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Physica B 393 (2007) 204–212

www.elsevier.com/locate/physb

Decorated ferrimagnetic Ising model with a random crystal field

A. Benyoussef1, A. El Kenz�, M. El yadari

Laboratoire de Magnetisme et de physique des Hautes Energies, Departement de physique, B.P. 1014, Faculte des sciences, Rabat, Morocco

Received 14 September 2006; received in revised form 8 December 2006; accepted 11 December 2006

Abstract

A mean-field approximation is used to study the effects of random crystal field on the critical behaviour of decorated ferrimagnetic

Ising model, in which the two magnetic atoms A and B have spins sA ¼ 12and SB ¼ 1, respectively. The results indicate that there may

exist some interesting phenomena in the system, such as the appearance of a new ferrimagnetic phase, namely partly ferrimagnetic phase,

and the possibility of one or two compensation temperatures. Re-entrant phenomena can be seen for appropriate ranges of crystal field.

Phase diagrams and magnetization curves are investigated in details.

r 2007 Elsevier B.V. All rights reserved.

PACS: 05.50.+q; 75.10.Hk; 75.50.Gg; 04.20.Jb

Keywords: Decorated Ising model; Ferrimagnetism; Compensation temperature; Re-entrance

1. Introduction

Many efforts have been devoted to study the magneticproperties of ferrimagnetic systems. This is due to theimportance of magnetic materials for technological appli-cations [1], such as garnets and ferrites, which areferrimagnetic. Experimentally, important advances havebeen made in the synthesis of two- and three-dimensionalferrimagnets, such as 2d organometallic ferrimagnets [2],2d networks of the mixed-metal material ½P(Phenyl)4�½MnCr(oxalate)3�n [4–6]. Theoretically, the magnetic prop-erties of these systems have been extensively studied by avariety of techniques, as, for example, mean-field approx-imation (MFA) [7], effective-field theory [8,9] finite clusterapproximation [10], renormalization group [11,12] andMonte-Carlo simulations [13,14].

Ferrimagnets are structured in several sublattices with afinite resultant moment and show paramagnetic behaviourabove the transition temperature T c. In contrast, with aferromagnet, there is an interesting possibility of theexistence, under certain conditions, of a compensation

front matter r 2007 Elsevier B.V. All rights reserved.

ysb.2006.12.081

ng author. Tel./fax: +212 35778973.

ss: elkenz@fsr.ac.ma (A.E. Kenz).

he Hassan II Academy of Science and Technology.

temperature Tcmp ðTcmpoT cÞ, at which the resultantmagnetization vanishes [15,16]. Recently, the possibilityof many compensation points in a variety of ferrimagneticsystems has been clarified theoretically [17–19].Decorated Ising spin models, which were originally

introduced into the literature by Syozi [20], have beenstudied many years ago as models exhibiting ferrimagnet-ism [21]. In fact, the decorated ferrimagnetic Ising system ismade up of two sublattices L1 and L2, where the sublatticeL1 is occupied by, spin 1

2, A atoms and the sublattice L2,

which is composed of one decorating point on every bondof L1, is occupied by, spin SðS41

2Þ, B atoms. The behaviour

of various decorated models has been explored by varietyof mathematical techniques, with some exact results[20–27]. Indeed, the effect of crystal field on the thermo-dynamic properties of a two-dimensional decorated ferri-magnetic Ising model has been explored and discussedexactly and it has been showed that the model can exhibittwo compensation temperatures when the decorating spinS is integer [23,24,26]. However, only one compensationtemperature has been found when S is half-integer. Theseresults have been also found in other works [25,27], whichprovide exact results for the ferrimagnetic transverse Isingmodel on decorated planar lattices that does not exhibitmore than one compensation temperature. In addition, the

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Fig. 1. The two-dimensional decorated Ising spin system consisting of two

kinds of magnetic atoms A and B with spins values sA ¼ 12(black points)

and SB ¼ 1 (white points), respectively.

A. Benyoussef et al. / Physica B 393 (2007) 204–212 205

compensation temperature surprisingly turned out to becompletely independent of the transverse field though itdepends on the spin of the decorating atoms. Thesedecorated ferrimagnetic systems have been also studiedby the use of mean-field theory [28] and effective-fieldtheory [26,27,29–32].

