Thermodynamical properties of the mixed spin Blume–Capel model with four-spin interactions

Thermodynamical properties of the mixed spin Blume – Capel model with four-spin interactions

N. Benayad ⁿ , M. Ghliyem

Laboratory of High Energy Physics and Scientific Computing, HASSAN II University—Casablanca, Faculty of Sciences Aïn Chock, B.P: 5366 Maarif, Casablanca 20100, Morocco

a r t i c l e i n f o

Article history:

Received 30 March 2012 Received in revised form 28 March 2013

Available online 29 April 2013 Keywords:

Mixed spin Four-spin interactions Crystal field interaction Thermodynamical properties

a b s t r a c t

A mixed spin Blume–Capel model consisting of spin-1/2 and spin-1 with four-spin interactions and crystal field interaction is studied by the use of the finite cluster approximation based on a single-site cluster theory. In this approach, the state equations are derived for the two dimensional square lattice. In addition to the phase diagrams and magnetizations which show a variety of interesting features, the influence of the four-spin interaction on the internal energy, specific heat and zero-field magnetic susceptibility is also examined.

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1. Introduction

The presence of higher-order spin couplings in Ising models has been received special attention both theoretically and experi- mentally. The origin of such interactions found its theoretical explanation in the theories of the superexchange interaction, the magnetoelastic effect, the perturbation expansion, and the spin- phonon coupling [1]. The increasing interest in investigating models with higher-order interactions arises from the fact that, on the one hand, they may exhibit rich phase diagrams and can describe phase transitions in some physical systems. On the other hand, they show physical behaviours not observed in the usual spin systems. For instance, they display the nonuniversal critical phenomena [2,3].

From the theoretical point of view, monoatomic Ising models with multispin interactions have been studied within different methods, such as mean fi eld approximation [4,5], effective fi eld theory [6 – 8], series expansions [9,10], renormalization group methods [11], Monte Carlo simulations [12] and exact calculations [13]. Experimentally, the models with multipsin interactions can be used to describe different physical systems such as classical fl uids [14], solid

³

He [15], lipid bilayers [16], metamagnets [17] and rare gases [18]. Moreover, it has been shown that for certain materials, these interactions play a signi fi cant role and they are comparable or even much important than the bilinear ones.

Indeed, the models with pair and quartet interactions have been used to study and explain the existence of fi rst-order phase transition in squaric acid crystal H

2

C

2

O

4

[19]. Such models have been also applied to describe thermodynamical properties of hydrogen-bonded ferroelectrics PbHPO

4

and PbDPO

4

[20], copoly- mers [21], and optical conductivity [22] observed in the cuprate ladder La

x

Ca

_14−x

Cu

24

O

41

. It is worthy to note here that the four spin interaction plays an important role in the two-dimensional antiferromagnet La

₂

CuO

₄

[23], the parent material of high-T

_c

superconductors.

Another problem of growing interest is associated with the magnetic properties of two-sublattice mixed spin Ising systems.

Their investigations are important since they have less transla- tional symmetry than their single-spin counterparts, and they are well adopted to study a certain type of ferrimagnetism [24].

Experimentally, it has been shown that the MnNi(EDTA) – 6H

2

O complex is an example of a mixed spin system [25]. The mixed Ising spin model consisting of spin-1/2 and spin-1 with only two- bilinear interaction has been studied by the renormalization group technique [26], by high temperature series expansions [27], by free-fermion approximation [28] and by fi nite cluster approxima- tion [29]. The in fl uence of the crystal- fi eld interaction on the phase transition has been also investigated using exact calcula- tions (for a honeycomb lattice) [30], mean fi eld approximation [31], fi nite cluster approximation [32], the Bethe lattice solution [33], Monte Carlo simulation [34] and renormalization group method [35] both in two and three dimensions. The presence of multispin interactions in these systems certainly modi fi es their magnetic properties. Indeed, in the case of four-spin interaction, we have shown, in previous works [36,37], that this kind of Contents lists available at SciVerse ScienceDirect

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Journal of Magnetism and Magnetic Materials

0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved.

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n

Corresponding author. Tel.: +212 5 22 23 06 84; fax: +212 5 22 23 06 74.

