2 and spin-2 Ising ferrimagnetic system on the Bethe lattice

Author’s Accepted Manuscript Monte Carlo study of alternate mixed spin-5/2 and spin-2 Ising ferrimagnetic system on the Bethe lattice A. Jabar, R. Masrour, A. Benyoussef, M. Hamedoun www.elsevier.com/locate/jmmm

PII: DOI: Reference:

S0304-8853(15)30523-0 http://dx.doi.org/10.1016/j.jmmm.2015.08.098 MAGMA60580

To appear in: Journal of Magnetism and Magnetic Materials Received date: 21 May 2015 Revised date: 27 July 2015 Accepted date: 23 August 2015 Cite this article as: A. Jabar, R. Masrour, A. Benyoussef and M. Hamedoun, Monte Carlo study of alternate mixed spin-5/2 and spin-2 Ising ferrimagnetic system on the Bethe lattice, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.08.098 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Monte Carlo study of alternate mixed spin-5/2 and spin-2 Ising ferrimagnetic system on the Bethe Lattice A. Jabar1, R. Masrour1,2,*, A. Benyoussef2,3,4, M. Hamedoun3 1)

Laboratoire de Magnétisme et Physique des Hautes Energies L.M.P.H.E.URAC 12, Université Mohammed V, Faculté des Sciences, B.P. 1014, Rabat, Morocco. 2) Laboratory of Materials, Processes, Environment and Quality, Cady Ayyed University, National School of Applied Sciences, PB 63 46000, Safi, Morocco. 3) Institute of Nanomaterials and Nanotechnologies, MAScIR, Rabat, Morocco. 4) Hassan II Academy of Science and Technology, Rabat, Morocco. Corresponding authors: [email protected] Abstract: The magnetic properties of alternate mixed spin-5/2 and spin-2 Ising model on the Bethe lattice have been studied by using the Monte Carlo simulations. The ground state phase diagrams of alternate mixed spin-5/2 and spin-2 Ising model on the Bethe lattice has been obtained. The thermal total magnetization and magnetization of spins-5/2 and spin-2 with the different exchange interactions, external magnetic field and temperatures have been studied. The critical temperature have been deduced. The magnetic hysteresis cycle on the Bethe lattice has been deduced for different values of exchange interactions, for different values of crystal field and for different sizes. The magnetic coercive field has been deduced. Keywords: Mixed alternate spins; Magnetic properties; Transition temperature; Magnetic coercive field. PACS:73.63.Fg; 75.75.-c; 71.15.Mb; 75.60.-d.

1

1. Introduction The mixed-spin Ising ferrimagnetic models with interactions between the spins of the two sublattice system and external agencies have been studied with exact or many approximate methods. It is clear that there may be many possibilities of mixing the spin-σA and spin-σB with σA  σB in obtaining the Ising ferrimagnetic systems to propose some models for physical materials. Maybe the highest mixed-spin system is the one with spin-2 and spin-5/2 and which has been studied with many techniques for different physical reasons such as: The magnetic properties of the ferrimagnetic system with dilution was investigated on the basis of the effective-field theory with correlations [3]. The compensation temperatures induced by longitudinal fields in a mixed spin Ising ferrimagnet are obtained by Ref.[4]. The first orderphasetransitionofmixedspin-2andspin-5/2Isingsysteminhoneycomblatticewithtwo single ionanisotropiesisstudiedbyMonteCarlo(MC)simulation [5]. The dynamic magnetic properties in the kinetic mixed spin-2 and spin-5/2 Ising model under a time-dependent magnetic field has been studied by Ref.[6]. The stationary states of the kinetic mixed spin-2 and spin-5/2 Ising ferrimagnetic system with repulsive biquadratic coupling are examined within a meanfield approach under the presence of a time varying (sinusoidal) magnetic field [7]. The Nonequilibrium magnetic properties in a two-dimensional kinetic mixed spin-2 and spin-5/2 Ising system in the presence of a time-varying (sinusoidal) magnetic field are studied within the effective-field theory with correlations [8]. The magnetic properties of a nonequilibrium mixed spin-2 and spin-5/2 Ising ferrimagnetic system with a crystal-field interaction (D) in the presence of a time-varying magnetic field on a hexagonal lattice are studied by using the Glauber-type stochastic dynamics [9]. The influence of the layer thickness and the surface intralayer exchange coupling on the magnetic and thermodynamic properties of the mixed spin-2 and spin-5/2 ferrimagnetic systems are studied in detail by the Monte Carlo simulation [10]. The mixed spin-1/2 and spin-S (S>1/2) Ising ferrimagnetic systems with a crystal field 2