The purpose of this work is to study, via a (MFA) theinfluence of crystal field disorder on the magnetic proper-ties (phase diagrams and magnetizations) of a decoratedtwo sublattice ferrimagnetic Ising system consisting of twomagnetic atoms A and B with spins sA ¼ 1

2and SB ¼ 1,

respectively. In fact, the effect of a random crystal field hasbeen investigated in different models; a spin 1 Ising model[33–36], Blume-Emery–Griffiths model [37] and mixed spinsystem [38]. The most interesting result emerging fromthese studies is the appearance of new types of diagrams.For example, in the particular two-valued distribution ofcrystal field, given in Refs. [35,36], a new phase (partlyferromagnetic phase) has been found. In our case, thesystem can exhibit a new phase, called partly ferrimagneticphase. Also, we discuss the existence of one and twocompensation points. Finally, an interesting feature canoccur which is the re-entrant phenomenon.

The outline of this paper is as follows: In Section 2, weintroduce the model and give the details of the MFA. InSection 3, we give the ground-state phase diagrams. InSection 4, we present and discuss our results, and finally inSection 5 we summarize our conclusions.

2. Model and method

Although MFA neglects all spin correlations, it is oftenquite interesting to study the magnetic behaviour of thecomplex spin system, such as the ferrimagnetic decoratedIsing system, within the mean-field approach because of itsconceptual simplicity. The whole lattice is divided into twosublattices L1 and L2. Every point of L1 is always occupiedby an A atom with the fixed spin sA ¼ 1

2. That of L2, which

is composed of one decorating point on every bond of L1,is always occupied by a B atom with a fixed spin SB ¼ 1.

The exchange interaction between A and B atoms isassumed to be antiferromagnetic. Furthermore, we assumethat there exists a ferromagnetic exchange interactionbetween every nearest-neighbour pair of A atoms. The two-dimensional system is depicted in Fig. 1. The system isdescribed by the following Hamiltonian:

H ¼ �J1

Xhiji

sisj þ J2

Xhiji

siSj �X

i

DiS2i , (1)

where the spins si ¼ �12 and S ¼ �1; 0 are localized in

sublattices L1 and L2, respectively. J1(J140) and J2(J240)are the exchange interactions. The summations are carriedout only over nearest-neighbour pairs of spins and Di is arandom crystal field distributed according to the law:

PðDiÞ ¼12½dðDi � Dð1þ aÞÞ þ dðDi � Dð1� aÞÞ�, (2)

with aX0. To write the mean-field equation let hs and hS

denote the molecular fields associated with order para-meters ms ¼ hsi and mS ¼ hSi, respectively:

hs ¼ 4J1ms � 4J2mS,

hS ¼ 2J2ms. (3)

The effective Hamiltonian of the system is

H0 ¼ �XN

i¼1

hssi þX2N

i¼1

hSSi �X2N

i¼1

DiS2i . (4)

It generates the following partition function:

Z0 ¼ Tr exp�H0

T

� �� �,

Z0 ¼YNi¼1

2 coshhs

2T

� �� �YNi¼1

1þ 2 expDi

T

� ��

�coshhS

T

� ��. ð5Þ

We note that Bolzmann’s constant has been set to unity.The variational principle based on the Gibbs–

Bogoliubov inequality for the free energy per site isdescribed by [39]

FpF ¼ �T lnðZ0Þ þ hH �H0i0 (6)

and the order parameters which are the spin averages aregiven by

ms ¼1

2tanh

2

TðJ1ms � J2mSÞ

� �, (7)

ARTICLE IN PRESS

02

-2

2

d

r

d+r=0(1/2,0)

(1/2,-1)

2

-2

0

2

r+d(1-α)=0

d

r

(1/2,0)(1/2,-1/2)

(1/2,-1)

r+d(1+α)=0

α=0

α=0.3

A. Benyoussef et al. / Physica B 393 (2007) 204–212206

mS ¼� sinhð2J2ms=TÞ expðDð1þ aÞ=TÞ

1þ 2 coshð2J2ms=TÞ expðDð1þ aÞ=TÞ

�sinhð2J2ms=TÞ expðDð1� aÞ=TÞ

1þ 2 coshð2J2ms=TÞ expðDð1� aÞ=TÞ. ð8Þ

Then the total free energy can be written as

F ¼ � T lnðZ0Þ � J1

Xhiji

hsiihsji þ J2

Xhiji

hsiihSji

þXN

i¼1

hshsii �X2N

i¼1

hShSii. ð9Þ

Usually the solutions of Eq. (6) will not be unique,the stable ones are those which minimize the free energy(Eqs. (7) and (8)), while the others are the unstable ones. Ifthe order parameters are continuous (discontinuous), thetransition is of second (first) order.

