the stable branch of S, namely Tc. If one compares Fig. 3(a) withFig. 7, one can see that the critical temperatures found using both calculations are ex-actly the same. However, the critical temperatures are found more easily and preciselywith the Hessian determinant calculation.On the other hand, the �rst-order phase transition temperatures for the stable branch

of S and Q are found by using the free energy values while increasing and decreasingthe temperature. The temperature at which the free energy values equal to each otheris the �rst-order phase transition temperature, Tt for stable S and Q, seen in Fig. 8marked with the dot. Finally, Tt2 and Tt3 are the same as Tu2 and Tu3, the upper limit


Fig. 7. The values of the Hessian determinant as a function of temperature for � = 1:2 and = −1:6. Thechanging signs of the determinant corresponds to the critical temperatures of the stable and unstable dipolemoment order parameters Tc and Tc3, respectively.

Fig. 8. The free energy as a function temperature for � = 0:8 and =−1:6. F1 corresponds to free energyvalues while decreasing the temperatures (arrow 1) and F2 corresponds increasing of temperatures (arrow2). The temperature where both free energies equal each other is the �rst-order phase transition temperatureTt for the stable state of order parameters, marked with dot. The peak corresponds to the upper limit of thestability temperature Tu.

of stability temperatures in the case of a �rst-order phase transition temperature for themetastable and unstable branches of S and Q. Tt ; Tt2 and Tt3 are represented by thick,thin and dotted arrows in the �gures.

4. Summary and discussion

The lowest approximation of the cluster variation method was used to study thebehavior of thermal variations of the dipolar (magnetization) and the quadrupolar orderparameters of the spin-1 Ising BEG model in the absence of the external magnetic�eld. Besides the stable branches of the order parameters, we establish the metastableand unstable parts of these curves and their phase transitions are also found. Thecritical temperature in the case of the second-order phase transition are obtained by


the calculation of the Hessian determinant. The �rst-order phase transition temperaturefor the stable branches of the order parameters is found using the free energy valueswhile increasing and decreasing the temperature. The temperature where both the freeenergies equal each other is the �rst-order phase transition temperature.Thermal variation of the order parameters for several values of � and are plotted in

Figs. 1–5. The behavior of the temperature dependence of the order parameters dependson � and values and, by comparing the free energy values of the solutions of theorder parameters, the following results have been found.(a) For � = 0 and ¡ 0, the thermal variation of the quadrupole moments order

parameter is illustrated in Fig. 1. In this case for 6−2:0 only the stable values of Q,namely Q1 is present and it increases from zero to Q=2=3 as the temperature increasesand therefore, there is no phase transition in stable values of Q corresponding to thethick line in Fig. 1(a). However, for −2:0¡ ¡ 0 there also exists the metastable andunstable branches of Q, namely Q2 and Q3, respectively, besides the stable branch ofQ. The metastable branch of Q decreases to zero discontinuously as the temperatureincreases, hence a �rst-order phase transition is found in Q2. The �rst-order phasetransition temperature, Tt2 is marked with the thin arrow in the �gure. The unstablebranch of Q, namely Q3, also occurs below Tt2. We should also mention that the similarbehavior of the stable branch of the quadrupolar order parameter can be obtained fromthe BC model for D=J ¿ 1. However, the metastable and unstable branches of Q do notexist in the BC model [6]. On the other hand for −1:3466 ¡−1 the �rst-order phasetransition Tt for Q is found. Below Tt , the other unstable branch of Q also occurs,seen in Fig. 1(b). In the �gure, Tu is the upper limit of stability temperature for Q1,and the thick and thin arrows represent the �rst-order phase transition temperature, Ttand Tt2 for Q1 and Q2, respectively. The stable branch of Q becomes the metastableafter Tt . In Fig. 1(b), one can observe that for = −4=3 all the branches of thequadrupolar order parameter intersect at the point Q=2=3 and kT=K=0:465, illustratedwith thick lines in the �gure. Moreover, for ¿− 4=3, the gap between the branchesincreases and Tt2 occurs at the lower temperature, illustrated with thin lines in Fig.1(b). Finally, for ¿− 1 the stable values of Q decreases from Q = 1 to Q = 2=3 asthe temperature increases and therefore there exists no phase transition for Q1. HoweverQ3 varies discontinuously and the system undergoes the �rst-order phase transition forthe unstable branch of the quadrupole order parameter, Q3, is shown by the thick linesin Fig. 1(c). The �rst-order phase transition temperature Tt3, is represented by a dottedarrow in the �gure. On the other hand for ¿0 there exist only the stable branchof Q and the data for Q in this case suggests the absence of phase transition at anytemperature, corresponding to the thin line in Fig. 1(c).(b) For �¿ 0 and ¿ 0, the stable branch of the magnetization S1, decreases to zero

