On the existence of compensation temperature in 2d mixed-spin Ising ferrimagnets: an exactly solvable model

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On the existence of compensation temperature in 2d mixed-spin Ising ferrimagnets: an exactly solvable model

A. Dakhama*, N. Benayad

GMS - Laboratoire de Physique The&orique, Faculte&des Sciences An(n Chock, Universite Hassan II Ain Chock, B.P. 5366, MaLarif, Casablanca, Morocco

Received 25 May 1999; received in revised form 3 August 1999

Abstract

We introduce an exactly solvable mixed spinsp"and arbitrary spinsS (S') Ising ferrimagnet. Although the system with only antiferromagnetic interactionJ

between pairs of nearest-neighbor spins pandS and crystal-"eld interaction D does not exhibit a compensation point, the system with additional interactions between pairs of nearest-neighbor spinsp, does, in agreement with Buendia and Novotny's conjecture. Finally, we discuss the origin of compensation phenomenon. 2000 Elsevier Science B.V. All rights reserved.

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

Keywords: Compensation point; Ising model; Exact solution; Ferrimagnetism

1. Introduction

Recently, there has been great interest in stable crystalline room-temperature magnets with spon- taneous moments because of their potential device application in thermomagnetic recording, elec- tronic and computer technologies [1]. In these materials, ferrimagnetic ordering is of fundamen- tal relevance. Important advances have been made in the synthesis of two and three-dimensional (3d) ferrimagnets, such as 2d organometallic ferri- magnets [2,3], 2d networks of the mixed-metal material

+

[P(Phenyl)

][MnCr(oxalate)

]

,L

[4], the amorphous V(TCNE)

V)y(solvent) [5], N(n-

C

L

H

L>

)

Fe

''

Fe

'''

(C O

)

with

n

" 3}5 [6], and

*Corresponding author.

ferrimagnetic amorphous oxides containing Fe

>

ions [7,8]. Intensive e!ort is required in the theoret- ical study of these materials in order to clarify their very interesting and sometimes unusual behaviors.

Ferrimagnets consist of several sublattices with inequivalent moments interacting antiferromag- netically. Under certain conditions, the sublattice magnetizations compensate each other, then the resultant magnetization vanishes at a compensa- tion temperature ¹

!

below the critical temper- ature ¹

!

[9]. The occurrence of a compensation point is of great technological importance, since at this point only a small driving

"eld is required to

change the sign of the resultant magnetization. This property is very useful in thermomagnetic record- ing.

Mixed-spin Ising systems provide good models to study ferrimagnetism. The magnetic properties

PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 6 0 6 - X

(2)

Fig. 1. The two-dimensional mixed-spin ferrimagnetic Ising system consisting of two kinds of magnetic atoms A (z) and B (䢇) with spin valuesp"andS "S(S'), respectively.

of these systems have been investigated by various methods. In particular, the mixed spin- and spin-1 Ising model, has been studied by the mean-"eld approximation [10], free-fermion approximation [11], e!ective-"eld theory [12,13],

"nite-cluster ap-

proximation [14], high-temperature series expan- sions [15], renormalization group technique [16,17], and Monte-Carlo simulation [18}20].

Mixed-spin Ising systems are of interest because they have less translational symmetry than their single-spin counterparts since they consist of two interpenetrating inequivalent sublattices. The latter property has a great in#uence on the magnetic properties of the mixed-spin systems and causes them to exhibit unusual behavior not observed in single-spin Ising models [21}24]. Recently, Buen- dia and Novotny considered a classical mixed-spin Ising ferrimagnetic system on a square-lattice [20].

Their model has two interpenetrating square sub- lattices, one with spins

p

" and the other with spins

S

" 1, in which pairs of nearest-neighbor spins

p

and

S

are coupled with antiferromagnetic coupling

J

. Using Monte-Carlo and numerical transfer-matrix techniques, they showed that com- pensation points are induced by the presence of an interaction

J

between the spins

p

(next-nearest neighbors in the lattice). They found that the sys- tem with only nearest-neighbor interaction

J

and crystal-"eld interaction

D

does not have any non- zero compensation temperature. The latter result is in contradiction with those obtained from mean-"eld [10] and e!ective-"eld theory [13].

