Stable, metastable and unstable solutions of a spin-1 Ising system obtained by the molecular-field approximation and the path probability method

Physica A 157 (1989) 1000-1017

North-Holland, Amsterdam

STABLE, METASTABLE AND UNSTABLE SOLUTIONS OF A SPIN-1

ISING SYSTEM OBTAINED BY THE MOLECULAR-FIELD

APPROXIMATION AND THE PATH PROBABILITY METHOD*

Mustafa KESKiN, Mehmet AR1

Fizik Bdiimii, Erciyes cniversitesi, 38039 Kayseri, Turkey

Paul H.E. MEIJER Physics Department of lhe Catholic University of America, Washington. D.C. 20064. USA

Received 14 June 1988

Revised manuscript received 15 November 1988

The spin-l lsing model Hamiltonian with arbitrary bilinear (J) and biquadratic (K) pair

interactions has been studied for zero field using the lowest approximation of the cluster

variation method. The spin-l or three-state system will undergo a first- or second-order phase

transition depending on the ratio of the coupling parameters a = J/K. The critical tempcra-

tures and, in the case of the first-order phase transition, the limit of stability temperatures are

obtained for different values of cy calculating by the Hessian determinant. The first-order

phase transition temperatures are found by using the free-energy values while increasing and

decreasing the temperature. Besides the stable branches of the order parameters, we establish

also the metastable and unstable parts of these curves.

The dynamics of the same model is studied by means of the path probability method and

the resulting set of equations is solved at temperatures below the critical temperature for

three different cases: a system with second-order transition. a system with a first-order

transition both in the quadrupole moment and the spin moment, at a temperature in between

the upper and lower stability points, and a system that undergoes a first-order transition in the

quadrupole moment only. The stable, metastable and unstable solutions are shown explicitly

in the flow diagrams. The unstable solutions for the first-order phase transitions are found by

solving the dynamic equations and the role of these solutions in the phase diagram is also

described. Finally a discussion is given of how metastable phases could be obtained.

1. Introduction

The thermodynamical behaviour of many cooperative physical systems can

be simulated by the spin-l Ising model. The spin-l Ising model is also known as

*This work is based on a presentation which was given at the IXth National Physics Meeting of

the Turkish Physical Society, Bursa 1987.

037%4371/89/$03.50 0 Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

M. Keskin et al. I Solutions of a spin-l

the Blume-Emery-Griffiths model [l]. The most model with nearest-neighbour pair interaction is

Ising system 1001

general Hamiltonian of the

x=-& s,-.lc s;s,-KZ s:s;+Af s; j=l (ii) (ii) j=l

- H3 c SiSj(Si + Sj) )

(ii)

where each spin S, at a site k = 1,2,3, . . . , N of a regular lattice can take the values 1, 0, - 1, and (ii) indicates summation over nearest-neighbour pairs k = i and k = j. The parameter H represents the magnetic field, J and K are the bilinear and biquadratic exchange interaction respectively, A is the crystal field and H3 denotes a third-order magnetic perturbation. Blume, Emery and Griffiths [l] introduced this model to describe phase separation and superfluid ordering in He3-He4 mixtures along the A line and near the critical mixing point. A special case of this Hamiltonian, namely K = H3 = 0, known as the Blume-Cape1 model, was first introduced by Blume [2] and independently by Cape1 [3]. One of the first molecular-field treatments of the spin-l Ising model Hamiltonian with arbitrary bilinear (J) and biquadratic (K) pair interactions was done by Chen and Levy [4] in order to determine the dependence of the magnetothermal behaviour on the coupling parameter of the system. Keskin and Meijer [5] and Keskin [6] gave a simple molecular-field treatment of the dynamics of a spin-l Ising model, Hamiltonian with J and K, without and with an external magnetic field, respectively, in order to study the behaviour of the metastable equilibrium state and the phenomenon of quenching via hidden thermodynamical variables. Other mean-field studies of the spin-l Ising or related models have been reported by Bernasconi and Rys [7], Sivardiere et al. [8], Yamashita and Nakano [9] and Tanaka and Takahashi [lo].