E-mail addresses: n.benayad@fsac.ac.ma,

noureddine_benayad@yahoo.fr (N. Benayad).

7 1. The fi rst summation is carried out only over nearest- neighbour pair of spins. The second term represents the four- spin interaction, where the summation is over all alternate squares, shaded in Fig. 1. The parameter D describes the uniform crystal fi eld interaction. To this end, we use the fi nite cluster approximation [38,39] within the framework of a single-site cluster theory. The second goal is to examine the internal energy, speci fi c heat and zero- fi eld magnetic susceptibility of the system.

The outline of this paper is as follows: In Section 2, we describe the theoretical framework and calculate the state equations. In Section 3, we investigate and discuss the magnetic properties of the system. In Section 4 other relevant thermodynamical quan- tities are presented and analysed. Finally, our concluding remarks are given in Section 5.

2. Theoretical framework

The theoretical framework that we adopt in the study of the mixed spin-1/2 and spin-1 Ising model with four-spin interaction and crystal fi eld interaction described by the Hamiltonian (1) is the fi nite cluster approximation (FCA) [38,39] based on a single- site cluster theory. We have to mention that this method has been successfully applied to a number of interesting pure and disor- dered spin Ising systems [32,40 – 42]. It has also been used for transverse Ising models [43 – 45] and semi-in fi nite Ising systems [46 – 49]. In all these applications, it was shown that the FCA improves qualitatively and quantitatively the results obtained in the frame of the mean- fi eld theory. In this approach, attention is focused on a cluster comprising just a single selected spin s

0

(S

0

) and its neighbour spins { s

1

, s

2

,S

1

,S

2

,S

3

,S

4

} ({S

1

,S

2

, s

1

, s

2

, s

3

, s

4

}) with which it directly interacts (see Fig. 2).

fi s

and S

j

(i, j ≠ 0) are kept fi xed). 〈s

0

〉

c

and 〈 S

0n

〉

c

are given by

〈s 0 〉 c ¼ Tr

s

0

s 0 expð −β H _0s Þ Tr s

0

expð −β H 0 s Þ ð4Þ

〈 S ⁿ ₀ 〉 ^c ¼ Tr s

0

S ⁿ ₀ expð −β H 0s Þ Tr s

0

expð −β H 0s Þ ð5Þ

where Tr s

0

(or TrS

0

) means the trace performed over s

0

(or S

0

) only. As usual ß ¼1/T where T is the absolute temperature. The sublattice magnetizations m , m and the quadrupolar moment q are then given by

μ≡〈〈s 0 〉 c 〉 ¼ Tr

s

0

s 0 expð −β H _0s Þ Tr 0s expð −β H 0 s Þ

* +

ð6Þ

m ≡〈〈 S 0 〉 c 〉 ¼

Tr s

₀

S 0 expð −β H 0s Þ Tr s

₀

expð −β H 0s Þ

* +

ð7Þ

q ≡〈〈 S ² ₀ 〉 c 〉 ¼

Tr s

₀

S ² ₀ expð −β H 0s Þ Tr s

₀

expð −β H 0s Þ

* +

ð8Þ

where 〈y〉 denotes the average over all spin con fi gurations.

Performing the inner traces in (6) – (8) over the states of the selected spin s

0

(S

0

), we obtain the following exact relations μ ¼ 1

2 tanh K

2 fðS 1 þ S 2 þ S 3 þ S 4 Þ þ α ðS 1 S 2 s 1 þ S 3 S 4 s 2 Þg

ð9Þ

m ¼ 2sinh½Kfð s 1 þ s 2 þ s 3 þ s 4 Þ þ α ð s 1 s 2 S ₁ þ s 3 s 4 S ₂ Þg expð β DÞ þ 2cosh½Kfð s 1 þ s 2 þ s 3 þ s 4 Þ þ α ð s 1 s 2 S ₁ þ s 3 s 4 S ₂ Þg

ð10Þ

q ¼ 2cosh½K fð s 1 þ s 2 þ s 3 þ s 4 Þ þ α ð s 1 s 2 S 1 þ s 3 s 4 S 2 Þg expð βDÞ þ 2cosh½K fð s 1 þ s 2 þ s 3 þ s 4 Þ þ α ð s 1 s 2 S 1 þ s 3 s 4 S 2 Þg

ð11Þ where K¼ßJ

2

and α ¼J

4

/J

2

.