are studied within the framework of the exact recursion relations on the Bethe lattice by Ref.[11]. A number of characteristic behaviors for field variations are obtained especially for antiferromagnetic interaction in the Ising model with crystal-field under uniform longitudinal magnetic field by Ref.[12]. The antiferromagnetic and ferrimagnetic Ising model on the twofold Cayley tree graph with fully q-coordinated sites is investigated in an external magnetic field by Ref.[13,14]. the phase diagrams for the Ising model on a Cayley tree-like lattice, called Triangular Chandelier, with competing nearest-neighbour interactions, prolonged nextnearest-neighbour interactions and one-level next-nearest-neighbour quadruple interactions have been investigated by Refs.[15,16]. The critical behaviors of the half-integer mixed spin3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system, have been studied by the exact recursion relations on the Bethe lattice [17]. The kinetic spin-5/2 Blume–Capel model with bilinear exchange interaction and single-ion crystal field on a square lattice is studied by Ref.[18]. The Phase diagrams of a nonequilibrium mixed spin-3/2 and spin-2 Ising system in an oscillating magnetic field has been investigated by Ref.[19]. The phase diagrams for a nonequilibrium mixed spin-1/2 and spin-2 Ising ferrimagnetic system on a square lattice in the presence of a time dependent oscillating external magnetic field is given by Ref.[20]. The outline of the rest of the present Letter is as follows. In Section 2, the model and its formulation, namely, Ising model. In Section 3, the Monte Carlo simulations are presented. In section 4, the results and discussion are given. Finally, we give a conclusion in section 5. 2. Model and formulations The Hamiltonian of the mixed spins-5/2 and 2 Ising model on the Bethe lattice with different spins such as in see Fig.1 includes nearest neighbors interactions, the crystal field and external magnetic field is given as:

H   J



i , j 

i

j

 J S

 S

i , k 

i

k

        i2   Sk2   h   i   Sk  k k  i   i 

3

(1)

where i, j stand for the first nearest neighbor spins i and j,  represent the crystal field and h is the external magnetic field. The J and JS are the exchange interactions between the first nearest-neighbor magnetic atoms with spins - and S-, respectively. The spins moment S and  are: ±5/2, ±3/2, ±1/2 and ±2, ±1, 0, respectively. In full text the J has been taken 1. 3. Monte Carlo simulations The mixed spins-S and  Ising model on the Bethe lattice is assumed to reside in the unit cells and the system consists of the total number of spins N=N+NS, with N=9 and NS=15 spins. We apply a standard sampling method to simulate the Hamiltonian given by Eq. (1). Cyclic boundary conditions on the lattice were imposed and the configurations were generated by sequentially traversing the lattice and making single-spin flip attempts. The flips are accepted or rejected according to a heat-bath algorithm under the Metropolis approximation. Our data were generated with 105 Monte Carlo steps per spin, discarding the first 104 Monte Carlo simulations. Starting from different initial conditions, we performed the average of each parameter and estimate the Monte Carlo simulations, averaging over many initial conditions. Our program calculates the following parameters, namely: The magnetizations of each atoms with spins  and S on Bethe lattice are:

M 

MS 

1 N



1 NS

S

(2)

i

i

(3)

i

i

The total magnetization of Ising ferrimagnetic on Bethe lattice is: M

15M S  9M  24

(4)

with E is the internal energy per site,

4

E

1 H N

(5)

where N=NS+N

with E 

1  J N



i , j 

 i j  J S



i , k 







 

  i



 i Sk      i2   Sk2   h    i   Sk    i

k

k

 

