ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 321 (2009) 3905–3912

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Phase diagrams of a nonequilibrium mixed spin-3/2 and spin-2 Ising systemin an oscillating magnetic field

Mustafa Keskin a,�, Yasin Polat b

a Department of Physics, Erciyes University, 38039 Kayseri, Turkeyb Institutes of Science, Erciyes University, 38039 Kayseri, Turkey

a r t i c l e i n f o

Article history:

Received 11 June 2009

Received in revised form

16 July 2009Available online 3 August 2009

PACS:

05.50.+q

05.70.Fh

64.60.Ht

75.10.Hk

75.50.Gg

Keywords:

Mixed spin-3/2 and spin-2 Ising

ferrimagnetic system

Glauber-type stochastic dynamic

Dynamic phase transition

Dynamic phase diagram

53/$ - see front matter & 2009 Elsevier B.V. A

016/j.jmmm.2009.07.052

esponding author.

ail address: keskin@erciyes.edu.tr (M. Keskin)

a b s t r a c t

The phase diagrams of the nonequilibrium mixed spin-3/2 and spin-2 Ising ferrimagnetic system on

square lattice under a time-dependent external magnetic field are presented by using the Glauber-type

stochastic dynamics. The model system consists of two interpenetrating sublattices of spins s ¼ 3/2 and

S ¼ 2, and we take only nearest-neighbor interactions between pairs of spins. The system is in contact

with a heat bath at absolute temperature Tabs and the exchange of energy with the heat bath occurs via

one-spin flip of the Glauber dynamics. First, we investigate the time variations of average order

parameters to find the phases in the system and then the thermal behavior of the dynamic order

parameters to obtain the dynamic phase transition (DPT) points as well as to characterize the nature

(first- or second-order) phase transitions. The dynamic phase diagrams are presented in two different

planes. Phase diagrams contain paramagnetic (p), ferrimagnetic (i1, i2, i3) phases, and three coexistence

or mixed phase regions, namely i1+p, i2+p and i3+p mixed phases that strongly depend on interaction

parameters.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Over the last few decades, magnetic properties of twosublattice mixed-spin systems with a crystal-field interactionor single-ion anisotropy have been very intensively studiedboth experimentally and theoretically. Experimentally, theAk[B(CN)6]lnH2O [1], MnNi(EDTA)-6H2O [2] and AFeII FeIII

(C2O4)3 [A ¼ N(n-CnH2n+1, n ¼ 3–5] [3] complexes have beenshown to be examples of mixed-spin systems. These molecular-based magnetic materials are considered to be possible usefulmaterials for magneto-optical recordings [3–5]. Theoretically,mixed-spin systems have been extensively studied by the well-known methods in equilibrium statistical physics due to thereason that they exhibit the much richer critical behavior thantheir single-spin counterparts. Another aspect that started toattract an appreciable interest towards mixed-spin systems is thatthese systems are well adapted to study a certain type offerrimagnetism which are of great interest because of their

ll rights reserved.

.

interesting and possible useful properties for technologicalapplications as well as academic researches.

Four well known and most studied mixed-spin Ising systemsare spins (1, 1/2), spins (1/2, 3/2), spins (1, 3/2) and spins (1, 2)systems. Equilibrium behavior of these four mixed-spin systemshave been investigated by variety of techniques such as the mean-field approximation (MFA), the effective-field theory (EFT), thecluster variation method (CVM), Bethe–Peierls approximation(BPA), Monte-Carlo (MC) simulations (see [6] and referencestherein). Moreover, the exact solutions of these systems on certainlattices, such as a honeycomb lattice, a bathroom-tile, dicedlattices, the Bethe lattice and two-fold Cayley tree, and severaldecorated planer lattices, have been also studied in detail (see [7]and references therein). While the equilibrium properties of thesemixed-spin systems have been investigated extensively, the none-quilibrium behavior of these systems have not been as thoroughlyexplored. An early attempt to study dynamics of mixed spin-1/2and spin-1 Ising system was made by Buendıa and Machado [8]within the Glauber-type stochastic dynamics. The nonequilibriumbehavior of this system was also studied by using the dynamicalpair approximation [9], the dynamic MC simulations and finite-size scaling arguments [10], the MC simulations and the dynamicalpair approximation [11–13], the pair approximation with point

ARTICLE IN PRESS

M. Keskin, Y. Polat / Journal of Magnetism and Magnetic Materials 321 (2009) 3905–39123906

distribution [14] and Glauber-type stochastic dynamics [15]. Veryrecently, we studied the dynamics of mixed-spins Ising systemswith spins (1/2, 3/2) [16], with spins (1, 3/2) [17], with spins (1, 2)[18] as well as with spins (1, 5/2) [6].