In the frame of mean fi eld approximation, Eqs. (9) – (11) reduce to μ ¼ 1

2 tanh Kð2m þ αμ m ² Þ

ð12Þ

m ¼ 2sinh½Kð4 μ þ 2 α m μ ² Þ

expð β DÞ þ 2cosh½Kð4 μ þ 2 α m μ ² Þ ð13Þ

q ¼ 2cosh½Kð4 μ þ 2 α m μ ² Þ

expð β DÞ þ 2cosh½Kð4 μ þ 2 α m μ ² Þ ð14Þ To calculate the average on the right-hand sides of Eqs. (9) – (11) over all spin con fi gurations, we use the procedure described in our Fig. 1. Part of the square lattice. and X correspond to s and S-sublattice sites,

respectively.

previous work [36]. This leads to the following coupled equations.

μ ¼ μ 2A 1 q ² þ 4A 2 q ³ þ 2A 3 q ⁴ þ m½4A 4 þ 4A 5 q þ 4A 6 q ² þ4A 7 q ³ þ m μ ² ½4A 8 q ³ þ μ m ² ½2A 9 þ 4A 10 q þ 4A 11 q ²

þm ³ ½4A 12 þ 4A 13 q þ μ m ⁴ ½2A 14 þ μ ² m ³ ½4A 15 q ð15Þ m ¼ μ ½4B 1 þ 8B 2 q þ 4B 3 q ² þ m½2B 4 þ 2B 5 q

þ μ ³ ½4B 6 þ 8B 7 q þ 4B 8 q ² þ m μ ² ½2B 9 þ 2B 10 q þ μ m ² ½4B 11 þm μ ⁴ ½2B 12 þ 2B 13 q þ m ² μ ³ ½4B 14 ð16Þ q ¼ ½C 1 þ 2C 2 q þ C 3 q ² þ μ ² ½6C 4 þ 12C 5 q þ 6C 6 q ²

þm ² ½C 7 þ μ m½4C 8 þ 4C 9 q þ μ ⁴ ½C 10 þ 2C 11 q þ C 12 q ²

þm μ ³ ½4C 13 þ 4C 14 q þ μ ² m ² ½6C 15 þ m ² μ ⁴ ½C 16 ð17Þ Expressions for non-zero coef fi cients quoted in Eqs. (15) – (17) can be taken from Appendix in [36], where only the de fi nition of the functions g(x) and h(x) must be changed, respectively, to gðxÞ ≡ 2sinhðKxÞ

expð β DÞ þ 2coshðKxÞ hðxÞ ≡ 2coshðKxÞ

expð β DÞ þ 2coshðKxÞ

After some algebraic manipulations (see [36]), we obtain an equation for μ of the form

μ ¼ a μ þ b μ ³ þ ⋯ ð18Þ

a and b are given by.

a ¼ 2A 1 q ² ₀ þ 4A 2 q ³ ₀ þ 2A 3 q ⁴ ₀

þ ð4B 1 þ 8B 2 q ₀ þ 4B 3 q ² ₀ Þð4A 4 þ 4A 5 q ₀ þ 4A 6 q ² ₀ þ 4A 7 q ³ ₀ Þ 1 − ð2B 4 þ 2B 5 q ₀ Þ

b ¼ 4A 1 q ₀ q ₄ þ 12A 2 q ₄ q ² ₀ þ 8A 3 q ₄ q ³ ₀ þ Að4A ₅ q ₄ þ 8A 6 q ₄ q ₀ þ 12A 7 q ₄ q ² ₀ Þ B

