-~ ~ H journal of magnetism

~ i and magnetic materials

ELSEVIER Journal of Magnetism and Magnetic Materials 146 (1995) L247-L250

Letter to the Editor

Stable, metastable and unstable solutions of a spin-1 Ising system in the presence of magnetic fields due to the dipole

and quadrupole moments

Mustafa Keskin a,* Hiiseyin Arslan b

Fizik Bfliimii, Gaziosmanpa~a Oniversitesi, 60110 Tokat, Turkey b Fizik Bi~liimii, Kahramanmara~ SiJtgii Imam Onicersitesi, 46100 Kahramanmara{, Turkey

Received 10 October 1994; in revised form 2 February 1995

Abstract

The spin-1 Ising model with bilinear (J) and biquadratic (K) interactions is studied for magnetic fields H s and He_) representing dipole and quadrupole moments, respectively, by using the lowest approximation of the cluster variation method. The influence of H s, HQ and a = J / K on metastable and unstable solutions, which are very important for many theoretical and experimental works, is investigated and it is found that metastable and unstable solutions occur at high temperatures when o~ and HQ are increased. However if H s is increased, metastable and unstable solutions can be found only at low temperatures.

The spin-1 Ising model Hamiltonian with arbi- trary bilinear, biquadratic and crystal-field interac- tions was introduced by Blume, Emery and Griffiths [1] to describe phase separation and superfluid order- ing in H e3 -He 4 mixtures. With disappearing bi- quadratic exchange interactions the model is known as the Blume-Capel model [2]. The model was subsequently reinterpreted to describe phase transi- tions in simple and multicomponent fluids [3]. It has become very attractive because of its simplicity and rich fixed-point structure. The model has been slud- ied by the mean-field approximation [1-3], high- temperature series expansions [4], the constant-cou-

* Corresponding author. Fax: +90-356-214 1127. Permanent adress: Fizik B/iliimii, Erciyes Llniversitesi, 38039 Kayseri, Turkey.

pling approximation [5], Monte Carlo methods [6], renormalizat ion group methods [7] and the effective-field theory [8] and Monte Carlo renormal- ization group techniques [9]. Nonequilibrium proper- ties of a spin-1 Ising system have been also studied by the path probability method [10], a multidimen- sional kinetic model based on the Glauber model [11] and the real-space renormalization group tech- nique [12].

Recently stable, metastable and unstable solutions of a spin-1 Ising model with arbitrary J and K pair interactions have been studied for zero magnetic field [13] as well as with an external magnetic field present [14].

The purpose of the present letter is twofold: first, is to report on the study of the spin-1 Ising model with bilinear ( J ) and biquadratic ( K ) interactions for magnetic fields due to the dipole and quadrupole

0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01597-X

L248 M. Keskin, H. Arslan /Journal of Magnetism and Magnetic Materials 146 (1995) L247-L250

moments, H s and HQ respectively and second, to investigate the effect of H s, H e and a = J / K on metastable and unstable solutions which are very important for many experimental and theoretical cases, such as metallic glasses, binary alloys, super- fluids, superconductors, gels, lasers, magnetic sys- tems, astrophysics, glasses and crystalline ceramics, etc. [15].

The spin-1 Ising system is a three-state system and the average value of each of the spin states will be indicated by X 1, X 2 and X 3 which are called the state or point variables. X 1 is the average fraction of spins with value + 1, X 2 is the average fraction of spins that have the value 0, and X 3 is the average fraction of spins that have the value - 1. Two long- range order parameters are introduced as follows: (1) the average magnetization (S) , which is the excess of one orientation over the other orientation, also called dipole moment, and (2) the quadrupole mo- ment Q, which is a linear function of the average of the squared magnetization (S 2), written as

Q = 3 ( S 2) - 2 . (1)

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

S --- ( S ) = X t - X~, (2a)

Q - ( Q ) = X 1 - 2 x 2 + x 3. (2b)

The model here studied is described by the spin-1 Hamiltonian,

/ 3 x - 1

_ _ -2 ~_~ (JSiSj + KQiQj ) (ij)

_ y" ( n s S , + HoQi ), (3)

where J is the exchange coupling and K is the quadrupole coupling. We have introduced coupling parameters J and K, and also fields H s and H o which depend on the temperature J ~ J / k T , K

K / k T , H s ~ H s / k T and HQ ~ H o / k T where T is the absolute temperature and k is the Boltzmann factor and /3 = 1 / k T . H s is the magnetic field which corresponds to S and H e is the field corresponding to Q.