(6) The magnetic susceptibilities of atoms with spins  and S are given by:

       

 1     i   N i  

  S     

 1   S  i   NS i  

2 1  N

2 1  NS

2   i   i 

(7)

2   Si   i 

(8)

Total susceptibility is :

tot 

9   15 S 14

where  

(9)

1 , T denotes the absolute temperature and k B is the Boltzmann’s constant. k BT

4. Results and discussion In previously work [21], we have studied the magnetic properties of a core shell mixed spins5/2 and 2 Ising model on the Bethe lattice and in the present work we have investigated the alternate spins-5/2 and 2 Ising model on the Bethe lattice by using the Monte Carlo simulation. Fig.1(a)-d, shows the ground state phase diagrams of alternate mixed spin-5/2 and spin-2 Ising model on the Bethe lattice in (Δ, JS), (h,JS), (h, Δ) for JS =1.0 and (h, Δ) planes for JS =-1.0, respectively. The obtained configurations are presented in Figs. 1(a) to 1(d) and 5

are given in table 1, where more than one phase can coexist. It should be mentioned that the ground state phase diagram is important in classifying the different phase regions of the model for the phase diagrams at higher temperatures. This behavior is observed in previously work [22]. The I, II, V, VI, XV, XVI, XVII, XVIII, XIX, XX, XXIII, XXIV phases represent the usual ferromagnetic ordering. The VII, VIII, IX, X, XI, XII phases presents the paramagnetic ordering. The others phases are presented in table 1 with the antiferromagnetic ordering. For h>0 the ferromagnetic (FM) and antiferromagnetic (AFM) phases are observed with JS>0 and JS<0, respectively. The mixed spins (2, 5/2) Ising system displays the dynamic reentrant behavior for the FM/FM, AFM/FM and AFM/AFM interactions. For h<0 the ferromagnetic and antiferromagnetic phases are observed with JS>0 and JS<0, respectively (see Fig.1c). When I change the sign of the coupling between spin- and spin-S, the nature and number of phase changes its change (see Figs. 2(d) and 2(e)). We have given in Fig.2(a)-b, the thermal magnetization and magnetic susceptibility for J S =1.0 and JS =-1.0 with Δ=0.0 and h=0.75. The obtained transitions temperatures for JS >0 and JS <0 are 5 and 3.3, respectively. These values are comparable with those given by effective-field theory and the Glauber-type stochastic dynamics approach [9,18,21]. The second-order phase transition temperature is observed in these figures. This behavior is observed in the previously work [23]. In the previously study [21] the temperature compensation and transition temperatures are observed and in the present study we have obtained only the transition temperature. The thermal total magnetization and magnetic susceptibility for JS =1, 1.5, 2 and JS =-1, -1.5, -2 are presented in Figs.3(a) and 3b, respectively with Δ=0.0 and h=0.75. The obtained values of transitions temperatures for JS=1, 1.5, 2 are: 5.70, 7 and 8.1, respectively. The obtained values for JS=-1, -1.5, -2 are: 3.45, 4.3, 5.4, respectively. The transitions temperatures increases with increasing the absolute value of exchange interactions JS. These values are comparable with those given by Refs.[24,6]. The obtained values for N=24 spins in the non 6