þ 4A 8 q ³ ₀ A B þ EA ²

B ² þ FA ³

B ³ þ ð4A 4 þ 4A 5 q 0 þ 4A 6 q ² ₀ þ 4A 7 q ³ ₀ Þ 8B 2 q ₄ þ 8B 3 q ₀ q ₄ þ C

B þ AD þ 2AB 5 q ₄

B ² þ 4B 11 A ² B ³

" #

where q

₀

is the solution of q ₀ ¼ C 1 þ 2C 2 q ₀ þ C 3 q ³ ₀

and

A ¼ 4B 1 þ 8B 2 q _o þ 4B 3 q ² ₀ ; B ¼ 1 ð2B 4 þ 2B 5 q ₀ Þ C ¼ 4B 6 þ 8B 7 q ₀ þ 4B 8 q ² ₀ ; D ¼ 2B 9 þ 2B 10 q ₀ E ¼ 2A 9 þ 4A 10 q ₀ þ 2A 11 q ² ₀ ; F ¼ 4A 12 þ 4A 13 q ₀

q ₄ ¼ q ₁ þ A B 2

q ₂ þ A B

q ₃

with

q ₁ ¼ 6C 4 þ 12C 5 q 0 þ 6C 6 q ² ₀

1 − 2C 2 − 2C 3 q ₀ ; q ₂ ¼ C 7

1 − 2C 2 − 2C 3 q ₀ ; q3 ¼ 4C 8 þ 4C 9 q ₀

1 − 2C 2 − 2C 3 q ₀

In the vicinity of the second-order transition, the sublattice magnetization m is given by

μ ² ¼ 1 − a

b ð19Þ

The right-hand side of (19) is positive since we are in the long- ranged ordered regime. This means that the signs of 1 − a and b are the same.

When b changes sign (1 − a keeping its sign), m

²

becomes negative. It means that we are not in the vicinity of a second- order transition line. So, the transition is of the fi rst order.

Therefore the point at which

aðK ; α; DÞ ¼ 1 and bðK ; α; DÞ ¼ 0 ð20Þ characterizes the tricritical points.

3. Magnetic properties

Let us fi rst plot the phase diagram of the system, described by the Hamiltonian (1), in the absence of the crystal fi eld (D¼0). In this case, the system reduces to two sublattice mixed spin-1/2 and spin-1 Ising model with four-spin interaction.

Fig. 3 represents the variation of the critical temperature T

_c

with α ¼J

4

/J

2

by solving (1 ¼a) as well as the tricritical condition (20) numerically. In this fi gure, the solid line represents the second-order transition which ends in a tricritical point (black point). This latter is found at ð α t ¼ ðJ ₄ = J ₂ Þ ¼ 1 : 114 ; ðT c = J ₂ Þ ¼ 1 : 196Þ which should be compared with the mean fi eld result

ð α ^t ¼ ðJ 4 = J 2 Þ ¼ 1 : 428 ; ðT c = J 2 Þ ¼ 1 : 633Þ obtained by using Eqs. (12) and (13). We note that the FCA gives better results than the mean- fi eld theory, since the former neglects only correlations between quantities pertaining to the different sites, and takes exactly into account all spin self-correlations; while the mean- fi eld uses a very crude approximation which neglects all spin correlations. It is worthy to note here that the phase diagram plotted in Fig. 3 is qualitatively similar to that predicted recently by Monte Carlo simulation [50].

Fig. 2. (a) Neighbours of spin s

0

with which it directly interacts. (b) Neighbours of spin S

0

with which it directly interacts.

Now, let us present the effects of the four-spin interaction J

4

on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with uniform crystal fi eld interaction, discussed previously by one of us (N.B) in [32]. In this case, we have found (see Fig. 4(a) for J

4

¼0) that the transition temperature decreases with increasing values of the crystal fi eld D and ends in a tricritical point. The results of the in fl uence of J

4

are summarized in Fig. 4. These give the sections of the critical surface T

c

(J

4

,D) with planes of fi xed values of the four-spin interaction less than α

t

. In Fig. 4(a) we plot various transition lines when the system keeps its tricritical behaviour, namely in the range − 0.901 ≤α≤α

t

. The T-component of the tricritical point decreases with the four-spin interaction and therefore there exists a tricritical line ending the second-order transition surface T

c

( α , D). For the remaining range − 4 oαo

− 0.901 of the four-spin interaction, the tricritical behaviour dis- appears and all transitions are always of second-order for any value of the crystal fi eld D, as is plotted in Fig. 4(b). An interesting behaviour of the system can be seen in this fi gure when the four- spin interaction has an important negative value. Indeed, when α belongs to the range − 4 o αo− 3.305, the system exhibits an

