Journal of Magnetism and Magnetic Materials 246 (2002) 177–183

An effective-field study of the mixed spin-1 and spin-32Ising

ferrimagnetic system

A. Bob!ak*, O.F. Abubrig, D. Horv!ath

Department of Theoretical Physics and Geophysics, Faculty of Science, P.J. $Saf !arik University, Moyzesova 16,

041 54 Ko$sice, Slovak Republic

Received 30 October 2001; received in revised form 11 December 2001

Abstract

An effective-field theory with correlations, that correctly incorporates the single-site kinematic relations of the spin

operators, is applied to the mixed spin-1 and spin-32Ising system with different single-ion anisotropies. The phase

diagrams are investigated numerically for the square and simple cubic lattices and results in some aspects are crucially

different to those obtained by using an approximate Van der Waerden identity for spin-32as well as in the case of the

honeycomb lattice. The thermal dependence of the magnetization is studied, and the existence and dependence of a

compensation point on single-ion anisotropies is also investigated for the square lattice. r 2002 Elsevier Science B.V.

All rights reserved.

PACS: 75.10.Hk; 75.30.Kz; 75.50.Gg

Keywords: Mixed-spin Ising model; Ferrimagnet; Tricritical point; Compensation point

1. Introduction

Over recent years there has been considerableinterest in the theoretical study of mixed-spin Isingsystems. These are studied not only out of purelytheoretical interest but also because they have beenproposed as possible models to describe a certaintype of molecular-based magnetic materials stu-died experimentally [1–4].Although the majority of studies have focused

on mixed spin-12and spin-s (s > 1

2) Ising models, the

mixed-spin Ising systems consisting of higher spinsare not without interest. Indeed, the magnetic

properties of a mixed spin-1 and spin-32Ising

ferrimagnetic system with different single-ionanisotropies [5] have recently been studied bythe use of the effective-field theory with correla-tions [6] that accounts correctly for the single-site kinematic relations of the spin operators.In particular, a variety of topologies of thefinite-temperature phase diagrams, with severalmulticritical points, resulting from different aniso-tropies, have been obtained for the system on ahoneycomb lattice. The highest-order multicriticalentity was the fourth-order point, while the phasediagram of the same system with an uniformanisotropy included only a tricritical point.On the other hand, this model on a square

lattice has been studied very recently in Ref. [7] by*Corresponding author. Fax: +38-4295-6222124.

E-mail address: abobak@kosice.upjs.sk (A. Bob!ak).

0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 0 4 8 - 3

the use of the effective-field treatment based on ageneralized but approximate Van der Waerdenidentity [8] for spin-3

2: In that work it was pointed

out that the results for the square lattice do showsome subtile qualitative differences from those forthe honeycomb lattice. However, whether thisdifference is really due to the difference incoordination number of the lattice or due to theadditional approximation used to treat the squarelattice over that used in the original effective-fieldtheory with correlations to treat the honeycomblattice is still an unresolved question. In this paper,we therefore apply the original effective-fieldtheory with correlations to a study of the mixedspin-1 and spin-3

2Ising system on a square lattice,

and we shall clarify whether the same theorypredict different qualitative features or not for thesystems with different coordination numbers. Toget a convincing answer, we also extend our studyto the system on a simple cubic lattice.

2. Model

The model we investigate is the mixed spin-1and spin-3

2Ising ferrimagnetic system described by

the Hamiltonian

H ¼ �JX

ði;jÞ

sAi sB

j � DA

X

i

ðsAi Þ

2 � DB

X

j

ðsBj Þ

2; ð1Þ

where sAi and sB

j refer to spins of magnitude 1 and 32

located on sublattices A and B; respectively. Thefirst summation is carried out only over nearest-neighbour pairs of spins on different sublatticesand JðJo0Þ is the nearest-neighbour exchangeparameter. The DA and DB are different single-ionanisotropies acting on the spin-1 and spin-3