The con#icting results of numerical simulation and approximate methods have animated a heated debate as to the existence of a compensation tem- perature in two-dimensional pure ferrimagnetic systems. Very recently, the exact solution of a mixed spin- and spin-1 ferrimagnetic Ising model on Bethe lattice has been obtained by Tucker [25]. The author showed, in particular, that a compensation point is possible in a Bethe lattice with coordination number

z

* 4. This compensa- tion point is induced by the crystal

"eld interaction.

The latter fact suggests that Buendia and Novotny's conjecture [20] is not veri"ed for the Bethe lattice. Our goal is to give an answer to the still unresolved question of whether a compensa- tion point is possible or not in some two-dimen-

sional mixed-spin Ising ferrimagnets with only anti- ferromagnetic interaction

J

between pairs of near- est-neighbor spins

p

and

S

and a crystal-"eld interaction

D

. We will also verify whether interac- tion between pairs of nearest-neighbor spins

p

, does really induce a compensation temperature in these systems as predicted by Buendia and Novotnys conjecture [20]. However our study is more general than Ref. [20] since we will consider arbitrary spins

S

' . To get a convincing answer we introduce an exactly solvable mixed spin

p

" and spin

S

(S ' ) Ising ferrimagnet.

This paper is organized as follows. The model is de"ned in Section 2 and the exact equations for the critical temperature, compensation temperature and magnetization are derived. Numerical results are presented in Section 3, where the occurrence of a compensation temperature is discussed for both integer and half-integer spin

S. In Section 4, some

conclusions are drawn.

2. Formulation of the model and its exact solution

The mixed-spin ferrimagnetic Ising system, we

are interested in, consists of two-dimensional sub-

lattices

L

and

L

as depicted in Fig. 1. The sites

of

L

are occupied by A atoms with the

"xed spin p

" . The sublattice

L

is occupied by B atoms

with the spin

S

' . Our lattice

L6L

has been

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Fig. 2. Ground-state phase diagram of the mixed spin- and spin-1 ferrimagnetic Ising system.

Fig. 3. Ground-state phase diagram of the mixed spin-and spin-ferrimagnetic Ising system.

constructed from a honeycomb lattice

L

where we added bonds between

p

spins in order to test Buendia and Novotny's conjecture.

Note that only magnetic atoms are represented.

Indeed there can exist non-magnetic atoms which are periodically localized on the lattice. The system is described by the following Hamiltonian

H

"

j

6GL7pXGSXL

!

J

6GH7pXGpXH

!

D L

(S

XL

) , (1) where the

"rst two summations are carried out only

over nearest-neighbor pairs of spins.

pXG

is the

z-

component of the spin of an A atom and takes the values $ . The spin component

SXL

, corresponds to a B atom, and takes one of the (2S # 1) allowed values. In the following, we will drop the subscript

z.J

(J

' 0) and

J

are the exchange interactions and

D

is the crystal-"eld interaction acting on B atoms.

The ground-state structures of our system can be easily found by comparing the energies of the corre- sponding con"gurations. The results are shown in Figs. 2 and 3 for an integer spin

S

" 1 and a half- integer spin

S

" , respectively. The ground states are denoted by (S ;

pG

,

pH

,

pI

) where

pG

,

pH

and

pI

are the nearest-neighboring spins of a spin

S

.

Note that only typical con"gurations are shown in Figs. 2 and 3. But, due to the symmetries of Hamil- tonian (1), other con"gurations coexist with those shown in the ground-state phase diagrams. Indeed, one can obtain other con"gurations which have the same ground-state energies by permuting two spins

pG

and

pH

of opposite signs. For example states ( ! 1; , ! , ) have the same energy as ( ! 1; , , ! ). Also, con"gurations like ( !