3. Ground-state

Before studying the phase diagrams, let us first discussthe ground-state phase diagrams for the particulardistribution (2). In fact, for T ¼ 0K and aX0, Eq. (6)has three solutions (ms ¼

12, mS ¼ 0), (ms ¼

12, mS ¼

�12)

called partly ferrimagnetic phase and the ferrimagneticphase ðms ¼

12;mS ¼ �1Þ. The energies of all possible

solutions can easily be calculated. By comparing theseenergies, the type of the ground-state diagrams is thendetermined as we see from Figs. 2a–c where we drew theground-state phase diagram in ðd ¼ D=J1; r ¼ J2=J1Þ fora ¼ 0, a ¼ 0:3 and a ¼ 1:2, respectively. Indeed, three casescan be distinguished:

2

α=1.2(1/2,-1/2)

(1)

r+d(1-α)=0

For a ¼ 0, a first-order transition lines between the(ms ¼

12, mS ¼ 0) phase and the ferrimagnetic phase

(ms ¼12, mS ¼ �1) occurs at line d þ r ¼ 0.

(2)

20d

r

(1/2,-1)

For 0oao1, two first-order transition between the(ms ¼

12,mS ¼ 0) phase and the (ms ¼

12, mS ¼

�12) phase

and between the (ms ¼12, mS ¼

�12 ) phase and the

(ms ¼12, mS ¼ �1) phase occur at rþ dð1� aÞ ¼ 0 and

rþ dð1þ aÞ ¼ 0, respectively.

r+d(1-α)=0

(3)

(1/2,-1/2)

For a41, two first-order transition lines between the(ms ¼

12, mS ¼

�12) phase, the (ms ¼

12, mS ¼ �1) phase

and the (ms ¼12, mS ¼

�12) phase occur at rþ dð1�

aÞ ¼ 0 and rþ dð1þ aÞ ¼ 0, respectively.

-2

Fig. 2. The ground-state phase diagrams in the ðd; rÞ plane for selected

values of a ¼ 0 (Fig. 2a), a ¼ 0:3 (Fig. 2b) and a ¼ 1:2 (Fig. 2c). The full

line represents the first-order phase transition. We have three situations

depending on the value of a.

For Ta0K, the phase diagrams for different values of aand r were determined numerically.

4. Results and discussions

4.1. Phase diagrams

In this section, let us examine the phase diagrams ofthe two-dimensional decorated ferrimagnetic Ising model(Fig. 1). At finite temperature, Eqs. (7) and (8) aresolved numerically. However, the phase diagrams in

(tc ¼ Tc=J1; d) and (tc; r) planes, obtained from MFA,are shown, for r ¼ 0:4 in Figs. 3 and 4, respectively. Thus,in Fig. 3a, for a ¼ 0:3, we can show a first-order transition,

ARTICLE IN PRESS

-4 -2 0 2 4

1.12

1.16

1.20

1.22

t c t c

d

(0,0)

(1/2,-1/2) (1/2,-1) (1/2,-1/2)

-5 -3 -1 1 3 5

1.00

1.05

1.10

1.15

1.20

1.25

t c

t c

r=0.4, α=0.3

(0,0)

(1/2,0) (1/2,-1)(1/2

,-1/

2)

d

r=0.4, α=1.2

-0.60 -0.55 -0.45 -0.35 -0.30

0.000

0.005

0.010

0.015

0.020

(1/2,-1)(1/2,-1/2)(1/2,0)

d

-1.0 0.0 1.0 2.00.000

0.010

0.020

0.030

0.040

(1/2,-1/2)(1/2,-1)(1/2,-1/2)

d

Fig. 3. The phase diagram in the (tc; d) plane for the two-dimensional decorated ferrimagnetic system ðz ¼ 4Þ with (r ¼ 0:4, a ¼ 0:3) and (r ¼ 0:4, a ¼ 1:2)in Figs. 3a and b, respectively. The insets show the details of the first-order transition lines.

