Hysteresis Characteristics of Generalized Spin-S Magnetic Binary Alloys
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Hysteresis Characteristics of Generalized Spin-S Magnetic Binary Alloys
G¨ul¸sen Karakoyun The Graduate School of Natural and Applied Sciences, Dokuz Eyl¨ul University, Tr-35160 ˙Izmir,Turkey ¨Umit Akıncı Department of Physics, Dokuz Eyl¨ul University, TR-35160 Izmir, Turkey
In this study, hysteresis characteristics of the generalized spin-S binary alloy represented by theformula A c B − c have been investigated within the framework of effective field approximation. Thebinary system consists of type A (spin-S) and type B (spin-S) atoms which are randomly distributedon a honeycomb lattice. Both integer and half-integer spin models of two magnetic atom types areexamined. By detailed investigation on hysteresis loops, multiple hysteresis behaviors are obtainedfor a given set of Hamiltonian parameters. Besides, the quantities of hysteresis characteristicsas the hysteresis loop area, remanent magnetization, and coercive field have been investigated asfunctions of concentration. Keywords:
Hysteresis characteristics, binary alloy, effective field theory
The development and characterization of the transition metal alloys and rare earth alloys arehigh interest due to the emergence of successful applications in this rapidly evolving field [1].Multicritical points of rare earth alloys such as
T b − Er , Dy − Er , T b − T m , Dy − T m and Gd − Er have been investigated in the presence of crystal field effects [2]. Phase diagrams of magneticand non-magnetic transition metal alloys have been studied both experimentally and theoreticallybased on Ising-type phenomenological models [3].Disordered binary alloys represented by A c B − c can be modeled theoretically using well-knownIsing-like models. Binary alloy systems consisting of different spin values, such as half integer -integer spin valued models are investigated by means of the effective field theory (EFT) [4–6], meanfield theory (MFT) [7–9] and Monte Carlo (MC) simulations [10]. Models with half integer - halfinteger spins are also examined by means of MFT [11] and also within the two frameworks EFT andMFT [12,13]. Besides, several spin systems are modeled such as S A = 1 / S by use of MFT [14] and within EFT and MFT [12]. Generalized of bothspin variables of binary ferromagnetic alloy has been investigated with competing anisotropies bymeans of MFT [15]. Besides, F e p Al q alloys [16] and N iBi alloys [17] have been constructed on theIsing model within the framework of EFT. Site-diluted Ising spin model for
F e − q Al q alloys havebeen examined by use of EFT [18, 19] and pair approximation [20]. The bond disordered Blume-Capel model for ternary ( F e . N i . ) − x M n x and F e p Al q M n x alloys have been studied withmean-field renormalization group (MFRG) method [21]. The Potts-like model has been utilized todescribe Gd − x C x alloy on basis of MC method [22].It is crucial to emphasize that magnetic materials used in important technological applicationsare represented by higher spin systems. AB p C − p ternary alloy system corresponding to themagnetic Prussian blue analogs of( N i
IIp
M n II − p ) . [ Cr III ( CN ) ] · zH O type consisting of S A = 3 / S B = 1, S C = 5 / F e
IIp
M n II − p ) . [ Cr III ( CN ) ] · nH O typeanalog consisting of S A = 3 / S B = 2, S C = 5 / S A = 1 / S B = 1, S C = 3 / S A = 1, S B = 3 / S C = 1 / [email protected] [email protected]
1n the other hand, hysteresis behavior of magnetically ordered organic and molecule-based ma-terials has been inspected extensively [28,29]. A molecular-based magnetic material
AF e II F e
III ( C O ) which corresponds to ferrimagnetic mixed spin-2 and spin-5 / S + 1 magnetic plateaus in the presence of single-ion anisotropies and at low tempera-tures. Triple and double hysteresis behaviors have been found by means of MC [30] and EFT [31]for the given system.In Ref. [32], Akıncı concluded that crystal field diluted S − S ( S >
1) Blume Capel [33]and Heisenberg models [34] have been realized. These models display 2 S -windowed hysteresis loopin this region. These works also focused on the difference between integer and half integer spinmodels. While half integer spin model displays central loop (which is symmetric about the origin),integer spin model does not exhibit central loop.Similar results for other systems have been reported in the literature. For instance, the binaryalloy system consisting of spin-1 / < c < .
