ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 322 (2010) 2789–2796

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

n Corr

E-m

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

Dynamic compensation temperatures in a mixed spin-1 and spin-3/2 Isingsystem under a time-dependent oscillating magnetic field

Mustafa Keskin a,n, Ersin Kantar b

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

a r t i c l e i n f o

Article history:

Received 10 February 2010

Received in revised form

7 April 2010Available online 22 April 2010

Keywords:

Mixed spin Ising system

Dynamic phase transition

Dynamic compensation temperature

Oscillating magnetic field

Glauber-type stochastic dynamic

53/$ - see front matter & 2010 Elsevier B.V. A

016/j.jmmm.2010.04.029

esponding author. Tel.: +90 352 4374901x33

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

a b s t r a c t

We study the existence of dynamic compensation temperatures in the mixed spin-1 and spin-3/2 Ising

ferrimagnetic system Hamiltonian with bilinear and crystal–field interactions in the presence of a time-

dependent oscillating external magnetic field on a hexagonal lattice. We employ the Glauber transitions

rates to construct the mean-field dynamic equations. We investigate the time dependence of an average

sublattice magnetizations, the thermal behavior of the dynamic sublattice magnetizations and the total

magnetization. From these studies, we find the phases in the system, and characterize the nature

(continuous or discontinuous) of transitions as well as obtain the dynamic phase transition (DPT) points

and the dynamic compensation temperatures. We also present dynamic phase diagrams, including the

compensation temperatures, in the five different planes. A comparison is made with the results of the

available mixed spin Ising systems.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

In our preceding paper [1], we investigated dynamical aspectsof the mixed spin-1 and spin-3/2 Ising ferrimagnetic modelHamiltonian with bilinear and biquadratic nearest-neighborexchange interactions and a crystal–field interaction in thepresence of a time-dependent oscillating external magnetic fieldon a square lattice. We use the Glauber-type stochastic dynamicsto describe the time evolution of the system and to obtain themean-field dynamic equations. The nature (continuous or dis-continuous) of transition is characterized by studying the thermalbehaviors of dynamic order parameters. The dynamic phasetransition (DPT) points are obtained, and the dynamic phasediagrams are presented in the three different planes. We foundthat dynamic phase diagrams contain the disordered (d),ferrimagnetic (i), the antiquadrupolar or staggered (a) phases,and the four coexistence regions or mixed phases, namely the i+d,i+a, a+d and i+a+d, that strongly depend on interactionparameters. The system also exhibits the dynamic tricriticalbehavior in most cases and reentrant behavior in a few cases. Inthis paper, we study the existence of dynamic compensationtemperatures in the mixed spin-1 and spin-3/2 Ising ferrimag-netic system under a time-dependent oscillating external mag-netic field on a hexagonal lattice. We use the Glauber-typestochastic dynamics to describe the time evolution of the system.

ll rights reserved.

105; fax: +90 352 4374931.

.

We also investigate the time variations of the average sublatticemagnetizations in order to find the phases in the system and thetemperature dependence of the average sublattice magnetizationsin a period, which is also called the dynamic sublatticemagnetization, to obtain the DPT points as well as to characterizethe nature (continuous or discontinuous) of transitions. The totalmagnetization is investigated as a function of reduced tempera-ture to find the compensation temperatures and to determine thetype of behavior. Finally, dynamic phase diagrams, including thecompensation temperatures, are presented in the five differentplanes.

It is worthwhile mentioning that the choice of the mixed spin-1and spin-3/2 system is stipulated by a possible prototype of themodel. For example, the metal-organic compound [Co(hfac)2].BNOn studied by Numata et al. [2], which embodies Co(II) ionswith spin-3/2 and the chiral triplet biradical ligands BNOn withspin-1. Recently, Bukharov et al. [3] is used this mixed systemunder a time-dependent oscillating magnetic field in order toanalyze a magnetic hysteresis of Co-based quasi-1D ferrimagneticmagnets within the Glauber-type stochastic dynamics by con-sidering fluctuations of local fields in the spirit of the generalizedmean-field theory [4].

We should also mention that the compensation temperature isa temperature, where the total magnetization vanishes below thecritical temperature. The occurrence of a compensation point isbecause the magnetic moments of sublattices compensate eachother completely at T¼Tcomp owing to different temperaturedependences of the sublattice magnetizations. The existence of acompensation temperatures near room temperature in some

ARTICLE IN PRESS

Fig. 1. The sketch of the spin arrangement on the hexagonal lattice. The lattice is

formed by alternate layers of s (open circles) and S (solid circles) spins.