A. Benyoussef et al. / Physica B 393 (2007) 204–212 207

at d ¼ �0:57, between the phases (ms ¼12, mS ¼ 0) and

(ms ¼12, mS ¼

�12) (partly ferrimagnetic phase) and another

one, at d ¼ �0:307, between the phases (ms ¼12, mS ¼

�12)

and (ms ¼12, mS ¼ �1) (ferrimagnetic phase). Remarkably,

in Fig. 3b, for a ¼ 1:2, the partly ferrimagnetic phase canoccur at small and great value of reduced crystal field d.Indeed, these phase appears for do� 0:182 and d42. Inboth figures, these first-order transition lines whichterminate at isolated critical points are drawn in the insets.However, in the inset of Fig. 3b, only the first-ordertransition line separating the partly ferrimagnetic phase

from the ferrimagnetic phase has been plotted. In fact, theisolated critical temperature in the second first-order line istoo small. Therefore, a continuous passage occurs at verylow temperature (see magnetization curves). A second-order transition line separates the above phases from theparamagnetic phase (ms ¼ 0, mS ¼ 0) is depicted by a solidline. Furthermore, the system exhibits the re-entrantphenomena. This observed re-entrance occurs upon changeof the crystal field. It is due to a competition betweenpositive and negative values of crystal field. By contrast, inFig. 3a this phenomenon is absent. Indeed, for a ¼ 0:3, in

ARTICLE IN PRESS

0.0 0.5 1.0 1.5 2.0

1.0

1.5

2.0

2.5

r

d=1, α =0.3

(1/2,-1)

(0,0)

0.0 0.5 1.0 1.5 2.0

1.0

1.5

2.0

2.5

3.0

d=-0.1, α =0.3

r

(1/2,-1/2)

(0,0)

0.0 0.5 1.0 1.5 2.0 2.5

1.0

1.5

2.0

2.5

3.0

t ct c

t c

r

(1/2,0)

(0,0)

d=-2, α =0.3

3.0

Fig. 4. The phase diagram in the (tc; d) space for the decorated ferrimagnetic

system with a¼0:3 and d¼�2, d¼�0:1 and d¼1 in Figs. 4a–c, respectively.

-5 -4 -3 -2 -1 0

0.0

0.5

1.0

1.5

2.0

2.5

d

r=0.5

r=1.3

r=2

r=2.5

r=3

t cm

p

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

r

d=-7

d=-5

d=-3

d=-2

d=-1

d=0

d=5.45

t cm

p

α=0.3

α=0.3a

b

Fig. 5. The variation of compensation temperature of r and d is given in

Figs. 5a and b, respectively, for the value of a is fixed as a ¼ 0:3. In both

cases we can show double compensation temperatures.

A. Benyoussef et al. / Physica B 393 (2007) 204–212208

concordance with probability law (Eq. (2)), the crystal fieldhas two positive values: Di ¼ 1:3D and Di ¼ 0:7D.

Figs. 4a–c express the second-order transition lineseparating the paramagnetic phase (ms ¼ 0, mS ¼ 0) fromthe phases (ms ¼

12, mS ¼ 0), (ms ¼

12, mS ¼

�12) and

(ms ¼12, mS ¼ �1) for d ¼ �2, �0:1 and 1, respectively.

We remark that the behaviour of this line is linear for greatvalues of r.In addition to critical temperature, compensation points

appear in the system. The reduced compensation tempera-ture tcmp ¼ T cmp=J1 is located below the reduced criticaltemperature tc. This compensation phenomenon, which isbeyond Neel theory has been discovered experimentally inðNiII0:22MnII0:6Fe

II0:18Þ15ðCr

IIIðCnÞ6Þ7:6H2O [40]. tcmp is deter-

mined from the condition M ¼ 0 for tcmpotc. Where M isthe total magnetization of the system, given by

m ¼M

NA¼ ms þ 2mS.

ARTICLE IN PRESS

0.1

0.3

0.5

etiza

tio

n

r=0.4, α =0.3, d=-0.7mσ

M

a

A. Benyoussef et al. / Physica B 393 (2007) 204–212 209

With m the magnetization per site and NA is the number ofsites with lattice L1.