“ horizontal ” re-entrance for relatively small values of the crystal fi eld which becomes more important when α-− 4.Thus, by increasing the value of D from D¼0, the transition temperature T

c

( α ,D) increases and passes by a smooth maximum and then reduces rapidly to zero at the critical value D

_c

¼2J

₂

of the crystal fi eld. We have to note that the all second-transition lines vanish at that critical value D

c

which is independent of the value of four-spin interaction ( − 4 ≤α≤− 0.901). On the other hand, we note the existence of a “ vertical ” re-entrance in a narrow range of the crystal fi eld. This phenomena can be explained by a competition of energy E and entropy S in the free energy F¼E − TS; which is due in fact to the competition between pair interaction, quartet interac- tion and crystal fi eld, when these latters take appropriate signs.

Since the system under study exhibits interesting phase dia- gram, it is worthy to investigate the temperature dependence of the sublattice magnetizations m and m as well as the total magnetization de fi ned by M¼( m+ m)/2 by solving numerically the coupled Eqs. (15) – (17). First, let us examine the in fl uence of the four-spin interaction on the magnetic properties of the system in the absence of the crystal fi eld interaction. Thus, we plot in Fig. 5, the magnetizations m , m and M for D¼0 and various values of J

4

/J

2

. As seen in the fi gure, at absolute zero, the sublattice ( m , m)

and the total M magnetizations have their saturation values. As the temperature is increased, the effects of thermal agitation, which favors random orientation of the magnetic moment, begin to be felt. Thus, the spontaneous magnetizations decrease with increas- ing temperatures, gradually at fi rst, and then more and more rapidly to vanish at a α -dependent critical temperature T

c

( α ).

Further, we note from the fi gure that the long range ferromagnetic order domain becomes less and less wide with decreasing values of α . This means that the ordering of m , m and M is disturbed by decreasing the values of the four-spin interaction. To analyse the in fl uence of the crystal fi eld on the behaviors of the sublattice and total magnetizations, we use the state Eqs. (15) – (17). Fig. 6, show how the thermal dependences of m , m and M depend on the value of D, when the strength of the four-spin interaction is kept fi xed. In particular, we have to point out that below critical temperature, each magnetization decreases with increasing values of the crystal fi eld. Therefore, the ferromagnetic order domain becomes more and more reduced with increasing values of D. As can be expected, for T just less than the critical temperature T

c

, we can show from Eq. (18) that the magnetizations vary as (T

_c

− T)

^1/2

. This gives us the critical exponent β ¼1/2, as in the mean fi eld-theory.

-4 -3 -2 -1 0 1

0.0

J

₄

/J

₂

Fig. 3. The phase diagram in (J

4

/J

2

,T/J

2

)plane for the mixed spin-1/2 and spin-1 Ising model with four- spin interaction on a square lattice. The black point denotes the tricritical point.

0.0 0.5 1.0 1.5 2.0

0.0 0.2

D/J

₂

0.0 0.5 1.0 1.5 2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

J

₄

/J

₂

=-2.0

T

c

/J

2

D/J

₂

J

₄

/J

₂

=-1.0

-3.0 -2.5

-3.5 -3.8 -3.95

Fig. 4. The phase diagrams in D–T plane. (a) When the system exhibits a tricritical behaviour (−0.901J

2

oJ

4

o1.114J

2

). The black point denotes the tricritical point.

(b) When all transition lines are of second order (−4J

2

oJ

4

o−0.901J

2

). The number

accompanying each curve denotes the value of J

4

/J

2

.

4. Other thermodynamical properties

Let us examine other relevant thermodynamical quantities such as the internal energy U, speci fi c heat C and zero- fi eld magnetic susceptibility of the present system.

4.1. Internal energy and speci fi c heat

In the spirit of the FCA, Eqs. (9) and (10) can be generalized by the following relations:

〈 g ₀ s 0 〉 ¼ g ₀ 1 2 tanh K

2 fðS 1 þ S 2 þ S 3 þ S 4 Þ þ α ðS 1 S 2 s 1 þ S 3 S 4 s 2 Þg

ð21Þ

〈G

0

S

0

〉 ¼ G

0

2sinh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þg expð β DÞ þ 2cosh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þg

ð22Þ

where g

0

and G

0

represent arbitrary functions of spin variables, except s

0

and S

0

, respectively.