continuously as the temperature increases, therefore a second-order phase transitionoccurs in S1 seen in Fig. 2. On the other hand, the stable branch of the quadrupoleorder parameter, Q1 smoothly decreases to Q = 2=3. Moreover, after certain values of� and , there also starts to exist the unstable branch of Q, namely Q3 and this branchterminates at the critical temperature Tc, e.g. for � = 1:5 and = 0:5; Q3 disappears at


Tc. In addition, the solution of S = 0 below Tc is unstable. The increasing of � and only changes the critical temperature Tc which occurs for the higher values of thetemperature.(c) For �¿ 0 and ¡ 0; in this case, the following three main di�erent behaviors

of the order parameters have been observed.(i) The stable and unstable branches of the magnetization, S1 and S3, respectively,

decrease to zero continuously as the temperature increases, therefore a second-orderphase transition occurs in S1 and S3. On the other hand, the stable values of thequadrupolar order parameters, Q1, decreases until Tc, as the temperature increases,then starts to rise to Q = 2=3. A similar behavior has been observed for the unstablebranch of the quadrupolar order parameter, Q3. Below the Tc3 which is the criticaltemperature for S3, there also exists a metastable state with S2 = 0 and Q2¿ 0, seenin Fig. 3(a) (� = 1:2 and = −1:6). For the small values of � there are two moreunstable branches of the Q and one of them undergoes a �rst-order phase transition.It should be mentioned that the same thermal variations of the order parameters arealso seen in the BC model for 06D=J60:463 [6]. The �rst-order phase transitiontemperature is the same as the critical temperature for S3, namely Tt3 = Tc3 and belowthis temperature there exist also the metastable branches of S2 =0 and Q2¿ 0, seen inFig. 3(b) (�=0:7 and =−4=3). In Fig. 3(c) one also observes that the metastable andone of the unstable branches of the quadrupole moment order parameter again undergoa �rst-order phase transition, but Tt3 is not the same as Tc3 in this case, Tt3¡Tc3.(ii) The stable values of the order parameters S and Q decrease zero discontinuously,

hence a �rst-order phase transition occurs, seen in Fig. 4. The �rst-order phase transitiontemperature Tt for the stable branch of S and Q is indicated by a thick arrow and Tu isthe upper limit of the stability temperature for the stable branch of S and Q in Fig. 4.The stable branches of the order parameters become the metastable after Tt . On theother hand, the behavior of the metastable and unstable branches of S and Q isvery complicated. For example, below Tu the metastable and unstable branches of theorder parameters occur and they do not undergo any phase transitions, seen inFig. 4(a), � = 0:8 and = −1:6. It is worthwhile to mention that the same behaviorof the order parameters are also obtained from the BC model for 0:463¡D=J ¡ 0:5 [6].For small values of � and the larger values of , there are two more unstablebranches of the Q and S below Tu. One of the unstable branches of Q undergoesa �rst-order phase transition, Tt3 and one of the unstable branches of S undergoes asecond-order phase transition, Tc3, the other is S3 = 0 below Tc3. Furthermore, themetastable branch of Q, namely Q2, also undergoes a �rst-order phase transition,seen in Fig. 4(b) (� = 0:4 and = −1:34). For � = 0:4; = −4=3, the unstable so-lutions of Q exchange their branches, and the second-order phase transition for S3and the �rst-order phase transition for Q3 occur at the same temperature, namelyTc3 = Tt3, illustrated in Fig. 4(c). Finally, for � = 0:36 and = −1:34 the behav-ior of the system looks like that in Fig. 4(b), but the second-order phase transi-tions for the unstable branch of the magnetization, namely S3 disappears and thisbranch meets the metastable branch of S and also, one of the unstable branches of