2;

respectively.We are here interested in studying the phase

diagrams and the thermal variation of the sub-lattice, mA ¼ /sA

i S; mB ¼ /sBj S; and total, M ¼

ðmA þ mBÞ=2; magnetizations per site, where/?S denotes the thermal average. Within theeffective-field theory with correlations, the magne-tizations and the phase diagrams for the squareand simple cubic lattices can be easily obtained bythe use of the formulation in Ref. [5]. On the otherhand, the compensation temperature, if it exists in

the system, can be determined from the conditionM ¼ 0:

3. Numerical results and discussion

3.1. Phase diagrams

The resulting phase diagrams in theðDB=jJ j; kBT=jJ jÞ plane, for the square z ¼ 4 andsimple cubic z ¼ 6 lattices are shown in Figs. 1(a)and (b), respectively, for selected values of DA=jJ j:In the figures the solid lines are used for thesecond-order transition, while the dashed curverepresents the positions of tricritical points. Fromthese figures we can see that the phase diagramsfor the square and simple cubic lattices arequalitatively the same, however, unlike the hon-eycomb lattice [5] they do not include unstabletricritical points as well as fourth-order points. Itmeans that the multicritical behaviour of themixed spin-1 and spin-3

2Ising system with different

single-ion anisotropy fields, at least within theeffective-field theory with correlations, depends onthe coordination number. Further, it should bepointed out also that the non-existence of thefourth-order points for the square lattice is incontradiction with the previous effective-fieldstudy [7]. Indeed, in that work one of the presentauthors (AB) obtained fourth-order points evenfor the square lattice. The reason for thisdifference is that in Ref. [7] an approximate Vander Waerden identity [8] for the spin-3

2which

accounts less accurately for the single-site correla-tions was used. Therefore, this discrepancy in-dicates that the effective-field treatment based onthe use of an approximate Van der Waerdenidentity is not appropriate for analysing the mixed-spin Ising systems with higher values of spins sinceit may give defective multicritical behaviour.From our calculations we find that for DA=jJ j >

�1:9768ðz ¼ 4Þ and DA=jJ j > �2:9453ðz ¼ 6Þ thereis no tricritical point and the phase diagrams aretopologically equivalent to the phase diagram forthe spin-3

2Blume–Capel model [8,9]. There exist

two ordered ferrimagnetic phases denoted as O1

(with mB ¼ 32at T ¼ 0 K) and O2 (with mB ¼ 1

2at

T ¼ 0 K) that are separated, at low temperatures,

A. Bob !ak et al. / Journal of Magnetism and Magnetic Materials 246 (2002) 177–183178

by a first-order transition line, starting atDB=zjJ j ¼ �0:5 at T ¼ 0 K (see Refs. [10,11]).However, the structure of the first-order transitionbetween the ordered phases is not shown in these

figures since to solve this problem one needs tocalculate the free energy for the O1 and O2 phasesand to find a point of intersection. As far as weknow, the complete function for the free energyhas not yet been constructed for this model (in theframe of the effective-field theory with correla-tions). Therefore, for obtaining the phase dia-grams we have confined our calculations only tothe second-order phase transitions, including thetricritical points.The values of the transition temperature in the

absence of anisotropies (i.e. for DA ¼ DB ¼ 0) arekBTc=jJ j ¼ 3:0217 and 4.8457 for z ¼ 4 and 6,respectively, and may be compared to the values3.6515 and 5.4772 of mean-field theory [11]. Forz ¼ 4 the result may also be compared to the valuekBTc=jJ j ¼ 3:0334 of effective-field treatment [7]based on the use of an approximate Van derWaerden identity for the spin-3

2; and is seen to be

slightly lower. In the limits DB=jJ j-7N atDA=jJ j ¼ 0 the boundary lines tend to the asymp-totic values kBTc=jJ j ¼ 3:8964 and 1.2988 (forz ¼ 4), 6.3347 and 2.1116 (for z ¼ 6), respectively.In these limits the spins on the sublattice B behavelike simple doublets having states sB

j ¼ 732and71

2;

respectively. For this reason the values of transi-tion temperatures in the limit of large positive DB

are three times higher than those for large negativeDB: The same difference in these limits holdsbetween coordinates ðDA=jJ j; kBT=jJ jÞ of thetricritical points. Indeed, for DB-þN thecoordinates of the tricritical points areð�5:9303; 1:2893Þ and ð�8:8255; 2:5536Þ for z ¼4 and 6, respectively, while for DB-�N thetricritical points are located at ð�1:9768; 0:4298Þfor z ¼ 4 and ð�2:9418; 0:8512Þ for z ¼ 6 (whichare the effective field coordinates of the mixed-spinsystems with sA

i ¼ 0;71 and sBj ¼ 73

2or sB

j ¼ 712).