557 range within the framework of EFT [5]. The effects of the symmetric doubleGaussian random field distribution on this system has also been investigated by use of EFT, anddouble hysteresis character has been obtained depending on Hamiltonian parameters [35]. Hys-teresis behavior of quenched disordered binary alloy cylindrical nanowire consisting of the samespin set as the previous model has been examined by MC [36]. The effects of the concentrationand temperature on disordered
F e x Al − x alloys have been studied by using first-principle theoryand MC simulation [37]. There are also studies in the literature involving the hysteresis features ofhigher spin models. For example, mixed spin-1 / / / / / / / F e − x B x alloys [42], F e -doped Au x P d − x alloys [43], M n x Zn − x F alloys in the presence of random field [44], T b x Y − x and T b x Gd − x alloys in the presence of single-ion anisotropy [45], F e − x N i x and M n − x N i x alloys [46], F e − q Al q [47] and F e x Al − x M n x alloys [48].In our recent work, we have investigated the hysteresis behavior of the binary magnetic alloythat consist of spin-1 and spin-1 / S of type- A and type- B atoms consisting of different spin values. For this aim, the outline of this paper as follows: In Sec.3 we briefly present the model and formulation. The results and discussions are presented in Sec.4, and finally Sec. 5 contains our conclusions. The system consists of randomly distributed A type atoms that have spin- S A and B type atomsthat have spin- S B within the Ising model. The concentrations of the A type atoms are denoted as c , and the B type atoms are denoted as 1 − c . Therefore the chemical formula is given by A c B − c .Note that, the lattice has no vacancies. The Hamiltonian of the binary Ising model is given by H = − J X ( ξ i ξ j σ i σ j + ξ i δ j σ i s j + δ i ξ j s i σ j + δ i δ j s i s j ) − D X i (cid:0) ξ i σ i + δ i s i (cid:1) − H X i ( ξ i σ i + δ i s i ) , (1)where σ i , s i are the z components of the spin- S A and spin- S B operators and they take the values σ i = − S A , − S A + 1 , . . . , S A − , S A and s i = − S B , − S B + 1 , . . . , S B − , S B , respectively. J > D is the crystal2eld (single ion anisotropy) and H is the longitudinal magnetic field. Here, ξ i = 1 means thatlattice site i is occupied by type-A type atoms and δ i = 1 means that lattice site i is occupied bytype-B type atoms. The site occupation number holds the relation ξ i + δ i = 1. The first summationin Eq. (1) is over the nearest-neighbor pairs of spins and the other summations are over all thelattice sites.In a typical EFT approximation, we consider specific site (denoted by 0) and nearest neighborsof it. All interactions of this site can be represented by H A / H B if the site 0 occupied by type-A/Batoms, respectively. These terms can be treated as local fields acting on a site 0, H A = − ξ σ J z X j =1 ( ξ j σ j + δ j s j ) + H − ξ ( σ ) D, (2) H B = − δ s " J z X δ =1 ( ξ j σ j + δ j s j ) + H − δ ( s ) D. (3)By defining spin-spin interaction part of these local fields as, E A = J z X j =1 ( ξ j σ j + δ j s j ) , E B = J z X δ =1 ( ξ j σ j + δ j s j ) (4)we can write Eqs. (2) and (3) more compact form as, H A = − ξ σ (cid:2) E A + H (cid:3) − ξ ( σ ) D, (5) H B = − δ s (cid:2) E B + H (cid:3) − δ ( s ) D. (6)For obtaining magnetizations ( m A , m B ) and quadrupolar moments ( q A , q B ) for the system, wecan use the exact identities [49] which are given by m A = hh ξ σ ii r h ξ i r = 1 h ξ i r ** T r ξ σ exp (cid:0) − β H A (cid:1) T r exp (cid:0) − β H A (cid:1) ++ r ,q A = (cid:10)(cid:10) ξ σ (cid:11)(cid:11) r h ξ i r = 1 h ξ i r ** T r ξ σ exp (cid:0) − β H A (cid:1) T r exp (cid:0) − β H A (cid:1) ++ r ,m B = hh δ s ii r h δ i r = 1 h δ i r ** T r δ s exp (cid:0) − β H B (cid:1) T r exp (cid:0) − β H B (cid:1) ++ r , (7) q B = (cid:10)(cid:10) δ s (cid:11)(cid:11) r h δ i r = 1 h δ i r ** T r δ s exp (cid:0) − β H B (cid:1) T r exp (cid:0) − β H B (cid:1) ++ r . where T r is the partial trace over the site 0, β = 1 / ( k B T ), k B is Boltzmann constant and T is the temperature. We have two averages here, thermal averages (inner bracket) and randomconfigurational averages (bracket with subscript r ). This last average should be taken into accountto include the effect of the random distribution of the atoms in the system.Since all relations in Eq. (7) are in the same form, it is enough to derive one of them forobtaining the final form of the equation. Let us choose the equation related to m A from Eq. (7).By writing Eq. (5) in Eq. (7) and performing partial trace operations by using identity e ξx = ξe x + 1 − ξ, (where x is any real number and ξ = 0 ,
1) we can obtain expression in a closedform as hh ξ σ ii r h ξ i r = (cid:10)(cid:10) f Am (cid:0) E A (cid:1)(cid:11)(cid:11) r , (8)where the function is given by [50]. All definitions of these functions will be given at the end ofthis section. By using differential operator technique [51], Eq. (8) can be written as3 h ξ σ ii r h ξ i r = DD e E A ∇ EE r f Am ( x ) | x =0 , (9)where ∇ represents the differential with respect to x . The effect of the differential operator ∇ onan arbitrary function F is given byexp ( a ∇ ) F ( x ) = F ( x + a ) , (10)with arbitrary constant a .At this stage of the derivation, we have to convert the exponential operator in average bracesto a polynomial form. For this aim using using approximated van der Waerden identities [52] istypical. This identity is exp ( aS ) = cosh ( aη ) + Sη sinh ( aη ) , (11)where η = (cid:10) S (cid:11) and S is the spin eigenvalue. By using E A of Eq. (4) in Eq. (9) we can obtainpolynomial form of the operator. Then, by using Eq. (10) with the identity e ξx = ξe x + 1 − ξ, wecan obtain equation for m A m A = z X p =0 z − p X q =0 p X r =0 z − q − r X s =0 q + r X t =0 C pqrst ( − t c z − p (1 − c ) p (cid:18) m A η A (cid:19) q (cid:18) m B η B (cid:19) r f Am ([ z − s − t ] J, ) (12)where η A = q A = (cid:10) σ (cid:11) , and C pqrst = 12 z (cid:18) zp (cid:19) (cid:18) z − pq (cid:19) (cid:18) pr (cid:19) (cid:18) z − q − rs (cid:19) (cid:18) q + rt (cid:19) . (13)By using the same steps between Eqs. (8)-(12) to other relations in Eq. (7), we can obtainequations for other quantities as, q A = z X p =0 z − p X q =0 p X r =0 z − q − r X s =0 q + r X t =0 C pqrst ( − t c z − p (1 − c ) p (cid:18) m A η A (cid:19) q (cid:18) m B η B (cid:19) r f Aq ([ z − s − t ] J, ) (14) m B = z X p =0 z − p X q =0 p X r =0 z − q − r X s =0 q + r X t =0 C pqrst ( − t c z − p (1 − c ) p (cid:18) m A η A (cid:19) q (cid:18) m B η B (cid:19) r f Bm ([ z − s − t ] J, ) (15) q B = z X p =0 z − p X q =0 p X r =0 z − q − r X s =0 q + r X t =0 C pqrst ( − t c z − p (1 − c ) p (cid:18) m A η A (cid:19) q (cid:18) m B η B (cid:19) r f Bq ([ z − s − t ] J. ) (16)Here the functions are defined as [50], f Am ( x, H, D ) = S A X k = − S A k exp (cid:0) βDk (cid:1) sinh [ βk ( x + H )] S A X k = − S A exp ( βDk ) cosh [ βk ( x + H )] , (17) f Aq ( x, H, D ) = S A X k = − S A k exp (cid:0) βDk (cid:1) cosh [ βk ( x + H )] S A X k = − S A exp ( βDk ) cosh [ βk ( x + H )] . (18)4 Bm ( x, H, D ) = S B X k = − S B k exp (cid:0) βDk (cid:1) sinh [ βk ( x + H )] S B X k = − S B exp ( βDk ) cosh [ βk ( x + H )] , (19) f Bq ( x, H, D ) = S B X k = − S B k exp (cid:0) βDk (cid:1) cosh [ βk ( x + H )] S B X k = − S B exp ( βDk ) cosh [ βk ( x + H )] . (20)Eqs. (12) and (14)-(16) constitute a system of coupled nonlinear equations. The coefficientsin this system are given by Eq. (13). By using functions from Eqs. (17)-(19) we can solve thissystem by numerical procedures. After getting the values of m A , m B , q A , q B from this solution wecan calculate the total magnetization ( m ) and quadrupolar moment ( q ) of the system via m = cm A + (1 − c ) m B , q = cq A + (1 − c ) q B . (21)The hysteresis curves