The lattice gas version of the spin-l Ising model has been investigated in mean-field theory by Lajzerowicz and Sivardiere [ll] and Sivardiere and Lajzerowicz [12]. The global phase diagram of this lattice gas in the five- dimensional parameter space was described in detail by Furman, Dattaguta and Griffiths [13].

The spin-l Ising model has also been studied with more sophisticated methods that the mean-field approximation. Allan and Betts [14] use high- temperature series of a spin-l Ising system to determine the effects of biquadratic exchange on the magnetothermal properties. A similar expansion for related problem was used Chen and Joseph [15] to find the Schrodinger exchange interaction for arbitrary spin. Monte Carlo calculation of the spin-l Ising model in two and three dimensions have been studied by Arora and

1002 M. Keskin et al. I Solutions of a spin-l Ising system

Landau [16] and Jain and Landau [17]. Harris [ 181 has also been reporting the Monte Carlo simulations of a spin-l Ising-like model in order to obtain information about metastable states. More general renormalization-group (191 studies of a spin-l Ising model on the square lattice with nearest-neighbour ferromagnetic exchange interactions (both bilinear and biquadratic) and crystal-field interaction were presented by Berker and Wortis [20]. Recently Chakraborty and Morita [21] studied properties of a spin-l Ising model on the Bethe lattice, in which the number of nearest neighbours appear in the expressions for the free-energy and the order parameters, with dipolar and quadupolar interactions. The equilibrium and the non-equilibrium behaviour of a spin-l Ising model Hamiltonian with J and K with the pair correlation was studied by Keskin and Meijer [22]. A general review of three-state systems can be found in the paper of Lawrie and Sarbach [23].

In spite of these studies, not all aspects of the phase transitions and the thermal variation of the order parameters have been studied in the spin-l Hamiltonian with arbitrary bilinear and biquadratic pair interactions. Especial- ly, the studies on the lower and upper limit of stability temperatures, T, and T,, respectively, as well as the transition temperatures T,, in the case of the first-order phase transition have not yet been reported. Therefore one of the aims of the present paper is to study these temperatures. For this purpose, the equilibrium behaviour of the model is studied in the lowest approximation of the cluster variation method of Kikuchi [24]. T,, T, and the critical temperatures in the case of the second-order phase transition are calculated by using the Hessian determinant [22,25], T,, is obtained calculation of the free-energy values while increasing and decreasing the temperature. The reason to find T,, T,, T,, and the critical temperatures is to establish the unstable, the metastable and stable region of this cooperative system.

The dynamic behaviour of the spin-l Ising model is also studied by the path probability method [26] in order to obtain the unstable solutions in the case of the first-order phase transition as well as to discuss the role of these solutions, as separators between the metastable and the stable points in the phase diagram.

The plan of our presentation is as follows: in section 2 the model description is given. In section 3 the set of self-consistent equations is obtained in order to analyze the equilibrium properties of the system. Also the Hessian determinant is calculated to find the critical and limit of stability temperatures. In section 4 the rate equations are obtained using the path probability method and the non-equilibrium behaviour of the model is investigated by means of the flow diagram [27,28]. Finally the discussion of the solutions is presented in section 5.

