Thermodynamical properties of the random field mixed spin Ising model

Thermodynamical properties ofthe random !eld mixed spin Ising model

N. Benayad ^∗ , A. Fathi, L. Khaya

Groupe de M ecanique Statistique, Laboratoire de Physique Th eorique, Facult e des Sciences, Universit e Hassan II - A!"n Chock, B.P. 5366 Maarif, Casablanca, Morocco

Received 13 April 2001

Abstract

A theoretical study ofa mixed spin Ising system consisting ofspin-1=2 and spin-1 with a random !eld is studied by the use ofthe !nite cluster approximation within the framework ofa single-site cluster theory. In this approach, the phase diagrams and magnetizations are investigated for the simple cubic lattice when the random !eld is bimodally and trimodally distributed. The internal energy, speci!c heat and susceptibility are also calculated for both kinds ofthe probability distributions. c 2001 Elsevier Science B.V. All rights reserved.

PACS: 75.10.Hk; 75.40.Cx; 05.45.−a; 05.70.−a

Keywords: Mixed spins; Random !elds; Phase diagrams; Magnetizations; Speci!c heat;

Susceptibility

1. Introduction

The random !eld Ising model (RFIM) on mono-atomic lattices where the local quenched random !eld H _i is assumed to have zero average H _{i r} = 0 and is uncorrelated at di;erent sites has been a subject ofmuch theoretical [1–5] and experimental [6–8]

investigations ofthe statistical physics ofrandom and frustrated systems. This is be- cause it helps to simulate many interesting but complicated problems. The best ex- perimental realization ofan RFIM has been the diluted uniaxial-antiferromagnet in a uniform !eld applied along the spin ordering axis (DAFF) such as Rb 2 Co x Mg _1−x F 4

and Fe _x Zn _1−x F ₂ in magnetic !eld, which correspond, respectively, to two- and three- dimensional proto-typical systems. It has been shown [9–12] that in the presence of a uniform !eld, the random exchange interactions give rise to local random staggered

∗

Corresponding author. Tel.: +212-22-230684; fax: +212-22-230674.

E-mail address: benayad@facsc-achok.ac.ma, nbenayad@hotmail.com (N. Benayad).

0378-4371/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.

PII: S 0378-4371(01)00322-3

!elds and that such a system should be isomorphic to the RFIM. The applied uniform

!eld and the e;ective random !eld generated by it are proportional [9]. Indeed, many experiments on DAFF’s have con!rmed some ofthe theoretical predictions derived for the RFIM [13,14]. Recently, it has been shown experimently [15] and through Monte Carlo (MC) simulations [16] that Fe x Zn 1−x F 2 in an applied uniform !eld exhibits the equilibrium critical behaviour ofthe RFIM for magnetic concentration x ¿ 0:75. We note that the trimodal distribution can be relevant for the study of diluted antiferro- magnets [8]. The RFIM can also be used to describe other processes such as the phase separation ofa two-component Juid mixture in porous material or gelatine and the solution ofhydrogen in metallic alloys [17]. Recent experiments on the liquid–vapour transition in aerogel [18] reveal a coexistence curve similar to the pure Ising model but slow dynamics that are suggestive ofrandom-!eld e;ects. Since the aerogel creates a random network which attracts the Juid, the RFIM should be an appropriate model [19] for this system.

One of the main points for which attention has been focused was the lower criti- cal dimension d _l . This is the dimension above which long range order ferromagnetic order can exist. It has been shown rigorously that in one dimension (d = 1) [20–22]

and d = 2 [23] there is no long range order at any temperature T , including T = 0.

Furthermore, it is also established rigorously that the low temperature phase in d ¿ 2 is ferromagnetic [24–26]. Thus, these theoretical studies have concluded that d _l = 2.

Also, experiments support the existence oflong range order in the three-dimensional RFIM (for instance in Fe x Zn 1−x F 2 ) [27–29], while such an order does not exist in d = 2 RFIM (for instance in Rb ₂ Co _x Mg _1−x F ₄ ) [30]. On the other hand, it is mentioned that the critical properties in three dimensions are controlled by a zero temperature

!xed point with modi!ed scaling behaviour [31–36]. It is worthy to mention here that Schwartz et al. [37] have suggested another scaling relation for the RFIM in which case there would again be only two independent exponents, as at conventional transi- tions. There is some evidence for this from series expansions [38,39]. Some theories suggest that ifa second-order transition occurs at all, it will be in an entirely new universality class [31–35]. However, the critical behaviour is still not fully under- stood because, although there is some agreement between MC [16,40,41] and equilib- rium experimental [15] data concerning the susceptibility and correlation length critical exponents, the situation ofthe speci!c heat and the parameter order critical exponents is not completely achieved in experiments [15,42] and MC simulations [16,40,41].