can be obtained by sweeping the magnetic field from − H to H andcalculating magnetization in each step. The reverse sweep will yield other branch of the curve, ifpresent. We consider the following scaled (dimensionless) quantities in this work d = DJ , t = k B TJ , h = HJ . (22)Results have been investigated on a honeycomb lattice (i.e. z = 3).In this study, we will discuss the effects of the crystal field and the concentration on thehysteresis properties of generalized spin-S binary alloy system. Note that for different selected spinvalues of these atoms we can choose both types of atoms as integer and half integer. We havelimited this work by examining the case of S A < S B . Note also that, the concentration of c in thecase of S A < S B corresponds to the concentration of 1 − c in the S A > S B case.The hysteresis loop can be obtained by calculating the magnetization by sweeping the magneticfield from − h to h direction and vice versa. The system prefers the non-magnetic s = 0 state whichis the disordered phase in the ground state, at large negative large crystal fields, for integer spinatoms. Besides, the system is exposed to the magnetic s = ± / s to the next occupied s + 1 groundstate, at low temperatures. If the magnetic field further increases, s + 1 → s + 2 → ... → S transitions occur, for both of integer and half integer spin values. This is the well known plateaulike ground state structure. If we examine these transitions for both positive and negative directionsof the external magnetic field, the magnetization response of the system could display interestinghysteresis characteristics. These characteristics are the main investigation area of this work.Firstly, we choose the spin values of two different types of atoms as half integer spins. Thehysteresis loops are depicted in Fig. 1 with selected values of the concentrations c = 0 . c = 0 . c = 0 .
8. Spin variables of the binary alloy system are S A = 1 / S B = 3 / S A = 5 / S B = 7 / c = 0 and c = 1 cases correspondto a situation where all lattice sites are occupied by spin-3 / / / / spin − / c = 0 . t = 0 .
45 and d = − c ), the outer windows that appears large negativeand positive values of magnetic field of hysteresis curve, disappear. The central loop continueswith the same structure, i.e. single hysteresis (SH) is observed. When the majority of the binaryalloy composed of type-A atoms (i.e. c = 0 . s = ± / c = 0 . c = 0 . h = 2 (see Figs. 1 (a) and (b)). As seen in Fig. 1 (b), for thebinary alloy system which consists of higher spin values, the number of windows of the hysteresisloops increase for a low temperature t = 0 . d = −
2. While c = 0 . S B -windowed) windows of hysteresis (7H) loops, c = 0 . c = 0 . S A -windowed) windows of hysteresis (5H) loops . Inset of graph in Fig. 1 (b) showsthe central loop which represent an ordered phase that exists for all concentration values. Asan important result, we can emphasize that, while the concentration decreases, two additionaloutermost symmetric windows of hysteresis loops appear. The magnetization of the system alsoincreases for large longitudinal magnetic field, as expected. These results are compatible with theresults presented in Ref. [33].In Fig. 2, we consider a system where both of the spin variables of binary alloy are integer-values. Hysteresis properties of the system are given for selected values of concentrations c = 0 . c = 0 . c = 0 .
8. Spin values are chosen as S A = 1 and S B = 2 in Fig. 2 (a) and S A = 1 and S B = 3 in Fig. 2 (b) for t = 0 . d = −
2. In the case of c = 0 . H ) loops which have no central loop (see Fig. 2 (a)). This means, the systemhas disordered phase which corresponds to the system with non-magnetic s = 0 ground state. Thephysical mechanism is as follows: with rising field in positive direction, s = 0 state begins to evolveinto s = ± s = ±
2, with rising magnetic field. For theconcentration values of c = 0 . c = 0 .