The internal energy U of the system described by the Hamilto- nian (1) is given by

U ¼ − J 2

N 2 〈∑

s S 0 〉− J 4

N 4 〈∑

sS S 0 〉 þ D N

2 〈 S ² ₀ 〉 ð23Þ

where Σ s ¼ Σ ⁴ i ¼ 1 s i ;Σ s S ¼ s ¹ s ² S 1 þ s ³ s ⁴ S 2 and N is the total number of magnetic atoms. In order to calculate

〈Σ s S 0 〉 and 〈Σ sS S 0 〉 , we substitute G

0

in (22) by ∑

s

and ∑

sS

, respectively. Then, they can be written as

〈∑

s

S

0

〉 ¼ ∑

s

2sinh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þg expð βDÞ þ 2cosh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þg i

ð24Þ

〈∑

sS

S

0

〉 ¼

∑

sS

2sinh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þg expð βDÞ þ 2cosh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þg

* +

ð25Þ Fig. 5. The thermal dependence of the sublattice and total magnetizations (m, m, M)

in the absence of the crystal field (D¼0), for positive (a) and negative (b) values of J

4

/J

2

.

Fig. 6. The thermal dependence of the sublattice and total magnetizations (m, m, M)

when the value of D is changed, with (a) J

4

/J

2

¼0.8 and (b) J

4

/J

2

¼−1.

contribution to the speci fi c heat of the system under investigation, via the standard relation

CðK ; α; DÞ ¼ ∂ UðK ; α; DÞ

∂ T ð28Þ

First, let us examine the temperature dependence of the internal energy U and the magnetic speci fi c heat C in the absence of the crystal fi eld (D¼0), for selected values of the four-spin interactions.

As seen from Fig. 7, U exhibits a singularity at the transition temperature. At low temperatures, this fi gure shows that the absolute value of the internal energy | U | increases with increasing strength of the four-spin interactions; while it seems that this latter has not a signi fi cant in fl uence on U at high temperatures. Further, we note here that the ground state energy of the system is very sensitive to the value of the four-spin interaction. On the other hand, we have plotted in the same fi gure the magnetic contribution C to the speci fi c heat of the system. As shown in this fi gure, its

of curves showing how their thermal behaviour depends on the strength of the crystal fi eld D. From these fi gures, one can observe that the internal energy is very sensitive to the value of D and its absolute value | U | is an increasing function of the crystal fi eld at any values of the temperature and the four-spin interaction. Here again, the speci fi c heat shows a discontinuity at the transition tempera- ture T

c

( α , D) and the amplitude of the jump decreases with increasing the value of D for any strength of the four-spin interac- tion. Further, below the critical temperature, Fig. 8 shows that the thermal behaviour of C depends qualitatively and quantitatively on the range of the four-spin interaction strength J

4

/J

2

. Indeed, for J

4

/ J

₂

4 − 3.5 the speci fi c heat is sensitive to the value of the crystal fi eld and it is remarkably depressed by increasing D. But for J

4

/J

2

≤− 3.5, C seems to be not so sensitive to the value of D. We also note that, in all cases, the internal energy increases with the increase of the temperature, while the speci fi c heat decreases with the increase of T when the system is in disordered state (T 4 T

c

); whereas in ordered one (T o T

c

) it increases with the increase of the tempera- ture. It is worthy to note, from Eq. (28), that C is proportional to 1/T

²

in the limit T -∞ for any values of D and α .