Other important questions concerning the RFIM have been the existence ofa tri- critical point and the order oflow-temperature phase transitions in three dimensions.

This does not seem to be completely elucidated. Indeed, mean !eld theories [43–

45] predicted that the existence ofa tricritical point depends on the form ofthe

random !eld distribution. In particular, the bimodal distribution (±H) yields a tri-

critical point and the transition was found to be of !rst-order for suLciently strong

random !eld, while the Gaussian one led to a second-order phase transition down to

T = 0. Using renormalization group calculations [46,47], it has been discussed that an

appropriate distribution function may lead to a tricritical point. Houghton et al. [48,49]

have performed detailed numerical studies based on the analysis of the high-temperature series for the static susceptibility of the RFIM with bimodal and Gaussian distribu- tions. They con!rmed the predictions ofthe mean !eld theory for d ¿ 4. In dimension d ¡ 4 however, the analysis suggests that, even for a Gaussian distribution, it has been observed to have Juctuation-driven !rst-order transitions at low temperatures (suL- ciently strong random !eld) implying that d = 4 is a critical dimension for the RFIM.

Furthermore, Monte Carlo results ofRFIM with a bimodal distribution have been ei- ther, inconclusive: some were interpreted in support ofa !rst-order transition at !nite low temperature [50], others favoured a continuous transition [16,51], or suggest an unusual behaviour [40,41]: a jump in the order parameter but with an in!nite correla- tion length at the transition. We have to point out here that the experiments [27–29,52]

on diluted MnF 2 are diLcult to clearly interpret but are suggestive ofa !rst-order transition. We note that there is no general consensus, experimentally, on whether the transition is continuous or !rst-order.

On the other hand, some authors [53,54] have studied the case ofspin-S (greater than 1=2) Ising model in a random !eld within mean-!eld like approximations for a bimodal distribution. They have predicted a long-range order at T = 0 up to S -dependent critical value ofthe !eld H, and the existence oftricritical point (T _t ; H _t ), where H _t =S is found to decrease very slowly with increasing S . Such a model for any S in a random !eld which is trimodally distributed has also been investigated within the mean-!eld theory [55]. The obtained phase diagram (for any S) is qualitatively similar to that found earlier [56] for spin 1=2 and it exhibits a rich multicritical behaviour. In the limit of very large values of S , the results go asymptotically to ones obtained for classical (S → ∞) model.

Recently, attention has been directed to the study ofthe magnetic properties of two-sublattice mixed spin Ising systems. They are of interest for the following main reasons. They have less translational symmetry than their single spin counterparts, and are well adapted to study a certain type offerrimagnetism [57]. Experimentally, it has been shown that the MnNi(EDTA)–6H 2 O complex is an example ofa mixed spin system [58]. The mixed Ising spin system consisting ofspin-1=2 and spin-1 has been studied by renormalization group technique [59,60], by high-temperature series expan- sions [61], by free-fermion approximation [62], by !nite cluster approximation [63,64], and Monte Carlo simulation [65]. Recently, one ofus (N,B) investigated [66] the crit- ical properties ofthe mixed spin-1=2 and spin-1 Ising model in a random transverse

!eld using the mean !eld renormalization group combined with the discretized path integral representation.

As far as we know, no works have been concerned with thermodynamical properties ofthe two-sublattice mixed spin-1=2 and spin-S Ising system in a random longitudi- nal !eld with symmetric probability distribution Q(H ) = Q(−H ). This system can be described by the Hamiltonian

H = −

ij

J _{ij i} S _j −

i

H _{i i} −

j

H _j S _j ; (1)

where i and S j are Ising spins ofmagnitude 1=2 and S at sites i and j, respectively.

J _ij is the exchange interaction, H _i and H _j are the random !elds and the !rst summation is carried out only over the nearest-neighbour pairs ofspins. In the present work, we limit our study to the case S = 1 and J _ij = J. The !elds H _i and H _j are assumed to be uncorrelated variables and obey the trimodal probability distribution,

Q(H i ) = p(H i ) + (1 − p)

2 [(H i − H ) + (H i + H )] ; (2)

where the parameter p measures the fraction of spins in the system not exposed to the !eld. One notes that at p = 1 or H = 0, the system reduces to the simple mixed spin-1=2 and spin-1 Ising model.