M. Keskin, E. Kantar / Journal of Magnetism and Magnetic Materials 322 (2010) 2789–27962790

ferrimagnetic materials is of crucial importance in the area ofthermomagnetic recording devices [5–8]. Moreover it has beenfound that some physical properties show peculiar behavior atthis point. For example, the coercivity field is strongly tempera-ture dependent only in the proximity of the compensationtemperature, in which it is a maximum at Tcomp, falling tominimum below Tcomp, before rising again at low temperatures[9–11]. The presence of a compensation temperature in the mixedspin Ising systems has been studied by a variety of techniques inequilibrium statistical physics (see [12–14] and references there-in). On the other hand, the existence of a dynamic compensationtemperature in the spin-1/2 and spin-1 Ising ferrimagnetic systemhas been studied on a hexagonal lattice [15,16], and also on ahexagonal substrate [17]. Recently, the appearance of a dynamiccompensation point in the mixed spin-1/2 and spin-3/2 [18], themixed spin-2 and spin-5/2 [19] and the kinetic mixed spin-3/2and spin-5/2 [20] Ising ferrimagnetic systems have also beeninvestigated. Finally, it is worthwhile mentioning that thecompensation temperature has been observed in differentsystems experimentally. Chern et al. [21] have illustrated somemeasurements of the compensation temperatures and phasediagram of Fe3O4 and Mn3O4 superlattices, which is a systemgrown by a deposition of alternate layers of Fe3O4 and Mn3O4

coupled antiferromagnetically. Kageyama et al. [22] have studiedmagnetic properties of the nickel II formate dehydrate Ni(H-COO)2 �2H2O. They found that Ni(HCOO)2 �2H2O is a weak ferri-magnet at low temperatures. Moreover at specific temperature,this compound undergoes a transition to a magnetically orderedstate exhibiting peculiar magnetic properties: a weak sponta-neous ferromagnetic like moment, compensation temperatureand the phenomenon of magnetization reversal.

This paper is organized as follows. In Section 2, we present themodel and its formulation, namely the derivation of the set ofmean-field dynamic equations by using the Glauber-type sto-chastic dynamics in the presence of a time-dependent oscillatingexternal magnetic field. In Section 3, the time variations of theaverage sublattice magnetizations are studied to find the phasesin the system. The temperature dependence of the dynamicsublattice magnetizations and the total magnetization areinvestigated to determine DPT points and the compensationtemperatures. Finally, the dynamic phase diagrams, including thecompensation temperatures, are presented in five differentplanes, and followed by a brief remark.

2. Model and formulations

We consider a mixed spin-1 and spin-3/2 Ising ferrimagneticsystem on a hexagonal lattice. The lattice is formed by alternatelayers of s and S spins; hence, the two different types of spins aredescribed by Ising variables, which can take the values s¼71, 0and S¼73/2, 71/2. s and S spins are distributed in alternatelayers of a hexagonal lattice, seen in Fig. 1. In Fig. 1, open and solidcircles represent s and S spins, respectively. The Hamiltonianmodel for the system is

H¼�J1

X/ijS

siSj�J2

X/ijS

sisj�J3

X/ijS

SiSj

�DX

i

s2i þX

j

S2j

0@

1A�H

Xi

siþX

j

Sj

0@

1A ð1Þ

where the summation index o ij4 denotes a summation over allpairs of nearest-neighbor spins. J1, J2 and J3 are the exchangecouplings between the nearest-neighbor pairs of spins s�S, s�sand S�S, respectively. It is seen from Fig. 1, J1 that the interaction

is restricted to the z1 nearest-neighbor pair of spins that z1¼4,and J2 and J3 are restricted to the coordination numbers of z2 andz3, respectively, in which z2¼z3¼2. The parameter J1 will be takennegative in all the subsequent analyses, i.e., the intersublatticecoupling is antiferromagnetic to have a simple but interestingmodel of a ferrimagnetic system. D is the crystal–field interactionor a single-ion anisotropy constant. We take the same single-ionanisotropy constant for both magnetic moments in order toavoid one more interaction constants, as in Refs [23,24]. Onecan also use the different anisotropy constants for both magneticmoments as in Refs [13,14,25–27]. H is an oscillating magneticfield of the form

HðtÞ ¼H0cosðwtÞ, ð2Þ

where H0 and w¼2pn are the amplitude and the angularfrequency of the oscillating field, respectively. The system is incontact with an isothermal heat bath at an absolute temperatureTA.