However, it is important to notice that theseasymptotic values for the tricritical point on thesquare lattice are at variance with Monte Carlocalculations [12,13] and a numerical transfermatrix study [13] which predict the absence of atricritical point for the such mixed-spin system. Inthe light of these results, it is difficult to saywhether the existence of the tricritical points forthe square lattice between these two limits is realor not. In any case, more rigorous techniques are

Fig. 1. The phase diagram in the ðDB;TÞ plane for the mixedspin-1 and spin-3

2Ising ferrimagnet, when the value of DA is

changed on (a) square (z=4) and (b) simple cubic (z=6)

lattices. The solid lines indicate second-order phase transitions,

while the dashed line represents the positions of tricritical

points. O1 and O2 are ordered ferrimagnetic phases and P is the

paramagnetic phase.

A. Bob !ak et al. / Journal of Magnetism and Magnetic Materials 246 (2002) 177–183 179

needed in order to clarify this tricritical behaviourof the present mixed-spin system on the squarelattice.Further, with a very small difference between

square and simple cubic lattices one can observe

the variation of tricritical temperature with DB: Inthe case of z ¼ 6; the tricritical temperature is anincreasing function of DB; whereas for the z ¼ 4 itexhibits a broad maximum in the region of DB

below DB ¼ 0 and then smoothly decreases to itsconstant value for large positive DB: This differ-ence in the tricritical behaviour between the twolattices can be also seen from Figs. 2(a) and (b)where the phase diagrams are shown in theðDA=jJ j; kBT=jJ jÞ plane. Here, it is worth mention-ing that the tricritical line for the z ¼ 6 isqualitatively the same as that recently obtainedby mean-field theory [11].

3.2. Magnetization and compensation temperature

In this subsection, let us at first examine thetemperature dependences of the sublattice magne-tizations mA and mB for the square lattice. Theresults are depicted in Fig. 3 for the system withDA=jJ j ¼ �0:5; when the anisotropy field strengthof DB=jJ j is changed. Notice that the selection ofDB=jJ j corresponds to the crossover from the O1 tothe O2 phase (see the ground-state phase diagramin Ref. [11]), therefore the ground state is always

Fig. 2. The phase diagram in the ðDA;TÞ plane for the mixedspin-1 and spin-3

2Ising ferrimagnet, when the value of DB is

changed on (a) square (z=4) and (b) simple cubic (z=6)

lattices. The solid lines indicate second-order phase transitions,

while the dashed line represents the positions of tricritical

points. O1 and O2 are ordered ferrimagnetic phases and P is the

paramagnetic phase.

Fig. 3. The temperature dependences of the sublattice magne-

tizations mA;mB for the mixed spin-1 and spin-32

Ising

ferrimagnet on the square lattice with DA=jJ j ¼ �0:5; whenthe value of DB=jJ j is changed.

A. Bob !ak et al. / Journal of Magnetism and Magnetic Materials 246 (2002) 177–183180

ordered. Moreover, the present system withDA=jJ j ¼ �0:5 exhibits only a second-order transi-tion, consequently, the sublattice magnetizationsat a critical point vanish continuously.As shown in Fig. 3, for DB=jJ jX� 1:0; the

sublattice magnetization mB has a standardcharacteristic convex shape. As DB=jJ j decreasefrom DB=jJ j ¼ �1:0; however, the temperaturedependence of mB may exhibit a rather rapiddecrease from its saturation value at T ¼ 0 K: Thephenomenon is further enhanced when the valueof DB=jJ j approaches the boundary between theO1 phase and the O2 phase, i.e. DB=jJ j ¼ �2: Inparticular, when DB=jJ j ¼ �2:0; the saturationvalue of mB is 1.0, which indicates that the halfof the spins on sublattice B are equal þ3