Q3 joins Q2, shown in Fig. 4(d). Furthermore, the metastable branch of Q jumps tothe stable branch of Q1, therefore it undergoes a �rst-order phase transition with thesame Tt .(iii) For �¡ 0:66; =−1:6 the metastable branches of S and Q undergo a �rst-order

phase transition and below the phase transition temperature the unstable branches ofS and Q occur, seen in Fig. 5(a). In the �gure one can observe that there is nophase transition for the stable value of Q and the stable value of S = 0. We shouldalso mention that the same thermal behavior of the order parameters are also seen inthe BC model for 0:56D=J61 [6]. For the small values of � and the larger valuesof , namely � = 0:2 and = −1:30, the stable branch of Q undergoes a �rst-orderphase transition, but the metastable and one of the unstable branches of S undergoa second-order phase transition, seen in Fig. 5(b). Moreover, below Tu there is onemore metastable branch of Q, which meets Q1 at Tt and three more unstable branchesof Q exist below Tc3, illustrated in Fig. 5(b). In addition, below Tu there exists onemore unstable branch of Q which meets the metastable branch at Tu and the unstablesolution of Q=2=3. For �=0:2 and =−4=3, all the branches of the quadrupolar orderparameter intersect at the point Q = 2=3 and kT=K = 0:465, seen in Fig. 5(c). On theother hand, for �= 0:2 and =−1:34 the behavior of the system looks like Fig. 5(c)except that one of the metastable branches of the quadrupolar order parameters alsoundergoes a �rst-order phase transition and the unstable branch of Q with Q¿ 2=3meets this branch at the transition temperature, Tt2 seen in Fig. 5(d). The stable branchof Q does not meet the unstable branch of Q, it meets again the metastable branch ofQ, after the �rst-order phase transition, shown in the �gure. The stable branch of Qbecomes the metastable after Tt .Finally, it is worthwhile to mention that metastable and unstable solutions are very

important in many theoretical and experimental works, such as simple uids (gas–liquids transitions), binary uids, binary alloys, super uids, superconductors, physicorp-tion and chemisorption systems, intercalation compounds, polymer blends, gels, lasers,electron–hole condensation in semiconductors, geological systems (minerals), chemi-cally reacting systems, order–disorder systems, coherent hydrogen–metal systems, met-als, glasses and crystalline ceramics, magnetic systems and astrophysics [69,70]. Alsoit is well known that if the systems stay in their metastable state or phase, propertiesof the systems either improve or deteriorate drastically. For example, the rapid coolingof alloys or metals leads to amorphous structures, namely metastable phases, and it isknown that the properties of alloys or metals improve signi�cantly, such as strength,sti�ness, fatigue behavior, corrosion, toughness, density modules, etc. [71–73]. On theother hand, the metastability is becoming a serious problem in high-performance verylarge scale integration (VLSI) in complimentary metal-oxide-semiconductor (CMOS)dynamic D-latch, mainly due to the relative high probability of error when a bistablecircuit operates at high frequencies [74]. Furthermore, metastable liquids play an im-portant role in both nature and technology, such as sap ascent in trees under tension,supercooled water in clouds, mineral inclusions, phase separation in polymer mixtures,explosive boiling, caviation in turbulent ow, etc. [75,76]. Since we have found that


the metastable and unstable states exist besides the stable state in the BEG model, onecan use this model to study how a system can be forced to relax into the metastableor the stable state. Moreover, one can also investigate how a system escapes from themetastable state to the stable state. Because metastability is a dynamical phenomenon[77], in order to carry out the above-mentioned works, one may need to study thedynamic of the BEG model. It is worthwhile to mention that the BEG model has beenused to study semiconductor alloy systems [9,10]. One can extend these works to thedynamic case to investigate and illustrate the entrapment or release from the metastablestate. On the other hand, unstable solutions play the role of separators between the sta-ble and metastable solutions.

Acknowledgements

We wish to thank Prof. Dr. A. Nihat Berker of MIT for many illuminating dis-cussions. A part of this work was supported by NATO Grant No. (CRG. 970008)1036=97.

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Stable, metastable and unstable solutions of the Blume–Emery–Griffiths model