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2 Ising model on the Bethe lattice
Accepted Manuscript
The Bimodal Random Crystal Field and Biquadratic Exchange Interaction Effects for the Spin-3/2 Ising Model on the Bethe Lattice M. Karimou, E. Albayrak, A. Tessilimy, F. Hontinfinde, R. Yessoufou PII: DOI: Reference:
S0577-9073(17)31040-7 10.1016/j.cjph.2017.10.005 CJPH 363
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
17 August 2017 20 October 2017 20 October 2017
Please cite this article as: M. Karimou, E. Albayrak, A. Tessilimy, F. Hontinfinde, R. Yessoufou, The Bimodal Random Crystal Field and Biquadratic Exchange Interaction Effects for the Spin-3/2 Ising Model on the Bethe Lattice, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.10.005
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Highlights • The phase transition properties of the BEG model for spin3/2 on the BL is considered. • Both effects of random crystal field and biquadratic exchange interactions are examined.
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• D (K) is either turned on with probability 1p (q) or turned off with probability p (1q).
• Phase diagrams are obtained on the (K/J, kT/J) and (D/J,kT/J) planes on honeycomb lattice.
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• The model presents secondand firstorder phase transitions, and also the tricritical points.
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The Bimodal Random Crystal Field and Biquadratic Exchange Interaction Effects for the Spin-3/2 Ising Model on the Bethe Lattice
M. Karimou1 , E. Albayrak2 , A. Tessilimy1 , F. Hontinfinde1,3 and R. Yessoufou1,3 1
Institute of Mathematic and Physical Sciences (IMSP), Republic of Benin 2 Erciyes University, Department of Physics, 38039, Kayseri, Turkey 3 University of Abomey-Calavi, Department of Physics, Republic of Benin
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October 24, 2017 Abstract
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The phase transition properties of Blume-Emery-Griffiths (BEG) model for the spin-3/2 system are investigated on the Bethe lattice (BL) when the system is under the effect of both random crystal field (D) and biquadratic exchange interaction (K). These randomization effects are either turned on with probability 1 − p (q) or turned off with probability p (1 − q) for D and K, respectively. The phase diagrams are obtained on the (K/J, kT /J) and (D/J, kT /J) planes for given values of p and q when z = 3.0 corresponding to honeycomb lattice. Both attractive (K > 0) and repulsive (K < 0) biquadratic exchange interaction values are considered to examine its effects on the BL. It was found that the model presents either second- or first-order phase transitions and also tricritical points. It is found that the second-order phase lines follow the phase lines of regular spin-3/2 BEG model as K → ±∞ for the phase diagrams on the (K/J, kT /J) planes.
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Keywords: Spin-3/2; Randomness; BEG model; Tricritical; Bimodal; Bethe Lattice. PACS:75.10.Hk;75.30.Kz;75.10.-b
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Introduction
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The phase transitions and the existence of critical points are an indispensable part of science and technology, therefore, the inexhaustible interest for the study of phase transitions and critical phenomena of the spin models on different lattices are also important [1]. The spin-3/2 Ising model is such a spin model with four-states and two-order parameters system and thus its phase diagrams are very interesting including the second- and first-order phase transition lines 2
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in addition to some critical points. Since its discovery, it has revolved to many variations. The spin-3/2 BEG model was introduced earlier to explain the phase transition [2] in DyVO4 [3], and tricritical properties in ternary mixtures by using the mean-field approximation (MFA) [4]. The simple spin-3/2 BEG model has been studied within the MFA and by the Monte-Carlo (MC) simulation [5], a renormalization-group transformation in position space based on the MigdalKadanoff recursion relations [6], the Honmura-Kaneyoshi differential operator technique [7], the effective-field theory (EFT) [8, 9] and on the BL by using the exact recursion relations (ERR) [10]. In addition, various properties of the model were considered by using numerous theoretical techniques under random biquadratic exchange interactions such as: The phase diagrams of spin-1 BEG model in the ±K distribution on the BL [11], the random J-model with biquadratic interaction [12], the phase diagrams of the spin-3/2 transverse Ising model with random interaction [13], the theoretical study of the random mixed-bond spin-3/2 Ising model [14], the random BEG model for the spin-1 system on the BL [15], the dilute spin-3/2 Ising model with a probability-distribution technique in the EFT [16] and so on. The applications of random crystal field (RCF) on different spin systems have been getting a lot of attention nowadays. The spin-1 case was studied by using the MFA [17, 18, 19, 20], the EFT [21], the EFT and MFA [22], an expansion technique for cluster identities of localized spin systems [23], within the framework of a real-space renormalization-group approximation [24], through a real-space renormalization-group approach and the MFA [25], the exact solution on the honeycomb lattice [26] and by means of ERR’s on the BL [27]. The effects of RCF were also studied for higher spin systems, i.e. the EFT was used for the case with spin-3/2 [28, 29, 30, 31] and the MFA [32] and, the MFA for the case of spin-2 [33]. It should be mentioned that there are also some works where more than one random variables are used in different forms of random distributions. A few examples of these may be given as follows: The tricritical points in the Ising model with random bond and crystal field interactions were investigated by the use of the EFT with correlations [34], the bond diluted spin-1 transverse ferromagnetic Ising model with RCF was studied in the framework of the EFT [35] and the effects of RCF and random magnetic field in the phase diagrams and in the thermodynamic properties of spin-3/2 Blume-Capel model via the Curie-Weiss MFA was investigated [36]. In this study, the magnetic properties of spin-3/2 Ising model is going to be examined under the effect of bimodal random field distributions of both D and K: the values of D (K) are turned on with probability 1 − p (q) and turned off with probability p (1 − q), respectively. The order-parameters are obtained on the BL in terms of the ERR’s considering only the nearest-neighbor (NN) interactions between the spins with the coordination number z = 3 corresponding to honeycomb lattice. The phase diagrams are calculated exactly on the (D/J, kT /J) planes for given values of K/J when p = 0.0 with 0 ≤ q ≤ 1, when q = 1.0 with 0 ≤ p ≤ 1 and for K/J = 1.0 when q = 0.0 with 0 ≤ p ≤ 1 and on 3
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the (K/J, kT /J) planes for given values of D/J when p = 0.0 with 0 ≤ q ≤ 1, when q = 1.0 with 0 ≤ p ≤ 1 and for D/J = 1.0 when p = 1.0 with 0 ≤ q ≤ 1. It is found that the model generates both second- and first-order phase transitions at the temperatures Tc and Tt respectively and tricritical points (TCP). The remainder of this work is built as follows: The ERR’s for the spin3/2 BEG model on the BL, the random distributions of D and K and the order-parameters, i.e. the magnetization and quadrupolar moment, in terms of the ERR’s are specified in Section 2. The phase diagrams are illustrated and discussed in Section 3. The last section is devoted to the conclusions.
The Formulations in terms of ERR’s on the Bethe Lattice
hi,ji
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The spin-3/2 BEG model is given by the Hamiltonian X X X Kij Si2 Sj2 − Di Si2 , Si Sj − H = −J
(1)
i
hi,ji
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where Si takes the values ±3/2 and ±1/2 and J is the bilinear exchange interaction parameter between the NN spins. The system is under the influence of both random crystal and random biquadratic exchange interaction which are denoted by Di for all sites and Kij for the NN biquadratic interactions. In order to obtain the phase diagrams, one has to obtain the order-parameters first. Since the spin-3/2 model is a four-state and two order-parameter system, the order-parameters are the magnetization (dipole moment) and quadrupole moment. Thus to obtain their exact formulation on the BL, one should start from a central spin S0 which may be called as the first generation spin. S0 has z NN’s which form the second generation spins S1 . Each spin in the second generation is connected to (z − 1) NN’s. Therefore, in total, the second generation has [z(z − 1)] NN’s which form the third generation spins S2 and so on to infinity. Each generation of spins is called as the shells of the BL and the number of generations goes to infinity in the thermodynamic limit, i.e n −→ ∞, as illustrated in Fig.1. It should also be mentioned that the BL is an infinite Cayley or regular tree, i.e., a connected graph without circuits, and historically gets its name from the fact that its partition function is exactly that of an Ising model in the Bethe approximation [37, 38]. Please see also the details of the BL calculations with ERRs [39], the demonstration of the behavior on Bethe or Bethe-like lattices are qualitatively correct even when conventional meanfield theories fail [40], the entropy of polydisperse chains placed on a lattice was considered and it was solved exactly on the BL [41] and on Husimi [42] lattices. In order to obtain the order parameters in terms of ERR’s, one usually starts with the partition function which is defined as X X Z= e−βH = P (Spc), (2) All Conf igurations
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Spc
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Xn(i,ij) =
gn (+ 32 ) 9 81 z−1 z−1 β( −9 β( 49 J+ 94 Di + 81 16 Kij ) X 4 J+ 4 Di + 16 Kij ) Y n−1 + e n−1 1 = [e gn (− 2 )
3
1
9
−3 1 9 4 J+ 4 Di + 16 Kij )
]/[eβ(
3
9
9
−1 1 1 4 J+ 4 Di + 16 Kij )
z−1 Zn−1 + eβ( 4 J+ 4 Di + 16 Kij ) ],
z−1 + eβ( + eβ( 4 J+ 4 Di + 16 Kij ) Zn−1
Yn(i,ij) =
9
3
9
1
1
9
−1 1 1 4 J+ 4 Di + 16 Kij )
−3 9 9 4 J+ 4 Di + 16 Kij ) 1
1
z−1 Xn−1
1
z−1 Zn−1 + eβ( 4 J+ 4 Di + 16 Kij ) ]
gn (+ 21 ) −3 3 9 9 9 9 z−1 z−1 = [eβ( 4 J+ 4 Di + 16 Kij ) Xn−1 + eβ( 4 J+ 4 Di + 16 Kij ) Yn−1 gn (− 12 )
1
1
−1 1 1 4 J+ 4 Di + 16 Kij )
]/[eβ(
9
9
−1 1 1 4 J+ 4 Di + 16 Kij )
z−1 Zn−1 + eβ( 4 J+ 4 Di + 16 Kij ) ].