alternate mixed spins [21] is inferior to those obtained in the presented work. From the figure, one can see that the sublattice with spin-S is more ordered than the sublattice with spin- below TC, i.e. |MS|>|M|. We have presented in Fig.4(a)-d, the total magnetization versus the crystal field for JS =0.5, 1.5, 2.5, 3, T=1, JS =-1, -2, -3, -3.5, T=1, T= 1, 2.5, 3.5, 4.5, JS =0.5 and JS=-0.5, respectively with Δ=0.0 and h=1. The total magnetization increase with increasing of the absolute value of exchange interactions JS( see Figs. 4(a) and b) for -4<0. This behavior is confirmed by the results given in Fig.1b. The total magnetization is independent of exchange interaction JS for >0. The total magnetization decrease with increasing of the temperature absolute value of exchange interactions JS(see Figs.4c and 4d). Fig.5 illustrate, the total magnetization versus the exchange interaction JS for T= 1, 2.5, 3.5, 4.5 with Δ=0.0 and h=1. The total magnetization decrease with increasing of the temperature for a fixed value of exchange interactions JS>0. The inverse behavior is observed for JS<0. We have presented in Fig.6(a,b)-(e,f), the variation of total magnetization and magnetization of each superlattice with spin-5/2 and 2 versus the external magnetic field for =0, -0.5, -1, respectively, with T=1 and JS =1. The increasing of absolute value of the crystal field decreasing the magnetic coercive field for JS>0. The systems becomes superparamagnetic for =-1, T=1 and JS=1 such as in Fig.6e. When JS=-1, the system becomes superparamagnetic -1 (see Fig7e). Finally, we have given in Fig.7(a,b)-(e,f), the variation of total magnetization and magnetization of each superlattice with spin- and S versus the external magnetic field for =0, -0.5, -1 with, temperature T=1and JS=-1. The increase of absolute value of the crystal field decreasing the magnetic coercive field for JS<0. This behavior is similar to those obtained by Refs.[21,25]. Figs.6e and 7e show that the coercive field is very low in the ferromagnetic case and the system remain superparamagnitc rapidly when

7

compared with the case of antiferromagnetic case. The saturation magnetization and remanent magnetization changing rapidly with the change of sign of JS. 5. Conclusions The magnetic properties of mixed spins-5/2 and 2 Ising model on the Bethe lattice have been obtained by using the Monte Carlo simulation. The transition temperature is deduced for different values of exchange interactions. The obtained values increase with increasing of the absolute value of exchange interactions. The total magnetization versus the crystal field has been investigated for different exchange interactions. The effect of ferro and antiferromagnetic in coercive field is studied. The coercive magnetic field decrease with decreasing the crystal field for a positive and for a negative of exchanges interactions in alternate and non alternate mixed spins. The magnetic coercive field decreasing when the absolute value of the crystal field increases for JS<0 and for JS>0. References: [1] T. Kaneyoshi, Y. Nakamura, S. Shin, J. Phys: Condens. Matter 10 (1998) 7025. [2] Y. Nakamura, S. Shin, T. Kaneyoshi, Phys. B 284–288 (2000) 1479. [3] Y. Nakamura, J. Phys: Condens. Matter 12 (2000) 4067. [4] Hadey K. Mohamad, E.P. Domashevskaya, A.F. Klinskikh. Solid State Commun. 150 (2010) 1253-1257. [5] YasenHou, QiZhang, YuJia. Physica. B. 442 (2014) 52–56. [6] Mehmet Ertaş, Mustafa Keskin, Bayram Deviren. Physica A 391 (2012) 1038–1047. [7] Mehmet Erta¸s · Mustafa Keskin · Bayram Deviren. J Stat Phys (2012) 146:1244–1262. [8] Mehmet Ertas¸ Bayram Deviren, and Mustafa Keskin. Phys. Rev. E 86, 051110 (2012). [9] Mustafa Keskin, and Mehmet Ertaş. Phys. Rev. E. 80 (2009) 061140. [10] Wei Wang, Wei Jiang, and Dan Lv. IEEE Trans. Magn. 47 (2011) 3943. 8