The !rst purpose ofthis paper is to investigate the phase diagrams and the thermal behaviours ofthe magnetizations ofthe mixed spin-1=2 and spin-1 in a random longitu- dinal !eld which is bimodally (p = 0) and trimodally (p = 0) distributed. To this end, we use the !nite cluster approximation [67,68] within the framework of a single-site cluster theory. We note that the latter and the e;ective !eld theory with correlations give the same results [69] by two di;erent mathematical techniques. The second goal is to examine the internal energy, speci!c heat and zero-!eld magnetic susceptibility ofthe system described by the Hamiltonian (1).

The outline ofthis work is as follows. In Section 2, we give a description ofthe theoretical framework. In Section 3, the phase diagrams and magnetizations of the system are investigated and discussed. In Section 4, other relevant thermodynamical quantities are presented and analysed.

2. Theoretical framework

The theoretical framework that we adopt in the study of the random !eld mixed spin Ising model described by the Hamiltonian (1), is the !nite cluster approximation (FCA) [44], based on a single-site cluster theory. In this approach, we consider a particular spin o (S o ) and denoted by 0 c (S o c ) as the mean value of o (S o ) f or a given con!guration c ofall other spins, i.e., when all other spins _i and S _j (i; j = 0) are kept in a !xed state. o c and S o c are then given by

o c = 1 2 tanh



 K 2

z j=1

S j + h _o 2



 (3) and

S _{o c} = 2 sinh[K _z

i=1 i + h o ] 1 + 2 cosh[K _z

i=1 _i + h _o ] ; (4)

where z is the nearest-neighbour coordination number ofthe lattice, K = J; h o = H o

and = 1=T .

{S ₁ ; S ₂ ; : : : ; S _z } ({ ₁ ; ₂ ; : : : ; _z }) are the nearest-neighbours spins ofspin _o (S _o ) with

which it directly interacts. For a !xed con!guration ofthe random !elds H i and H j ,

the sublattice magnetizations per site = and m s = S are given by the following exact relations

=

1

2 tanh



 K 2

z j=1

S _j + h _o 2





(5)

and

m _s = 2 sinh[K _z

i=1 i + h o ] 1 + 2 cosh[K _z

i=1 i + h o ]

; (6)

where · · · denotes the canonical thermal average.

It is a formidable task to average the right-hand side of Eqs. (5) and (6) over all spin con!gurations. The FCA has been designed to treat all spin self-correlations exactly while still neglecting correlations between di;erent spins. For our mixed spin-1=2 and spin-1 Ising system described by Eq. (1), the appropriate distributions are

P({ _i }) =

i

1 2 +

_i − 1

2

+ 1

2 −

_i + 1 2

; (7)

R({S j }) =

j

1

2 (m s + x s )(S j − 1) + (1 − x s )(S j ) + 1

2 (−m s + x s )(S j + 1)

; (8) where x s = S ² .

When calculating the thermal average on the right-hand side of(5) and (6), it is however, easier to observe that any function such as f() and g(S ) of or S can be written as the linear superposition

f() = f ₁ + f ₂ ; (9)

g(S ) = g 1 + g 2 S + g 3 S ² ; (10)

with appropriate coeLcients f 1; 2 and g 1; 2; 3 . Applying this to all spins i and S j in Eqs. (3) and (4), their right-hand sides are decomposed as

_{o c} = ^z

n=0

z−n l=0

{S ² ; S} _z;l;n A _l;n (K; h _o ) ; (11)

S o c = z

n=0

{} z;n B n (K; h o ) ; (12)

where {S ² ; S} _z;l;n denotes the superposition ofall the terms containing l di;erent factors of S _j ² and n di;erent factors of S j

, with j = j . These factors are selected from the set {S ₁ ; S ₂ ; : : : ; S _z ; S ₁ ² ; S ₂ ² ; : : : ; S _z ² }. For example, if z = 4; l = 2 and n = 1, then

{S ² ; S} 4;2;1 = S 1 (S ₂ ² S ₃ ² + S ₂ ² S ₄ ² + S ₃ ² S ₄ ² ) + S 2 (S ₁ ² S ₃ ² + S ₁ ² S ₄ ² + S ₃ ² S ₄ ² )

+ S 3 (S ₁ ² S ₂ ² + S ₁ ² S ₄ ² + S ₂ ² S ₄ ² ) + S 4 (S ₁ ² S ₂ ² + S ₂ ² S ₃ ² + S ₁ ² S ₃ ² ) : (13)