On the other hand, to date there has been little work dealingwith the mixed high-spin Ising systems. One of the mixed high-spin Ising systems is the mixed spin-3/2 and spin-2 Ising system.As far as we know, there are two recent papers in which theequilibrium behavior of this mixed system is studied [19,20].Bob�ak and Dely [19] investigated the effect of single-ionanisotropy on the phase diagram of the mixed spin-3/2 andspin-2 Ising system by the use of a mean-field theory based onBogoliubov inequality for the free energy. Albayrak [20] examinedthe critical and compensation temperature for this mixed systemon the Bethe lattice by using the exact recursion equations. To ourknowledge, the nonequilibrium properties of the mixed spin-3/2and spin-2 Ising system have not been investigated. Therefore, theaim of this present paper is to investigate dynamical aspect of themixed spin-3/2 and spin-2 Ising system Hamiltonian with acrystal-field interaction in the presence of a time-dependentoscillating external magnetic field by using the Glauber-typestochastic dynamics. We employ the Glauber transition rates toconstruct the set of mean-field dynamic equations. We investigatethe time variations of average magnetizations to find the phasesin the system. We also study the thermal behavior of the dynamicmagnetizations to characterize the nature (continuous or dis-continuous) of the phase transitions and obtain the DPT points,and finally present the dynamic phase diagrams in two differentplanes.

The rest of the paper is organized as follows. In Section 2, themodel and its formulations, namely the derivation of the set ofmean-field dynamic equations, are given by using Glauber-typestochastic dynamics in the presence of a time-dependentoscillating external magnetic field. In Section 3, the numericalresults for average order parameters, the DPT points and phasediagrams are studied in detail. Finally, we give a summary andconclusion in Section 4.

2. Model and formulations

The mixed spin-3/2 and spin-2 Ising model is described as atwo-sublattice system, with spin variables si ¼73/2, 71/2 andSj ¼72, 71, 0 on the sites of sublattices A and B, respectively. Thesystem has two long-range order parameters, namely the averagemagnetizations /sS and /SS for the A and B sublattices,respectively, which are the excess of one orientation over theother, also called the dipole moments. The Hamiltonian of themixed spin-3/2 and spin-2 Ising model with the bilinear (J)nearest-neighbor pair interaction and a single-ion potential orcrystal-field interaction (D) in the presence of a time-dependentoscillating external magnetic field is

H ¼ �JX/ijS

sAi SB

j � D�X

i

½ðsAi Þ

2� 5=4� þ

Xj

½ðSBj Þ

2� 2�

�

� HðX

i

sAi þ

Xj

SBj Þ; ð1Þ

where /ijS indicates a summation over all pairs of nearest-neighboring sites, and H is an oscillating magnetic field of theform

HðtÞ ¼ H0cos ðwtÞ; ð2Þ

where H0 and w ¼ 2pn are the amplitude and the angularfrequency of the oscillating field, respectively. The system is incontact with an isothermal heat bath at absolute temperature.

Now, we apply Glauber-type stochastic dynamics [21] toobtain the mean-field dynamic equation of motion. Thus, thesystem evolves according to a Glauber-type stochastic process at arate of 1/t transitions per unit time. Leaving the S spins fixed, wedefine PA(s1,s2,y,sN;t) as the probability that the system has thes-spin configuration, s1,s2,y,sN, at time t, also, by leaving the sspins fixed, we define PB(S1,S2,y, SN;t) as the probability that thesystem has the S-spin configuration, S1,S2,y, SN, at time t. Then,we calculate Wi