[11] C. Ekiz. Physica A 353 (2005) 286–296. [12] C. Ekiz. Physics Letters A 349 (2006) 21–26. [13] C. Ekiz. Physics Letters A 327 (2004) 374–379 [14] C. Ekiz. J. Magn. Magn. Mater. 293 (2005) 759–767. [15] S. Uğuz, H. Akin. Physica A 389 (2010) 1839-1848. [16] N. Ganikhodjaev, S. Uğuz. Physica A 390 (2011) 4160–4173. [17] E. Albayrak, A. Yigit. Physics Letters A 353 (2006) 121–129. [18] Mehmet Ertas, Mustafa Keskin, Bayram Deviren. J. Magn. Magn. Mater. 324 (2012) 1503–1511. [19] Mustafa Keskin, Yasin Polat. J. Magn. Magn. Mater. 321 (2009) 3905–3912. [20] M. Keskin, O. Canko, S. Güldal. Physics Letters A 374 (2009) 1–7. [21] R. Masrour, A. Jabar, A. Benyoussef, M. Hamedoun. J. Magn. Magn. Mater. 393 (2015) 151-156. [22] E. Albayrak. Physica. B 391 (2007) 47–53. [23] E. Albayrak, S. Yilmaz. Physica A 387 (2008) 1173–1184. [24] E. Albayrak, S. Yilmaz, S. Akkaya. J. Magn. Magn. Mater. 310 (2007) 98–106. [25] Y. Hu, K.C. Chan, L. Liu, Y.Z. Yang. J. Magn. Magn. Mater. 322 (2010) 2567–2570.

9

S

S 

S

S

 S JS 

S

J



S

 

S

S 

S  

S

S

S S

S

Fig. 1. Bethe lattice for mixed spin-5/2 and spin-2 Ising model on the Bethe with N=9 and NS=15 spins numbers.

10

Regions I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI XXVII XXVIII XXIX XXX

Phases (-2,-5/2) (+2,+5/2) (-2,+5/2) (+2,-5/2) (-1,-5/2) (+1,+5/2) (0,-5/2) (0,+5/2) (0,-3/2) (0,+3/2) (0,-1/2) (0,+1/2) (-1,+5/2) (+1,-5/2) (-1,-5/2) (+1,+5/2) (-1,-1/2) (+1,+1/2) (-2,-1/2) (+2,+1/2) (-2,+1/2) (+2,-1/2) (-2,-3/2) (+2,+3/2) (-2,+3/2) (+2,-3/2) (-1,+1/2) (+1,-1/2) (-1,+3/2) (+1,-3/2)

Table 1: The ground state configurations for alternate mixed spin-5/2 and spin-2 Ising model on the Bethe lattice.

11

1.0

(a)

I, II 0.5

V, VI IX, X XI, XII

JS0.0

VII, VIII XIII, XIV

-0.5

III, IV

-1.0 -5

1.0

-4

(b)

-3

J

XX

XVII, XVIII

III,

-2

XX

0.5

JS 0.0

XX -1.0 -3.0

-2.4

0

I, II IV

XIX,XX XXI,XXII

XI, XII

-0.5

1.0

-1

V

-1.8

X II, X

VII

I

III, IV

XXV, XXVI



-1.2

-0.6

0.0

(c)

I

II

IV

III

0.5

JS 0.0 -0.5

-1.0 -3

-2

-1

0

h 12

1

2

3

-1.0

(d)

II

I -1.5



-2.0

XXIV

XXIII

-2.5

XVII

XVIII

-3.0 -3

-1.0

(e)

-2

-1

0

1

h

IV

2

3

III

-1.5

XXVI

-2.0



XXX

XXV XXIX

XXVII XXVIII

-2.5

-3.0

XI

XII

-3.5 -4

-2

0

h

2

4

Fig.1(a)-d: The ground state phase diagrams of alternate mixed spin-5/2 and spin-2 Ising model on the Bethe lattice on the (Δ, JS), (h,JS), (h, Δ) for JS=1.0 and (h, Δ) planes for JS =-1.0.

13

3.0

Mtot,tot

(a)

M

2.5

MS, S 2.0

1.5

MS,S

1.0

0.5

0.0 0

3

5

10

T

15

20

(b)

25

Mtot,tot M

2

MS, S

MS,S1 0

-1

-2 0

3

6

T

9

12

15

Fig.2: The thermal magnetization and magnetic susceptibility for JS =1.0 (a), JS=-1.0 (b) with Δ=0.0 and h=0.75.