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z−1 + eβ( 4 J+ 4 Di + 16 Kij ) Zn−1 + eβ( 3
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and
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z−1 Zn−1 + eβ( 4 J+ 4 Di + 16 Kij ) ]/[eβ(
z−1 + eβ( 4 J+ 4 Di + 16 Kij ) Yn−1 + eβ(
Zn(i,ij) =
1
z−1 Xn−1
gn (− 32 ) 9 81 z−1 z−1 β( −9 β( 94 J+ 94 Di + 81 4 J+ 4 Di + 16 Kij ) X 16 Kij ) Y n−1 + e n−1 1 = [e gn (− 2 )
−3 1 9 4 J+ 4 Di + 16 Kij ) 3
−3 9 9 4 J+ 4 Di + 16 Kij )
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z−1 + eβ( + eβ( 4 J+ 4 Di + 16 Kij ) Yn−1
+ eβ(
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where P (Spc) is the unnormalized probability distribution and β = 1/(kT ) with k and T are the Boltzmann constant and absolute temperature, respectively. It should be noted that one of us have already obtained the ERR’s on the BL [10], therefore, they are directly taken form there without going into any details. The ERR’s are represented as the ratios of the partial partition functions of each of the separate branches on the BL, i.e. four gn (S)’s exist, therefore, there are only three ERR’s present for the spin-3/2 case. The ratio of gn (S)’s, the ERR’s, is actually a measure of how one distribution is stronger than the other for given system parameters. Thus, they are given for the spin-3/2 BEG model on the BL as follows:
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The RCF distribution influences all the spin-3/2 sites and it is taken as P (Di ) = pδ(Di ) + (1 − p)δ(Di − D),
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where the first term pδ(Di ) imposes that an amount of p spins are free from the influence of D while the second term indicates that an amount of 1 − p spins are under the influence of the crystal field. In order to add the effects of RCF into the ERR’s, one should integrate Eq.(3) over the P (Di ) distribution to obtain the RCF implemented ERR’s which is done as Z (ij) Xn = Xn(i,ij) P (Di )dDi Z = Xn(i,ij) [pδ(Di ) + (1 − p)δ(Di − D)]dDi , (5) 5
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=
Yn(i,ij) P (Di )dDi
=
Z
Yn(i,ij) [pδ(Di ) + (1 − p)δ(Di − D)]dDi
=
Z
Zn(i,ij) P (Di )dDi
and Zn(ij)
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Z
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Yn(ij)
Zn(i,ij) [pδ(Di ) + (1 − p)δ(Di − D)]dDi .
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In order to include also the random distribution effects of K, one should consider another bimodal random distribution having the same form as the bimodal RCF distribution and given as P (Kij ) = qδ(Kij − K) + (1 − q)δ(Kij ),
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where K is assumed to be distributed throughout the BL with probabilities q and 1 − q for which K is either turned on attractively or repulsively and turned off, i.e K = 0, respectively. We also need to take the second integrations over the P (Kij ) for Eqs.(5)-(7). Thus all the randomness of crystal field and biquadratic interactions included ERR’s can be given for spin-3/2 as Z Z (ij) Xn = Xn P (Kij )dKij = Xn(ij) [qδ(Kij − K) + (1 − q)δ(Kij )]dKij
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81 9 81 3 9 3 9 z−1 z−1 h eβ( 49 J+ 16 K) z−1 X + eβ(− 4 J+ 16 K) Yn−1 + eβ( 4 J+ 16 K) Zn−1 + eβ(− 4 J+ 16 K) i = qp β(− 3 J+ 9 K) n−1 3 9 1 1 1 1 z−1 z−1 z−1 4 16 e Xn−1 + eβ( 4 J+ 16 K) Yn−1 + eβ(− 4 J+ 16 K) Zn−1 + eβ( 4 J+ 16 K) h 9 9 81 9 9 81 3 1 9 z−1 z−1 z−1 + q(1 − p) eβ( 4 J+ 4 D+ 16 K) Xn−1 + eβ(− 4 J+ 4 D+ 16 K) Yn−1 + eβ( 4 J+ 4 D+ 16 K) Zn−1 i h 1 9 3 9 9 3 9 9 3 z−1 z−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 + eβ(− 4 J+ 4 D+ 16 K) / eβ(− 4 J+ 4 D+ 16 K) Xn−1 i 1 1 1 1 1 1 z−1 + eβ(− 4 J+ 4 D+ 16 K) Zn−1 + eβ( 4 J+ 4 D+ 16 K)
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3βJ z−1 z−1 z−1 h e 9βJ − 9βJ − 3βJ 4 X 4 Y 4 Z 4 i n−1 + e n−1 + e n−1 + e p(1 − q) 3βJ 3βJ βJ βJ z−1 z−1 z−1 e− 4 Xn−1 + e 4 Yn−1 + e− 4 Zn−1 +e 4 h 9 9 9 9 3 1 z−1 z−1 z−1 + eβ(− 4 J+ 4 D) Yn−1 + eβ( 4 J+ 4 D) Zn−1 (1 − p)(1 − q) eβ( 4 J+ 4 D) Xn−1 i h 3 1 3 9 3 9 z−1 z−1 eβ(− 4 J+ 4 D) / eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 i 1 1 1 1 z−1 eβ( 4 J+ 4 D) Zn−1 + eβ(− 4 J+ 4 D) ,
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Yn(ij) [qδ(Kij − K) + (1 − q)δ(Kij )]dKij
9 3 9 z−1 z−1 h eβ(− 94 J+ 81 β( 49 J+ 81 β(− 34 J+ 16 K) z−1 16 K) X 16 K) Y Zn−1 + eβ( 4 J+ 16 K) i n−1 + e n−1 + e = qp β(− 3 J+ 9 K) z−1 3 9 1 1 1 1 z−1 z−1 4 16 e Xn−1 + eβ( 4 J+ 16 K) Yn−1 + eβ(− 4 J+ 16 K) Zn−1 + eβ( 4 J+ 16 K)
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h 9 9 81 9 9 81 3 1 9 z−1 z−1 z−1 + q(1 − p) eβ(− 4 J+ 4 D+ 16 K) Xn−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 + eβ(− 4 J+ 4 D+ 16 K) Zn−1 i h 3 3 1 9 9 9 9 9 3 z−1 z−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 + eβ( 4 J+ 4 D+ 16 K) / eβ(− 4 J+ 4 D+ 16 K) Xn−1 i 1 1 1 1 1 1 z−1 + eβ( 4 J+ 4 D+ 16 K) + eβ(− 4 J+ 4 D+ 16 K) Zn−1
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Z
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Zn(ij) [qδ(Kij − K) + (1 − q)δ(Kij )]dKij
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Zn
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9 3 9 1 1 1 1 z−1 z−1 h eβ( 43 J+ 16 K) z−1 Xn−1 + eβ(− 4 J+ 16 K) Yn−1 + eβ( 4 J+ 16 K) Zn−1 + eβ(− 4 J+ 16 K) i qp β(− 3 J+ 9 K) z−1 3 9 1 1 1 1 z−1 z−1 4 16 e Xn−1 + eβ( 4 J+ 16 K) Yn−1 + eβ(− 4 J+ 16 K) Zn−1 + eβ( 4 J+ 16 K) h 3 9 9 3 9 9 1 1 1 z−1 z−1 z−1 + eβ(− 4 J+ 4 D+ 16 K) Yn−1 + eβ( 4 J+ 4 D+ 16 K) Zn−1 q(1 − p) eβ( 4 J+ 4 D+ 16 K) Xn−1 i h 1 1 1 9 9 3 9 9 3 z−1 z−1 eβ(− 4 J+ 4 D+ 16 K) / eβ(− 4 J+ 4 D+ 16 K) Xn−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 i 1 1 1 1 1 1 z−1 eβ(− 4 J+ 4 D+ 16 K) Zn−1 + eβ( 4 J+ 4 D+ 16 K)
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9βJ 3βJ z−1 z−1 z−1 h e− 9βJ − 3βJ 4 X 4 Y 4 Z 4 i n−1 + e n−1 + e n−1 + e + p(1 − q) 3βJ βJ 3βJ βJ z−1 z−1 z−1 e− 4 Xn−1 + e 4 Bn−1 + e− 4 Zn−1 +e 4 h 9 9 9 1 9 3 z−1 z−1 z−1 + (1 − p)(1 − q) eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 + eβ(− 4 J+ 4 D) Zn−1 i h 3 3 1 9 9 3 z−1 z−1 + eβ( 4 J+ 4 D) / eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 i 1 1 1 1 z−1 + eβ(− 4 J+ 4 D) Zn−1 + eβ( 4 J+ 4 D)
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βJ z−1 z−1 z−1 h e 3βJ − 3βJ − βJ 4 X 4 Y 4 Z 4 i n−1 + e n−1 + e n−1 + e + p(1 − q) 3βJ 3βJ βJ βJ z−1 z−1 z−1 e− 4 Xn−1 + e 4 Yn−1 + e− 4 Zn−1 +e 4 h 3 9 3 9 1 1 z−1 z−1 z−1 + (1 − p)(1 − q) eβ( 4 J+ 4 D) Xn−1 + eβ(− 4 J+ 4 D) Yn−1 + eβ( 4 J+ 4 D) Zn−1 i h 1 1 3 9 3 9 z−1 z−1 + eβ(− 4 J+ 4 D) / eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 i 1 1 1 1 z−1 + eβ(− 4 J+ 4 D) Zn−1 + eβ( 4 J+ 4 D) .