A(si-s0 i) and WjB(Sj-S0j), the probabilities per

unit time that the ith s spin changes from si to s0 i ( while thespins on B sublattice momentarily fixed) and the jth S spinchanges from Sj to S0j (while the spins on A sublattice momentarilyfixed), respectively. Thus, if the spins on the sublattice Bmomentarily fixed, the master equation for the sublattice A canbe written as

d

dtPAðs1;s2; . . . ;sN ; tÞ ¼

�X

i

� Xsias0 i

WAi ðsi-s0iÞ

�PAðs1;s2; . . . ;si; . . .sN; tÞ

þX

i

� Xsias0 i

WAi ðs

0i-siÞP

Aðs1;s2; . . . ;s0i; . . .sN; tÞ�; ð3Þ

where WiA(si-s0i) is the probability per unit time that the ith

spin changes from the value si to s0i, and in this sense the Glaubermodel is stochastic. Since the system is in contact with a heat bathat absolute temperature Tabs, each spin can change from the valuesi to s0 i with the probability per unit time;

WAi ðsi-s0 iÞ ¼

1

texpð�bDEAðsi-s0iÞÞP

s0 iexpð�bDEAðsi-s0 iÞÞ

; ð4Þ

whereP

s0 i is the sum over the four possible values of s0 i, 73/2,71/2, and

DEAðsi-s0 iÞ ¼ �ðs0i � siÞðH þ JX

j

SBj Þ � ½ðs

0iÞ

2� ðsiÞ

2�D; ð5Þ

gives the change in the energy of the system when the si-spinchanges. The probabilities satisfy the detailed balance condition

WAi ðsi-s0 iÞ

WAi ðs0 i-siÞ

¼PAðs1;s2; . . . ;s0 i; . . . ;sNÞ

PAðs1;s2; . . . ;si; . . . ;sNÞ; ð6Þ

and substituting the possible values of si, we get

WAi

3

2-�

3

2

� �¼WA

i

1

2-�

3

2

� �¼WA

i �1

2-�

3

2

� �

¼1

2texpð�bDÞexpð�3by=2Þ

expðbDÞcoshð3by=2Þ þ expð�bDÞcoshðby=2Þ;

ð7aÞ

WAi

3

2-�

1

2

� �¼WA

i

1

2-�

1

2

� �¼WA

i �3

2-�

1

2

� �

¼1

2texpð�bDÞexpð�by=2Þ

expðbDÞcoshð3by=2Þ þ expð�bDÞcoshðby=2Þ;

ð7bÞ

WAi

3

2-

1

2

� �¼WA

i �1

2-

1

2

� �¼WA

i �3

2-

1

2

� �

¼1

2texpð�bDÞexpðby=2Þ

expðbDÞcoshð3by=2Þ þ expð�bDÞcoshðby=2Þ;

ð7cÞ

ARTICLE IN PRESS

M. Keskin, Y. Polat / Journal of Magnetism and Magnetic Materials 321 (2009) 3905–3912 3907

WAi

1

2-

3

2

� �¼WA

i �1

2-

3

2

� �¼WA

i �3

2-

3

2

� �

¼1

2texpðbDÞexpð3by=2Þ

expðbDÞcoshð3by=2Þ þ expð�bDÞcoshðby=2Þ;

ð7dÞ

where y ¼ H+JP

jSjB. Notice that, since Wi

A(si-s0 i) does notdepend on the value si. We can therefore write Wi

A(si-

s0 i) ¼WiA(s0i), then the master equation becomes

d

dtPAðs1;s2; . . . ;sN; tÞ ¼

�X

i

� Xsias0 i

WAi ðs

0iÞ

�PAðs1;s2; . . . ;si; . . . ;sN ; tÞ

þX

i

WAi ðsiÞ

� Xsias0 i

PAðs1;s2; . . .s0 i; . . . ;sN; tÞ�: ð8Þ

Since the sum of probabilities is normalized to one, bymultiplying both sides of Eq. (8) by si for mA and taking theaverage, we obtain

t d

dt/siS ¼ �/siS

þ/3expðbDÞsinhð3by=2Þ þ expð�bDÞsinhðby=2Þ

2expðbDÞcoshð3by=2Þ þ 2expð�bDÞcoshðby=2ÞS:

ð9Þ

This dynamic equation can be written in terms of a mean-fieldapproach; hence the second mean-field dynamical equation of thesystem in the presence of a time-varying field is:

Od

dxmA ¼ �mA

þ3expðd=TÞsinh½3ðmB þ h cosxÞ=2T� þ expð�d=TÞsinh½ðmB þ h cosxÞ=2T�

2expðd=TÞcosh½3ðmB þ h cosxÞ=2T� þ 2expð�d=TÞcosh½ðmB þ h cosxÞ=2T�;

ð10Þ

where mA ¼ /sS, mB ¼ /SS, x ¼ wt, T ¼ (bzJ)�1, h ¼ H0/zJ, d ¼ D/zJ, O ¼ tw.