8, the outer symmetric two windows of the hysteresiscurve disappear and the system exhibits double hysteresis (DH) character. Magnetization decreaseswith increasing concentrations at large applied magnetic field. As seen in Fig 2 (b), if we fix thespin value of A atoms as S A = 1, and increase the spin value of B atoms such that S A = 3, thenthe binary alloy system with concentration c = 0 . H ) loops,without central loop. A considerable noteworthy result is that the system displays DH behaviorfor c = 0 . c = 0 .
8, due to the higher spin value of B atoms. The system eliminates theoutermost four symmetric loops to adapt from the 6-windowed hysteresis loop behavior to the DHloop behavior. Regardless of the difference between the spin values of the binary alloys, the numberof outer windows that disappeared will be twice that difference for concentration that close to thesmall spin value. To clarify, difference between the spin values of type-A and type-B atoms are 1(2), the number of outer windows that disappear will be two (four) with increasing components ofA atoms in Fig 2 (a) (Fig. 2 (b)), respectively.In the previous examinations, the binary alloy system had only one of the ordered (half in-teger spin valued binary alloys) or disordered (integer spin valued binary alloys) phases at lowtemperatures and large negative crystal field values. To find out which phase does the systemexhibit in binary alloys consisting of spin values integer and half integer, we choose spin value ofA atoms as integer and spin value of B atoms as half integer in Fig 3. Hysteresis behaviors ofbinary alloy consisting of (a) S A = 1, S B = 3 /
2, (b) S A = 1, S B = 7 / S A = 2, S B = 5 / c = 0 . c = 0 . c = 0 . c = 0 . t = 0 .
46 and d = − .
8. For thecase of c = 0 .
1, namely majority of the lattice sites consist of type B atoms, the system has THbehavior with central loop (see the curves related to the c = 0 . c = 0 .
3, the central loop disappears and the outer symmetric two loopsbecome narrower. Since the central loop is lost, the system should exhibit a phase transition inthis concentration ranges. For the value of c = 0 .
3, the system does not have a magnetic groundstate at low magnetic field values. As the applied magnetic field increases, smaller magnetic field6nduces the transition between the ground states of the system. Contrary to the common belief,the difference between the new ground states dominated by the B atoms are less than one. Thetransition from TH to DH can be arranged systematically as follows: The number of windows thatdisappeared will be two times of difference between two spin values of type A and type B atoms,i.e. 2( S B − S A ) windows disappear. The difference between S A = 1 and S B = 3 / / c = 0 .
5, the systemexhibits paramagnetic hysteresis (PH) which prefers s = 0 state (see the curves for c = 0 . c = 0 .
8, due to the spin value ofA atoms, 2 S A -windowed (namely DH) behavior appears. Depending on the concentration of thebinary alloy, DH for c = 0 . c = 0 . T H − DH − P H − DH hysteresis behaviors occur. The firstone (TH) corresponds to the ordered phase and the others (DH, PH and other DH) are disorderedphase. Phase transition between two phases causes 2( S B − S A ) = 1 windows to disappear. In Fig 3(b) Hamiltonian parameters are set as t = 0 .
46 and d = − c = 0 . c = 0 .
3, the system exhibits DH behavior which is demonstrated in lowerinset. The observed DH loops are also narrower than the concentric loops. So, transition fromordered to a disordered phase occurs between these concentration ranges. As the concentration oftype-A atoms increases in the system, we observe that some loops disappear according to the newground states, which are dominated by the B atoms. From the general result mentioned above,2( S B − S A ) = 5 windows should disappear. Four of them are the outermost symmetric windowsand the other is the central loop as can be seen in Fig 3 (b). For the value of c = 0 .
5, the systemexhibits disordered phase and then for c = 0 . H − DH − P H − DH hysteresis behaviors are observed as the concentrationincreases. In Fig 3 (c), the system exhibits 5 H − H − P H − H hysteresis behaviors (whilethe concentration increases) for t = 0 .
47 and d = −
2, respectively. Transition from the orderedphase ((5 H ) windowed loop) to the disordered phase ((4 H ) windowed loop) causes 2( S B − S A ) = 1window disappears, which is the central loop. The other 4-windowed loops get narrower. For thevalue of c = 0 . c = 0 .