2(or �3

2)

and the other half are equal to þ12(or �1

2). Note

that this mixed state persists as long as DB=jJ j ¼�2:0 and DA=jJ j > �2:0: In this case, the totalmagnetization for the ferrimagnetic system is M ¼0 at T ¼ 0 K; and hence there is a compensationpoint at which the two sublattice magnetizationscancel. On the other hand, for the ferromagneticsystem ðJ > 0Þ with DB=J ¼ �2:0 the saturatedtotal magnetization is M ¼ 1:0: The result may becompared to the value M ¼ 0:985 of Ref. [14],obtained using an effective-field treatment basedon an approximate Van der Waerden relation [8]for spin-3

2: This difference obviously stems from

the additional approximation used in Ref. [14]over that used in the present effective-fieldformulation. A similar difference in the values ofthe saturated total magnetization obtained fromthe two effective-field methods on can find in Refs.[5,15] for the same mixed-spin Ising system on ahoneycomb lattice ðz ¼ 3Þ: By further decreasingDB=jJ j the ground state becomes O2; with mB ¼0:5 at T ¼ 0 K: However, even in the regionDB=jJ jo� 2:0; the thermal variation of mB

exhibits different behaviours depending on thevalue DB=jJ j: An interesting feature is the initialrise of mB with temperature in the region�4:0oDB=jJ jo� 2 before decreasing to its zerovalue at the critical point. On the other hand, forall values of DB=jJ j the sublattice magnetizationmA may show a normal behaviour, even though itis coupled to mB: Here it is worth mentioning thatthe mixed state also emerges at the boundary

between the D1 and DB disordered phases (seeFig. 1 in Ref. [11]). Indeed, at T ¼ 0 K and forDA=zjJ jo� 1:5 and DB=zjJ j ¼ 0; we have foundthat mB ¼ 0 and qB ¼ 5

4: It implies that in this

mixed state half of the spins on sublattice B arerandomly in the sB

j ¼ 732states and the other half

are randomly in the sBj ¼ 71

2states.

Finally, let us investigate whether the presentmixed-spin Ising ferrimagnetic system may alsoexhibit a compensation point (or points) at Ta0when the single-ion anisotropies are changed. Thevariation of the compensation temperature Tk asfunction DB=jJ j for the square lattice is shown inFig. 4 for several values of DA=jJ j: As seen fromthe figure, all the Tk curves emerge from DB=jJ j ¼�2:0 at T ¼ 0 K and for the values of DA=jJ j > 0increase monotonically with DB; to terminate atthe corresponding phase boundaries (dashedlines). As DA=jJ j is reduced, the range of DB=jJ jover which the compensation point occurs gradu-ally becomes small, but the compensation tem-perature still reaches the corresponding transitionline even for DA=jJ jo0: However, for DA=jJ j near

Fig. 4. Dependence of the compensation temperature (solid

lines) on the single-ion anisotropy DB=jJ j in a mixed spin-1 and

spin-32Ising ferrimagnet on the square lattice, when the value of

DA=jJ j is changed. The dashed lines show part of the second-

order lines separating the paramagnetic and ordered ferrimag-

netic phases.

A. Bob !ak et al. / Journal of Magnetism and Magnetic Materials 246 (2002) 177–183 181

to �2:0 a new type of compensation curves appear:the Tk curves are extended to DB=jJ j-�N

below the corresponding transition lines. Thecurve labelled �1:95 is an example of suchbehaviour of Tk: Moreover, we can see from thefigure that in a restricted region of DB; close toDB=jJ j ¼ �2:0; some compensation temperaturelines exhibit bulges, which implies the existence oftwo compensation points in the system with anappropriate negative value of DA: It is also worthmentioning that the dependence of the compensa-tion temperature on the single-ion anisotropies forthe simple cubic lattice is qualitatively the same asthat presented above for the square lattice, there-fore, it is not shown here. Of course, in this case allthe Tk lines rise from DB=jJ j ¼ �3:0 at T ¼ 0 Kand, particularly, for DA=jJ j near to �3:0 they areextended to DB=jJ j-�N below the correspond-ing transition lines.Typical sublattice magnetization curves, which