M. Keskin et al. ! Solutions of a spin-l Ising system 1003

2. Description of the spin-l Ising model

The spin-l Ising system is a three-state and two-order-parameter system. The average value of each of the spin state will be indicated by Xi, X, and X,, which are also called the state or point variables. X, is the fraction of spins value +l, X, is the fraction of spins that have the value 0, and X, is the fraction of spins that have the value -1. These variables obey the following normalization relation:

(2)

Two order parameters are introduced as follows: (1) the average magnetization (S) , which is the excess of one orientation over the other orientation, also called the dipole moment, (2) the quadrupole moment Q, which is a linear function of the average of the square of the magnetization (S’), written as

The

bY

Q=3(S’)-2. (3)

order parameters expressed in terms of these internal variables are given

S-(S)=X*-x,,

Q=(Q)=X,-2X,+X,. (4)

Using (2) and (4), the internal variables can be expressed as linear combina- tions of the order parameters

X,=$+;.S+bQ,

X, = +(I - Q> , (5)

The Hamiltonian X of such a model is

X3=$-;S+tQ.

Px = - c f (JV, + KQ,cQl> , (kl)

(6)

where J is the bilinear exchange interaction and K is the biquadratic exchange interaction. We have introduced J and K dependent on the temperature J-, Jlk,T and K+ Klk,T, where T is the absolute temperature and k, is the Boltmann factor and p = 1 lk, T.


Besides the spin interpretation, one can also use a description in terms of the lattice gas, introducing both vacant and occupied sites. Particles in the occupied sites can have two internal states, say orientations, corresponding to spin values of plus and minus one, and the spin value zero corresponds to vacant sites. Furthermore, K is related to the interparticle interaction which is independent of the relative orientation and J is the orientation-dependent part of the coupling.

3. Solutions for the system at equilibrium

The equilibrium properties of the system are determined by means of the lowest approximation of the cluster variation method. The method consists of the following three steps:

(1) consider a collection of weakly interacting systems and define the internal variables;

(2) obtain the weight factor W in terms of the internal variables; (3) find the free-energy expression and minimize it. The internal variables X,, X, and X, have been defined in section 2. The

weight factor W can be expressed in terms of the internal variables as

w= N!

,fi GW’)! ’ where N is the number of lattice points. A simple expression for the internal energy of such a system is found by working out eq. (6) in the mean-field approximation. This leads to:

PE -= N

-4~9 - ;KQ'

= -; J(X, - X,)’ - f K(X, - 2X, + X$

The entropy S, and the free-energy

S, = k, In W , F= E- TS,.

Using the eqs. (6)-(9) and making free-energy F can be now written as

(8)

F are given by

(9)

use of the Stirling approximation, the

M. Keskin et al. I Solutions of a spin-l Ising system 1005

=;J(X,-X,)‘+fK(X,-2X*+X,)2-i:Xi(lnxi-1) i=l

where A is introduced to obtain the normalization condition. The minimization of eq. (10) with respect to Xi gives

d@ -0 ay- (i = 1,2,3)

Using eqs. (2), (10) and (ll), the internal variables are found to be

Xi = e,lZ ,

where

e,=exp(-E) and Z=gei (i=1,2,3),

(11)

(12)

(13)

Z represents the partition function. e,, e2, e3 are calculated using eq. (8) as follows:

e, = exp(JS + KQ) ,

e2 = exp(-2KQ),

e3 = exp(-JS + KQ) .

(14)

One can find easily the following set of self-consistent equations by using eqs. (3), (12), (13) and (14):

S= 2 sinh(a KS)

exp(-3KQ) + 2 cosh(cl!KS) ’

Q = cosh(aKS) - exp(-3KQ)

cosh(cl!KS) + 1 exp(-3KQ) ’

(15)

where (Y = J/K is called the ratio of the coupling constants or the relative energy barrier.


These two nonlinear algebraic equations are solved by the Newton-Raphson method [29]. Thermal variations of the S and Q for several values of CY are plotted in figs. l-3. The discussion of these solutions will be given in the last section. However, at this point it should be noticed that for (Y > 6 a second- order phase transition exists and that for 3 6 CY < 6 a first-order phase transition occurs. For cy = 6 the system has a tricritical point.