{} z;n denotes the superposition ofall the terms containing n di;erent factors of i , selected from the set { ₁ ; ₂ ; : : : ; _z }. Using the probability distributions (7) and (8) to treat Eqs. (5) and (6) or averaging Eqs. (11) and (12) over all spin con!gurations and neglecting correlations between quantities pertaining to di;erent spins, we obtain

= z

n=0

z−n l=0

C _z ⁿ C _z−n ^l x ^l _s m ⁿ _s A l;n (K; h o ) ; (14) m _s = ^z

n=0

C _z ^{n n} B _n (K; h _o ) ; (15)

with C _a ^b = _b!(a−b)! ^a! which is a combinatorial factor. The parameter x s de!ned by x s = S _i ² has to be evaluated in the spirit ofthe FCA. Similar to Eq. (6), we have the exact relation

x _s = 2 cosh[K _z

i=1 i + h o ] 1 + 2 cosh[K _z

i=1 _i + h _o ]

: (16)

Expanding the right-hand side ofEq. (16) by using Eq. (9) and neglecting correlations between quantities pertaining to di;erent spins or using the probability distribution (7), we obtain

x _s = ^z

n=0

C _z ^{n n} D _n (K; h _o ) : (17)

The next step is to carry out the con!gurational averaging over the random !elds H i

according to the probability distribution function Q(H _i ) given by (2). The ordering parameters ; m and x are then de!ned as = P ; m = P m s and x = P x s , where the bar denotes the random !eld average. Thus, using the probability distribution (2), we obtain the following set of coupled equations for ; m and x

= ^z

n=0

z−n l=0

C _z ⁿ C _z−n ^l x ^l m ⁿ A P _l;n (K; h;p) ; (18) m = ^z

n=0

C _z ^{n n} B P _n (K; h;p) ; (19)

x = ^z

n=0

C _z ^{n n} D P _n (K; h;p) ; (20)

where h = H . The non-zero coeLcients P A _l;n ; B P _n and P D _n for z = 6 are listed in the appendix.

3. Phase diagrams and magnetizations

The sublattice magnetizations ; m and the quadrupolar moment x are given by

Eqs. (18)–(20). They are valid for any lattice (arbitrary coordination number z), and

constitute a set ofrelations, according to which we can study the magnetic properties ofthe present system. In this paper, we are !rst concerned with the investigation of the phase diagrams when the structure ofthe system is a simple cubic lattice (z = 6).

To calculate the transition temperature T _c , we substitute m and x in (18) with their expressions taken from (19) and (20). Then we obtain an equation for ofthe form

= a + b ³ + · · · ; (21)

where

a = 6 P B 1 A P o;1 + 5 P D o B P 1 A P 1;1 + 10 P D ² _o B P 1 A P 2;1 + 10 P D ³ _o B P 1 A P 3; 1 + 5 P D ⁴ _o B P 1 A P 4;1 + P D ⁵ _o B P 1 A P 5; 1 ; (22) and

b = 6 P B ₃ A P _o;1 + 5 P D ₂ B P ₁ A P _1;1 + 5 P D _o B P ₃ A P _1;1 + 20 P D _o D P ₂ B P ₁ A P _2;1 + 10 P D ² _o B P ₃ A P _2;1

+ 30 P D ² _o D P 2 B P 1 A P 3; 1 +10 P D ³ _o B P 3 A P 3;1 +20 P D ³ _o D P 2 B P 1 A P 4;1 +5 P D ⁴ _o B P 3 A P 4;1 +5 P D ⁴ _o D P 2 B P 1 A P 5; 1

+ P D ⁵ _o B P 3 A P 5;1 + 20 P B ³ ₁ A P o;3 + 60 P B ³ ₁ D P o A P 1;3 + 60 P B ³ ₁ D P ² _o A P 2; 3 + 20 P B ³ ₁ D P ³ _o A P 3; 3 : (23) It follows from (21) that the second-order transition line, in T –H plane for a given value of p, is determined by the equation

1 = 6 P B ₁ A P _o;1 + 5 P D _o B P ₁ A P _1;1 + 10 P D ² _o B P ₁ A P _2;1 + 10 P D ³ _o B P ₁ A P _{3; 1} + 5 P D ⁴ _o B P ₁ A P _4;1 + P D ⁵ _o B P ₁ A P _{5; 1} : (24) The magnetization in the vicinity ofthe second-order transition is given by

² = 1 − a

b : (25)