refer to the compensation temperatures presentedin Fig. 4 are shown in Fig. 5, for the case whenDA=jJ j ¼ �1:95: The curve labelled �1:975 corre-sponds to the value of DB=jJ j where two compen-

sation points occur at nonzero temperatures. Onthe other hand, the sublattice magnetizationslabelled �2:0 and �2:5 are for the value ofDB=jJ j corresponding to the single compensationpoint at Ta0 K: However, as mentioned above,the former has also an additional compensationpoint at T ¼ 0 K: The existence of compensationpoint (or points) for the system with DA=jJ j ¼�1:95 and assumed values of DB=jJ j can also beclearly seen from the temperature dependences ofthe total magnetization jM j in Fig. 6. In particular,one can observe that the magnitude of the totalmagnetization between the compensation pointand transition temperature is always smaller thanthat between the two compensation points.

4. Conclusions

In this work we have studied magnetic proper-ties of the mixed spin-1 and spin-3

2Ising ferri-

magnet by an effective-field theory withcorrelations. In particular, for the square andsimple cubic lattices we have obtained finite-temperature phase diagrams with tricritical pointsonly, when the single-ion anisotropies are includedin the system. This feature of the phase diagrams

Fig. 5. The thermal dependences of the sublattice magnetiza-

tions jmA j (dashed lines) and jmBj (solid lines) for the mixed

spin-1 and spin-32Ising ferrimagnet with z ¼ 4 and DA=jJ j ¼

�1:95; when the value of DB=jJ j is changed.

Fig. 6. The thermal dependences of the total magnetization jMjfor the mixed spin-1 and spin-3

2Ising ferrimagnet with z ¼ 4 and

DA=jJ j ¼ �1:95; when the value of DB=jJ j is changed.

A. Bob !ak et al. / Journal of Magnetism and Magnetic Materials 246 (2002) 177–183182

has also been predicted recently in Ref. [11] byusing a mean-field approximation. On the otherhand, the same system on a honeycomb lattice [5]within effective-field theory with correlationsexhibits fourth-order points. These results indicatethat the multicritical behaviour of the mixed spin-1and spin-3

2Ising system with different single-ion

anisotropies depends on the lattice coordinationnumber. Of course, independent confirmation ofthis fact, by Monte Carlo simulations or renorma-lization-group approach, would be desirable.It is also important to notice that the fourth-

order points appear in the effective-field treatment,however, based on an approximate Van derWaerden identity for spin-3

2for the same system

on a square lattice [7], although this fact wasoverlooked in Ref. [14]. Since the present effective-field theory is based on the use of the exact Vander Waerden identities for both the spin-1 andspin-3

2; we are forced to conclude that the existence

of the fourth-order points in Ref. [7] is due tothe additional approximation over that used in theeffective-field theory with correlations. Thus, thetheory based on the approximate Van der Waer-den identities may fail to describe the multicriticalbehaviour of complex systems, although it leads torelatively good results for the single-ion models[8,9].Finally, we have shown that this mixed-spin

ferrimagnetic system may exhibit one or even twocompensation points, in agreement with previouseffective-field [5] and mean-field [11] studies. Ourstudy suggests that there is a strong dependencebetween the compensation temperature and thesingle-ion anisotropies DA and DB: Not only thevalue of the compensation temperature depend onthe anisotropies DA and DB but even its existence.Since the present system may exhibit a compensa-tion point even for DA ¼ 0; we can conclude thatan anisotropy DB is mainly responsible for theappearance of the compensation phenomenon, at

least for the system with DAX0: We consider thatthe existence of the compensation points can be ofrelevance for experimental physicists, due to therole of the compensation temperature, as a way toachieve temperature-dependent coercivities [16,17].

Acknowledgements

This work was supported by the Scientific GrantAgency of Ministry of Education of SlovakRepublic (No. 1/6020/99).

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