4.2. Susceptibility

Let us now investigate the zero- fi eld isothermal magnetic susceptibility. To this end, we add to the Hamiltonian (1) the term – B( ∑

i

s

i

+∑

j

S

j

), where B is an external magnetic fi eld. Thus, iden- tities (9) – (11) are generalized into

μ ¼ 1 2 tanh K

2 fðS 1 þ S 2 þ S 3 þ S 4 Þ þ α ðS 1 S 2 s ¹ þ S 3 S 4 s ² Þg þ b 2

ð29Þ

m¼ 2sinh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þg þ b expð β DÞ þ 2cosh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

₁

þ s

3

s

4

S

₂

Þg þ b

ð30Þ

q ¼ 2cosh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

₁

þ s

3

s

4

S

₂

Þg þ b expð βDÞ þ 2cosh½Kfð s

1

þ s

2

þ s

3

þ s

4

Þ þ α ð s

1

s

2

S

₁

þ s

3

s

4

S

₂

Þg þ b

ð31Þ where b¼ βВ .

In order to obtain the state equations, we follow the same procedure described in detail previously in the theoretical frame- work. We fi nd equations similar to those expressed in Eqs. (15) – (17), except that in the obtained equations, the coef fi cients are not only functions of K, α and D, but also depend on b.

Deriving the obtained equations with respect to B at the point B¼0, and eliminating ∂ q =∂ Bj, we obtain the local susceptibilities pertaining to μ and m:

χ μ ¼ ∂μ

∂ B

B ¼ 0 and χ m ¼ ∂ m

∂ B

B ¼ 0 ð32Þ

Fig. 7. The temperature dependence of the internal energy U and the magnetic

specific heat C in the absence of the crystal field (D¼0). The number accompanying

each curve denotes the value of J

4

/J

2

.

Therefore, the initial susceptibility χ per site is calculated by

χ ¼ ¹ ₂ ð χ μ þ χ m Þ ð33Þ

First, we have reported, in Fig. 9, the in fl uence of the four-spin interactions J

4

/J

2

in the absence of D. As we can see from this fi gure, the magnetic susceptibility exhibits a singularity at the critical temperature T

_c

, due to the second order phase transitions from the ordered phase to the disordered paramagnetic one.

Second, the temperature dependence of the susceptibility χ has been examined for the Blume – Capel model (D ≠ 0). This study has been carried out for various values of the four-spin interactions and the crystal- fi eld. The corresponding thermal behaviour is presented in Fig. 10. These latters show that for a fi xed value of D/J

2

, the thermal behaviours of the susceptibility depend quanti- tatively on the strength of the four-spin interaction. Close to the critical temperature, we fi nd that the susceptibility χ depends on temperature as | T − T

_c

|

⁻¹

. Therefore, it diverges at the critical point T

c

and gives the critical exponent γ ¼1as in the mean- fi eld theory.

As can be observed from Figs. 9 and 10, the susceptibility keeps this behaviour even if the system approaches its tricritical point, when this latter exists. As expected, χ presents 1/T behaviour in the limit T -∞ .

5. Conclusion

In this work, we have studied the mixed spin Blume – Capel model, consisting of spin-1/2 and spin-1, with four-spin interac- tions J

4

and uniform crystal fi eld interaction D on the square lattice. We have used the fi nite cluster approximation within the

framework of a single-site cluster theory. In this approach, we have derived the state equations.

First, we have investigated in detail the phase diagrams and magnetizations for various values of J

4

. In the presence of the crystal fi eld, we have found that the system exhibits tricritical behaviour in the range − 0.901 ≤α≤α

t

. For the remaining range of J

₄

( − 4 o αo− 0.901) the tricritical behaviour disappears and all Fig. 8. The temperature dependence of the internal energy U and the specific heat C for positive (a) and negative (b) value of J

4

/J

2

. The number accompanying each curve denotes the value of D/J

2

.

Fig. 9. The temperature dependence of the initial magnetic susceptibility χ in the

absence of the crystal field (D¼0). The number accompanying each curve denotes

the value of J

4

/J

2

.

transitions are of second-order. We have to note the existence of the reentrant behaviour for important negative values of J