The right-hand side of(25) must be positive. Ifthis is not the case, the transition is

ofthe !rst-order and the point at which a = 1 and b = 0 characterizes the tricritical

point. In Fig. 1, we represent the phase diagram in the (T; H) plane for various values

of p. When the random !eld is bimodally distributed (p = 0), the critical temperature

decreases gradually from its value T c (H = 0) in the pure mixed spin system, to end in

a tricritical point. As shown in the !gure, when we consider a trimodal random !eld

distribution (i.e., p = 0), the system keeps a tricritical behaviour for a relatively small

range of p (0 6 p 6 0:2899). The T -component ofthe tricritical point decreases with

increasing values of p and vanishes at p = 0:2899. So, a vestigial critical point which

exists in the mono-atomic RFIM [56] does not occur in the random !eld mixed spin

Ising model. When p belongs to the range 0:2899 ¡ p 6 1, the tricritical behaviour

disappears and all transitions are always ofsecond-order for any value ofthe !eld

H. Moreover, we note the existence ofa critical value p ^∗ = 0:5259, indicating two

qualitatively di;erent behaviours ofthe system which depend on the range ofp. Thus

for p ¡ p ^∗ , the system exhibits at the ground state a phase transition at a !nite critical

value H _c of H. But for p ^∗ ¡ p ¡ 1, there is no critical !eld, and therefore, at very

low temperatures, the ordered state is stable for any value of the !eld strength. This

Fig. 1. The phase diagram in T–H plane ofthe random !eld mixed spin Ising model on simple cubic lattice (z = 6). The number accompanying each curve denotes the value of p.

latter behaviour is qualitatively similar to that obtained in random !eld spin 1=2 Ising model [56]. It is also worthy to note here that such critical value p ^∗ has been found [70] by two ofus (N. B and A. F) in the study ofa mixed spin Ising model in a transverse random !eld. As expected, we can see in Fig. 1 that for a !xed value of H, the critical temperature is an increasing function of p. As clearly shown from the !gure, the system exhibits a reentrant behaviour in the narrow ranges of H and p. This phenomena is due to the competition between the exchange interaction and the frustration induced by quenched local !elds. It is interesting to note here that such reentrance does not exist in the spin-1=2 random !eld Ising model for a trimodal distribution [55,56].

Let us now examine the temperature dependence ofsublattice magnetizations and

m as well as the total magnetization de!ned by M = (+m)=2. To this end, we have to

solve the coupled equations (18)–(20) for a simple cubic lattice (z = 6). In Fig. 2a, we

plot the magnetizations ; m and M in the case ofthe bimodal (p = 0) and trimodal

(p = 0:2) distributions when H = 2 and 1:5. As indicated in Fig. 1, this corresponds

to second-order transition. The sublattice and the total magnetizations decrease con-

tinuously in the vicinity ofthe transition temperature and vanish at T = T c (p; H). On

Fig. 2. The thermal dependence ofthe sublattice and total magnetizations (; m; M ) for the simple cubic

lattice when the system exhibits (a) second-order transition and (b) !rst-order transition.

Fig. 3. The thermal dependence ofthe total magnetization for the simple cubic lattice when H=J = 2. The number accompanying each curve denotes the value of p.

the other hand, when p belongs to 0 6 p 6 0:2899, the system undergoes a !rst-order transition in a p-dependent narrow range of H , and the magnetizations ; m and M exhibit a gap at that transition. Such thermal behaviours are depicted in Fig. 2b when p = 0 for appropriate values of the !eld strength. Since the concentration (p = 0) of the sites not exposed to the !eld has a quantitative inJuence on the transition temperature, it is interesting to study the thermal dependences ofthe total magnetization M with p, for a given value of H. In Fig. 3, we plot typical results for selected values of p and it shows that the long range ferromagnetic order domain becomes more and more wide with increasing values of p. As shown in Fig. 1, the role ofthe local random

!eld (according to (2)) is to inhibit the ordering of , m and M . So, it is also worthy

to examine the variation ofthe thermal behaviour ofM with the !eld H, when p

is kept !xed. The results are represented in Fig. 4 for various values of H when the

random !eld is bimodally (p = 0) and trimodally (p = 1=3) distributed. In the former

case, the magnetization M exhibits a discontinuity at the transition when H belongs to

a narrow range. While in the latter case, with the increase of T; M decreases from its

saturation value 0.75 to vanish at a critical temperature T _c which depends on the value

of H .

Fig. 4. The thermal dependence ofthe total magnetization when the value ofH is changed from 0 to 2, with p = 0 (solid lines) and p = 1=3 (dashed lines).