14

2.5

JS=1.0

(a)

JS=1.5

2.0

JS=2.0

1.5

Mtot,tot 1.0

0.5

0.0 0

5

10

15

20

25

A

2.5

JS=-1.0

(b)

JS=-1.5

2.0

JS=-2.0

1.5

2xMtot,tot 1.0

0.5

0.0 0

3

6

T

9

12

15

Fig.3: The thermal total magnetization and magnetic susceptibility for JS=1, 1.5, 2 (a), JS=1, -1.5, -2 (b) with Δ=0.0 and h=0.75.

15

(a)

JS=0.5

2.5

JS=1.5 2.0

JS=2.5 JS=3.0

1.5

Mtot 1.0

0.5

0.0 -5

-4

-3

-2



-1

0

1

(b)

JS=-1.0

0.8

2

JS=-2.0 JS=-3.0

0.6

JS=-3.5

Mtot 0.4 0.2

0.0 -5

-4

-3

-2



16

-1

0

1

2

2.5

(c)

T=1.0 T=2.5 T=3.5 T=4.5

2.0

1.5

Mtot 1.0

0.5

0.0 -4

1.0

-2



0

2

(d)

T=1.0 T=2.5 T=3.5 T=4.5

0.8

4

0.6

Mtot 0.4

0.2

0.0 -4

-2



0

2

4

Fig.4: The total magnetization versus the crystal field for JS =0.5, 1.5, 2.5, 3, T=1 (a), JS =1, -2, -3, -3.5, T=1(b), T= 1, 2.5, 3.5, 4.5, JS=0.5 (c) and JS=-0.5 (d) with Δ=0.0 and h=1.

17

2.5

T=1.0 T=2.5 T=3.5 T=4.5

2.0

1.5

Mtot

1.0

0.5

0.0 -4

-2

0

2

JS

4

Fig.5: The total magnetization versus the exchange interaction (JS) between spin-5/2 and 2 for T= 1, 2.5, 3.5, 4.5 with Δ=0.0 and h=1.

3

(b)

M

(a)

MS

2

3

2

1

1

Mtot0

0

M,S

-1

-1

-2

-2

-3 -1.0

-3 -0.5

0.0

h

0.5

-1.0 1.0

18

-0.5

0.0

h

0.5

1.0

3

(c)

(d)

M MS

2

3

2

1

1

Mtot0

0

M,S

-1

-1

-2

-2

-3 -1.0

2

-3 -0.5

0.0

h

0.5

-1.0 1.0

(e)

-0.5

0.0

h

0.5

1.0

(f)

M MS

3

2

1 1

Mtot

0

0

M,S -1 -1 -2 -2 -3 -1.0

-0.5

0.0

h

0.5

-1.0 1.0

-0.5

0.0

h

0.5

1.0

Fig.6: The variation of total magnetization and magnetization of each superlattice with spin5/2 and 2 vs the external magnetic field for =0 (a,b), -0.5 (c,d), -1 (e,f) with temperature T=1 and JS=1.

19

1.0

(a)

(b)

M

3

MS 2

0.5

1

0

B

Mtot0.0

M,S

-1 -0.5 -2

-1.0 -1.0

-3 -0.5

0.0

0.5

-1.0 1.0

h

-0.5

0.0

h

20

0.5

1.0

0.8

(c)

(d)

M

3

MS 2

0.4

1

M,S

0.0 Mtot

0

-1 -0.4 -2

-0.8 -1.0

0.6

-3 -0.5

0.0

h

0.5

-1.0 1.0

-0.5

0.0

h

0.5

M

(e)

1.0

(f)

MS

3

2

0.3 1

Mtot0.0

0

M,S

-1 -0.3 -2

-0.6 -1.0

-3 -0.5

0.0

0.5

-1.0 1.0

h

-0.5

0.0

0.5

1.0

h

Fig.7: The variation of total magnetization and magnetization of each superlattice with spin5/2 and 2 vs the external magnetic field for =0 (a,b), -0.5 (c,d), -1 (e,f) with temperature T=1 and JS=-1.

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

The alternate mixed spin-5/2 and spin-2 Ising ferrimagnetic system on the Bethe Lattice are studied.



The critical temperature have been deduced.



The magnetic coercive filed has been deduced for a ferro and antiferromagnetic phase.

22