Using the lowest approximation of the cluster variation method, the free energy F can be found as

• = - / 3 F / N 3

I KQ2 = ½ Js2 ÷ 5 ÷ OsS ÷ HQQ - E Xi(ln X i 1 ) i=1

where A is introduced to maintain the normalization condition and N is the number of lattice points. The first four terms of Eq. (4) are the internal energy of the system and the fifth term gives the entropy of the system in this approximation.

The minimization of Eq. (4) with respect to X i and using Eq. (2), gives the set of self-consistent equations

2 sinh( aKS + Hs) S =

Q =

e x p [ - 3 ( K Q + 14o) ] + 2 cosh( aKS + H s ) '

(5a)

cosh( aKS + Hs) - exp[ - 3 (KQ + Ho) ]

cosh( aKS + Hs) + ½exp[ - 3 (KQ + H Q ) ] '

(5b)

where a = J / K is called the ratio of the coupling constants or relative energy barrier. The two nonlin- ear algebraic equations are solved by using the New- ton-Raphson method. Thermal variations of the or-

" '.... ~. .

0.5- o3 ~ . . . .

$1 o 0 . . . . . . . . . . . . . . . . . . . 1 Tqc

S3 " V')

-0.5' /

$2,,.,

kT

Fig. 1. Thermal variations of order parameters S and Q as a function of the reduced temperature. Subscript 1 indicates the stable solutions (drawn lines,) 2 the metastable solutions (dashed- dotted lines) and 3 the unstable solutions (dotted lines). Tqc is the quasicritical temperature, a = 7.5, H s = HQ = 0.1.

M. Keskin, H. Arslan /Journal of Magnetism and Magnetic Materials 146 (1995) L247-L250 L249

der parameters S and Q for a = 7.5, H s = HQ = 0.1 are plotted in Fig. 1. In the figure, subscript 1 indicates the stable solutions (drawn lines), 2 the metastable solutions (dashed-dotted lines) and 3 the unstable solutions (dotted lines). This classification is done by comparing the free energy values of these solutions. It can be seen from the figure that below a certain reduced temperature two more solutions, namely metastable and unstable solutions of S and Q exist. This temperature is called the quasicritical temperature, Tqc. Thermal variations of stable, metastable and unstable solutions of S and Q as a function of the reduced temperature for various val- ues of a, H s and HQ are given and discussed in Ref. [16] extensively.

Now we should investigate the influence of a, H s and HQ on the occurring and the termination of metastable and unstable solutions which is one of the main purposes of the present work. Metastable and unstable state solutions are very important in many theoretical and experimental works such as simple fluids (gas-liquid transitions), binary fluids, binary alloys, superfluids and superconductors, physisorp- tion and chemisorption systems, intercalation com- pounds, polymer blends, gels, lasers, electron-hole condensation in semiconductors, geological systems (minerals), chemically reacting systems, metals, glasses and crystalline ceramics, order-disorder sys- tems, coherent hydrogen-metal systems, magnetic systems and astrophysics [15]. For example, the rapid cooling of alloys or metals leads to amorphous struc- tures, namely metastable phases, and it is known that the properties of the alloys or metals improve drasti- cally such as strength, stiffness, fatigue behaviour, corrosion, toughnes, density modules, etc. [17]. For this reason, we plotted first a as a function o f Tqc for several values of H s and HQ i n Fig. 2, then Tqc

is plotted as a function of H s for a constant a and HQ in Fig. 3, and as a function of HQ in Fig. 4 for constant values of a and H s. Fig. 2 illustrates that a is a linear function of Tq~ and slopes are small for small values of H s and big values of HQ. Tq~ is an exponentially decaying function of H s for constant values of a and HQ, seen in Fig. 3. Fig. 4 shows a plot of HQ as a function of Tq~ for a = 1 . 5 and H s = 0.1 and it is seen that Tqc increases with HQ but around Tqc = 1.155, H a asymptotically increases. Finally, from the theoretical viewpoint we conclude

/ / . " .... el/b/ [ / d

~ O I / l / I .......

/ / / /

2c ......

./...-

i

20 40

Tqc Fig. 2. The quasicritical temperature as a function of the ratio of

the coupling parameter. Metastable and unstable solutions occur

below Tqc. (a) H s = 0.5; HQ = 0.1; (b) H s = HQ = 0.5; (c) H s = HQ = 0 . 1 ; (d) H s =O.1, HQ =0 .5 .