4. Other thermodynamical properties

Next, we evaluate other relevant thermodynamical quantities such as the internal energy U, speci!c heat C and zero-!eld magnetic susceptibility * ofthe present system.

4.1. Internal energy and speci:c heat

In the spirit ofthe FCA, Eqs. (5) and (6) can be generalized by the following exact relations

g _{o o} =

g _o 1 2 tanh



 K 2

z j=1

S _j + h _o 2





; (26)

G _o S _o = G _o 2 sinh[K _z

i=1 i + h o ] 1 + 2 cosh[K _z

i=1 _i + h _o ]

; (27)

where g o and G o represent arbitrary functions of spin variables except o and S o ,

respectively.

For a given con!guration ofthe random !elds, the internal energy U S ofthe system described by the Hamiltonian (1) is given by

U S = − J N 2

S o

− N

2 H o o − N

2 H o S o ; (28)

where

= _z

i=1 _i , and N is the total number ofmagnetic atoms. In order to cal- culate

S o , we substitute G o =

into (27). Then it can be written as

S _o

=

2 sinh[K _z

i=1 i + h o ] 1 + 2 cosh[K _z

i=1 i + h o ]

: (29)

Following the same procedure as for the evaluation of and m _S , namely using the probability distribution (7), we !nd that

S _o

= ^z

n=0

ⁿ L _n (K; h _o ) : (30)

For the simple cubic lattice (z = 6), L ₁ = L ₃ = L ₅ = 0 ;

L o (K; h o ) = 3

2 B 1 (K; h o ) ; L ₂ (K; h _o ) = 6

5B ₁ (K; h _o ) + 5

2 B ₃ (K; h _o )

; L 4 (K; h o ) = 6

10B 3 (K; h o ) + 5

4 B 5 (K; h o )

; L ₆ (K; h _o ) = 6B ₅ (K; h _o ) :

Using Eqs. (14) and (15), the second and third terms in (28) are given by H o o =

z n=0

z−n l=0

C _z ⁿ C _z−n ^l x ^l _s m ⁿ _s E l;n (K; h o ) ; (31)

H o S o = z

n=0

C _z ^{n n} I n (K; h o ) ; (32)

with

E l;n (K; h o ) = H o A l;n (K; h o ) ; (33)

I _n (K; h _o ) = H _o B _n (K; h _o ) : (34)

Now, we have to perform the con!gurational averaging. According to the probability distribution function Q(H _i ), the internal energy U = P U _S is given by

− U NJ = 1

2 z n=0

ⁿ L P _n (K; h; p) + 1 2

z n=0

z−n l=0

C _z ⁿ C _z−n ^l x ^l m ⁿ E P _l;n (K; h; p)

+ 1 2

z n=0

C _z ^{n n} I P _n (K; h; p) : (35)

The non-zero coeLcients P L n ; E P l;n and P I n for z = 6 are listed in the appendix. Using this expression of U (T; H; p), we can determine the magnetic contribution to the speci!c heat via the standard relation

C(T; H; p) = @U (T; H; p)

@T : (36)

Figs. 5(a–c) represent a variety ofcurves showing the temperature dependence ofthe internal energy U and the speci!c heat C for the simple cubic lattice in the case ofbimodal (p = 0) and trimodal (p = 0) distributions. As seen from these !gures, U exhibits a singularity at the transition temperature. At low temperatures, these !gures show that (i) for a given relatively small value of p, the absolute value ofthe internal energy |U | ofthe system decreases with increasing strength ofthe !eld; and (ii) for a selected non-zero value of H (less than the critical value), |U | is an increasing func- tion ofthe concentration p ofthe site which is not exposed to the !eld. Moreover, we note a remarkable behaviour ofthe energy U(T = 0; H; p) ofthe ordered state at zero-temperature. Indeed, there exists a critical value p c = 0:374 of p indicating two qualitatively di;erent behaviours ofthe ground state energy as a function ofthe !eld H. Thus, when p belongs to the range 0 6 p ¡ p _c , the energy ofthe ordered state, at T = 0, does not depend on the value of H (any H ¡ H c ); while for p c ¡ p ¡ 1, the ground state energy ofthe ordered system still does not depend on relatively small values of H, but for larger H it becomes sensitive to the value of!eld strength. These behaviours are illustrated in Fig. 5(a–c) for selected values of p. On the other hand, we have plotted in the same !gures the magnetic contribution C to the speci!c heat of the system. For both the distributions (p = 0 and p = 0), its thermal behaviour shows a discontinuity at the transition temperature T _c (H; p) and the amplitude ofthe jump gradually decreases with increasing value of H. We also note, from Eq. (36), that C is proportional to 1=T ² in the limit T → ∞. Moreover, as seen from the !gures, C is remarkably depressed by decreasing the !eld strength H . It is worthy to note here that such a depression has been found recently [42] in the high magnetic concentration Ising antiferromagnet Fe 0:93 Zn 0:07 F 2 in the magnetic !eld using optical linear birefringence techniques.