Now assuming that the spins on sublattice A remainmomentarily fixed and that the spins on the sublattice B change,we obtain the mean-field dynamical equation of mB for the Bsublattice by using a similar calculation as before, except we takeSj ¼72, 71, 0 instead of si ¼73/2, 71/2. The set of mean-fielddynamical equations for the B lattice are obtained as

OdmB

dx¼ �mB

þ2exp½4d=T�sinh½2ðmA þ hcosxÞ=T� þ exp½d=T�sinh½ðmA þ hcosxÞ=T�

exp½4d=T�cosh½2ðmA þ hcosxÞ=T� þ exp½d=T�cosh½ðmA þ hcosxÞ=T� þ 0:5:

ð11Þ

Thus, a set of mean-field dynamical equations of the systemare obtained. We fixed z ¼ 4 and O ¼ 2p. In the next section, wewill give the numerical solution and a discussion of the set ofcoupled mean-field dynamical equations.

3. Numerical results and discussions

3.1. Time variations of the average order parameters

In order to investigate the behaviors of time variations of orderparameters, first we have to study the stationary solutions of the

set of coupled mean-field dynamical equations, given in Eqs. (10)and (11), when the parameters T, d and h are varied. The stationarysolutions of these equations will be periodic functions of x withperiod 2p; that is,

mAðxþ 2pÞ ¼ mAðxÞ and mBðxþ 2pÞ ¼ mBðxÞ: ð12Þ

Moreover, they can be one of two types according to whetherthey have or do not have the property

mAðxþ pÞ ¼ �mAðxÞ and mBðxþ pÞ ¼ �mBðxÞ: ð13Þ

The first type of solution satisfying both Eq. (13) is called asymmetric solution and corresponds to a paramagnetic (p)solution. In this solution, the submagnetizations mA and mB areequal to each other, they oscillate around zero and are delayedwith respect to the external magnetic field. The second type ofsolution, which does not satisfy Eq. (13), is called a nonsymme-trical solution, but this solution corresponds to a ferrimagnetic (i)solution because the submagnetizations mA and mB are not equalto each other (mAamB) and they oscillate around a nonzerovalues. Hence, if mA(x) and mB(x) oscillate around 72 and 73/2,respectively, this solutions called the ferrimagnetic-I (i1) phase, ifmA(x) and mB(x) oscillate around 71 and 73/2, respectively, thesolutions called the ferrimagnetic-II (i2) phase and mA(x) andmB(x) oscillate around 71 and 71/2, respectively, the solutioncalled the ferrimagnetic-III (i3) phase. These facts are seenexplicitly by solving Eqs. (10), and (11) within the Adams–Moul-ton predictor-corrector method for a given set of parameters andinitial values and presented in Fig. 1.

From Fig. 1, one can see following seven different solutions orphases, namely the p, i1, i2, and i3 fundamental phases orsolutions, and three coexistence phases or solutions, namely thei1+p in which i1 and p solutions coexist; the i2+p in which i2 and p