Now, we apply the Glauber-type stochastic dynamics [28] toobtain the set of mean-field dynamic equations. Thus, the systemevolves according to a Glauber-type stochastic process at a rate of1/t transitions per unit time; hence, the frequency of spin flipping,f, is 1/t. Leaving the S spins fixed, we define Ps(s1,s2, y, sN; t) asthe probability that the system has the s-spin configuration,s1,s2, y, sN, at time t, also, by leaving the s-spins fixed, we definePS(S1,S2,y,SN; t) as the probability that the system has the S-spinconfiguration, S1,S2, y, SN, at time t. Then, we calculateWs

i ðsi-s0iÞ and WSj ðSj-S0jÞ, the probabilities per unit time that

the ith s spin changes from si to s0 i (while the S spinsmomentarily fixed) and the jth S-spin changes from Sj to S0j(while the s spins momentarily fixed), respectively. Thus, if the S

spins are momentarily fixed, the master equation for the s spinscan be written as

d

dtPsðs1,s2,:::,sN; tÞ ¼�

Xi

Xsi as0

i

Wsi ðsi-s0iÞ

0@

1APsðs1,s2, :::,si,:::sN; tÞ

þP

i

Psi as0

iWs

i ðs0i-siÞ

� �Psðs1,s2,:::,s0i,:::sN ; tÞ

ð3Þ

ARTICLE IN PRESS

M. Keskin, E. Kantar / Journal of Magnetism and Magnetic Materials 322 (2010) 2789–2796 2791

where Wsi ðsi-s0iÞ is the probability per unit time that the ith spin

changes from the value si to s0i. Since the system is in contactwith a heat bath at absolute temperature TA, each spin can changefrom the value si to s0i with the probability per unit time

Wsi ðs-s0 iÞ ¼

1

texpð�bDEsðsi-s0 iÞPs0

iexpð�bDEsðsi-s0iÞ,

ð4Þ

where b¼1/kBTA, kB is the Boltzmann factor, Ss0 i is the sum overthe three possible values of s0i¼71, 0, and

DEsðsi-s0iÞ ¼�ðs0i�siÞ J1

Xj

Sjþ J2

Xj

sjþH

0@

1A�½ðs0 iÞ2�ðsiÞ

2�D,

ð5Þ

gives the change in the energy of the system when the si-spinchanges. The probabilities Ws

i ðsi-s0iÞ are given in appendix.The probabilities satisfy the detailed balance condition. SinceWs

i ðsi-s0iÞ does not depend on the value si, we can writeWs

i ðsi-s0iÞ ¼Wsi ðs

0iÞ, then the master equation becomes

d

dtPsðs1,s2, . . .,sN ; tÞ ¼�

Xi

Xsi as0

i

Wsi ðs

0iÞ

0@

1APsðs1,s2, . . .,si, . . .,sN ; tÞ

þX

i

Wsi ðsiÞ

Xsi as0

i

Psðs1,s2, . . .,s0i, . . .,sN; tÞ

0@

1A ð6Þ

Since the sum of the probabilities is normalized to one, bymultiplying both sides of Eq. (6) by sk and taking the average, weobtain

t d/skSdt

¼�/skSþ2sinhbðJ1

PjSjþ J2

PjsjþHÞ

2coshbðJ1P

jSjþ J2P

jsjþHÞþexpð�bDÞ

* +

ð7Þ

or, in terms of a mean-field approach

t d/sSdt¼�/sSþ 2sinhbðJ1z1/SSþ J2z2/sSþH0cosðwtÞÞ

2coshbðJ1z1/SSþ J2z2/sSþH0cosðwtÞÞþexpð�bDÞ

� �,

ð8Þ

where oy4 denotes the canonical thermal average. The systemevolves according to the differential equation given by Eq. (8) andcan be written in the form

Od

dxms ¼�msþ

2sinha1

T

� �

2cosha1

T

� �þexp �

d

T

� �* +

, ð9Þ

where T¼kBTA/9J19, d¼D/9J19, a1¼(�z1mS+ J2/9J19z2ms+h0 cos(x)),h0¼H0/9J19, ms¼/sS, mS¼/SS, x¼wt, and O¼tw.