S

;

!

pG

, !

pH

, !

pI

) are equivalent to (S

;

pG

,

pH

,

pI

), since the Hamiltonian (1) is invariant under reversal of all spins. The boundaries between the regions are obtained by pairwise equating the ground-state energies. The partition function is given by

Z

"

+N1^,

exp( !

bH),

where

b

is (k ¹ )\. This function can be evaluated by decimation. For this purpose we perform the partial trace over all the spins

S

L

, to give

Z

"

+N^,X

(

p

)e

)

6GH7N^GN^H

, (2)

(4)

where

X

(

p

) is given by

X

(

p

) "

+1^,

e

\)

6GL7N^G1^L>"_L Q^L

"

,

LuL

(

pG

,

pH

,

pI

) (3)

and

uL

is the weight function de"ned by

uL

(

pG

,

pH

,

pI

) "

>1

K\1

e

"K

cosh(K

m(pG

#

pH

#

pI

)), (4) where the sum should be performed over the (2S # 1) possible values of a spin

S

L

. For each site

n, uL

(

pG

,

pH

,

pI

) is a function of the nearest-neighbor spins

pG

,

pH

and

pI

of

S

L

, and

K

"

bJ

,

K

"

bJ

and

D

"

bD

. It is easy to verify that Eq. (4) can be written as

uL

(

pG

,

pH

,

pI

) " e

E>)N^GN^H>N^HN^I>N^IN^G

, (5) with

K

" ln

^u^uL

(

^L

! ⁽

^,

, ^,

, ⁾ ) ⁽⁶⁾

and

g

" ln[

uL

( , , )[

uL

( ! , , )] ].

Note that

g

is an analytic function.

On substituting into Eq. (2) Eq. (3) with

uL

(

pG

,

pH

,

pI

) replaced from Eq. (5), one can show that the partition function

Z

is related to that of a spin- Ising model (

p

" $ ) on triangular lattice with an e!ective exchange interaction

K'

given by

K'

"

K

#

K

, (7)

through the relation

Z

" e

,EZ'

. (8)

Here

N

is the total number of A atoms. From Eq. (8) we can get all the thermodynamic properties since the partition function

Z

'

of the spin- Ising model on triangular lattice is known exactly (See for example Ref. [26]). Eq. (8) shows that any ther- modynamic function of the model under study can

be related to the corresponding function of the spin- Ising model.

2.1. Sublattice and total magnetizations

Let us

"rst consider the sublattice magnetization mN

de"ned by

mN

"

1pG2

" 1

Z₊

N1^,pG

e

\@&

,

where the angular brackets denote the usual ther- mal average. Decimating the spins

S, following the

same procedure as for the partition function, and taking account of Eq. (8), one can easily show that

mN

"

1pG2'

, (9)

where

1pG2'

is the magnetization/site of the spin- Ising model on triangular lattice with an e!ective exchange interaction

K

'

, Eq. (7) [26].

To calculate the sublattice magnetization

m 1

of a spin

S, we should proceed di!erently. By de"ni-

tion

m1

"

1S

L2

"

^>1^>1^K\1K\1^m

e

⁾\)

^e

^\)KN^KN^G>N^G^>N^H>N^H^>N^I>"K^I^>"K

^.

The function between brackets can be written as a linear combination of the variables

pG

,

pH

and

pI

. If this is done one gets

m1

" 3a

1pG2'

#

b1pGpHpI2'

. (10) When deriving Eq. (10), account has been taken of

1pG2

"

1pH2

"

1pI2

"

1pG2'

, since the system is translationally invariant so that the magnetization should be site independent. The coe$cients a and b are given by

a

" [h( , , ) #

h(

! , , )],

b

" 2[h( , , ) ! 3h( ! , , )], with

h(pG

,

pH

,

pI

)

"!

>1K\1me"K

sinh(K

m(pG

#

pH

#

pI

))

>1K\1

e

"K

cosh(K

m(pG

#

pH

#

pI

)) .

It has been shown that the three-spin correlation

function

1pGpHpI2'

of the spin- triangular Ising

(5)

model is equal to the corresponding function of the honeycomb lattice model which is known exactly [27]. However for our purpose, we shall express

1pGpHpI2'

in terms of the interaction parameter

K'

of the triangular lattice. Thus we obtain

1pGpHpI2'

" 1 ! 3u

v 1pG2'

, (11) where the coe$cients

u

and

v

are given by

u

" 1 # e

\)^'

2 ^e ^e

⁾⁾^'^'

^! ^# ¹ ³ ^,

v

" 2(e

\)^'

! 1) ^e ^e

⁾⁾^'^'

^! ^# ¹ ³ ^.

The total magnetization/spin of the system at any temperature is the sum of contributions from the sublattices and it is de"ned by

M

"

m N

#

m

2

1

. (12)

Using Eqs. (9) and (10)}(12),

M

can be expressed as

M

" ¹ ^# ^3a ^# ¹ ^!