solutions coexist; the i3+p in which i3 and p solutions coexist, havebeen found. In Fig. 1(a) only the symmetric solution is alwaysobtained, in this case mA ¼ mB oscillate around zero value(mA(x) ¼ mB(x) ¼ 0). Hence, we have a paramagnetic (p) solutionor phase. On the other hand in Fig. 1(b)–(d) only the nonsym-metric solutions are found; therefore, we have the i1, i2, and i3solutions, respectively. In Fig. 1(b), mA(x) oscillates around 73/2and mB(x) oscillates around 72, this solution corresponds to theferrimagnetic (i1) phase (mA(x)amB(x)a0). In Fig. 1(c), mA(x)oscillates around 73/2 and mB(x) oscillates around 71, thissolution corresponds to the ferrimagnetic (i2) phase(mA(x)amB(x)a0). In Fig. 1(d), mA(x) oscillates around 71/2and mB(x) oscillates around 71, this solution corresponds to theferrimagnetic (i3) phase (mA(x)amB(x)a0). In Fig. 1(e), mA(x)oscillates around 73/2 and mB(x) oscillates around 72, whichcorresponds to the i1 phase, and also mA(x) and mB(x) are equal toeach other and they oscillate around zero value, this solutioncorresponds to the p phase; hence we have the coexistencesolution (i1+p). In Fig. 1(f), mA(x) oscillates around 73/2 and mB(x)oscillates around 71, which corresponds to the i2 phase, and alsomA(x) and mB(x) are equal to each other and they oscillate aroundzero value, this solution corresponds to the p phase; hence wehave the coexistence solution (i2+p). In Fig. 1(g), mA(x) oscillatesaround 71/2 and mB(x) oscillates around 71, which correspondsto the i3 phase, and also mA(x) and mB(x) are equal to each otherand they oscillate around zero value, this solution corresponds tothe p phase; hence we have the coexistence solution (i3+p). Weshould also mention that the symmetric solution does not dependon the initial values, but the other solutions depend on the initialvalues.

ARTICLE IN PRESS

Fig. 1. Time variations of the average magnetizations (mA, mB): (a) exhibiting a paramagnetic (p) phase: d ¼ 0.25, h ¼ 1.25 and T ¼ 2.0. (b) Exhibiting a ferrimagnetic (i1)

phase: d ¼ 0.25, h ¼ 0.5 and T ¼ 1.0. (c) Exhibiting a ferrimagnetic (i2) phase: d ¼ �0.5, h ¼ 0.375 and T ¼ 0.25. (d) Exhibiting a ferrimagnetic (i3) phase: d ¼ �0.625,

h ¼ 0.375 and T ¼ 0.375. (e) Exhibiting a coexistence region (i1+p): d ¼ 0.25, h ¼ 1.25 and T ¼ 0.625. (f) Exhibiting a coexistence region (i2+p): d ¼ �0.5, h ¼ 0.875 and

T ¼ 0.0375. (g) Exhibiting a coexistence region (i3+p): d ¼ �1.0, h ¼ 0.25 and T ¼ 0.05.

M. Keskin, Y. Polat / Journal of Magnetism and Magnetic Materials 321 (2009) 3905–39123908

ARTICLE IN PRESS

M. Keskin, Y. Polat / Journal of Magnetism and Magnetic Materials 321 (2009) 3905–3912 3909

3.2. Thermal behavior of the dynamic order parameters

In this subsection, we investigate the behavior the averagemagnetizations in a period or the dynamic magnetizations as afunction of the reduced temperature. This investigation leads us toobtain the dynamic phase transition points and characterize thenature (first- or second-order) dynamic phase transitions. The

Fig. 2. The reduced temperature dependence of the dynamic magnetizations, MA and MB

phase and Tt represents the first-order phase transition temperature from the i3 or i1 pha

phase to the p phase for d ¼ 0.25 and h ¼ 0.5; 1.94 is found TC. (b) Exhibiting a first-or

0.0975 is found Tt. (c) Exhibiting a second-order phase transition from the i1 phase to th

TC ¼ 1.26. (d) Exhibiting two successive phase transitions: the first one is a first-order an

the initial values of MA ¼ 0 and MB ¼ 0 and; 0.085 and 1.26 are found to be Tt and TC,

dynamic sublattice magnetizations (MA, MB) are defined as

MA ¼1

2p

Z 2p

0mAðxÞdx and MB ¼

1

2p

Z 2p

0mBðxÞdx ð14Þ

The behaviors of MA and MB as a function of the reducedtemperature for several values of d and h are obtained by solving

. The TC is the second-order phase transition temperature from the i1 phase to the p

ses to the p phase and Tt. (a) Exhibiting a second-order phase transition from the i1der phase transition from the i3 phase to the p phase for d ¼ �1.0 and h ¼ 0.125;

e p phase for d ¼ �0.4375, h ¼ 0.075 and the initial values of MA ¼ 3/2 and MB ¼ 2;

d the second one is a second-order phase transition for d ¼ �0.4375, h ¼ 0.075 and

respectively.