Now, assuming that the spins s remain momentarily fixed andS spins change, we obtain the second mean-field dynamicequation by using similar calculation

Od

dxmS ¼�mSþ

3expd

T

� �sinh 3

a2

T

� �þexp �

d

T

� �sinh

a2

T

� �

2expd

T

� �cosh 3

a2

T

� �þexp �

d

T

� �cosh

a2

T

� �* +

,

ð10Þ

where a2¼(�z1ms+ J3/9J19z3mS+h0 cos(x)). We fixed J1¼ �1 thatthe intersublattice interaction is antiferromagnetic and O¼2p. Inthe next section, we will give the numerical results of theseequations.

3. Numerical results and discussion

3.1. Time variations in average order parameters

In order to investigate the behaviors in time variations of orderparameters, first, we have to study the stationary solutions of theset of coupled mean-field dynamical equations, given in Eqs. (9)and (10), when the parameters T, J2, J3, d and h0 are varied. Thestationary solutions of these equations will be periodic functionsof with period 2p; i.e., ms(x+2p)¼ms(x) and mS(x+2p)¼mS(x).Moreover they can be one of the three types according to whetherthey have or do not have the properties.

msðxþpÞ ¼�msðxÞ ð11aÞ

and

mSðxþpÞ ¼�mSðxÞ ð11bÞ

The first type of solution satisfies both Eqs. (11a) and (11b),and is called a symmetric solution, which corresponds to aparamagnetic (p) phase. In this solution, the sublattice averagemagnetizations, ms(x) and mS(x), are equal to each other. Theyoscillate around the zero value and are delayed with respect tothe external magnetic field. The second type of solution, whichdoes not satisfy Eqs. (11a) and (11b), is called a nonsymmetricsolution, but this solution corresponds to a ferrimagnetic (i)solution because the sublattice magnetizations ms(x) and mS(x)are not equal to each other (ms(x)amS(x)), and they oscillatearound a nonzero value. Hence, if ms(x) and mS(x) oscillatesaround 71 and 73/2, respectively, this solution is called theferrimagnetic-1 (i1) phase; if ms(x) and mS(x) oscillates around71 and 71/2, respectively, the solution is called the ferrimag-netic-2 (i2) phase. The third type of solution, which satisfies Eq.(11a) but does not satisfy Eq. (11b), corresponds to anon-magnetic solution (nm). In this case, ms(x) oscillates aroundthe zero value and is delayed with respect to the externalmagnetic field and mS(x) does not follow the external magneticfield anymore, but instead of oscillating around a zero value, itoscillates around a nonzero value. We should also mention thatthe non-magnetic (nm) phase has also been experimentallyobserved in URu2Si2 [29] and brass–iron electron [30] compoundsand theoretically in the Blume–Emery–Griffiths spin-1 film [31],the mixed spins (2, 5/2) [19] and the mixed spins (1, 1/2) [16]Ising system under the oscillating magnetic field. Moreover thenanometer phase appears a very low external magnetic field andtemperature that has also been observed in Refs. [16,19,31]. Thesefacts are seen explicitly by solving Eqs. (9) and (10), numerically.These equations are solved by using the numerical method ofthe Adams–Moulton predictor corrector method for a given setof parameters and initial values. A few explanatory examplesare given in Fig. 2. Fig. 2(a)–(e) represents the paramagnetic,non-magnetic, ferrimagnetic-1, ferrimagnetic-2, fundamentalsolutions or phases, and the i1+p mixed phase, respectively. Inaddition to this mixed phase, three more mixed phases, namelythe i1 in which i2, p solutions coexist; the i1+nm in which i1, nmsolutions coexist and the i2+nm in which i1, nm solutions coexist,exist the system.

3.2. Thermal behavior of dynamic sublattice magnetizations and the

total magnetization

We investigate the behavior of the average sublattice magnetiza-tions in a period or the dynamic sublattice magnetizations as afunction of the reduced temperature. This investigation leads us tocharacterize the nature (a first- or a second-order) of dynamic phasetransitions and to find the DPT points. We also study the totaldynamic magnetization as a function of temperature to estimate the

ARTICLE IN PRESS

0

mσ

(ξ),

mS (ξ

)

mσ (ξ) = mS (ξ)

mσ

(ξ),

mS (ξ

)

mσ

(ξ),

mS (ξ

)-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

60

0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

20 40 20 40 60 80 10060 80 100

20 40 60 80 100 50 100 150 200 250

20 40 60 80 100ξ

70 80

mS (ξ)

mS (ξ)

mσ (ξ)

mS (ξ)

mS (ξ)

mσ (ξ)

mS (ξ)

mS (ξ)

mσ (ξ)

mσ (ξ)

mS (ξ)

mσ (ξ) = mS (ξ)

mS (ξ)

mσ (ξ)

mσ (ξ)

mσ (ξ)