^v

^3u

^b

^1p^G²^'

^. ⁽¹³⁾

Since the explicit expression for

1pG2'

is known exactly [26], we obtain the following explicit for- mulas for the sublattice magnetizations

mN

and

m1

mN

" (1 !

k

) , (14)

m1

" ^3a ^# ¹ ^!

^v

^3u

^b

⁽¹ ^!

^k

⁾

^, ⁽¹⁵⁾

where

k

is given by

k

" (1 !

t

)

16t (1 #

t

)(1 #

t)

with

t

" tanh(K

'

/4).The explicit expression for the total magnetization/spin follows from Eqs. (12), (14) and (15).

The compensation temperature ¹

!

of a fer- rimagnetic system is a temperature at which the total magnetization/spin of the system vanishes be- low the critical temperature ¹

!

. According to

Eq. (13), the compensation temperature ¹

!

is a solution of the following equation

1 # 3a # 1 ! 3u

v b

" 0, (16)

such that ¹

!

(¹

!

.

From the exact solution of the partition function Eq. (8), we can easily get an explicit expression for the speci"c heat of our model. If this is done we can show that the speci"c heat is logarithmically diver- gent at the second-order phase transition line as in the case of the speci"c heat of the spin- Ising model [26]. Furthermore from Eqs. (14) and (15), we see that the critical exponent

b

is equal to , i.e.

the same as for the spin- Ising model [26]. In fact, our model belongs to the same universality class of the spin- Ising model. The exact solution of the present model with

J

" 0 and

S

" 1 have been recently given in Ref. [25]. The results are in accord with the present calculation.

3. Results and discussions 3.1. Phase diagrams

Since the model is related to the spin- Ising model for any value of the parameters, it undergoes a second-order transition when

K

,

a

"

J

/J

and

d

"

D

/J

are solutions of the Eq. (7) with

K'

"

K

'!

" ln(3), i.e. the critical frontier equation

aK

#

K

"

K

'!

, (17) where

K

given by Eq. (6) and

K

'!

" ln(3) is the critical value of the spin- triangular Ising model [26]. We solved numerically the Eq. (17) for di!er- ent values of the spin

S.

In Figs. 4 and 5, we show, for di!erent values of

a

,

the phase diagrams for a spin

S

" 1 and a spin

S

" , respectively. For both integer and half-inte-

ger spins, and

a!

)

a

)

a!

, there is no long-range

order for a certain region of

d

(

d!

(For example,

d

(

d!

"! 1.5 for

S

" 1 and

a

" 0, and

d

(

d!

"! 0.75 for

S

" and

a

"! 0.5). The

critical value

d!

depends on the values of

a

and

S,

and it is given for

S

" 1 by

d!

"! (

a

# ) and for

S

" by

d!

"! (

a

# 1), (for

a!

)

a

)

a!

). For

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Fig. 4. The phase diagram of the mixed spin- and spin-1 ferrimagnetic Ising system for di!erent values ofa"J

/J .

Fig. 5. The phase diagram of the mixed spin-and spin-ferrimagnetic Ising system for di!erent values ofa"J/J.

a

'

a!

, the ordered state exists at low temperatures for all values of

d

while for

a

(

a!

, the system is paramagnetic for all values of the temperature and

d

. The critical values

a!

and

a!

are spin-dependent, for example (

a!

,

a!

) " ( ! 1, 0) and ( ! 1.5, ! 0.5) for

S

" 1 and

S

" , respectively. The values of

a!

,

a!

and

d!

may be obtained from the analysis of the ground state phase diagram (see for example, Figs. 2 and 3 corresponding to

S

" 1 and

S

" , respectively). Note that the phases (m, , , ! ), with

m

" 0 and ! 1 for the spin

S

" 1 and

m

"! and ! for the spin

S

" , do not support any kind of long-range order while the phases (m, , , ) with the above values of

m, do support

ferrimagnetic order.

If we compare integer and half-integer phase diagrams, we see that they are qualitatively similar except that

a!

is negative for systems with half- integer spins

S. In particular, fora

" 0 the system with half-integer spins

S

is magnetically ordered at low temperatures for all values of

d

, while for the system with integer spins

S, there is a lower bound

of

d

, below which there is no magnetic order. So, we conclude that the critical behavior of mixed-spin Ising models may be qualitatively di!erent for inte- ger and half-integer spins for certain values of the interactions. This property has been found in a ran- dom bond Ising and a decorated ferrimagnetic Ising systems [21,24].