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After having obtained the random D and K implemented ERR’s, the orderparameters, i.e. magnetization and quadrupolar moment, can be calculated in terms of ERR’s. Note that all the sites of the BL are equivalent deep inside in the thermodynamic limit. Therefore, one can pick a central spin, S0 , and calculate its order-parameters accordingly. The magnetization or dipolar moment for the central spin or any other spin deep inside the BL can be given as 1 h 3e 4 (Xnz − Ynz ) + e 4 (Znz − 1) i βD 2 e 9βD 4 (X z + Y z ) + e 4 (Z z + 1) n n n 9βD
M =< S0 >=
βD
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and similarly the quadrupolar moment of the central spin is found as 1 h 9e 4 (Xnz + Ynz ) + e 4 (Znz + 1) i . βD 4 e 9βD 4 (X z + Y z ) + e 4 (Z z + 1) n n n 9βD
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βD
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The Phase Diagrams
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These equations can be written in terms of the reduced quantities. To do that one simply sets J = 1 and replaces D, K and β = 1/kT by D/J, K/J and βJ = J/kT , respectively. In the next section, we present the phase diagrams of the model obtained from the analysis of thermal variations of the orderparameters.
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The phase diagrams of the random spin-3/2 BEG model are studied by the analysis of thermal variations of the order-parameters. They are presented on the (K/J, kT /J) and on the (D/J, kT /J) planes for given values of the rest of the system parameters. Our calculations are carried out for z = 3 which corresponds to honeycomb lattice. It should be mentioned that our model reduces to four specific models depending on the p and q values as: (i) when p = 0 and q = 1, the model reduces to the regular spin-3/2 BEG model [10], (ii) the p = 0 and q = 0 values reduce the model to the Blume-Capel model see for example [43], (iii) when p = 1 and q = 0, D and K are turned off and the Hamiltonian only includes the bilinear exchange interaction and lastly (iv) when p = 1 and q = 1, our Hamiltonian contains only J and K with zero crystal field. For the intermediate values of p and q, we have the case of random K and random D model. In the phase diagrams, the second- and first-order phase transition lines are shown with solid and dotted lines and they are simply named as the Tc - and Tt -lines, respectively. The TCP’s are the points at which the Tc - and Tt -lines meet are shown with filled circles. The ferromagnetic and paramagnetic phase regions are indicated by (F ) and (P ), respectively. Note also that the places of the second- and first-order phase transition temperatures are determined from the thermal analysis of the order-parameters: The magnetization M goes to zero continuously and quadrupole moment Q makes a little cusp at the Tc ’s, while at the Tt ’s, M presents a jump discontinuity to zero and Q makes a discontinuity. The first phase diagrams are obtained on the (K/J, kT /J) planes for given values of D/J = 3.0, 1.0, 0.0, −1.0, −3.0, −5.0 and p = 0 when 0 ≤ q ≤ 1 with ∆q = 0.1 as shown in Figs.2(a)-(f). They correspond to the cases (i) and (ii) for q = 1 and 0, respectively, and to the random K and D model for the intermediate values of q. It is clear that the sites are under the influence of D/J when p = 0 and K/J values are turned off when q = 0. In this case, the Tc -lines are independent from K/J, i.e. they are straight lines on these planes of phase diagrams. The temperatures of these lines decrease as D/J reduces to lower values. This is caused by the competition between J and D as expected. The model becomes the well-known BEG model when q = 1. As seen from the figures, only the Tc -lines are seen for the first four values of D/J. They start 8
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from lower temperatures for higher q in the repulsive region, i.e. K/J < 0.0, then all the lines coincide at K/J = 0.0, and then their temperatures increase for higher q’s in the attractive region K/J > 0.0. In addition, they become constant as |K/J| becomes large. When D/J = −3.0, we see that the Tc - and Tt -lines combine at TCP’s for q = 0.8, 0.9 and 1.0. The first TCP’s are seen at higher temperatures for higher q values and the second ones are along the q = 0-line. The rest of the lines are in the form of the Tc -lines for the rest of the q values. Either form of the lines terminate on the straight line for q = 0.0 at higher K/J’s for lower q’s. For D/J = −5.0 we see the same behaviors. But now the TCP’s are also seen when q = 0.6 and 0.7 and the lines terminate at higher K/J’s. The wiggles of the Tc -lines for q = 0 in Figs.2 (a)-(c) caused by the competition when K is repulsive and D is positive. This competition is obvious from the Hamiltonian of our model. Since both has the squares of the spins which are always positive. So when K is repulsive, i.e. K < 0.0, and D is positive they oppose to each other which causes the lines wiggle. As seen these wiggles start disappearing as K and D gets closer to each other signwise and eventually disappear, see for example Fig.2(d) where both K and D are negative. The second phase diagrams are also obtained on the (K/J, kT /J) planes for given values of D/J=7.0, 1.0, -1.0, -1.75, -3.0, -7.0 and for q = 1 when 0 ≤ p ≤ 1 with ∆p = 0.1 as shown in Figs.3(a)-(f). They correspond to the cases (i) and (iv) and to the intermediate values of p. The Tc -lines start from the same lower constant temperatures and terminate at higher constant temperatures as |K/J| becomes large for each D/J’s. The region corresponding to 0 < p < 1 shrinks to one line when D/J = 0.0, but it grows as |D/J| becomes large. The Tt -lines are only seen when q = 0 for D/J = −1.75, −3.0, −7.0 which becomes longer as D/J becomes more negative. In this case, they start from higher TCP’s and end at lower TCP’s. The last phase diagram on this plane, i.e. Fig.4, is obtained for D/J = 1.0 and p = 1 when 0 < q < 1 as an example. In this case, they correspond to the cases (iii) and (iv), and for the intermediate values of q. As seen from the figure, they are very similar with the Fig.2(c). The next three phase diagrams are obtained on the (D/J, kT /J) planes for given values K/J. The first one is obtained for the case (i) and (ii) and for the intermediate values of q. The Tc -lines become constant as |D/J| becomes large for all K/J values. As seen from the Figs.5(a)-(c), the model presents both Tc - and Tt -lines combined at two TCP’s for higher q values for the attractive case. As seen, the TCP’s at higher temperatures go down in temperature as q is lowered. The second ones, which are seen at higher D/J’s for lower q’s, are lined up along the constant portion of q = 0 line. Figs.5(d)-(f) correspond to the repulsive case for which the lines are only in the form of Tc -lines. The q = 1 case is interesting, since the Tc -lines start from constant lower temperatures for K/J = −2.0 and -7.0, then as D/J increases they increase in temperature, then they become constant again, for the further increase of D/J they increase in temperature and then they become constant again. The first constant portion separate the F phase with ±1/2 state from the P phase, the last portion 9
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Conclusion
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separates the F phase with ±3/2 state from the P phase, and the middle portion separates the F phase with the intermediate spin states from the P phase, see for example the temperature variations of magnetization as shown in Fig.6. It should again be noted that the wiggles in Figs.5 (d)-(f) are caused by the competition between repulsive K and positive D values as explained in Fig.2. The next ones, i.e. Figs.7(a)-(f), correspond to the cases (i) and (iv) for the intermediate values of p and are obtained for K/J=3.0, 0.1, 0.0, -0.5, -2.0 and -3.0. The Tc -lines for p = 1 and q = 1 are just straight lines since they are independent from D/J values. Their temperatures decrease as K/J’s decrease. Again, the Tc -lines combined with the Tt -lines connected to two TCP’s are only seen for the case with p = 0 in the attractive case. The lines are just Tc -lines for the rest of the p values for all K/J’s. As seen, the Tc -lines are seen at lower temperatures for lower p’s at negative D/J’s in Fig.7(a) and (b). Then they all combine on a straight line as D/J’s increase in the negative region. It is clear that all the lines combine at D/J = 0.0. Then the Tc -lines increase in temperature for lower p’s for positive D/J values. As K/J is decreased, for example as given in Fig. 7(f), all the Tc -lines start from the same temperature as for the p = 1-line, then as D/J increase they spread and the Tc -lines are seen at higher temperatures for lower p’s again. Our last phase diagram is obtained for the cases (ii) and (iii) and for the intermediate values of p and K/J=1.0, see Fig.8. They are very similar with the Fig.7(f), i.e. the lines are in the form of Tc -lines In this section we have presented our phase diagrams and tried to justify their behaviors for the cases (i)-(iv) and for the intermediate values of p and q’s. The next section gives our conclusions, comparisons and some results.
