Kinetic Blume-Capel Model with Random Diluted Single-ion Anisotropy
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Kinetic Blume-Capel Model with RandomDiluted Single-ion Anisotropy
Gul Gulpinar a ∗ , Erol Vatansever a a Dokuz Eyl¨ul University, Department of Physics, 35160-Buca, ˙Izmir, Turkey
Abstract
This investigation employs a square lattice and Glauber dynamics methodologyto probe the effects of diluteness in the crystal-field interaction in a Blume-CapelIsing system under an oscillating magnetic-field. Fourteen different phase diagramshave been observed in temperature-magnetic-field space as the concentration of thecrystal-field interactions is varied. Besides, a comparison is given with the resultsof the pure spin-1 Ising systems.
Key words:
Quenched disorder, Random crystal field, Dynamical critical points,Kinetic Blume-Capel Model.
PACS:
Ising spin-1 systems, with density as an added degree of freedom, havebeen utilizied to investigate a diverse range of systems: materials with mobile ∗ corresponding author. Email address: [email protected] (Gul Gulpinar a ). Preprint submitted to Chinese Physics B 27 November 2018 efects, structural glasses [1], the superfluid transition in He − He mixtures[2], frustrated Ising lattice gas systems [3], binary fluids, binary alloys andfrustrated percolation [4]. Recently, Blume-Capel (BC) [6] and Blume-Emery-Griffiths (BEG) models have been used to probe the phenomenon of inversemelting, a phenomena observed in diverse class of systems such as colloids,polymers, miscelles, etc [5].On the other hand, it is well known that the effects upon criticality andresulting phase diagrams, due to underlying competing interactions in variousspin-1 Ising systems can be complicated. Since then, many previous investiga-tions have been focused on various novel types of competing interactions havebeen the focus of previous studies using the BEG model in conjunction withrenormalization-group [7,8,9,10] and/or mean-field methodologies [11,12,13].Besides, effects of disorder on magnetic systems have been systematically stud-ied, not only for theoretical interests but also for the identifications with ex-perimental realizations [14,15,16]. It has been shown by renormalization grouparguments that first-order transitions are replaced by continuous transition,consequently tricritical points and critical end points are depressed in temper-ature, and a finite amount of disorder will suppress them [17].An special magnetic system with disorder is spin-1 Ising model with quencheddiluted single ion anisotropy is used to model phase separations of superfluid-ity for helium mixtures in aerogel [18,19]. Due to this fact, various researchershave been motivated to study the effect of the crystal field disorder on themulticritical phase diagram of BC model via effective field theory [20] andmean field approach [21,22],cluster variation method [23], as well as by intro-ducing an external random field [24]. Whereas, Branco et al. considered theeffects of random crystal fields using real-space RG [8,9] and mean-field ap-2roximations [9,25] for both BC and BEG model Hamiltonians, respectively.Recently, Snowman has employed a hierarchical lattice and renormalization-group methodology to probe the effects of competing crystal-field interactionsin a BC model [26] . Finally, Salmon and Tapia have studied the multicriti-cal behavior of the BC model with infinite-range interactions by introducingquenched disorder in the crystal field ∆ i , which is represented by a superpo-sition of two Gaussian distributions [27].While the equilibrium properties of the BC model with random single ionanisotropy have been studied extensively, as far as we know, the kinetic aspectsof the model have not been investigated via Glauber dynamics. Therefore, thepurpose of the present paper is, to present a study of the kinetics of the spin-1BC model with a quenched two valued random crystal field in the presence ofa time-dependent oscillating external magnetic field. We make use of Glauber-type stochastic dynamics to represent the time evolution of the system [28].More precisely, we have obtained the dynamic phase transition (DPT) pointsand presented phase diagrams in constant crystal field and the reduced mag-netic field amplitude versus reduced temperature plane for various values ofthe crystal field concentration. This type of calculation for pure BC model, wasfirst performed by Buendia and Machado [29]. They have presented only twophase diagrams in the temperature-magnetic field plane for the pure spin-1BC model. Later, Keskin et. al. have shown that one of the two phase dia-grams in Ref [29] was incomplete; i.e., they