4

and appropriate values of D. Further, the effects of J

4

and D on the thermal behaviours of the sublattice and total magnetizations have also been examined. We have found that they decrease

expð β DÞ þ 2cosh ð Kx Þ

B 1 ¼ 1

4 g 2 þ α 4

− g 2 − α 4

n o

þ 3 2 gð α

4 Þ þ g 1 þ α 4

− g 1 − α 4 o n

B 2 ¼ 1

4 g 2 þ α 4

− g 2 − α 4

n o

þ 3 2 gð α

4 Þ − g 1 þ α 4

− g 1 − α 4 o n

B 3 ¼ 1

4 g 2 þ α 4

− g 2 − α 4

n o

− 1 2 gð α

4 Þ B 4 ¼ 1

8 g 2 þ α 2

− g 2 − α 2

n o

þ 3 4 g α

2 þ 1

2 g 1 þ α 2

− g 1 − α 2

n o

− 1

4 g 2 þ α 4

− g 2 − α 4 o

− 3 2 g α

4 − g 1 þ α 4

− g 1 − α 4

n o

B 5 ¼ 1 2 g 1 − α

2

− g 1 þ α 2

n o

− 1

4 g 2 þ α 4

− g 2 − α 4 o

− 3 2 g α

4

þ g 1 þ α 4

− g 1 − α 4

n o

þ 1

8 g 2 þ α 2

− g 2 − α 2

n o

þ 3 4 g α

2 B

₆

¼ 1

8 g 2 þ α 2

− g 2 − α 2

n o

− 1 4 g α

2 − 1 4 g 2 þ α

4

− g 2 − α 4

n o

þ 1 2 g α

4 B

₇

¼ 1

16 g 2 þ α 2

þ g 2 − α 2 n o

þ 1 4 g ð Þ 1 − 1

8 g ð Þ 2 − 1 8 g 1 þ α

2

þ g 1 − α 2 n o B 8 ¼ g 2 þ α

4

− g 2 − α 4

− 2g α 4 B 9 ¼ −g 2 þ α

4

þ g 2− α 4

þ 2g α 4 −g α

2 þ 1

2 g 2 þ α 2

−g 2− α 2

n o

B ₁₀ ¼ 1

4 g 2 þ α 2

þ g 2 − α 2

n o

− 1

2 gð2Þ − gð1Þ þ 1

2 g 1 þ α 2

þg 1 − α 2 o n

L 0 ¼ B 1 ; L 1 ¼ 2B 2 ; L 2 ¼ B 3 ; L 3 ¼ 8B 4 þ B 1 þ B 2 þ 4B 3

L 4 ¼ 8B 5 þ B 4 þ B 5 þ 4B 6 ; L 5 ¼ 12B 1 þ 3B 6

L 6 ¼ 24B 2 þ 6B 7 ; L 7 ¼ 12B 3 þ 3B 8 ; L 8 ¼ B 7 ; L 9 ¼ 4B 6 ; L 10 ¼ 8B 7

L 11 ¼ 4B 8 ; L 12 ¼ 4B 1 þ 4B 2 þ 16B 3 þ 2B 8 ; L 13 ¼ 4B 4 þ 4B 5 þ 16B 6 þ 2B 9

L 14 ¼ 12B 7 þ 3B 10 ; L 15 ¼ 4B 10

I 1 ¼ B 1

8 ; I 2 ¼ B 4

8 ; I 3 ¼ B 1 þ B 2 þ B 6

4 þ B 7

4 ; I 4 ¼ B 2 þ B 3 þ B 7 þ B 7

4 þ B 8

4 þ B 10

4 ; I 5 ¼ 2B 4 þ 2B 3 þ B 8

8 ; I 6 ¼ 2B 5 þ 2B 6 þ B 9

8 ; I 7 ¼ B 2

8 þ B 5

8 ; I 8 ¼ 2B 2 ; I 9 ¼ 2B 5 ;

I 10 ¼ 4B 1 þ 4B 2 þ B 6 þ B 7 ; I

₁₁

¼ 4B

₂

þ 4B

₃

þ B

₇

þ B

₈

þ 4B

₇

þ B

₁₀

; I 12 ¼ 2B 4 þ 2B 5 þ 2B 3 þ 2B 6 þ B 8

8 þ B 9

8 ; I 13 ¼ 2B 1 þ 2B 4 : Fig. 10. The thermal dependence of the initial magnetic susceptibility χ when the

value of D is changed, with (a) J

4

/J

2

¼0.0, (b) J

4

/J

2

¼1 and (c) J

4

/J

2

¼−1. The number

accompanying each curve denotes the value of D/J

2

.

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