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In this paper, we have studied the phase diagrams of the spin-3/2 Ising system with the disorders associated with the crystal field and biquadratic exchange interactions randomly. The ERR’s are employed on the BL for the calculation of the order-parameters. The disorders are associated with three ingredients of the physical system. The first is the temperature that induces order-disorder phase transitions due to thermal agitation on the lattice. The second one is associated with the random crystal field, which is governed by a probability distribution function of the bimodal type. And the last one corresponds to the random biquadratic exchange interaction, which is also governed by a probability distribution function in bimodal type. All together we have four parameters (J, T, D, K) competing with each other to maintain the system ordered or disordered. When our model reduces to the well-known models such as the BC and the BEG model, it is found that overall similarities are observed. It should be noted that this work is the first work considering both random D and K effects, thus a direct comparison is not possible.
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References [1] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press, New York, 1971; C.-K. Hu, Chin. J. Phys. 52 (2014) 1; N. Sh. Izmailian, C.-K. Hu, Physica A 254 (1998) 198; C.-K. Hu, N. Sh. Izmailian, Phys. Rev. E 58 (1998) 1644.
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[2] J. Sivardi´ere, M. Blume, Phys. Rev. B 5 (1972) 1126. [3] A. H. Cooke, D. M. Martin, M. R. Wells, J. Phys. (Paris), Colloq. 32 (1971) C1 488; A. H. Cooke, D. M. Martin, M. R. Wells, Solid State Commun. 9 (1971) 519. [4] S. Krinsky, D. Mukamel, Phys. Rev. B 11 (1975) 399.
[5] F. C . S´a Barreto, O. F. De Alcantara Bonfim, Physica A 172 (1991) 378.
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[6] A. Bakchich, A. Bassir, A. Benyoussef, Physica A 195 (1993) 188. [7] T. Kaneyoshi, M. Jaˇscur, Phys. Lett. A 177 (1993) 172. [8] T. Kaneyoshi, J. Phys. Soc. Jpn 56 (1987) 4199.
[9] I. P. Fittipalde, T. Kaneyoshi, J. Phys. Condens. Matter 1 (1989) 6513. [10] E. Albayrak, M. Keskin, J. Magn. Magn. Maters. 241 (2002) 249.
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[11] E. Albayrak, J. Magn. Magn. Maters. 386 (2015) 20.
[12] A. Yigit, E. Albayrak, J. Supercond. Nov. Magn. 29 (2016) 2535.
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[13] M. Kerouad, A. Elatri, A. Ainane, M. Saber, Phys. Status Solidi B 195 (1996) 519. [14] A. Ainane, A. Elatri, M. Kerouad, M. Saber, J. Magn. Magn. Maters. 146 (1995) 283.
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[15] E. Albayrak, Physica B 479 (2015) 107.
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[16] M. Kerouad, M. Saber, J. W. Tucker, J. Magn. Magn. Maters. 132 (1994) 223. [17] A. Benyoussef, T. Biaz, M. Saber, M. Touzani, J. Phys. C: Solid State Phys. 20 (1987) 5349.
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[18] C. E. I. Carneiro, V. B. Henriques, S. R. Salinas, J. Phys. Condens. Matter 1 (1989) 3687. [19] N. Boccara, A. Elkenz, M. Saber, J. Phys. Condens. Matter 1 (1989) 5721. [20] C. E. I. Carneiro, V. B. Henriques, S. R. Salinas, J. Phys A: Math. Gen. 23 (1990) 3383.
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[21] T. Kaneyoshi, J. Phys. C: Solid State Phys. 21 (1988) L679-L682. [22] T. Kaneyoshi, J. Mielnicki, J. Phys. Condens. Matter 2 (1990) 8773. [23] A. Benyoussef, H. Es-Zahraouy, J. Phys. Condens. Matter 6 (1994) 3411. [24] N. S. Branco, B. M. Boechat, Phys. Rev. B 56 (1997) 11673.
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[25] N. S. Branco, Phys. Rev. B 60 (1999) 1033. [26] V. Urumov, J. Phys. Condens. Matter 1 (1989) 7037. [27] E. Albayrak, Physica A 390 (2011) 1529.
[28] Y. -Q. Liang, G. -Z. Wei, Q. Zhang, Z. -H. Xim, J. Magn. Magn. Maters. 267 (2003) 275.
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[29] Y. -Q. Liang, G. -Z. Wei, Q. Zhang, Z. -H. Xim, J. Magn. Magn. Maters. 265 (2003) 305. [30] Y. -Q. Liang, G. -Z. Wei, Q. Zhang, Z. -H. Xim, G. -L. Song, J. Magn. Magn. Maters. 284 (2004) 47.
[31] Y. -Q. Liang, G. -Z. Wei, G. -L. Song Q, Phys. Stat. Solidi (b) 241 (2004) 3636.
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[32] L. Bahmad, A. Benyoussef, A. El Kenz, J. Magn. Magn. Maters. 320 (2008) 397. [33] L. Bahmad, A. Benyoussef, A. El Kenz, Phys. Rev. B 76 (2007) 094412. [34] T. Kaneyoshi, J. Phys. C: Solid State Phys. 19 (1986) L557-L561.
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[35] L. L. Deng, S. L. Yan, J. Magn. Magn. Maters 251 (2002) 138.
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[36] W. P. da Silva, P. H. Z. de Arruda, T. M. Tunes, M. Godoy, A. S. Arruda, J. Magn. Magn. Maters. 422 (2017) 367 [37] M. Kurata, R. Kikuchi, T. Watari, J. Chem. Phys. 21 (1953) 434. [38] C. Domb, Adv. Phys. 9 (1960) 283.
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[39] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.
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[40] P. D. Gujrati, Phys. Rev. Lett. 74 (1995) 809. [41] M. A. Neto, J. F. Stilck, J. Chem. Phys. 128 (2008) 184904. [42] M. A. Neto, J. F. Stilck, J. Chem. Phys. 138 (2013) 044902. ¨ [43] E. Albayrak, M. Keskin, J. Magn. Magn. Maters. 218 (2000) 121; O. Ozsoy, E. Albayrak, M. Keskin, Physica A 304 (2002) 443. 12
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Figure Captions Fig.1- The Bethe lattice of coordination number z = 3. The filled circles denote the spins and the dotted circles represent the shells of the BL.
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Fig.2- The phase diagrams on the (K/J, kT /J) planes for given values of D/J = 3.0, 1.0, 0.0, −1.0, −3.0, −5.0 and for p = 0 when 0 ≤ q ≤ 1 with ∆q = 0.1.