Fig. 2. Time variations in the average sublattice magnetizations [ms(x) and mS(x)]: (a) exhibiting a paramagnetic phase (p) such that ms(x)¼mS(x)¼0 for J1¼ �1.0, J2¼3.0,

J3¼0.8, h0¼0.1, d¼ �1.0 and T¼10. (b) Exhibiting a non-magnetic phase (nm) such that ms(x)¼0, mS(x)¼71/2 or ms(x)¼71/2, mS(x)¼0 for J1¼ �1.0, J2¼3.0, J3¼0.8,

h0¼0.1, d¼ �8.0 and T¼0.2. (c) Exhibiting a ferrimagnetic-1 phase (i1) such that ms(x)¼71, mS(x)¼73/2 for J1¼ �1.0, J2¼3.0, J3¼0.8, h0¼0.1, d¼ �1.0 and T¼2.

(d) Exhibiting a ferrimagnetic-2 phase (i2) such that ms(x)¼1, mS(x)¼71/2 for J1¼ �1.0, J2¼6.0, J3¼ �0.5, h0¼0.1, d¼ �1.0 and T¼2. (e) Exhibiting a coexistence region

or mixed phase (i1+p), J1¼ �1.0, J2¼5.0, J3¼0.8, h0¼3.0, d¼ �5.0 and T¼3.

M. Keskin, E. Kantar / Journal of Magnetism and Magnetic Materials 322 (2010) 2789–27962792

compensation temperatures and to determine the type of behavior.The dynamic sublattice magnetizations (Ms, MS) and the totaldynamic magnetization Mt ¼MsðxÞþMSðxÞ=2 are defined as

Ms ¼1

2p

Z 2p

0msðxÞdx, ð12aÞ

MS ¼1

2p

Z 2p

0mSðxÞdx, ð12bÞ

and the total dynamic magnetization

Mt ¼1

2p

Z 2p

0

mSðxÞþmsðxÞ2

� �dx: ð12cÞ

The behaviors of dynamic sublattice magnetizations (Ms, MS) andthe total dynamic magnetization (Mt) as functions of the tempera-ture for several values of interaction parameters are obtained bycombining the numerical methods of the Adams–Moulton predictorcorrector with the Romberg integration. The total magnetization(Mt) vanishes at the compensation temperature Tcomp. Then, thecompensation point can be determined by looking for the crossingpoint between the absolute values of the sublattice magnetizations.Therefore, at the compensation point, we must have

9MsðTcompÞ9¼ 9MSðTcompÞ9 ð13Þ

and

sgn ½MsðTcompÞ� ¼�sgn ½MSðTcompÞ� ð14Þ

ARTICLE IN PRESS

M. Keskin, E. Kantar / Journal of Magnetism and Magnetic Materials 322 (2010) 2789–2796 2793

We also require that TcompoTC, where TC is the critical pointtemperature. These conditions demonstrate that at Tcomp, the Ms

and MS cancel each other, whereas at TC both are zero. A fewexplanatory and interesting examples are plotted in Fig. 3(a)–(e),in order to illustrate the calculation of the DPT points andthe compensation temperatures. In these figures, TC and Tt arethe critical or the second- and first-order phase transitiontemperatures, respectively. Fig. 3(a) shows the behavior of 9Ms9,9MS9 and 9Mt9 as a function of the temperature for J1¼�1.0,

30

IMσI

, IM

SI,

IMtI

0.0

0.5

1.0

1.5

6420

IMσI

, IM

SI,

IMtI

0.0

0.5

1.0

1.5

Mt

Ms

Mσ

Mt

Ms

Mσ

T0.500.25

IMσI

, IM

SI,

IMtI

0.00

0.25

0.50

0.75

Ms

Mσ

TC

TCTcomp

Ms

Mσ

Mt

0.00

Fig. 3. The temperature dependence of the dynamic sublattice magnetizations (9Ms9, 9Morder phase transition temperatures, respectively. (a) Exhibiting a second-order phase t

d¼ �1.0 and TC is found 8.21. (b) Exhibiting a second-order phase transition from the

initial values are taken 9Ms9¼1 and 9MS9¼3/2; TC is found 5.11. (c) Exhibiting two suc

phase to the i1 phase and the second one is a second-order phase transition from the i

initial values of 9Ms9 and 9MS9 are taken zero; TC and Tt are found 5.11 and 0.40, respec

phase for J1¼ �1.0, J2¼6.0, J3¼0.8, h0¼0.1 and d¼ �8.0 and the initial values are t

transition from the i2 phase to the p phase for J1¼ �1.0, J2¼6.0, J3¼0.8, h0¼0.1 and d