ARTICLE IN PRESS

Fig. 3. Phase diagrams of the mixed spin-3/2 and spin-2 Ising ferrimagnetic model in the (T, h) plane. The paramagnetic (p), ferrimagnetic (i1, i2, i3) phases and three

different coexistence regions, namely the i1+p, i2 +p and i3+p regions, are found. Dashed and solid lines represent the first- and second-order phase transitions, respectively,

and the dynamic tricritical point is indicated with a filled circle, (a) d ¼ 0.25, (b)d ¼ �0.25, (c) d ¼ �0.375, (d) d ¼ �0.4375, (e) d ¼ �0.5, (f) d ¼ �0.625, (g) d ¼ �0.6875

and (h) d ¼ �1.0.

M. Keskin, Y. Polat / Journal of Magnetism and Magnetic Materials 321 (2009) 3905–39123910

ARTICLE IN PRESS

M. Keskin, Y. Polat / Journal of Magnetism and Magnetic Materials 321 (2009) 3905–3912 3911

Eq. (14). We solve these equations by combining the numericalmethods of the Adams–Moulton predictor corrector with theRomberg integration in order to illustrate the calculation of theDPT points and the dynamic phase boundaries among the phases.Four interesting results are plotted in Fig. 2(a)–(d). In thesefigures, TC and Tt represent second- and first-order phasetransition temperatures, respectively. Fig. 2(a) shows thebehavior of MA and MB as a function of the reduced temperaturefor d ¼ 0.25 and h ¼ 0.5. In this figure, MA ¼ 3/2 and MB ¼ 2 atzero temperature, and they decrease to zero continuously as thereduced temperature increases; therefore, a second-order phasetransition occurs at TC ¼ 1.94 and the dynamic phase transition isfrom the ferrimagnetic (i1) phase to the paramagnetic (p) phase.Fig. 2(b) shows the behavior of MA and MB as a function of thereduced temperature for d ¼ �1.0 and h ¼ 0.125. In this figure,MA ¼ 1/2 and MB ¼ 1 at zero temperature, and they decrease tozero discontinuously as the reduced temperature increases;therefore, a first-order phase transition occurs at Tt ¼ 0.0975and the transition is from the ferrimagnetic (i3) phase to theparamagnetic (p) phase. Fig. 2(c) and (d) illustrate the thermalvariations of MA and MB d ¼ �0.4375 and h ¼ 0. 075 for twodifferent initial values; i.e., the initial values are taken as MA ¼ 3/2and MB ¼ 2 for Fig. 2(c) and MA ¼ MB ¼ 0 for Fig. 2(d). Thebehavior of Fig. 2(c) is similar to that of Fig. 2(a); hence, thesystem undergoes a second-order phase transition at TC ¼ 1. 26. InFig. 2(d), MA ¼ MB ¼ 0 at zero temperature and the systemundergoes two successive phase transition as the temperatureincreases: the first one is a first-order phase transition because adiscontinuity occurs for the dynamic order parameters. Thetransition is from the p phase to the i1 phase at Tt ¼ 0.085. Thesecond one is a second-order phase transition at TC ¼ 1.26 and thetransition is from the i1 phase to the p phase as similar to Fig. 2(a)and (c). From Fig. 2(c) and (d) one can see that the i1+p

coexistence or mixed phase region also exists in the system,compare Fig. 2(c) and (d) with Fig. 3(d).

Fig. 4. Same as Fig. 3, but in the (d, T) plane for; (a) h ¼ 0.75 and (b) h ¼ 1.375.

3.3. Dynamic phase diagrams

Since we have obtained the DPT points and characterized thenature phase transitions in Section 3.2, now we can present thedynamic phase diagrams of the system. The calculated phasediagrams in the (h, T) plane are presented in Fig. 3 for variousvalues of d, and those in the (d, T) plane for various values of h areillustrated in Fig. 4. In these phase diagrams, the solid and thedashed lines represent the second- and the first-order phasetransition lines, respectively, and the dynamic tricritical points aredenoted by filled circles. QP, B and Z represent the dynamicquadruple point, the dynamic double critical end point and zero-temperature critical point, respectively.