4.2. Susceptibility

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

i i +

j S j ), where B is an external magnetic !eld. Consequently, identities (5), (6) and (16) are generalized into

=

1

2 tanh



 K 2

z j=1

S j + h _o 2 + b

2





; (37)

m s = 2 sinh[K _z

i=1 _i + h _o + b]

1 + 2 cosh[K _z

i=1 i + h o + b]

; (38)

Fig. 5. The temperature dependence ofthe internal energy U and the magnetic speci!c heat for the simple

cubic lattice (z = 6) for selected values of H=J with (a) p = 0, (b) p = 1=3 and (c) p = 0:6.

x s = 2 cosh[K _z

i=1 _i + h _o + b]

1 + 2 cosh[K _z

i=1 _i + h _o + b]

; (39)

where b = B. In order to obtain the state equations, we follow the procedure described in the theoretical framework. We !nd equations similar to those expressed in Eqs. (18)–(20), except that in the obtained equations, the coeLcients are not functions ofonly K, h and p but also depend on b. Thus, the state equations for the simple cubic lattice are given by

= 6

n=0 6−n

l=0

C ₆ ⁿ C _6−n ^l x ^l m ⁿ A P _l;n (K; h; p; b) ; (40)

m = 6

n=0

C ₆ ^{n n} B P _n (K; h; p; b) ; (41)

x = 6 n=0

C ₆ ^{n n} D P _n (K; h; p; b) : (42)

The initial susceptibility * per site is de!ned by

* = 1

2 (* + * m ) ; (43)

where

* = @

@B

B=0

and * _m = @m

@B

B=0

: (44)

Deriving the above Eqs. (40)–(42) with respect to B at the point B = 0, and eliminating

@x=@B, the local susceptibilities pertaining to and m are given by

* = a _o + a ₂ c _o + a ₁ b _o

1 − (a 2 c 1 + a 1 b 1 ) ; (45)

* m = b o + b 1 a o + a 2 c o + a 1 b o

1 − (a ₂ c ₁ + a ₁ b ₁ ) ; (46)

where

a _o = ⁶

n=0

6−n l=0

C ₆ ⁿ C _6−n ^l x ^l m ⁿ @ A P _l;n

@B B=0

; a ₁ = ⁶

n=0 6−n

l=0

C ₆ ⁿ C _6−n ^l nx ^l m ⁿ⁻¹ A P _l;n | _B=0 ;

a 2 = ⁶

n=0

6−n l=0

C ₆ ⁿ C _6−n ^l lx ^l−1 m ⁿ A P _l;n | B=0 ; b o = ⁶

n=0

C ₆ ^{n n} @ B P _n

@B B=0

;

b 1 = 6 n=0

C ₆ ⁿ n ⁿ⁻¹ B P _n | B=0 ;

c _o = ⁶

n=0

C ₆ ^{n n} @ D P _n

@B B=0

; c ₁ = ⁶

n=0

C ₆ ⁿ n ⁿ⁻¹ D P _n | _B=0 :

The temperature dependence of * is shown in Figs. 6(a–c) for both the distributions.

For the bimodal (p = 0) (Fig. 6a) and trimodal (p = 1=3) (Fig. 6b) cases, *(T ) is

plotted for selected values of H when the latter is less than the critical value H c (p).

When H _c does not exist, namely p ¿ p ^∗ , the system can exhibit a transition for any value of H. The corresponding thermal behaviour of * is shown in Fig. 6c for p = 0:6.

As expected, * diverges at the critical point T _c (p; H ) and presents the 1=T behaviour in the limit T → ∞. Here again, it is worthy to note that the behaviours of * with H at low and high temperatures is similar to those obtained recently [71] in Fe _0:93 Zn _0:07 F ₂ .