1.3

O9

0.5

01 . , , , ,'c'-

O. 5 0.7 0.9 1.1

Tqc Fig. 3. The quasicritical temperature as a function of H s for a = 1.5 and HQ = 0.1. Metastable and unstable solutions occur

below Tqc.

112 Ii3 I.i~ I~5 116 Tqc

Fig. 4. The quasicritical temperature as a function of H 0 for ot = 1.5 and H s = 0.1. Metastable and unstable solutions occur

below Tqc.

L250 M. Keskin, H. Arslan /Journal of Magnetism and Magnetic Materials 146 (1995) L247-L250

that if one wants to obtain metastable and unstable state solutions at high kT, one has to take large values of H• and small values of H s, seen in Figs. 2-4 .

References

[1] M. Blume, V.J. Emery and R.B. Griffiths, Phys. Rev. A 4 (1971) 1071.

[2] M. Blume, Phys. Rev. 141 (19661 517; H.W. Capel, Physica (Utrecht) 32 (19661 966; 33 (1967) 295; 37 (1967) 423.

[3] J. Lajzerowicz and J. Sivardi~re, Phys. Rev. A 11 (19751 2079; J. Sivardibre and J. Lajzerowicz, ibid. 11 (1975) 2090; 11 (1975) 2101; D. Mukamel and M. Blume, Phys. Rev. A 10 (1974) 610; D. Furman, S. Dattagupta and R.B. Griffiths, Phys. Rev. B 15 (1977) 441.

[4] D.M. Saul, M. Wortis, and D. Stauffer, Phys. Rev. B 9, (1974) 4964; R.I. Joseph and R.A. Farrell, Phys. Rev. B 14 (1976) 5121.

[5] K. Takahashi and M. Tanaka, J. Phys. Soc. Jpn. 46 (1979) 1428.

[6] A.K. Jain and D.P. Landau, Phys. Rev. B 22 (1973) 445; M. Tanaka and N, Nagahashi, Phys. Stat. Sol. (b) 98 (19801 KI7; M. Tanaka and Y. Fujiwara, Phys. Stat. Sol. (b) 143 (1987) K107.

[7] A.N. Berker and M. Wortis, Phys. Rev. B 14 (1976) 4946; O.F. de Alcantara Bonfim and F.C. S~Baretto, Phys. Lett. 109A (1985) 341; C.E.I. Carneiro, V.B. Henriques and S.R. Salinas, J. Phys. A: Math. Gen. 20 (1987) 189.

[8] I.P. Fittipaldi and A.F. Siqueira, J. Magn. Magn. Mat. 54-57 (1986) 694; J.W. Tucker, ibid. 104-107 (1992) 191.

[9] R.R. Netz, Europhys Lett. 17 (19921 373. [10] M. Keskin and P.H.E. Meijer, Physica A 129 (1983) 1; M.

Keskin, ibid. 135 (1986) 226; M. Keskin and P.H.E. Meijer, J. Chem. Phys. 85 (1986) 7324.

[11] T. Obokata, J. Phys. Soc. Jpn. 26 (1969) 895; G.L. Batten, Jr. and H.L. Lemberg, J. Chem. Phys. 70 (1978) 2934.

[12] Y. Achiam, Phys. Rev. B 31 (1985) 260. [13] M. Keskin, M. An and P.H.E. Meijer, Physica A 157 (1989)

1000; M. Keskin and S. Ozgan, Phys. Lett. A 145 (1990) 340; M. Keskin, M. An and S. 13zgan, Tr. J. of Phys. 15 (1991) 575.

[14] M. Keskin, Physica Scripta 47 (19931 328. [15] J.D. Gunton and M. Droz, Introduction to the Theory of

Metastable and Unstable States (Springer, Berlin 1983); J.D. Gunton, M. San Miquel and P.S. Sahni, in: Phase Transitions Critical Phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983).

[16] M. Keskin and H. Arslan, Turkish J. of Physics, to appear. [17] See, e.g., H. Jones, Rapid Solidification of Metals and Alloys

(Institution of Metallurgists, London, 1982); Amorphous Metallic Alloys, F.E. Luborsky, ed. (Butterworths, London, 19831; Rapid Solidification Technology, R.L. Ashbrook, ed. (American Society for Metals, Metals Park, OH, 1983).