Fig. 3 illustrates the dynamic phase diagrams in (T, h) plane forvarious values d, and eight main topological different types ofphase diagrams are found. From these phase diagrams thefollowing interesting phenomena have been observed: (1) thephase diagrams of Fig. 3(a)–(g) illustrate a dynamic tricriticalbehavior, but Fig. 3(h) does not. (2) The dynamic phase linesamong the fundamental phases are a second-order line, but phaselines between fundamental and mixed phases are a first-ordermostly. (3) The mixed phases always occur for low values of thereduced temperatures. (4) For very low values of T and h

(including T ¼ 0, and h ¼ 0), the i1 phase always occurs ford4�0.1847, the i1+p phase exists for �0.1847Zd4�0.48125, thei2+p phase exists for �0.48125Zd4�0.59875, the i3+p phaseexists for �0.59875Zd4�1.145. Finally, we should also mentionthat the system does not go any phase transitions for dr�1.145.

We also calculate the phase diagrams in the (d, T) planes andwe find nine main distinct topological types of phase diagrams inthe (d, T) plane. Since the most of the phase diagrams in this planecan be readily obtained from the phase diagrams in the (T, h)plane, especially for very high and low values of h, we give twointeresting phase diagrams in the (d, T) plane in which theycannot be readily obtained from the phase diagrams in the (T, h)plane. Fig. 4(a) is obtained for h ¼ 0.75 and does not display thedynamic tricritical point. The phase diagram exhibits the i1 and P

fundamental phases and the i1+p, i2+p and i3+p mixed phases. Italso contains a dynamic double critical end point (B), a dynamicquadruple point (QP) and a zero-temperature critical point (Z).The dynamic phase boundaries among the phases are first-orderphase lines except that the dynamic phase boundary between thei1 and p phases is a second-order line. Fig. 4(b) is obtained forh ¼ 1.375 and contains i1 and P fundamental phases and the i1+p

mixed phase. The dynamic phase boundaries among the phasesare first-order phase lines except that the dynamic phaseboundary between the i1 and p phases is a second-order line. It

ARTICLE IN PRESS

M. Keskin, Y. Polat / Journal of Magnetism and Magnetic Materials 321 (2009) 3905–39123912

als exhibits a dynamic tricritical point and a zero-temperaturecritical point (Z).

4. Summary and conclusion

We have analyzed, within a mean-field approach, the sta-tionary states of the kinetic mixed spin-3/2 and spin-2 Isingferrimagnetic model Hamiltonian with a single-ion potential or acrystal-field interaction under the presence of a time-varying(sinusoidal) magnetic field. We use a Glauber-type stochasticdynamics to describe the time evolution of the system. First, westudied the time variations of the order parameters in order tofind the phases in the systems. Then, the behavior of the averageorder parameters in a period or the dynamic magnetizations asfunctions of the reduced temperature and the crystal-fieldinteraction were investigated. This study led us to characterizethe nature (continuous and discontinuous) of the transitions andto obtain the DPT points. Finally, we have found that the behaviorof the system strongly depends on the values of the interactionparameters and eight different phase diagram topologies areobtained in the (h, T) plane and nine fundamental phase diagramsare found in the (d, T) plane. Since the phase diagrams in the (d, T)plane can be readily obtained from the phase diagrams in the (T,h) plane, we give two interesting phase diagrams in the (d, T)plane in which they cannot be seen easily from the phasediagrams in the (T, h) plane. The phase diagrams exhibit theparamagnetic (p), ferrimagnetic (i1, i2, i3) fundamental phases andthree coexistence or mixed phase regions, namely i1+p, i2+p andi3+p mixed phases, which strongly depend on interaction para-meters. Therefore, in the system zero, one or two dynamictricritical points; zero or one dynamic double critical end point(B); zero or one dynamic zero-temperature critical point (Z) andone dynamic quadruple point (QP) also occur that depend oninteraction parameters.