had missed a very important partof the phase diagram due to the reason that they did not make the calcula-tions for higher values of the amplitudes of the external oscillating magneticfield [30]. Keskin et. al. presented the phase diagrams in the reduced magneticfield amplitude (h) and reduced temperature (T) plane and calculated five dis-3inct phase diagram topologies. Recently, Elyadari et.al. has investigated thekinetic Blume Capel Model with a random crystal field distributed accordingto the following law: P (∆ i ) = pδ (∆ i − ∆(1 + α )) + (1 − p ) δ (∆ i − ∆(1 − α )) .This kind of random crystal field has been introduced to study the criticalbehavior of He − He mixtures in random media (aerogel) modeled by thespin-1 Blume-Capel model [31]. In their model, the negative crystal field valuecorresponds to the field at the pore-grain interface and the positive one is abulk field that controls the concentration of He atoms. In other words, inthis kind of randomness the crystal field value is finite for each site about itsamplitude takes one of the values ∆(1 + α ) or ∆(1 − α ) with equal and fixedprobabilities ( p = 1 / p of active local crystal fields. [20,21]. With thismotivation we have performed numerical calculations for various values of thecrystal field concentrations in order to observe the effect of the quenched va-cancy in the crystal field on the five different kinetic phase diagram topologiesfound by Keskin and co-workers [30].Meanwhile, it is worthwhile to stress that the DPT was first found in thestudy of the kinetic Ising system in an oscillating field [32], and it was fol-lowed by Monte Carlo simulation researches of kinetic Ising models [33,34].Further, Tutu and Fujiwara [35] represented a systematic method for obtain-ing the phase diagrams in DPTs, and constructed a general theory of DPTsnear the transition point based on a mean-field description, such as Landau ′ s4eneral treatment of the equilibrium phase transitions. DPT may also havebeen observed experimentally in ultrathin Co films on Cu(001) [36] by meansof the surface magneto-optic Kerr effect and in ferroic systems (ferromagnets,ferroelectrics and ferroelastics) with pinned domain walls [39] and ultra-thin[ Co/P t ] multilayer [37]. In addition, reviews of earlier research on the DPTand related phenomena are found in Ref [34].The paper is organized as follows: In Sec.2, we discuss the kinetic BC modelwith single ion isotropy briefly. Moreover, the derivation of the mean-field dy-namic equations of motion is given by using a master equation formalism inthe presence of an oscillating external magnetic field is also given in Sec.2.In Sec.3, the DPT points are calculated and the phase diagrams presented.Finally, Sec.4 represents the summary and conclusions. The generalization of the kinetic BC model for a quenched random crystalfield is given by the Hamiltonian,ˆ H = − J X h ij i S i S j − X { i } ∆ i S i − H X { i } S i , (1)where the spin variables S i = 0 , ± J > i is given by the following joint probability density: P (∆ i ) = pδ (∆ i − ∆) + (1 − p ) δ (∆ i ) . (2)5inally, H is a time-dependent external oscillating magnetic field and givenby, H ( t ) = H cos ( ωt ) , (3)here H and ω = 2 πν denote the amplitude and the angular frequency of theoscillating field respectively.When we put this system in contact with a heat reservoir at temperatureT, the spin variables S i can be considered as stochastic functions of time. Thesystem evolves according to a Glauber-type stochastic process at a rate of τ transitions per unit time. More precisely, we will follow the heat-bath prescrip-tion [38]: the new value of the spin variable at site i ( S i new ) is determined bytesting all its possible states in the heat-bath of its (fixed) neighbors (herefour on a square lattice): w i ( S i old → S i new ) = 1 τ exp {− β ∆ E ( S i old → S i new ) } P exp {− β ∆ E ( S i old → S i new ) } , (4)where β = kT and τ defines a time scale (characteristic mean time interval forone spin flip), and∆ E ( S i old → S i new ) = ( S i old − S i new ) J X h j i S j + H − (cid:16) S i old − S i new (cid:17) ∆ i , (5)give the changes in the energy of the system in the case of flipping of the i th spin in the lattice. If we define P ( S , S , ..., S N ; t ) as the probability that thesystem has the configuration { S , S , ..., S N } , at time t. Making use of masterequation formalism [28], one can write the time derivative of P ( S , S , ..., S N ; t )6s, ddt P ( S , ..., S N ; t ) = − P i P S i old = S i new w i ( S i old → S i new ) P ( S , ..., S i old , ...S N ; t )+ P i P S i old = S i new w i ( S i new → S i old ) P ( S , ..., S i new , ...S N ; t ) . (6)The detailed balance condition reads, w i ( S i old → S i new ) w i ( S i new → S i old ) = P ( S , S , ..., S i new , ..., S N ) P ( S , S , ..., S i old , ...S N ) . (7)In addition, substituting the possible values of S i new and S i old , one ob-tains: w i (1 →