J2¼5.0, J3¼0.8, d¼ �1.0 and h0¼0.1. In Fig. 3(a), 9Ms9¼1 and9MS9¼3/2 at the zero temperature, and they decrease to zerocontinuously until TC as the temperature increases; hence, asecond-order phase transition occurs at TC¼8.21. In this case, thedynamic phase transition is from the i1 phase to the p phase.Moreover from the behavior of the total dynamic magnetization,one can see that only one compensation temperature, or N-typebehavior occurs in the system that exhibits the same behaviorclassified after Neel [32] theory as the N-type [33]. Fig. 3(b) and

96

64200.0

0.5

1.0

1.5

Mt

Ms

Mσ

T

32100.00

0.25

0.50

0.75

1.00

1.25

Mt

Ms

Mσ

TC

TC

Tt

TcompTt

Tcomp

Ms

Mσ

Ms

Mσ

S9) and the total dynamic magnetization (9Mt9). TC and Tt are the second- and first-

ransition from the i1 phase to the p phase for J1¼ �1.0, J2¼5.0, J3¼0.8, h0¼0.1 and

i1 phase to the p phase for J1¼ �1.0, J2¼3.0, J3¼0.9, h0¼1.5 and d¼ �2.5 and the

cessive phase transitions, the first one is a first-order phase transition from the p

1 phase to the p phase for J1¼ �1.0, J2¼3.0, J3¼0.9, h0¼1.5 and d¼ �2.5 and the

tively. (d) Exhibiting a second-order phase transition from the nm phase to the p

aken 9Ms9¼0 and 9MS9¼ 1/2; TC is found 0.40. (e) Exhibiting a first-order phase

¼ �8.0 and the initial values are taken 9Ms9¼1 and 9MS9¼ 1/2; TC is found 2.10.

ARTICLE IN PRESS

T C, T

com

p

1

2

3

4

5

6

i1

p i2

i1+pi2+p

i1

i1+p

-10

T C, T

com

p

0

2

4

6

8

p

i1

i2

i2+p

i1

i1+p

-8 -6 -4 -2 0 2 4 6

M. Keskin, E. Kantar / Journal of Magnetism and Magnetic Materials 322 (2010) 2789–27962794

(c) illustrates the thermal variations of 9Ms9, 9MS9 and 9Mt9 forJ1¼ �1.0, J2¼3.0, J3¼0.9, d¼ �2.5 and h0¼1.5 for variousdifferent initial values. The behavior of Fig. 3(b), is similar toFig. 3(a); hence, the system undergoes a second-order phasetransition from the i1 phase to the p phase at TC¼5.11. In Fig. 3(c),the system undergoes two successive phase transitions. The firstone is a first-order, because the discontinuous occurs for thedynamic sublattice magnetizations at Tt¼0.40. Transition is fromthe p phase to the i1 phase. The second one is a second-order fromthe i1 phase to the p phase at TC¼5.11. This means that thecoexistence region i.e., the i1+p mixed phase exists in the system.Moreover the compensation temperature or the N-type behaviorexists in the system again. We should also mention that twosuccessive transitions have also been experimentally observed inDyVO4 [34]. Finally, Fig. 3(d) and (e) shows the behavior of thethermal variations of 9Ms9, 9MS9 and 9Mt9 for J1¼ �1.0, J2¼6.0,J3¼0.8, d¼ �8.0 and h0¼0.1 for various different initial values. InFig. 3(d), 9Ms9¼0 and 9MS9¼1/2 at the zero temperature and 9Ms9decrease zero continuously as the temperature increases; hence,the system undergoes a second-order phase transition from thenanometer phase to the p phase at TC¼0.40. In Fig. 3(e), 9Ms9¼1and 9MS9¼1/2 at the zero temperature and 9Ms9 decrease zerodiscontinuously as the temperature increases; hence, the systemundergoes a first-order phase transition from the i2 phase to the p

phase at Tt¼2.10. From Figs. 3(d) and (e), one can see that thei2+nm mixed phase exists until TC. There is no compensationtemperature in the system for this case.