I.6 2.0 3.0 50

Q, s,

v Fig. 1. The order parameters S and Q as functions of temperature exhibiting a second-order phase

transition. Subscript 1 indicates the stable state and 3 the unstable state. Dashed lines for (Y = 6 and

solid lines for (Y = 7.5.

Fig. 2. The order parameters S and Q as functions of temperature exhibiting a first-order phase

transition. Subscript 1 indicates the stable state, 2 the metastable state and 3 the unstable state. T,

and i”, are the lower and upper limit of stability temperatures, respectively and T,, is the first-order

phase transition temperature. (a) a = 4.0, (b) a = 3.0.


Fig. 3. The order parameters S and Q as functions of temperature for a = 2.0, exhibiting a first-order phase transition in Q. Subscript 1 indicates the stable state, 2 the metastable state and 3 the unstable state. Tql are the quasicritical temperatures. T, and TU are the lower and upper limit of stability temperatures for Q only.

The critical temperatures in the case of a second-order phase transition are given in fig. 1 for (Y = 6 and (Y = 7.5. In figs. 2a and b we indicate for two different values of 3 s (Y < 6, corresponding to the region of first-order phase transition, the lower and upper limits of stability temperatures T, and T,, respectively. They were found by calculation of the Hessian determinant which is the determination of the second derivatives of the free-energy with respect to Xi as follows:

a2?D ----= ax, ax, 0 (i, j = 1,2,3) .

The determinant can be written explicitly as

J+K-+ 1

A= -2K

-J+K

-2K

4K-+ 2

-2K

-J+K

-2K

J+K-f 3

(16)

(17)

The change in sign of the determinant corresponds to the critical temperatures while increasing and decreasing the temperature for the case (Y 2 6. In the case where 3 s CY < 6, the change of sign does not determine the temperature of the phase transition (where S and Q undergo a jump), but the upper and lower limits of stability. The upper limit is found with increasing the temperature coming from the ordered phase, and the lower limit is found with decreasing

1008 M. Keskin et al. I Solutions of a spin-l lsing system

the temperature from the disordered state. In the last case the determinant function crosses the line T = 0 for the second time, as seen in fig. 4. The first-order phase transition temperature, T,,, is between T, and T, and in order to find T,, we calculate the free-energy values while decreasing and increasing the temperature. The temperature where free-energies are equal to each other is the first-order phase transition temperature T,,.

Finally, fig. 5 shows the a-3/2 T, dependence of the critical temperature (solid line for cx 3 6.0) and the a-3 12 T,, dependence of the transition temperature (dashed line) in the case of the first-order phase transition. From

Fig. 4. The values of the determinant as a function of temperature for (Y = 4.0. D, corresponds to

the determinant values while decreasing the temperatures and D2 corresponds to increasing of

temperature.

P c--

1’

-031 T.&nperatu re

Fig. 5. The critical temperature (solid), and the first-order phase transition (dashed) as a function of (I - jT.


Table I The values of T,, T,, and T, for several values of the ratio of the coupling parameter (Y.

a T, Tt, T” 5.0 2.139 3.340 3.352 4.0 2.099 2.720 2.185 3.0 1.939 2.150 2.185

fig. 5 one can easily find T,, and the critical temperatures for different values of cr. We obtained the same critical and transition temperatures as Chen and Levy [4], but their fig. 1 is distorted in the region of 3 c (Y < 6 since the line of the first-order transition is not a straight extension of the line of the second-order transition, because the first-order transition has to end at the tricritical point. The tricritical point lies below the extension of the lines of the second-order phase transition. The dependence of T,, T,, and T,, for several values of (Y are also given in table I.