Fig.3- The phase diagrams on the (K/J, kT /J) planes for given values of D/J=7.0, 1.0, -1.0, -1.75, -3.0, -7.0 and for q = 1 when 0 ≤ p ≤ 1 with ∆p = 0.1.
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Fig.4- The phase diagrams on the (K/J, kT /J) plane for D/J = 1.0 and p = 1 when 0 ≤ q ≤ 1 with ∆q = 0.1. Fig.5- The phase diagrams on the (D/J, kT /J) planes for K/J=7.0, 2.0, 1.0, -1.0, -2.0, -7.0 and for p = 0 when 0 ≤ q ≤ 1 with ∆q = 0.1.
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Fig.6- The temperature variation of magnetization for K/J = −7.0, p = 0 and q = 1 when D/J=0.0, 20.0 and 50.0.
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Fig.7- The phase diagrams on the (D/J, kT /J) planes for K/J=3.0, 0.1, 0.0, -0.5, -2.0, -3.0 and for q = 1 when 0 ≤ p ≤ 1 with ∆p = 0.1.
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Fig.8- The phase diagrams on the (D/J, kT /J) plane for K/J = 1.0 and q = 0 when 0 ≤ p ≤ 1 with ∆p = 0.1.
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The Bimodal Random Crystal Field and Biquadratic Exchange Interaction Effects for the Spin-3/2 Ising Model on the Bethe Lattice M. Karimou, E. Albayrak, A. Tessilimy, F. Hontinfinde, R. Yessoufou PII: DOI: Reference:
S0577-9073(17)31040-7 10.1016/j.cjph.2017.10.005 CJPH 363
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
17 August 2017 20 October 2017 20 October 2017
Please cite this article as: M. Karimou, E. Albayrak, A. Tessilimy, F. Hontinfinde, R. Yessoufou, The Bimodal Random Crystal Field and Biquadratic Exchange Interaction Effects for the Spin-3/2 Ising Model on the Bethe Lattice, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.10.005
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Highlights • The phase transition properties of the BEG model for spin3/2 on the BL is considered. • Both effects of random crystal field and biquadratic exchange interactions are examined.
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• D (K) is either turned on with probability 1p (q) or turned off with probability p (1q).
• Phase diagrams are obtained on the (K/J, kT/J) and (D/J,kT/J) planes on honeycomb lattice.
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• The model presents secondand firstorder phase transitions, and also the tricritical points.
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The Bimodal Random Crystal Field and Biquadratic Exchange Interaction Effects for the Spin-3/2 Ising Model on the Bethe Lattice
M. Karimou1 , E. Albayrak2 , A. Tessilimy1 , F. Hontinfinde1,3 and R. Yessoufou1,3 1
Institute of Mathematic and Physical Sciences (IMSP), Republic of Benin 2 Erciyes University, Department of Physics, 38039, Kayseri, Turkey 3 University of Abomey-Calavi, Department of Physics, Republic of Benin
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October 24, 2017 Abstract
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The phase transition properties of Blume-Emery-Griffiths (BEG) model for the spin-3/2 system are investigated on the Bethe lattice (BL) when the system is under the effect of both random crystal field (D) and biquadratic exchange interaction (K). These randomization effects are either turned on with probability 1 − p (q) or turned off with probability p (1 − q) for D and K, respectively. The phase diagrams are obtained on the (K/J, kT /J) and (D/J, kT /J) planes for given values of p and q when z = 3.0 corresponding to honeycomb lattice. Both attractive (K > 0) and repulsive (K < 0) biquadratic exchange interaction values are considered to examine its effects on the BL. It was found that the model presents either second- or first-order phase transitions and also tricritical points. It is found that the second-order phase lines follow the phase lines of regular spin-3/2 BEG model as K → ±∞ for the phase diagrams on the (K/J, kT /J) planes.
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Keywords: Spin-3/2; Randomness; BEG model; Tricritical; Bimodal; Bethe Lattice. PACS:75.10.Hk;75.30.Kz;75.10.-b
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Introduction
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The phase transitions and the existence of critical points are an indispensable part of science and technology, therefore, the inexhaustible interest for the study of phase transitions and critical phenomena of the spin models on different lattices are also important [1]. The spin-3/2 Ising model is such a spin model with four-states and two-order parameters system and thus its phase diagrams are very interesting including the second- and first-order phase transition lines 2
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in addition to some critical points. Since its discovery, it has revolved to many variations. The spin-3/2 BEG model was introduced earlier to explain the phase transition [2] in DyVO4 [3], and tricritical properties in ternary mixtures by using the mean-field approximation (MFA) [4]. The simple spin-3/2 BEG model has been studied within the MFA and by the Monte-Carlo (MC) simulation [5], a renormalization-group transformation in position space based on the MigdalKadanoff recursion relations [6], the Honmura-Kaneyoshi differential operator technique [7], the effective-field theory (EFT) [8, 9] and on the BL by using the exact recursion relations (ERR) [10]. In addition, various properties of the model were considered by using numerous theoretical techniques under random biquadratic exchange interactions such as: The phase diagrams of spin-1 BEG model in the ±K distribution on the BL [11], the random J-model with biquadratic interaction [12], the phase diagrams of the spin-3/2 transverse Ising model with random interaction [13], the theoretical study of the random mixed-bond spin-3/2 Ising model [14], the random BEG model for the spin-1 system on the BL [15], the dilute spin-3/2 Ising model with a probability-distribution technique in the EFT [16] and so on. The applications of random crystal field (RCF) on different spin systems have been getting a lot of attention nowadays. The spin-1 case was studied by using the MFA [17, 18, 19, 20], the EFT [21], the EFT and MFA [22], an expansion technique for cluster identities of localized spin systems [23], within the framework of a real-space renormalization-group approximation [24], through a real-space renormalization-group approach and the MFA [25], the exact solution on the honeycomb lattice [26] and by means of ERR’s on the BL [27]. The effects of RCF were also studied for higher spin systems, i.e. the EFT was used for the case with spin-3/2 [28, 29, 30, 31] and the MFA [32] and, the MFA for the case of spin-2 [33]. It should be mentioned that there are also some works where more than one random variables are used in different forms of random distributions. A few examples of these may be given as follows: The tricritical points in the Ising model with random bond and crystal field interactions were investigated by the use of the EFT with correlations [34], the bond diluted spin-1 transverse ferromagnetic Ising model with RCF was studied in the framework of the EFT [35] and the effects of RCF and random magnetic field in the phase diagrams and in the thermodynamic properties of spin-3/2 Blume-Capel model via the Curie-Weiss MFA was investigated [36]. In this study, the magnetic properties of spin-3/2 Ising model is going to be examined under the effect of bimodal random field distributions of both D and K: the values of D (K) are turned on with probability 1 − p (q) and turned off with probability p (1 − q), respectively. The order-parameters are obtained on the BL in terms of the ERR’s considering only the nearest-neighbor (NN) interactions between the spins with the coordination number z = 3 corresponding to honeycomb lattice. The phase diagrams are calculated exactly on the (D/J, kT /J) planes for given values of K/J when p = 0.0 with 0 ≤ q ≤ 1, when q = 1.0 with 0 ≤ p ≤ 1 and for K/J = 1.0 when q = 0.0 with 0 ≤ p ≤ 1 and on 3
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the (K/J, kT /J) planes for given values of D/J when p = 0.0 with 0 ≤ q ≤ 1, when q = 1.0 with 0 ≤ p ≤ 1 and for D/J = 1.0 when p = 1.0 with 0 ≤ q ≤ 1. It is found that the model generates both second- and first-order phase transitions at the temperatures Tc and Tt respectively and tricritical points (TCP). The remainder of this work is built as follows: The ERR’s for the spin3/2 BEG model on the BL, the random distributions of D and K and the order-parameters, i.e. the magnetization and quadrupolar moment, in terms of the ERR’s are specified in Section 2. The phase diagrams are illustrated and discussed in Section 3. The last section is devoted to the conclusions.