We should also mention that we do indeed observe a phasediagram of Fig. 3(a) very similar to that found in earlier studies ofthe mixed Ising systems with spins (1/2, 1) [8,15], spins (1/2, 3/2)[16], spins (1, 3/2) [17], spins (1, 5/2) [6] and spins (1, 2) [18].Moreover, very similar phase diagrams with Fig. 3(b) and (e)–(f)have also been obtained in the kinetic mixed spin (1/2, 1) Isingsystem [8,15], the kinetic mixed spin (1, 3/2) Ising system [17] andthe kinetic mixed spin (1, 2) Ising system [18] (Ref. [17,18] exceptthat the i+p mixed phase becomes i+a mixed phase). Moreover, aphase diagram similar to Fig. 3(g) has also been obtained in thekinetic mixed spin (1, 2) Ising system [18], but the i3+p mixedphase becomes i+a+d mixed phase. Finally, Fig. 3(c), (d) and (h) are

new phase diagrams in which were not obtained previous studiesof kinetic mixed Ising systems.

Finally, it should be mentioned that this mixed high-spin Isingsystem spins with (3/2, 2) gives an interesting dynamic behavioras well as three new dynamic phase diagrams. Hence, we hopethat our detailed theoretical investigation may stimulate furtherworks to study the nonequilibrium or the dynamic phasetransition (DPT) in the mixed high-spin Ising systems by usingmore accurate techniques such as kinetic Monte-Carlo (MC)simulations. We should also mention that although the resultsof the dynamic MC method is rather reliable, it is alwaysconstrained by the speed of computer; hence our results will beinstructive for the time consuming process of searching criticalbehavior while using the dynamic MC simulations.

Acknowledgements

This work was supported by the Scientific and TechnologicalResearch Council of Turkey (TUB_ITAK) (Grant no: 107T533) andErciyes University Research Funds (Grant no: FBA-06-01). We arevery grateful to Bayram Deviren for useful discussions.

References

[1] T. Mallah, S. Thi�ebaut, M. Verdauger, P. Veillet, Science 262 (1993) 1554.[2] M. Drillon, E. Coronado, D. Beltran, R. Georges, J. Chem. Phys. 79 (1983) 449.[3] C. Mathoni�ere, C.J. Nuttall, S.G. Carling, P. Day, Inorg. Chem. 35 (1996) 1201.[4] J.M. Manriquez, G.T. Yee, R.S. McLean, A.J. Epstein, J.S. Miller, Science 252

(1991) 1415.[5] H. Iwamura, J. S. Miller (Eds.), Proceeedings of the Conference on

Ferromagnetic and High Spin Molecular-Based Materials, Mol. Cryst. Liq.Cryst. 232 (1993).

[6] M. Keskin, M. Batı, O. Canko, J. Korean Phys. Soc., submitted.[7] E. Albayrak, M. Keskin, J. Magn. Magn. Mater. 261 (2004) 196;

E. Albayrak, A. Yi˘git, Physica A 349 (2005) 471;

J. Strecka, Phys. Status Solidi (b) 243 (2006) 708;C. Ekiz, Physica A 387 (2008) 1185.

[8] G.M. Buendıa, E. Machado, Phys. Rev. E 58 (1998) 1260.[9] M. Godoy, W. Figueiredo, Phys. Rev. E 61 (2000) 218.

[10] M. Godoy, W. Figueiredo, Phys. Rev. E 65 (2002) 026111.[11] M. Godoy, W. Figueiredo, Phys. Rev. E 66 (2002) 036131.[12] M. Godoy, W. Figueiredo, Braz. J. Phys. 34 (2004) 422.[13] M. Godoy, W. Figueiredo, Physica A 339 (2004) 392.[14] C. Ekiz, M. Keskin, Physica A 317 (2003) 517.[15] M. Keskin, O. Canko, Y. Polat, J. Korean Phys. Soc. 53 (2008) 497.[16] B. Deviren, M. Keskin, O. Canko, J. Magn. Magn. Mater. 321 (2009) 458.[17] M. Keskin, E. Kantar, O. Canko, Phys. Rev. E 77 (2008) 051130.[18] M. Keskin, M. Ertas-, O. Canko, Phys. Scr. 79 (2009) 025501.[19] A. Bob�ak, J. Dely, J. Magn. Magn. Mater. 310 (2007) 1419.[20] E. Albayrak, Physica B 391 (1997) 47.[21] R.J. Glauber, J. Math. Phys. 4 (1963) 294.