In Figs. 5a and b, for a ¼ 0:3 we plot the compensationtemperature dependence on r and d, respectively. Inaddition of the one compensation point, the curves, inboth figures, exhibit double compensation temperatures.This last phenomenon appears for �2:37pdp0 and 0orp1:34 in Figs. 5a and b, respectively. However, thecompensation phenomenon appears in the range ofdp5:45 (Fig. 5a) and r40 (Fig. 5b). Furthermore, forpositive value of d the behaviour of compensation linesbecome linear (Fig. 5a). For a ¼ 1:2, as is shown in Figs. 6aand b, the phenomenon of two successive compensationpoints disappears and only one compensation temperaturecan occur. We conjecture that the phenomenon of doublecompensation temperature arises from the existence of the(ms ¼

12, mS ¼ 0) phase which appears for ao1.

-4 -2 0 2 4

0.0

0.2

1.0

1.2

1.4

1.6

d

t cm

p

r=0.1

r=0.3

r=0.5

r=0.9r=0.95

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.4

0.8

1.2

1.6

r

d=-3d=-7

d=-1

d=0d=1

t cm

p

0.8

0.4

0.6

α=1.2

α=1.2a

b

Fig. 6. The compensation temperature line in the (tcmp; r) (Fig. 6a) and(tcmp; d) (Fig. 6b) planes, for a ¼ 1:2. In both cases we can show only one

compensation point.

4.2. Magnetic properties

Now, let us examine the temperature dependencies of themagnetization ms and mS and total magnetization M inorder to complete the phase diagrams represented above.Indeed, as is shown in Fig. 7a, the double compensation

0.0 0.5 1.0 1.5

-0.3

-0.1

0.0ma

gn

t

0.0 0.5 1.0 1.5

-0.5

-0.3

-0.1

0.0

0.1

0.3

0.5

ma

gn

etiza

tio

n

t

r=0.4, α =0.3, d=-0.4

0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

ma

gn

etiza

tio

n

t

r=0.4, α =0.3, d=2.1mσ

mS

mσ

mS

mS

M

M

b

c

Fig. 7. The variation of magnetization ms, mS and M of temperature for

r ¼ 0:4, a ¼ 0:3 and selected values of d ¼ �0:7 (Fig. 7a), d ¼ �0:4 (Fig. 7b)and d ¼ 2:1 (Fig. 7c).

ARTICLE IN PRESS

0.15

0.25

r=0.4, α =1.2, t=1.116

atio

n

mσ

A. Benyoussef et al. / Physica B 393 (2007) 204–212210

phenomenon occurs for two successive temperatures,where the total magnetization M vanishes below thecritical temperature Tc. Furthermore, at T ¼ 0K themagnetizations ms and mS start at 1

2 and 0, respectively.This is in concordance with Fig. 3a. On the other hand,only one compensation point is present in both Figs. 7band c. In these last figures the magnetizations (ms, mS)

0.0 0.5 1.0 1.5

-0.5

-0.3

-0.1

0.0

0.1

0.3

0.5

magne

tization

t

r=0.4, α =1.2, d=-3

0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

magnetization

t

r=0.4, α =1.2, d=0.5

r=0.4, α =1.2, d=0.3

0.0 0.5 1.0 1.5

-0.5

-0.3

-0.1

0.0

0.1

0.3

0.5

ma

gn

etiza

tio

n

t

mS

mS

mS

mσ

mσ

mσ

M

M

M

a

b

c

Fig. 8. The variation of magnetization ms, mS and M of temperature for

r ¼ 0:4, a ¼ 1:2 and selected values of d ¼ �3 (Fig. 8a), d ¼ 0:5 (Fig. 8b)

and d ¼ 3 (Fig. 8c).