3.2. Compensation point

A compensation point corresponds to the tem- perature ¹

!

below the transition temperature such that the sublattice magnetizations compensate each other and the total magnetization

M

(Eq. (13)) vanishes. For given values of the parameters

a

and

d

, ¹

!

is solution of the Eq. (16). In the following, the line joining the solutions of Eq. (16) in the (K

\

,

d

) plane for a given value of

a

will be called the compensation point line.

In Fig. 6, we report the compensation point lines

for

a

" 4 and di!erent values of

S, namely S

" 1,

, 2 and . For this value of

a

the compensation

point line terminates at the corresponding critical

temperature for

S

" , 2 and , but extends to

d

"R for

S

" 1. At su$ciently higher values of

a

,

the lines for other values of

S

extend to R , whilst at

a su$ciently low value of

a

the line for

S

" 1 will

stop at the appropriate critical temperature. For

a given value of

a

we see from Fig. 6 that systems

with small values of

S

may exhibit compensation

points in a large region of

d

. We noticed that

the systems with greater values of

a

exhibit good

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Fig. 6. The compensation point lines of the mixed spin-and spin SIsing ferrimagnetic system for integer and half-integer spinsS, namelyS"1,, 2 and, whena"4.

Fig. 7. The temperature dependencies of the total magnetization Mfor the two-dimensional mixed spin-and spin-1 Ising ferrimagnetic system fora"4 and di!erent values ofd"D/J

, namelyd"!1.25,!1,0 and 1. The system exhibits compensation points for the assumed values ofaandd.

Fig. 8. The temperature dependencies of the total magnetization Mfor the two-dimensional mixed spin-and spin-Ising ferrimagnetic system fora"4 and di!erent values ofd"D/J

, namelyd"!1.25,!1,!0.75,!0.5 and 0. The system exhibits compensation points for the assumed values ofaandd.

compensation points, since in these cases the com- pensation points exist for positive values of

d

, for which the system is strongly ordered so the abso- lute value of the total magnetization is big. But when

a

is too small, compensation points are pos- sible for small values of

d

. In this case, the system is weakly ordered and the total magnetization re- mains too small. Note that the compensation tem- perature is almost insensitive to

a

as long as

d

is small enough. When

a

" 0 (J

" 0) the current mixed spin- and arbitrary spin

S

Ising ferrimag- netic systems do not present any compensation point in agreement with Buendia and Novotny's conjecture [20] i.e., compensation points are in- duced by the presence of interactions between the spins

p

(next-nearest neighbors in the lattice) and system with only nearest-neighbor interaction

J

and crystal-"eld interaction

D

does not have any non-zero compensation temperature. Furthermore, the existence of compensation points is possible only in the ferrimagnetic phase. In the ground state this corresponds to the phases (S, ! , ! , ! ) and ( !

S,

, , ) for half-integer spins

S

and to the same phases with

S

O 0 for integer spins

S.

The temperature dependencies of the total mag- netization

M

for the two-dimensional mixed spin- and spin

S

Ising ferrimagnetic system, for

a

" 4 and di!erent values of

d

"

D

/J

, are represented in Figs. 7 and 8 for

S

" 1 and

S

" , respectively.

These

"gures shows clearly the existence of com-

pensation points for the assumed values of

a

and

d

.

3.3. Origin of compensation phenomenon

Now, let us discuss the mechanism of the com- pensation phenomenon in the present system along the lines presented in Ref. [20] for the decorated square lattice. In the region where the system can exhibit a compensation point, the sublattice

L

is more ordered than the sublattice

L

below ¹

!

, i.e.

"m

1"

'

"m

N"

(see Figs. 9 and 10). These sublattice

magnetizations have opposite signs but the cancel-

lation is still incomplete so there is a residual mag-

netization in the system (M O 0). This is due to the

antiferromagnetic nearest-neighbor interaction

J

which tends to align neighboring spins in oppo-

site directions. As the system is heated up, the

direction of this residual magnetization can switch.