Appendix

For abbreviation, we de!ne new functions f(x; p; h) = P

Q(H o )f(x; h 0 ) dH o ; F P n (x; p; h) =

Q(H o )F n (x; h 0 ) dH o ; with

f(x; h _o ) = 1

2 tanh( x + h _o 2 ) ; F ₁ (x; h _o ) = 2 sinh(x + h _o )

[1 + 2 cosh(x + h o )] ; F 2 (x; h o ) = 2 cosh(x + h _o )

[1 + 2 cosh(x + h o )] :

The non-zero coeLcients appearing in (18)–(20) and (35) are given by A P o;1 = P f(K; p; h); A P 1;1 = − f(K; p; h) + P 1

2 f(2K; p; h) P ; A P 2; 1 = 5

4 f(K; p; h) P − f(2K; p; h) + P 1

4 f(3K; p; h) P ; A P 3; 1 = − 7

4 f(K; p; h) + P 7

4 f(2K; p; h) P − 3

4 f(3K; p; h) + P 1

8 f(4K; p; h) P ; A P _{4; 1} = 21

8 f(K; p; h) P − 3 P f(2K; p; h) + 27

16 f(3K; p; h) P

− 1

2 f(4K; p; h) + P 1

16 f(5K; p; h) P ; A P _{5; 1} = − 33

8 f(K; p; h) + P 165

32 f(2K; p; h) P − 45

16 f(3K; p; h) + P 11

8 f(4K; p; h) P

− 5

16 f(5K; p; h) + P 1

32 f(6K; p; h) P ;

←

Fig. 6. The temperature dependence ofthe initial magnetic susceptibility for the simple cubic lattice (z = 6)

for selected values of H=J with (a) p = 0, (b) p = 1=3 and (c) p = 0:6.

A P o;3 = − 3

4 f(K; p; h) + P 1

4 f(3K; p; h) P ; A P 1; 3 = 3

4 f(K; p; h) P − 1

4 f(2K; p; h) P − 1

4 f(3K; p; h) + P 1

8 f(4K; p; h) P ; A P 2; 3 = − 7

8 f(K; p; h) + P 1

2 f(2K; p; h) + P 3

16 f(3K; p; h) P

− 1

4 f(4K; p; h) + P 1

16 f(5K; p; h) P ; A P 3; 3 = 9

8 f(K; p; h) P − 27

32 f(2K; p; h) P − 1

16 f(3K; p; h) + P 3

8 f(4K; p; h) P

− 3

16 f(5K; p; h) + P 1

32 f(6K; p; h) P ; A P 0; 5 = 5

8 f(K; p; h) P − 5

16 f(3K; p; h) + P 1

16 f(5K; p; h) P ; A P 1; 5 = − 5

8 f(K; p; h) + P 5

32 f(2K; p; h) + P 5

16 f(3K; p; h) P − 1

8 f(4K; p; h) P

− 1

16 f(5K; p; h) + P 1

32 f(6K; p; h) P ; B P 1 = 5

16 F P 1 (K; p; h) + 1

4 F P 1 (2K; p; h) + 1

16 F P 1 (3K; p; h) ; B P 3 = − 3

4 F P 1 (K; p; h) + 1

4 F P 1 (3K; p; h) ;

B P ₅ = 5 P F ₁ (K; p; h) − 4 P F ₁ (2K; p; h) + P F ₁ (3K; p; h) ; D P _o = 5

16 F P ₂ (0; p; h) + 15

32 F P ₂ (K; p; h) + 3

16 F P ₂ (2K; p; h) + 1

32 F P ₂ (3K; p; h) ; D P 2 = − 1

4 F P 2 (0; p; h) − 1

8 F P 2 (K; p; h) − 3

4 F P 2 (2K; p; h) + 1

32 F P 2 (3K; p; h) ; D P 4 = P F 2 (0; p; h) − 1

2 F P 2 (K; p; h) − F P 2 (2K; p; h) + 1

2 F P 2 (3K; p; h) ; D P ₆ = − 20 P F ₂ (0; p; h) + 30 P F ₂ (K; p; h) − 12 P F ₂ (2K; p; h) + 2 P F ₂ (3K; p; h) ; L P _o (K; p; H ) = 3

2 B P ₁ (K; p; H ) ; L P 2 (K; p; H ) = 6

5 P B 1 (K; p; H ) + 5

2 B P 3 (K; p; H )

;

L P 4 (K; p; H ) = 6

10 P B 3 (K; p; H ) + 5

4 B P 5 (K; p; H )

; L P ₆ (K; p; H ) = 6 P B ₅ (K; p; H ) :

E P l;n =

Q(H o )H o A l;n (K; h 0 ) dH o ; I P _n =

Q(H _o )H _o B _n (K; h ₀ ) dH _o :

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