The Formulations in terms of ERR’s on the Bethe Lattice
hi,ji
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The spin-3/2 BEG model is given by the Hamiltonian X X X Kij Si2 Sj2 − Di Si2 , Si Sj − H = −J
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where Si takes the values ±3/2 and ±1/2 and J is the bilinear exchange interaction parameter between the NN spins. The system is under the influence of both random crystal and random biquadratic exchange interaction which are denoted by Di for all sites and Kij for the NN biquadratic interactions. In order to obtain the phase diagrams, one has to obtain the order-parameters first. Since the spin-3/2 model is a four-state and two order-parameter system, the order-parameters are the magnetization (dipole moment) and quadrupole moment. Thus to obtain their exact formulation on the BL, one should start from a central spin S0 which may be called as the first generation spin. S0 has z NN’s which form the second generation spins S1 . Each spin in the second generation is connected to (z − 1) NN’s. Therefore, in total, the second generation has [z(z − 1)] NN’s which form the third generation spins S2 and so on to infinity. Each generation of spins is called as the shells of the BL and the number of generations goes to infinity in the thermodynamic limit, i.e n −→ ∞, as illustrated in Fig.1. It should also be mentioned that the BL is an infinite Cayley or regular tree, i.e., a connected graph without circuits, and historically gets its name from the fact that its partition function is exactly that of an Ising model in the Bethe approximation [37, 38]. Please see also the details of the BL calculations with ERRs [39], the demonstration of the behavior on Bethe or Bethe-like lattices are qualitatively correct even when conventional meanfield theories fail [40], the entropy of polydisperse chains placed on a lattice was considered and it was solved exactly on the BL [41] and on Husimi [42] lattices. In order to obtain the order parameters in terms of ERR’s, one usually starts with the partition function which is defined as X X Z= e−βH = P (Spc), (2) All Conf igurations
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Xn(i,ij) =
gn (+ 32 ) 9 81 z−1 z−1 β( −9 β( 49 J+ 94 Di + 81 16 Kij ) X 4 J+ 4 Di + 16 Kij ) Y n−1 + e n−1 1 = [e gn (− 2 )
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z−1 Zn−1 + eβ( 4 J+ 4 Di + 16 Kij ) ],
z−1 + eβ( + eβ( 4 J+ 4 Di + 16 Kij ) Zn−1
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gn (+ 21 ) −3 3 9 9 9 9 z−1 z−1 = [eβ( 4 J+ 4 Di + 16 Kij ) Xn−1 + eβ( 4 J+ 4 Di + 16 Kij ) Yn−1 gn (− 12 )
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]/[eβ(
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gn (− 32 ) 9 81 z−1 z−1 β( −9 β( 94 J+ 94 Di + 81 4 J+ 4 Di + 16 Kij ) X 16 Kij ) Y n−1 + e n−1 1 = [e gn (− 2 )
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z−1 + eβ( + eβ( 4 J+ 4 Di + 16 Kij ) Yn−1
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where P (Spc) is the unnormalized probability distribution and β = 1/(kT ) with k and T are the Boltzmann constant and absolute temperature, respectively. It should be noted that one of us have already obtained the ERR’s on the BL [10], therefore, they are directly taken form there without going into any details. The ERR’s are represented as the ratios of the partial partition functions of each of the separate branches on the BL, i.e. four gn (S)’s exist, therefore, there are only three ERR’s present for the spin-3/2 case. The ratio of gn (S)’s, the ERR’s, is actually a measure of how one distribution is stronger than the other for given system parameters. Thus, they are given for the spin-3/2 BEG model on the BL as follows:
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The RCF distribution influences all the spin-3/2 sites and it is taken as P (Di ) = pδ(Di ) + (1 − p)δ(Di − D),
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where the first term pδ(Di ) imposes that an amount of p spins are free from the influence of D while the second term indicates that an amount of 1 − p spins are under the influence of the crystal field. In order to add the effects of RCF into the ERR’s, one should integrate Eq.(3) over the P (Di ) distribution to obtain the RCF implemented ERR’s which is done as Z (ij) Xn = Xn(i,ij) P (Di )dDi Z = Xn(i,ij) [pδ(Di ) + (1 − p)δ(Di − D)]dDi , (5) 5
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In order to include also the random distribution effects of K, one should consider another bimodal random distribution having the same form as the bimodal RCF distribution and given as P (Kij ) = qδ(Kij − K) + (1 − q)δ(Kij ),
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81 9 81 3 9 3 9 z−1 z−1 h eβ( 49 J+ 16 K) z−1 X + eβ(− 4 J+ 16 K) Yn−1 + eβ( 4 J+ 16 K) Zn−1 + eβ(− 4 J+ 16 K) i = qp β(− 3 J+ 9 K) n−1 3 9 1 1 1 1 z−1 z−1 z−1 4 16 e Xn−1 + eβ( 4 J+ 16 K) Yn−1 + eβ(− 4 J+ 16 K) Zn−1 + eβ( 4 J+ 16 K) h 9 9 81 9 9 81 3 1 9 z−1 z−1 z−1 + q(1 − p) eβ( 4 J+ 4 D+ 16 K) Xn−1 + eβ(− 4 J+ 4 D+ 16 K) Yn−1 + eβ( 4 J+ 4 D+ 16 K) Zn−1 i h 1 9 3 9 9 3 9 9 3 z−1 z−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 + eβ(− 4 J+ 4 D+ 16 K) / eβ(− 4 J+ 4 D+ 16 K) Xn−1 i 1 1 1 1 1 1 z−1 + eβ(− 4 J+ 4 D+ 16 K) Zn−1 + eβ( 4 J+ 4 D+ 16 K)
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3βJ z−1 z−1 z−1 h e 9βJ − 9βJ − 3βJ 4 X 4 Y 4 Z 4 i n−1 + e n−1 + e n−1 + e p(1 − q) 3βJ 3βJ βJ βJ z−1 z−1 z−1 e− 4 Xn−1 + e 4 Yn−1 + e− 4 Zn−1 +e 4 h 9 9 9 9 3 1 z−1 z−1 z−1 + eβ(− 4 J+ 4 D) Yn−1 + eβ( 4 J+ 4 D) Zn−1 (1 − p)(1 − q) eβ( 4 J+ 4 D) Xn−1 i h 3 1 3 9 3 9 z−1 z−1 eβ(− 4 J+ 4 D) / eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 i 1 1 1 1 z−1 eβ( 4 J+ 4 D) Zn−1 + eβ(− 4 J+ 4 D) ,
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Yn(ij) [qδ(Kij − K) + (1 − q)δ(Kij )]dKij
9 3 9 z−1 z−1 h eβ(− 94 J+ 81 β( 49 J+ 81 β(− 34 J+ 16 K) z−1 16 K) X 16 K) Y Zn−1 + eβ( 4 J+ 16 K) i n−1 + e n−1 + e = qp β(− 3 J+ 9 K) z−1 3 9 1 1 1 1 z−1 z−1 4 16 e Xn−1 + eβ( 4 J+ 16 K) Yn−1 + eβ(− 4 J+ 16 K) Zn−1 + eβ( 4 J+ 16 K)
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h 9 9 81 9 9 81 3 1 9 z−1 z−1 z−1 + q(1 − p) eβ(− 4 J+ 4 D+ 16 K) Xn−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 + eβ(− 4 J+ 4 D+ 16 K) Zn−1 i h 3 3 1 9 9 9 9 9 3 z−1 z−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 + eβ( 4 J+ 4 D+ 16 K) / eβ(− 4 J+ 4 D+ 16 K) Xn−1 i 1 1 1 1 1 1 z−1 + eβ( 4 J+ 4 D+ 16 K) + eβ(− 4 J+ 4 D+ 16 K) Zn−1
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9 3 9 1 1 1 1 z−1 z−1 h eβ( 43 J+ 16 K) z−1 Xn−1 + eβ(− 4 J+ 16 K) Yn−1 + eβ( 4 J+ 16 K) Zn−1 + eβ(− 4 J+ 16 K) i qp β(− 3 J+ 9 K) z−1 3 9 1 1 1 1 z−1 z−1 4 16 e Xn−1 + eβ( 4 J+ 16 K) Yn−1 + eβ(− 4 J+ 16 K) Zn−1 + eβ( 4 J+ 16 K) h 3 9 9 3 9 9 1 1 1 z−1 z−1 z−1 + eβ(− 4 J+ 4 D+ 16 K) Yn−1 + eβ( 4 J+ 4 D+ 16 K) Zn−1 q(1 − p) eβ( 4 J+ 4 D+ 16 K) Xn−1 i h 1 1 1 9 9 3 9 9 3 z−1 z−1 eβ(− 4 J+ 4 D+ 16 K) / eβ(− 4 J+ 4 D+ 16 K) Xn−1 + eβ( 4 J+ 4 D+ 16 K) Yn−1 i 1 1 1 1 1 1 z−1 eβ(− 4 J+ 4 D+ 16 K) Zn−1 + eβ( 4 J+ 4 D+ 16 K)
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9βJ 3βJ z−1 z−1 z−1 h e− 9βJ − 3βJ 4 X 4 Y 4 Z 4 i n−1 + e n−1 + e n−1 + e + p(1 − q) 3βJ βJ 3βJ βJ z−1 z−1 z−1 e− 4 Xn−1 + e 4 Bn−1 + e− 4 Zn−1 +e 4 h 9 9 9 1 9 3 z−1 z−1 z−1 + (1 − p)(1 − q) eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 + eβ(− 4 J+ 4 D) Zn−1 i h 3 3 1 9 9 3 z−1 z−1 + eβ( 4 J+ 4 D) / eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 i 1 1 1 1 z−1 + eβ(− 4 J+ 4 D) Zn−1 + eβ( 4 J+ 4 D)
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βJ z−1 z−1 z−1 h e 3βJ − 3βJ − βJ 4 X 4 Y 4 Z 4 i n−1 + e n−1 + e n−1 + e + p(1 − q) 3βJ 3βJ βJ βJ z−1 z−1 z−1 e− 4 Xn−1 + e 4 Yn−1 + e− 4 Zn−1 +e 4 h 3 9 3 9 1 1 z−1 z−1 z−1 + (1 − p)(1 − q) eβ( 4 J+ 4 D) Xn−1 + eβ(− 4 J+ 4 D) Yn−1 + eβ( 4 J+ 4 D) Zn−1 i h 1 1 3 9 3 9 z−1 z−1 + eβ(− 4 J+ 4 D) / eβ(− 4 J+ 4 D) Xn−1 + eβ( 4 J+ 4 D) Yn−1 i 1 1 1 1 z−1 + eβ(− 4 J+ 4 D) Zn−1 + eβ( 4 J+ 4 D) .