4. Dynamic equations and non-equilibrium behaviour

Since the time-dependent statistics of an Ising system proposed by Glauber [30] is difficult to extend to systems which have more than one order parameter [31], the path probability method (PPM) of Kikuchi [26] is used in order to construct a possible set of dynamic equations. The PPM is the natural extension into the time domain of the cluster variation method (CVM) and it provides a systematic derivation of the rate equations for successive approxi- mations which are well known in equilibrium statistical mechanics. It is remarkable that the PPM has received much less attention than the CVM which is widely used. The PPM was also used to study homogeneous and inhomogeneous stationary systems, such as: substitutional diffusion in ordered systems [32]; diffusion and superionic conductivity in solid electrolytes 1331; nucleation and growth in two dimesions [34]; and the kinetics of the order- disorder transformation in bee alloys [35,36] and inhomogeneous non-stationary systems [37]. Moreover the dynamic behaviour of the Pople and Karasz model [38] as well as fluctuation problems [39] were recently studied PPM.

by the

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


where X, is the path probability rate for the system to go from state i toj. The coefficients xii are the product of three factors: k, the rate constants with k,, = k,,, a temperature-dependent factor which guarantees that the time- independent state is the equilibrium state, and a third factor which is the fraction of the system that is in the state i, e.g. X,. Detailed balancing requires that:

x, = M,, (19)

The following two options were introduced by Kikuchi [26b]:

(A) X,i = k,Z-‘Xi exp

(B) X,j = k,tZ-‘X, exp

(204

(2Ob)

which both fulfill the necessary requirements expressed by eq. (19). Assump- tion A is called recipe II and assumption B is called recipe I by Kikuchi [26b]. The constants k, can be functions of the temperature. The simplest assumption for this temperature dependence is to use an Arrhenius factor: k, exp(- ul k,T), where U is the activation energy, which ensures that the rate will go to zero at T = 0 as required. The spin-l model can be used for a lattice gas containing molecules that have two orientations: using X, and X3 as occupation number and X, as hole. There are two rate constants in this model as follows: the first rate constant is k12 = kz3 = k, which is the insertion or removal of particles associated with the translation of particles through the lattice. The second rate constant k,, = k, is associated with reorientation of a molecule at a fixed site. It is assumed that double processes, the simultaneous insertion or removal or rotation of two particles do not take place, i.e. only single jumps are allowed. The occurrence of the rate constant is given in table II.

The formulas will be based on the recipe II because the general behaviour of the solution of the rate equations, namely flow diagrams, is not drastically

Table II The description of the rate

constants.

Xl X2 X,

k, k,

k,

k,

k, k


changed, i.e. the lines follow more or less the same pattern using either recipe

I or recipe II (see e.g. refs. [5,31]). Using eq. (4), the rate equations are written

dS dX, dX, _--- dt - dt dt ’

dQ dX,_2dX,+dX,= dX, P-1)

_- - dt- dt dt dt -3 dt ’

Inserting eq. (2Oa), namely the recipe II, into eq. (18) and using eq. (21), the rate equations for the order parameters are obtained:

Z & = (ke, + e2 + ke,)S - 5 (1 - k)(e, - e,)Q + y (e, - e3) y 1

dQ ’ k,dt

- = -(el + e, + e,)Q + (e, - 2e, f e3) , (22)

where k = k,lk,, ei and Z are given by eqs. (13) and (14). The solutions of the rate equations are expressed by means of a flow diagram

[27,28], which shows the solution of these equations in a two-dimensional phase space of S and Q, starting with intial values very close to the boundaries. As time progresses (by given small steps) the values of S and Q are computed, and the point representing them moves in the plane. A set of solution curves is obtained by considering all different initial values. The results are given, for fixed values of (Y, k,T and ki, in fig. 6 for the case of the second-order phase

Q

Fig. 6. The flow diagram of the systems for a = 7.5, k,T= 4.5 and two different sets of rate constants: (solid) k, = k, = 1.0 and (dashed) k, = 1.0, k, = 10.0. The open circle corresponds to the stable solution and the closed circle the unstable solution.