-12

T C, T

com

p

0

1

2

3

4

5

6

nm

p

i1

i2

i1+nm

i2+nm

-100

i1

d

-3.50.0

0.2

0.4

0.6

0.8

i2+nm

i1+nm

i1 i1

B EQP

QP

-8 -6 -4 -2 0 2

-10 -8 -6 -4 -2 0 2

-3.0 -2.5 -2.0

Fig. 4. Dynamic phase diagrams of the mixed spin-1 and spin-3/2 Ising

ferrimagnetic model in the (d, T) plane. Dashed and solid lines are the dynamic

first- and second-order phase boundaries, respectively. The dash-dot-dot line

illustrates the compensation temperatures. The special points are the dynamic

tricritical point represented by a filled circle, and the dynamic double-critical end

point (B), the dynamic critical end point (E) and the dynamic quadruple point (QP).

The filled triangle separates the i2 phase from the i1 for high value of T and the i2+p

mixed phase from the i1+p mixed phase for low value of T. (a) J1¼ �1.0, J2¼6.0,

J3¼0.99 and h0¼4.0. (b) J1¼ �1.0, J2¼6.0, J3¼0.99 and h0¼1.5. (c) J1¼ �1.0,

J2¼6.0, J3¼0.99 and h0¼0.1.

3.3. Dynamic phase diagrams including the dynamic compensation

temperatures

Since we have obtained DPT points and compensationtemperatures in Section 3.2, we can now present the dynamicphase diagrams of the system. The calculated phase diagrams inthe (d, T) plane are presented in Fig. 4, and (J2, T), (� J3, T), (d, J2),(d, � J3) planes are presented in Fig. 5 for various values ofinteraction parameters. In these dynamic phase diagrams, thesolid, dashed and dash-dot-dot lines represent the second-order,first-order phase transitions temperatures and the compensationtemperatures, respectively. The special points are the dynamictricritical point with a filled circle, double-critical end point (B),critical end point (E), quadruple point (QP) and the separatingpoint with the filled triangles.

Fig. 4 illustrates the dynamic phase diagrams in the (d, T) planeand three main topological different types of phase diagrams areseen. From these phase diagrams the following phenomena havebeen observed: (1) all figures contain the compensation tempera-tures. (2) Fig. 4(a) exhibits a reentrant behavior, i.e., as thetemperature is increased, the system passes from the paramag-netic (p) phase to the ordered phases, and back to the p phaseagain. (3) The system exhibits a dynamic tricritical behavior, seenin Fig. 4(a)–(c). Moreover Fig. 4(c) also shows the dynamic doublecritical end point (B), the dynamic critical end point (E) and thedynamic quadruple point (QP). (4) The system contains the p, nm,i1 and i2 fundamental phases, and the i1+p, i2+p and i1+nm mixedphases. (5) In Fig. 4(a) and (b), the filled triangle separates the i2phase from the i1 for high value of T and the i2+p mixed phasefrom the i1+p mixed phase for a low value of T. We found similarbehavior to the one seen in the phase diagrams of the two-sublattice spin-3/2 Ising model by using the renormalization-group calculation (RG) [35], the mixed ferrimagnetic ternary alloyon the Bethe lattice [36], the mixed spin-2 and spin-5/2 [19], thekinetic mixed spin-3/2 and spin-5/2 [20], the mixed ferrimagneticternary system with a single-ion anisotropy on the Bethe lattice[37] and the mixed spin-1/2 and spin-S Ising model on a

bathroom tile (4–8) lattice [38]. The phase diagrams ofFig. 4(a)–(c) are only observed in this system. The comparisonof the present results in the (d, T) plane with the results ofavailable mixed spin Ising systems, namely, the mixed spins (1/2,1) [16], spins (3/2, 5/2) [20] and spins (2, 5/2) [19] Ising systems:(1) the mixed spins (1, 3/2) and spins (1/2, 1) exhibit only onedynamic tricritical point, but spins (3/2, 5/2) and spins (2, 5/2)exhibit at most four and two tricritical points, respectively. (2)These four mixed spins Ising systems show reentrant behaviordepending on the values of interaction parameters. (3) The mixed

ARTICLE IN PRESS

1

T C, T

com

p

0

4

8

12

i1

i1

p

0.0

T C, T

com

p

0

1

2

3

4

5

i1

p

i1+p

i1

i1+p

d-3.0

J 2

2

4

6

p

i1i1+p

i1

i1+p

d-3.5

- J3

0.0

0.5

1.0

p

i1

i1

i1+p

J2

2 3 4 5 6 0.5 1.0- J3

-2.5 -2.0 -1.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5

Fig. 5. Same as Fig. 4, but (a) in (J2, T) plane for J1¼ �1.0, J3¼0.8, h0¼0.1 and d¼ �1.0. (b) In (� J3, T) plane for J1¼ �1.0, J2¼2.0, h0¼0.1 and d¼ �1.5. (c) In (d, J2) plane

for J1¼�1.0, J3¼0.1, h0¼0.1 and T¼0.5. (d) In (d, � J3) plane for J1¼ �1.0, J2¼1.5, h0¼0.1 and T¼1.0.