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Fig. 9. The temperature dependence of the sublattice magneti- zationsm

1andm

Nfor the two-dimensional mixed spin-and spin-1 Ising ferrimagnetic system fora"4 andd"!1.

Fig. 10. The temperature dependence of the sublattice magne- tizationsm

1andm

Nfor the two-dimensional mixed spin-and spin-Ising ferrimagnetic system fora"4 andd"!1.

Indeed, due to the entropy e!ect some spins can

#ip

their directions. However, if the ferromagnetic in- teraction

J

between the spins

pG

, which tends to align them in the same direction, is strong enough, it is only the spins

S

that are susceptible to

#ip.

Thus, the sublattice

L

becomes more ordered than the sublattice

L

for temperatures above

¹ , i.e.

"m 1"

(

"m

N"

. So there is an intermediate temperature ¹

!

such that the cancellation is complete (m

1

"!

m

N

and

M

" 0). When the ther- mal agitation overwhelms the magnetic interac- tions the system loses magnetism at ¹

!

. Note, that even for zero crystal-"eld

D

" 0, the system may exhibit a compensation point while this is not pos-

sible for

J

" 0. So we can conclude that

J

plays a relevant role in the appearance of the compensa- tion phenomenon while

D

in#uences the location of the compensation temperature.

Owing to these facts, we can say that in order for compensation to exist there must exist at least two relevant interactions of di!erent natures, one favor- ing ferrimagnetism and another one favoring bal- ance in the magnetic order between the sublattices.

In our system this corresponds to

J

and

J

, re- spectively. The crystal-"eld interaction does not induce any compensation point since it does not balance the magnetic order from one sublattice to the other. In fact, for

a

" 0, negative values of

D

only reduce the magnetic order in

L

-sublattice without increasing that of

L

-sublattice, and for integer spins

S

it favors paramagnetism (see phase diagram of Fig. 4).

In a previous study [23], we showed that a fer- rimagnetic surface, with nearest-neighbor interac- tion

J

and crystal-"eld interaction, interacting with a ferromagnetic substrate (bulk) may exhibit compensation phenomenon. In this case the bulk which acts on the surface as a temperature-depen- dent magnetic

"eld plays the role of the interaction J

.

4. Conclusions

In this paper we introduced an exactly solvable mixed spins

p

" and arbitrary spins

S

(S ' ) Ising ferrimagnet. We showed that the current sys- tem with only aniferromagnetic interaction

J

be- tween pairs of nearest-neighbor spins

p

and

S

and crystal-"eld interaction

D

does not exhibit any compensation point. However, with interaction be- tween pairs of nearest-neighbor

p

included, a com- pensation temperature is induced in the system, in line with Buendia and Novotny's conjecture [20].

For the lattice structure considered here, we showed that in order for a compensation phenom- enon to exist there must exist at least two relevant interactions of di!erent character, one favoring fer- rimagnetism (J

) and another (J

) favoring balance

in the magnetic order between the sublattices. The

crystal-"eld is not mainly responsible for the ap-

pearance of the compensation phenomenon but

(9)

may in#uence the existence and location of the compensation temperature. Monte-Carlo simula- tion results [20] agree with ours but those pre- dicted by mean-"eld and e!ective-"eld theories are not correct since they predict that a compensation phenomenon is possible even in the absence of the interaction (J

), and that the crystal-"eld interac- tion, by itself, may induce such a phenomenon [10,13]. This incorrect result arises from the de- coupling approximations used for treating the cor- relations between spins in those theories. Let us recall, however, that crystal-"eld interactions, by themselves, can induce a compensation point in the Bethe lattice [25]. Consequently, an important question arises: is the latter result peculiar to the Bethe lattice or a general feature for a certain class of lattice structures including Bethe lattice struc- ture?

In a previous paper one of us [24], used an exactly solvable two-dimensional decorated mixed- spin ferrimagnetic Ising system to show the possibility of two successive compensation temper- atures. This outstanding result which is beyond the Ne

H

el theory has been discovered experimentally in ( N i

''

M n

''

F e

''

) [ C r

'''

( C N ) ]

)

7 . 6 H O [28]. The theoretical prediction of the possibility of multiple compensation temperatures [24] and the recent design and preparation of materials with such unusual behavior [28] will certainly open a new area of research on such materials, so we hope that our present and previous works will stimulate further investigations of ferrimagnetic systems.

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