1012 M. Keskin et al. I Solutions of LI spin-l lsing system

a 3

1.0

0.8 Q

0.6

04

0.2

i 0 d2 o:L 016 S 018 b

1

Fig. 7. The flow diagram of the system for two different sets of values of the constants: (solid)

k, = kL = 1.0 and dashed k, = 1.0; kL = 10.0. The open circle corresponds to the stable solution,

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

Dash-dotted lines are the separators. (a) cy = 4.0. k,,T= 2.695; (b) (Y = 3.0, k,,T = 2.1.

._

0.5-

S

O-

-0.5--

-2.0 -1.5 -10 -05 0 05 1.0

a Q

10 I

S

0.0

i -1.0 ;

-2.0

b Fig. 8. (a) Same as fig. 7 but CI =2.0 and k,T= 1.25. (b) Same as (a) but k,T=2.J


transition, in fig. 7 for the case of the first-order phase transition, and in fig. 8 for the case where there is a first-order phase transition in Q, with S = 0. By comparing the coordinates of fixed points with the free-energy value at that point, we established whether these solutions were stable or metastable. The open circles correspond to the stable equilibrium solutions and the closed circles correspond to the unstable equilibrium solutions and the closed squares are the metastable equilibrium solutions. The discussion of these solutions will be given in the next section.

5. Discussion of results and summary

The equilibrium behaviour of the spin-l Hamiltonian with bilinear (J) and biquadratic (K) pair interaction has been studied for zero field in the lowest approximation of the cluster variation method. Thermal variations of S and Q for several values of (Y are plotted in figs. 1-3. The behaviour of the system depends on the (Y values and, by comparing the free-energy values of these solutions, the following results have been found:

(a) ff 2 6, the stable vales of S and Q are larger than zero and decrease to zero continuously as the temperature increases; therefore a second-order phase transition occurs [40]. The values of S = Q = 0 below the critical temperature are unstable solutions and they are found while decreasing the temperature

(fig. 1). (b) 3 < (Y < 6, the stable values of S and Q are again bigger than zero, but

decrease to zero discontinuously; hence a first-order phase transition occurs [40]. The metastable values of S and Q occurs between T, and T, displayed in fig. 2a and the values of S = Q = 0 below the T, are unstable solutions.

(c) (Y = 3 is a special case; it corresponds to total symmetry in the three variables X, , X,, X, [13]. The stable values are obtained as S = Q > 0 and they decrease to zero discontinuously. The metastable values of S and Q are found to be zero between T, and T,. Again the values of S = Q = 0 below the T, are also unstable solutions (see fig. 2b).

(d) cy > 3. In this case there is a first-order transition in Q with Q < 0 and so S = 0, as pointed out by Chen and Levy [4]. Below a certain temperature, the metastable values of S and Q occur with S = 0 and Q > 0. This temperature we called therefore the quasicritical temperature [6,35] and is indicated by T,, in fig. 3. Furthermore, there is one more quasicritical temperature, indicated by Tq2. In this case, below t,, there are two different metastable values of S and Q which are S > 0, Q > 0 and S < 0, Q > 0 as seen in fig. 3. The last two solutions of S have symmetry with respect to the temperature axis. More than one metastable phase can be seen also in some experimental works on the rapid


solidification technology [41]. There are also three different unstable values of

S and Q which are S = 0, Q = 0 below T,r, S > 0, Q < 0 and S < 0, Q < 0, as

well as S = Q = 0 below the temperature which is indicated with an arrow in

fig. 3. The analytical discussion of the last two set of these unstable state

solutions is given in the Appendix. Again, the last two unstable solutions of S

have symmetry with respect to the temperature axis.

We gave the (Y dependence of the critical temperature (solid line), and the CY

dependence of T,, (dashed line) in fig. 5. The dependence of T,, T,, and T,, for

several values of (Y are given in table 1.

The non-equilibrium behaviour of the same model was also studied by the

PPM, and the results were given by means of the following diagrams in figs.