M. Keskin, E. Kantar / Journal of Magnetism and Magnetic Materials 322 (2010) 2789–2796 2795

spins (1, 3/2) exhibits B, E and QP special points, but spins (1/2, 1)exhibits only a multicritical point (A), spins (3/2, 5/2) contains Band QP special points, and the spins (2, 5/2) exhibits triple (TP)and QP special points.

We have also calculated the dynamic phase diagrams, includ-ing the compensation behaviors in the (J2, T), (� J3, T), (d, J2) and(d, � J3) planes and three, five, three and two main topologicaldifferent types of dynamic phase diagrams are found, respec-tively. Since most of the phase diagrams in these planes can bereadily obtained from the phase diagram in (d, T) plane, especiallyhigh and low values of d and T. We give only one interesting phasediagram in each plane, see Fig. 5(a)–(d). Fig. 5(a) shows the phasediagram in the (J2, T) plane for J1¼ �1, J3¼0.8, h0¼0.1, andd¼ �1.0. The system exhibits the i1 and p fundamental phases,and the dynamic phase boundary between these fundamentalphases is second-order lines. Similar phase diagram with Fig. 5(a)was also obtained in the mixed spin-1/2 and spin-1 system[15,16], the mixed spin-2 and spin-5/2 system [19] and the mixedspin-3/2 and spin-5/2 system [20]. For J1¼ �1, J2¼1.0, h0¼0.1and d¼ �1.5, Fig. 5(b) represents the phase diagram in the (� J3,T) plane. In this phase diagram, the system exhibits one mixedphase, namely i1+p besides the i1 and p fundamental phases. Thedynamic phase boundary between the i1 and p is a second-orderline, but between the i1 and i1+p phases is a first-order line.Similar phase diagram with Fig. 5(b) was also obtained in themixed spin-2 and spin-5/2 system [19]. The phase diagram isconstructed for J1¼ �1, J3¼0.1, h0¼0.1 and T¼0.5, illustrated inFig. 5(c) and shows that the system exhibits the p and i1fundamental phases and the i1+p mixed phase. The dynamicphase boundaries among the phases are first-order phase lines.Fig. 5(d) shows the phase diagram in the (� J3, d) plane forJ1¼ �1, J2¼1.5, h0¼0.1 and T¼1.0. The system exhibits the p andi1 fundamental phases and the i1+p mixed phase, and thedynamic phase boundaries among the phases are first-order lines.The phase diagrams shown in Fig. 5(c) and (d) are only observedin this system.

Finally, we should emphasize that although the present studyis primarily of theoretical interest, the results we obtain may be ofinterest for another Co-based magnets. We also hope that thisstudy is able to stimulate theoretical physicist continue to getmore theoretical results about the DPT and the dynamiccompensation temperatures by using more accurate techniquessuch as the dynamic Monte-Carlo (MC) simulations. Moreover ourresults will be instructive in the time-consuming process ofsearching the critical behavior of this system while using thedynamic MC simulations, as well as they may give some light forexperimental works.

Acknowledgments

This work was supported by the Scientific and TechnologicalResearch Council of Turkey (TUB_ITAK), Grant no: 107T533 andErciyes University Research Fund, Grant no: FBA-06-01.

Appendix. The values of Wri ðri-r0iÞ

The probabilities Wsi ðsi-s0iÞ in Eq. (4) are calculated as

follows:

Wsi ð1-0Þ ¼Ws

i ð�1-0Þ ¼1

texpð�bDÞ

2coshðbxÞþexpð�bDÞ,

Wsi ð1-�1Þ ¼Ws

i ð0-�1Þ ¼1

texpð�bxÞ

2coshðbxÞþexpð�bDÞ,

Wsi ð0-�1Þ ¼Ws

i ð�1-1Þ ¼1

texpðbxÞ

2coshðbxÞþexpð�bDÞ,

where

x¼ J1

Xj

Sjþ J2

Xj

sjþH

ARTICLE IN PRESS

M. Keskin, E. Kantar / Journal of Magnetism and Magnetic Materials 322 (2010) 2789–27962796

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