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After having obtained the random D and K implemented ERR’s, the orderparameters, i.e. magnetization and quadrupolar moment, can be calculated in terms of ERR’s. Note that all the sites of the BL are equivalent deep inside in the thermodynamic limit. Therefore, one can pick a central spin, S0 , and calculate its order-parameters accordingly. The magnetization or dipolar moment for the central spin or any other spin deep inside the BL can be given as 1 h 3e 4 (Xnz − Ynz ) + e 4 (Znz − 1) i βD 2 e 9βD 4 (X z + Y z ) + e 4 (Z z + 1) n n n 9βD
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and similarly the quadrupolar moment of the central spin is found as 1 h 9e 4 (Xnz + Ynz ) + e 4 (Znz + 1) i . βD 4 e 9βD 4 (X z + Y z ) + e 4 (Z z + 1) n n n 9βD
Q =< S02 >=
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The Phase Diagrams
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These equations can be written in terms of the reduced quantities. To do that one simply sets J = 1 and replaces D, K and β = 1/kT by D/J, K/J and βJ = J/kT , respectively. In the next section, we present the phase diagrams of the model obtained from the analysis of thermal variations of the orderparameters.
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The phase diagrams of the random spin-3/2 BEG model are studied by the analysis of thermal variations of the order-parameters. They are presented on the (K/J, kT /J) and on the (D/J, kT /J) planes for given values of the rest of the system parameters. Our calculations are carried out for z = 3 which corresponds to honeycomb lattice. It should be mentioned that our model reduces to four specific models depending on the p and q values as: (i) when p = 0 and q = 1, the model reduces to the regular spin-3/2 BEG model [10], (ii) the p = 0 and q = 0 values reduce the model to the Blume-Capel model see for example [43], (iii) when p = 1 and q = 0, D and K are turned off and the Hamiltonian only includes the bilinear exchange interaction and lastly (iv) when p = 1 and q = 1, our Hamiltonian contains only J and K with zero crystal field. For the intermediate values of p and q, we have the case of random K and random D model. In the phase diagrams, the second- and first-order phase transition lines are shown with solid and dotted lines and they are simply named as the Tc - and Tt -lines, respectively. The TCP’s are the points at which the Tc - and Tt -lines meet are shown with filled circles. The ferromagnetic and paramagnetic phase regions are indicated by (F ) and (P ), respectively. Note also that the places of the second- and first-order phase transition temperatures are determined from the thermal analysis of the order-parameters: The magnetization M goes to zero continuously and quadrupole moment Q makes a little cusp at the Tc ’s, while at the Tt ’s, M presents a jump discontinuity to zero and Q makes a discontinuity. The first phase diagrams are obtained on the (K/J, kT /J) planes for given values of D/J = 3.0, 1.0, 0.0, −1.0, −3.0, −5.0 and p = 0 when 0 ≤ q ≤ 1 with ∆q = 0.1 as shown in Figs.2(a)-(f). They correspond to the cases (i) and (ii) for q = 1 and 0, respectively, and to the random K and D model for the intermediate values of q. It is clear that the sites are under the influence of D/J when p = 0 and K/J values are turned off when q = 0. In this case, the Tc -lines are independent from K/J, i.e. they are straight lines on these planes of phase diagrams. The temperatures of these lines decrease as D/J reduces to lower values. This is caused by the competition between J and D as expected. The model becomes the well-known BEG model when q = 1. As seen from the figures, only the Tc -lines are seen for the first four values of D/J. They start 8
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from lower temperatures for higher q in the repulsive region, i.e. K/J < 0.0, then all the lines coincide at K/J = 0.0, and then their temperatures increase for higher q’s in the attractive region K/J > 0.0. In addition, they become constant as |K/J| becomes large. When D/J = −3.0, we see that the Tc - and Tt -lines combine at TCP’s for q = 0.8, 0.9 and 1.0. The first TCP’s are seen at higher temperatures for higher q values and the second ones are along the q = 0-line. The rest of the lines are in the form of the Tc -lines for the rest of the q values. Either form of the lines terminate on the straight line for q = 0.0 at higher K/J’s for lower q’s. For D/J = −5.0 we see the same behaviors. But now the TCP’s are also seen when q = 0.6 and 0.7 and the lines terminate at higher K/J’s. The wiggles of the Tc -lines for q = 0 in Figs.2 (a)-(c) caused by the competition when K is repulsive and D is positive. This competition is obvious from the Hamiltonian of our model. Since both has the squares of the spins which are always positive. So when K is repulsive, i.e. K < 0.0, and D is positive they oppose to each other which causes the lines wiggle. As seen these wiggles start disappearing as K and D gets closer to each other signwise and eventually disappear, see for example Fig.2(d) where both K and D are negative. The second phase diagrams are also obtained on the (K/J, kT /J) planes for given values of D/J=7.0, 1.0, -1.0, -1.75, -3.0, -7.0 and for q = 1 when 0 ≤ p ≤ 1 with ∆p = 0.1 as shown in Figs.3(a)-(f). They correspond to the cases (i) and (iv) and to the intermediate values of p. The Tc -lines start from the same lower constant temperatures and terminate at higher constant temperatures as |K/J| becomes large for each D/J’s. The region corresponding to 0 < p < 1 shrinks to one line when D/J = 0.0, but it grows as |D/J| becomes large. The Tt -lines are only seen when q = 0 for D/J = −1.75, −3.0, −7.0 which becomes longer as D/J becomes more negative. In this case, they start from higher TCP’s and end at lower TCP’s. The last phase diagram on this plane, i.e. Fig.4, is obtained for D/J = 1.0 and p = 1 when 0 < q < 1 as an example. In this case, they correspond to the cases (iii) and (iv), and for the intermediate values of q. As seen from the figure, they are very similar with the Fig.2(c). The next three phase diagrams are obtained on the (D/J, kT /J) planes for given values K/J. The first one is obtained for the case (i) and (ii) and for the intermediate values of q. The Tc -lines become constant as |D/J| becomes large for all K/J values. As seen from the Figs.5(a)-(c), the model presents both Tc - and Tt -lines combined at two TCP’s for higher q values for the attractive case. As seen, the TCP’s at higher temperatures go down in temperature as q is lowered. The second ones, which are seen at higher D/J’s for lower q’s, are lined up along the constant portion of q = 0 line. Figs.5(d)-(f) correspond to the repulsive case for which the lines are only in the form of Tc -lines. The q = 1 case is interesting, since the Tc -lines start from constant lower temperatures for K/J = −2.0 and -7.0, then as D/J increases they increase in temperature, then they become constant again, for the further increase of D/J they increase in temperature and then they become constant again. The first constant portion separate the F phase with ±1/2 state from the P phase, the last portion 9
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separates the F phase with ±3/2 state from the P phase, and the middle portion separates the F phase with the intermediate spin states from the P phase, see for example the temperature variations of magnetization as shown in Fig.6. It should again be noted that the wiggles in Figs.5 (d)-(f) are caused by the competition between repulsive K and positive D values as explained in Fig.2. The next ones, i.e. Figs.7(a)-(f), correspond to the cases (i) and (iv) for the intermediate values of p and are obtained for K/J=3.0, 0.1, 0.0, -0.5, -2.0 and -3.0. The Tc -lines for p = 1 and q = 1 are just straight lines since they are independent from D/J values. Their temperatures decrease as K/J’s decrease. Again, the Tc -lines combined with the Tt -lines connected to two TCP’s are only seen for the case with p = 0 in the attractive case. The lines are just Tc -lines for the rest of the p values for all K/J’s. As seen, the Tc -lines are seen at lower temperatures for lower p’s at negative D/J’s in Fig.7(a) and (b). Then they all combine on a straight line as D/J’s increase in the negative region. It is clear that all the lines combine at D/J = 0.0. Then the Tc -lines increase in temperature for lower p’s for positive D/J values. As K/J is decreased, for example as given in Fig. 7(f), all the Tc -lines start from the same temperature as for the p = 1-line, then as D/J increase they spread and the Tc -lines are seen at higher temperatures for lower p’s again. Our last phase diagram is obtained for the cases (ii) and (iii) and for the intermediate values of p and K/J=1.0, see Fig.8. They are very similar with the Fig.7(f), i.e. the lines are in the form of Tc -lines In this section we have presented our phase diagrams and tried to justify their behaviors for the cases (i)-(iv) and for the intermediate values of p and q’s. The next section gives our conclusions, comparisons and some results.