6-8. The rate equations are mainly solved for temperatures between T, and T,,

because between T, and T, there are unstable solutions, which cannot be found

by the Newton-Raphson method. They have been obtained via the soluton of

the dynamic equations and are plotted in fig. 2 (dashed lines) and fig. 3 (only

dotted line). These unstable solutions are like a saddle point, as can be seen in

figs. 7 and 8. For in the neighbourhood the system goes into either the

metastable or the stable state. In the case of the second-order phase transition,

the system always relaxes into the stable state as seen in fig. 6, since there is no

metastable state, see fig. 1. This is also the case in the first-order phase

transitions below the T, since there is no metastable solution; again the system

always relaxes towards the stable state like fig. 6. In fig. Sa, there is more than

one metastable state which can also be seen in some experimental works on the

rapid solidification technology (RST) [41]. In this figure, the unstable equilib-

rium solutions cannot be seen very well since there are three different unstable

solutions close together, namely at S = 0, Q = 0; S = +0.206, Q = -0.058 and

S = -0.206, Q = -0.058 ( see fig. 3 for k,T = 1.25). Since the flow lines cannot

get close to each other, one cannot see the unstable solutions as well as in figs.

7 and 8b.

It should be mentioned that changing the k-values does not influence the

positions of the stable, metastable and unstable equilibrium solutions. How-

ever, changing the k-value does change the initial part of the flow lines,

although the final approach of the lines remains almost the same, as seen in

figs. 6-8. The reason for choosing k, > k, was that most systems have a shorter

relaxation time for a rotation than for a translation.

Finally, one of the aims, of this work was to study how to obtain the

metastable phases, since they are of importance in experimental works, such as

the RST. The rapid cooling of alloys or metals leads to amorphous structures,

namely metastable phases, and it is known that the properties of the alloys or

metals improve drastically such as strength, stiffness, fatigue behaviour, corro-

sion, toughness, density modules, etc. [42]. From the theoretical viewpoint we


conclude that to make it more likely for the system to relax into the metastable states, one needs the following conditions:

(a) For k, = k, and 3 < (Y < 6, the region in phase space leading to the metastable state is larger than for k, > k, as seen in fig. 7 by the help of the separator lines.

(b) In the case (Y < 3, one can obtain one metastable state below T, and two

metastable states below Tq2. For k, = k, the region in phase space leading to the metastable state is smaller than for k, > k, (see fig. 8).

(c) One can always attain the metastable phase between T, and T, in the case of the first-order phase transition and below the quasicritical temperature.

Acknowledgements

This work was supported by NATO Grant No. 86/708. We also would like to express our thanks to the Computer Center of Erciyes University.

Appendix

The unstable state found in the computation corresponds to the following conditions on the energy levels: one of the two spin levels, say the level with S < 0, lies so far above the two others that its Boltzmann factor can be ignored. The two remaining levels are nearly degenerate, that is aS = 3Q. Consequent- ly the two self-consistent equations become:

1 - 2r Q=-.-- l+r ’

s=-1 l+r’ 64.1)

with r = e,le, = exp(-x/T). Here x = 3Q - cuS. The value for this order parameter is determined by the root of

For small x this leads to

9+a 6(3+ o) _x+ ***

(A.2)

(A.3)

The result is that in the limit of x-0, T+ 0 under the condition that x/T

remains equal to ln(6/a + 3), one finds for the order parameter S and Q the values:

1016 M. Keskin et al. i Solutions of a spin-l Ising system

-6 s=- -2a

9+(Y) Q=-

9+ff. (A.41

The same result for Q can be obtained when S has the opposite sign. The energy at T = 0 is E = -(aS* + Q’). For the ground state with S = kl, Q = 1 this gives E = -(a + 1) = -3. But for the unstable state using (A.4) one finds E = -4a/(9 + US) = - fi, which is higher as it should be.

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Stable, metastable and unstable solutions of a spin-1 Ising system obtained by the molecular-field approximation and the path probability method