8, the innermost 4-windowed hysteresis loops (which have different centers from the other4-windowed loops) appear.When we generalized integer-half integer results, for low temperatures and negative large crystalfields, 2 S B − S A − P H − S A -windowed hysteresis loops are observed while the concentrationincreases for S A < S B . The first one is the ordered phase and the other three are in disorderedphase. The transition from 2 S B to 2 S A -windowed loop causes 2( S B − S A ) disappearing windows.The last 2 S A -windowed hysteresis appears as innermost loops due to the case of S A < S B .In Fig 4, we have investigated binary alloy model consisting of half integer - integer spins suchas (a) S A = 1 / S B = 1, (b) S A = 3 / S B = 2 and (c) S A = 3 / S B = 3. In Fig 4 (a), thesystem exhibits DH − P H − P H − SH hysteresis behaviors for the values of t = 0 .
46 and d = − . H − H − P H − H hysteresis behaviors are obtained for t = 0 . d = − . H − H − P H − H hysteresis behaviors are observed for t = 0 . d = − . S B -windowed hysteresis loops are observed for lower concentration values, as expected. Ifthe concentration of the system is increase from c = 0 . c = 0 .
3, the innermost DH disappearsdue to the new ground states dominated by atoms of type-B (see Fig. 4 (b)). The system exhibitsparamagnetic hysteresis behavior for c = 0 .
5. If the concentration of the system is increases from c = 0 . c = 0 .
8, the innermost 2 S A -windowed hysteresis appears with a central loop. The firstthree hysteresis behaviors (4 H − H − P H ) correspond to disordered paramagnetic phase and thelast one is (3 H ) the ordered phase. The phase transition occurs between these two phases.The effects of crystal field parameter are examined for the binary alloys consisting of integer-half integer spin model as, can be seen in Fig 5. Spin values of the system are selected as S A = 1, S B = 7 / c = 0 . t = 0 . c = 0 . t = 0 .
46 in Fig. 5(b). As an evolution of the hysteresis loops with changing crystal field, SH − H − H − H − H isobserved (see Fig. 5 (a)). We concluded by comparison of Fig. 3 (b) and Fig. 5 (a) from which we7ee that, by increasing the temperature, the central loop disappears for the value of d = −
2. Thesystem exhibits SH behavior, which is represented by ordered phase for d = −
1, and disorderedphase for lower values of the crystal field. As the crystal field increases to a negative large values,the outermost symmetric loops gradually disappear and the windows start to be separated fromeach other. As seen in Fig. 5 (b) SH − SH − DH − P H − P H hysteresis behaviors are observed forgiven values of the crystal field. Phase transition occurs when the crystal field parameter changesfrom d = − d = −
2. When the system consists of mostly type-A atoms, multiple hysteresisbehavior is replaced by SH and DH behaviors. In Fig. 6, we have investigated the effects ofcrystal field on binary alloy consisting of half integer-integer spins at the temperature t = 0 . S A = 3 / S B = 2, the system displays SH − H − H − H hysteresisbehaviors for c = 0 . d = − . d = − SH − H − DH − P H for c = 0 .
9, respectively can beseen in Fig. 6 (b). The central loop disappears when passing from 3 H to DH behavior, and phasetransition occurs in this range.In order to investigate the evolution of the hysteresis properties, we describe the quantitieshysteresis loop area (HLA), remanent magnetization (RM) and coercive field (CF). HLA is definedas the area covered by hysteresis loop in ( m, h ) plane, and it describes the loss of energy due tothe hysteresis. The RM is residual magnetization in the system after an applied magnetic field isremoved. CF is the intensity of the magnetic field needed to change the sign of the magnetization.In Figs. 7, 8 and 9 we demonstrate the variation of the quantities (HLA, RM and CF, respec-tively) with the temperature, for one or both of two spin variables chosen as integer or half integerspin. Chosen spin values at crystal field d = 0 are (a) S A = 1 / S B = 3 /
2, (b) S A = 1, S B = 2, (c) S A = 1, S B = 3 / S A = 3 / S B = 2. Besides, the selected values of the concentrations are c = 0 . c = 0 . c = 1 . c = 0 . c = 0 .