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In this paper, we have studied the phase diagrams of the spin-3/2 Ising system with the disorders associated with the crystal field and biquadratic exchange interactions randomly. The ERR’s are employed on the BL for the calculation of the order-parameters. The disorders are associated with three ingredients of the physical system. The first is the temperature that induces order-disorder phase transitions due to thermal agitation on the lattice. The second one is associated with the random crystal field, which is governed by a probability distribution function of the bimodal type. And the last one corresponds to the random biquadratic exchange interaction, which is also governed by a probability distribution function in bimodal type. All together we have four parameters (J, T, D, K) competing with each other to maintain the system ordered or disordered. When our model reduces to the well-known models such as the BC and the BEG model, it is found that overall similarities are observed. It should be noted that this work is the first work considering both random D and K effects, thus a direct comparison is not possible.
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References [1] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press, New York, 1971; C.-K. Hu, Chin. J. Phys. 52 (2014) 1; N. Sh. Izmailian, C.-K. Hu, Physica A 254 (1998) 198; C.-K. Hu, N. Sh. Izmailian, Phys. Rev. E 58 (1998) 1644.
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[2] J. Sivardi´ere, M. Blume, Phys. Rev. B 5 (1972) 1126. [3] A. H. Cooke, D. M. Martin, M. R. Wells, J. Phys. (Paris), Colloq. 32 (1971) C1 488; A. H. Cooke, D. M. Martin, M. R. Wells, Solid State Commun. 9 (1971) 519. [4] S. Krinsky, D. Mukamel, Phys. Rev. B 11 (1975) 399.
[5] F. C . S´a Barreto, O. F. De Alcantara Bonfim, Physica A 172 (1991) 378.
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[6] A. Bakchich, A. Bassir, A. Benyoussef, Physica A 195 (1993) 188. [7] T. Kaneyoshi, M. Jaˇscur, Phys. Lett. A 177 (1993) 172. [8] T. Kaneyoshi, J. Phys. Soc. Jpn 56 (1987) 4199.
[9] I. P. Fittipalde, T. Kaneyoshi, J. Phys. Condens. Matter 1 (1989) 6513. [10] E. Albayrak, M. Keskin, J. Magn. Magn. Maters. 241 (2002) 249.
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[11] E. Albayrak, J. Magn. Magn. Maters. 386 (2015) 20.
[12] A. Yigit, E. Albayrak, J. Supercond. Nov. Magn. 29 (2016) 2535.
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[13] M. Kerouad, A. Elatri, A. Ainane, M. Saber, Phys. Status Solidi B 195 (1996) 519. [14] A. Ainane, A. Elatri, M. Kerouad, M. Saber, J. Magn. Magn. Maters. 146 (1995) 283.
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[15] E. Albayrak, Physica B 479 (2015) 107.
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[16] M. Kerouad, M. Saber, J. W. Tucker, J. Magn. Magn. Maters. 132 (1994) 223. [17] A. Benyoussef, T. Biaz, M. Saber, M. Touzani, J. Phys. C: Solid State Phys. 20 (1987) 5349.
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[18] C. E. I. Carneiro, V. B. Henriques, S. R. Salinas, J. Phys. Condens. Matter 1 (1989) 3687. [19] N. Boccara, A. Elkenz, M. Saber, J. Phys. Condens. Matter 1 (1989) 5721. [20] C. E. I. Carneiro, V. B. Henriques, S. R. Salinas, J. Phys A: Math. Gen. 23 (1990) 3383.
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[21] T. Kaneyoshi, J. Phys. C: Solid State Phys. 21 (1988) L679-L682. [22] T. Kaneyoshi, J. Mielnicki, J. Phys. Condens. Matter 2 (1990) 8773. [23] A. Benyoussef, H. Es-Zahraouy, J. Phys. Condens. Matter 6 (1994) 3411. [24] N. S. Branco, B. M. Boechat, Phys. Rev. B 56 (1997) 11673.
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[25] N. S. Branco, Phys. Rev. B 60 (1999) 1033. [26] V. Urumov, J. Phys. Condens. Matter 1 (1989) 7037. [27] E. Albayrak, Physica A 390 (2011) 1529.
[28] Y. -Q. Liang, G. -Z. Wei, Q. Zhang, Z. -H. Xim, J. Magn. Magn. Maters. 267 (2003) 275.
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[29] Y. -Q. Liang, G. -Z. Wei, Q. Zhang, Z. -H. Xim, J. Magn. Magn. Maters. 265 (2003) 305. [30] Y. -Q. Liang, G. -Z. Wei, Q. Zhang, Z. -H. Xim, G. -L. Song, J. Magn. Magn. Maters. 284 (2004) 47.
[31] Y. -Q. Liang, G. -Z. Wei, G. -L. Song Q, Phys. Stat. Solidi (b) 241 (2004) 3636.
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[32] L. Bahmad, A. Benyoussef, A. El Kenz, J. Magn. Magn. Maters. 320 (2008) 397. [33] L. Bahmad, A. Benyoussef, A. El Kenz, Phys. Rev. B 76 (2007) 094412. [34] T. Kaneyoshi, J. Phys. C: Solid State Phys. 19 (1986) L557-L561.
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[35] L. L. Deng, S. L. Yan, J. Magn. Magn. Maters 251 (2002) 138.
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[36] W. P. da Silva, P. H. Z. de Arruda, T. M. Tunes, M. Godoy, A. S. Arruda, J. Magn. Magn. Maters. 422 (2017) 367 [37] M. Kurata, R. Kikuchi, T. Watari, J. Chem. Phys. 21 (1953) 434. [38] C. Domb, Adv. Phys. 9 (1960) 283.
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[39] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.
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[40] P. D. Gujrati, Phys. Rev. Lett. 74 (1995) 809. [41] M. A. Neto, J. F. Stilck, J. Chem. Phys. 128 (2008) 184904. [42] M. A. Neto, J. F. Stilck, J. Chem. Phys. 138 (2013) 044902. ¨ [43] E. Albayrak, M. Keskin, J. Magn. Magn. Maters. 218 (2000) 121; O. Ozsoy, E. Albayrak, M. Keskin, Physica A 304 (2002) 443. 12
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Figure Captions Fig.1- The Bethe lattice of coordination number z = 3. The filled circles denote the spins and the dotted circles represent the shells of the BL.
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Fig.2- The phase diagrams on the (K/J, kT /J) planes for given values of D/J = 3.0, 1.0, 0.0, −1.0, −3.0, −5.0 and for p = 0 when 0 ≤ q ≤ 1 with ∆q = 0.1.
Fig.3- The phase diagrams on the (K/J, kT /J) planes for given values of D/J=7.0, 1.0, -1.0, -1.75, -3.0, -7.0 and for q = 1 when 0 ≤ p ≤ 1 with ∆p = 0.1.
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Fig.4- The phase diagrams on the (K/J, kT /J) plane for D/J = 1.0 and p = 1 when 0 ≤ q ≤ 1 with ∆q = 0.1. Fig.5- The phase diagrams on the (D/J, kT /J) planes for K/J=7.0, 2.0, 1.0, -1.0, -2.0, -7.0 and for p = 0 when 0 ≤ q ≤ 1 with ∆q = 0.1.
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Fig.6- The temperature variation of magnetization for K/J = −7.0, p = 0 and q = 1 when D/J=0.0, 20.0 and 50.0.
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Fig.7- The phase diagrams on the (D/J, kT /J) planes for K/J=3.0, 0.1, 0.0, -0.5, -2.0, -3.0 and for q = 1 when 0 ≤ p ≤ 1 with ∆p = 0.1.
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Fig.8- The phase diagrams on the (D/J, kT /J) plane for K/J = 1.0 and q = 0 when 0 ≤ p ≤ 1 with ∆p = 0.1.
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