-4 -2 0 2 4

-0.15

-0.05

0.00

0.05

ma

gn

etiz

d

mS

M

Fig. 9. The variation of magnetizations ms, mS and M of reduced crystal

field d for r ¼ 0:4, a ¼ 1:2 and t ¼ 1:116.

start, at T ¼ 0K, at (ms ¼12, mS ¼

�12) (Fig. 7b) and

(ms ¼12, mS ¼ �1) (Fig. 7c), respectively. We note that

these figures are plotted for r ¼ 0:4 and a ¼ 0:3.In Figs. 8a–c, we plot the dependencies of magnetiza-

tions ms, mS and M on temperature with r ¼ 0:4, a ¼ 1:2and d ¼ �3, d ¼ 0:5 and d ¼ 3, respectively. The startingvalues of ms and mS, at T ¼ 0K, are in concordance withthose obtained in ground state diagrams. We note that thephase (ms ¼

12, mS ¼ 0) is absent in this case. As is

conjectured before, the curves exhibit only one compensa-tion temperature tcmp.In Fig. 9, the dependence of magnetizations ms, mS and

M on reduced crystal field are depicted for t ¼ 1:116. As,we can see from this figure, the magnetizations ms, mS

may exhibit two second-order phase transitions. The firstone is between the phases (ms ¼

12, mS ¼

�12) and (ms ¼ 0,

mS ¼ 0) and the second one is between the phases (ms ¼ 0,mS ¼ 0) and (ms ¼

12, mS ¼ �1). This behaviour is in

concordance with the re-entrant phenomenon, whichappears in Fig. 3b at low temperature. The samephenomenon can also occur at high temperature.Finally, let us examine the temperature dependence

of the magnetization near T ¼ 0K. In fact, in Fig. 10athe magnetizations ms and mS are depicted for r ¼ 0:4,a ¼ 0:3 and t ¼ 0:005. We can show a successively first-order transition between the phases (ms ¼

12, mS ¼ 0)

and (ms ¼12, mS ¼

�12) at d ¼ �0:571 and between the

phases (ms ¼12, mS ¼

�12) and (ms ¼

12, mS ¼ �1) at

d ¼ �0:307. Whereas, a continuous passage occursbetween these phases for t ¼ 0:01 (Fig. 10b). On theother hand, this continuous passage appears firstly betweenthe phases (ms ¼

12, mS ¼ �1) and (ms ¼

12, mS ¼

�12) at

very low temperature (Fig. 10c) and later betweenthe phases (ms ¼

12, mS ¼

�12) and (ms ¼

12, mS ¼ �1)

(Fig. 10d).

ARTICLE IN PRESS

-1 0 1 2 3

-1.5

-1.0

-0.5

0.0

0.5

r=0.4, α =1.2, t=0.04

magnetization

d

-1 0 1 2 3

-1.5

-1.0

-0.5

0.0

0.5

r=0.4, α =1.2, t=0.0092

magnetization

d

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2

-1.5

-1.0

-0.5

0.0

0.5

r=0.4,α =0.3, t=0.01

magnetization

d

-1.0 -0.8 -0.6 -0,4 -0.2 0.0 0.2

-1.5

-1.0

-0.5

0.0

0.5

r=0.4, α =0.3, t=0.005

magnetization

d

mσ

mS

mσ

mS

mσ

mS

mσ

mS

M M

MM

Fig. 10. The variation of magnetizations ms, mS and M for r ¼ 0:4, a ¼ 0:3 for selected values of t ¼ 0:005 (Fig. 10a) and t ¼ 0:01 (Fig. 10b) and r ¼ 0:4,(a ¼ 1:2) for selected values of t ¼ 0:0092 (Fig. 10c) and t ¼ 0:04 (Fig. 10d).

A. Benyoussef et al. / Physica B 393 (2007) 204–212 211

5. Conclusion

In this work, we have studied, using MFA, the effectsof a random crystal field on a decorated ferrimagneticIsing model, in which the two magnetic atoms A and Bhave spins sA ¼ 1

2and SB ¼ 1, respectively. Our investiga-

tion has revealed many interesting phenomena related tothe effects of this random crystal field on transitiontemperature and compensation points. Indeed, a newferrimagnetic phase, namely partly ferrimagnetic phase,appears. In this phase the magnetization ms and mS havethe values 1

2and �1

2, respectively. Furthermore, the existence

of compensation temperature depend on the values of a.In fact, for 0oao1, double compensation points mayexist. We have conjectured that the double compensationtemperature occurs if we have the phase (ms ¼

12, mS ¼ 0).

This last phase appears for 0oao1. By contrast, thisphenomenon disappears for a41, where we have acompetition between positive and negative values of crystalfield. Finally, we can observe the re-entrant phenomenonat low and high temperature. The phase diagramsand magnetization curves have been studied for theinteresting cases.

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Further Reading

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