5. In c = 1 . c = 1 concentration at low temperatures and the value of HLA is greater than Fig. 7 (a).Therefore, more energy dissipation occurs. The temperature values that HLA vanishes, increases.For c = 0 the value of HLA has remained constant at low temperatures as in Fig 7 (b). If we fixspin value of A atoms as S A = 1, and increase the spin value of B atoms such that S B = 3 /
2, HLAcurves of c = 0 . c = 0 . S B = 2, then increase the spin value of A atoms such that S A = 3 /
2, HLA curves of c = 0 . c = 1 . In conclusion, hysteresis characteristics of the generalized spin-S magnetic binary alloy systemrepresented by A c B − c have been investigated within the framework of effective field theory. Thesystem consists of type- A and type- B atoms with the concentrations c and 1 − c , respectively.Results of the generalized spin- S binary alloy model are discussed as one or both of the spins of A and B atoms are selected as integer or half integer spin values.The effects of the concentration and crystal field parameters of the magnetic binary alloy modelstrongly depend on whether the spins of the atoms are integer or half-integer. As consistently bythe related literature, the special cases ( c = 0, c = 1) of binary alloy system exhibit 2 S -windowedhysteresis character. The evolution of the multiple hysteresis loops are observed for higher spinvalued alloy system for the large negative value of crystal field at low temperatures. Integer (half-integer) spin valued binary alloy system exhibits disordered (ordered) phase in this region. It hasbeen found that the number of outer windows that disappeared will be twice that difference betweenspin values when majority of binary alloy composed of type A atoms in the case of S A < S B .The significantly remarkable point of our results is that the effect of concentration is one of themost important parameters affecting the hysteresis behavior in the system. Since the majority of8 c=0.1 c=0.3 c=0.8 m h d=-2t=0.45 S A =1/2S B =3/2(a) -4 -3 -2 -1 0 1 2 3 4-4-3-2-101234 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 m h m h d=-2t=0.5 c=0.1 c=0.3 c=0.8S A =5/2S B =7/2(b) Figure 1: Hysteresis evolutions of the binary alloy system which consist of half integer-half integerspin model such as (a) S A = 1 / S B = 3 / t = 0 .
45 and (b) S A = 5 / S B = 7 / t = 0 . c = 0 . c = 0 . c = 0 . d = − S B − S A − P H − S A -windowed hysteresis loops areobserved with increasing concentrations for integer-half integer system. The transition from 2 S B to 2 S A -windowed loop causes 2( S B − S A ) vanishing windows. When the majority of binary alloy iscomposed of integer spin values, inner (low magnetic field) ground states of the system disappearat low magnetic field as we started to add more half-integer spin to the half integer-integer system.As concentration rises, the magnetic ground states dominated by half-integer spin appear and thesystem is in an ordered phase anymore.It has been demonstrated that the outermost symmetric loops disappear gradually and win-dows are separated from each other, as the crystal field parameter is increased in the negativedirection. Besides, the quantities of hysteresis loops have been investigated with the variation ofthe temperature. Rising temperature drags the system into a disordered phase due to the thermalagitations. As the concentration increases from c = 0 to 1, HLA, RM and CF decreases for allbinary alloy system which is one or both of two spin variables chosen as integer or half integer spinmodel. These quantities increase as the spin value gets higher.We hope that the results obtained in this work may be beneficial form both theoretical andexperimental points of view. 9 m h d=-2t=0.5 c=0.1 c=0.3 c=0.8S A =1S B =2(a) -4 -3 -2 -1 0 1 2 3 4-4-3-2-101234 m h d=-2t=0.5 c=0.1 c=0.3 c=0.8S A =1S B =3(b) Figure 2: Hysteresis evolutions of the binary alloy system which consist of integer-integer spinmodel such as (a) S A = 1, S B = 2 and (b) S A = 1, S B = 3 spin variables for selected concentrationvalues c = 0 . c = 0 . c = 0 .
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2, (b) S A = 1, S B = 7 / S A = 2, S B = 5 / c=0.1 c=0.3 c=0.5 c=0.8S A =1/2S B =1 m h d=-1.6t=0.46(a) -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5 (b)S A =3/2S B =2 -0.2 -0.1 0.0 0.1 0.2-0.50.00.5 m h m h c=0.1 c=0.3 c=0.5 c=0.9d=-1.8t=0.5 -2 -1 0 1 2-3-2-10123 (c)S A =3/2S B =3 -0.2 -0.1 0.0 0.1 0.2-101 m h m h c=0.1 c=0.3 c=0.5 c=0.9d=-1.8t=0.5 Figure 4: Hysteresis evolutions of the binary alloy system which consists of half integer-integerspin model such as (a) S A = 1 / S B = 1, (b) S A = 3 / S B = 2 and (c) S A = 3 / S B = 3 spinvariables for a given set of Hamiltonian parameters.14
10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-4-3-2-101234 d=-1.0 d=-2.0 d=-3.0 d=-3.5 d=-4.5 -4 -3-4-3-2-10 m h m h S A =1S B =7/2c=0.1t=0.5(a) Figure 5: Hysteresis evolutions of the binary alloy system which consists of integer- half integerspin model such as S A = 1, S B = 7 / c = 0 . c = 0 . -4 -2 0 2 4-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5 S A =3/2S B =2 m h d=-0.5 d=-2.0 d=-3.0 d=-5.0c=0.1t=0.46(a) -5 -4 -3 -2 -1 0 1 2 3 4 5-2.0-1.5-1.0-0.50.00.51.01.52.0 S A =3/2S B =2(b) -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 m h m h d=-0.5 d=-2.0 d=-3.0 d=-5.0c=0.9t=0.46 Figure 6: Hysteresis evolutions of the binary alloy system which consists of half integer-integerspin model such as S A = 3 / S B = 2 for the concentrations (a) c = 0 . c = 0 . .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0012345678910 d=0.0 c=0.0 c=0.5 c=1.0 H L A t SA=1/2SB=3/2(a) d=0.0 c=0.0 c=0.5 c=1.0 H L A t SA=1SB=2(b) d=0.0 c=0.0 c=0.5 c=1.0 H L A t SA=1SB=3/2(c) d=0.0 c=0.0 c=0.5 c=1.0 H L A t SA=3/2SB=2(d)
Figure 7: Variation of HLA with the temperature of the binary alloy system which consists of (a) S A = 1 / S B = 3 /
2, (b) S A = 1, S B = 2, (c) S A = 1, S B = 3 / S A = 3 / S B = 2 spinvariables for selected values of concentrations c = 0 . c = 0 . c = 1 . dd
2, (b) S A = 1, S B = 2, (c) S A = 1, S B = 3 / S A = 3 / S B = 2 spinvariables for selected values of concentrations c = 0 . c = 0 . c = 1 . dd = 0 crystal fieldparameter. 16 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-1.5-1.0-0.50.00.51.01.5 c=0.0 c=0.5 c=1.0 d=0.0 R M t SA=1/2SB=3/2(a) c=0.0 c=0.5 c=1.0 d=0.0 R M t SA=1SB=2(b) c=0.0 c=0.5 c=1.0 d=0.0 R M t SA=1SB=3/2(c) c=0.0 c=0.5 c=1.0 d=0.0 R M t SA=3/2SB=2(d)
Figure 8: Variation of RM with the temperature of the binary alloy system which consists of (a) S A = 1 / S B = 3 /
2, (b) S A = 1, S B = 2, (c) S A = 1, S B = 3 / S A = 3 / S B = 2 spinvariables for selected values of concentrations c = 0 . c = 0 . c = 1 . dd
2, (b) S A = 1, S B = 2, (c) S A = 1, S B = 3 / S A = 3 / S B = 2 spinvariables for selected values of concentrations c = 0 . c = 0 . c = 1 . dd = 0 crystal fieldparameter. 17 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-2.0-1.5-1.0-0.50.00.51.01.52.0 c=0.0 c=0.5 c=1.0 C F t d=0.0SA=1/2SB=3/2(a) c=0.0 c=0.5 c=1.0 C F t d=0.0SA=1SB=2(b) c=0.0 c=0.5 c=1.0 C F t d=0.0SA=1SB=3/2(c) c=0.0 c=0.5 c=1.0 C F t d=0.0SA=3/2SB=2(d) Figure 9: Variation of CF with the temperature of the binary alloy system which consists of (a) S A = 1 / S B = 3 /
2, (b) S A = 1, S B = 2, (c) S A = 1, S B = 3 / S A = 3 / S B = 2 spinvariables for selected values of concentrations c = 0 . c = 0 . c = 1 . dd