0) = w i ( − →
0) = 1 τ exp ( − β ∆ i )2 cosh ( βδ ) + exp ( − β ∆ i ) ,w i (1 → −
1) = w i (0 → −
1) = 1 τ exp ( − βδ )2 cosh ( βδ ) + exp ( − β ∆ i ) ,w i (0 →
1) = w i ( − →
1) = 1 τ exp ( βδ )2 cosh ( βδ ) + exp ( − β ∆ i ) , (8)where δ = J P h j i S j + H . At this point one can notice that w i ( S i old → S i new )does not depend on the value S i old , we can write w i ( S i old → S i new ) = w i ( S i new ), then the master equation becomes: ddt P ( S , ..., S N ; t ) = − P i P S i old = S i new w i ( S i new ) P ( S , ..., S i old , ...S N ; t )+ P i w i ( S i old ) P S i old = S i new P ( S , ..., S i new , ...S N ; t ) . (9)On the other hand, the sum of probabilities is normalized to one so thatby multiplying both sides of Eq.(9) by S p and taking the average, one obtains, τ ddt h S p i = − h S p i + *Z P (∆ i ) 2 sinh (cid:16) β h J P h j i S j + H i(cid:17) cosh ( β h J P h j i S j + H i ) + exp ( − β ∆ i ) d ∆ i + . (10)7inally, after integration over the distribution of P (∆ i ) and making use ofmean field approximation, the kinetic equation of the magnetization becomes,Ω ddξ m = − m + p sinh ( m + hcos ( ξ ) T )2 cosh ( m + hcos ( ξ ) T ) + exp ( − dT ) + (1 − p ) 2 sinh ( m + hcos ( ξ ) T )2 cosh ( m + hcos ( ξ ) T ) + 1 , (11)where ξ = ωt, m = h S i , T = ( βzJ ) − , d = ∆ /zJ, h = H /zJ . In theseequations the variable Ω was defined as the ratio between the external fieldfrequency ω and the frequency of spin flipping ( f = 1 /τ ), i.e., Ω = ωτ = ω/f . Here we consider a cooperatively interacting many-body system, drivenby an oscillating external perturbation, an oscillating magnetic field so thatthe thermodynamic response of the system, the magnetization, will then alsooscillate with necessary modifications in its form [34]. Moreover, the timedependence of magnetization can be one of two types according to whetherthey have or do not have the property: m ( ξ + π ) = − m ( ξ ) . (12)A solution that satisfies Eq.(12) is called symmetric solution; it correspondsto a paramagnetic (P) phase. In this solution, the magnetization m ( ζ ) oscil-lates around the zero value and is delayed with respect to the external field.Solutions of the second type, which do not satisfy Eq.(12), are called non-symmetric solutions; they correspond to a ferromagnetic (F) phase. In thiscase, the magnetization does not follow the external magnetic field any more,but, instead, oscillates around a nonzero value. Eq. (11) is solved numericallyby using fourth order Runge-Kutta method for fixed values of T, d, p , and Ω.Throughout this study we have fixed Ω = 2 π , J = 1 and z = 4 for a given set ofparameters and initial values. The results are presented in Figs.1(a)-(c). Here,we can see three different solutions: F,P and coexistence of F and P (F+P).8n Fig.1(a), only the symmetric solution is always obtained, and, hence, wehave a paramagnetic (P) solution; but, in Fig.1(b), only the nonsymmetricsolutions are found, and we, therefore, have a ferromagnetic (F) solution. Onecan observe from these figures that these solutions do not depend on the initialvalues. On the other hand, in Fig.1(c), both the F and P phases exist in thesystem this case corresponds to the coexistence solution (F + P). As can beseen in Fig.1(c) explicitly, the solutions depend on the initial values. In order to obtain the dynamic phase boundaries between three phases orregions in that are given Figs.1(a)-(c), one should calculate the DPT points.The DPT points are obtained by investigating the behavior of the averagemagnetization in a period as a function of the reduced temperature. M = 12 π ξ +2 π Z ξ m ( ξ ) dξ, (13)Here m ( ξ ) is a stable and periodical function. In general our solution sta-bilizes after 6000 periods. In this manner, ξ can take any value after thistransient. In the high field and high temperature region time dependent stag-gered magnetization follows the reduced external magnetic field within a singleperiod which corresponds to vanishing time average of the dynamical orderparameter (paramagnetic phase). Whereas, at low field values the magneti-zation can not fully switch sign in a single period and the time average ofthe magnetization in a period is non zero and consequently ordered or ferro-9agnetic phase arises. Fig.2 represents the reduced temperature dependenceof the average magnetization (M) for various values of magnetic field ampli-tude h and crystal field concentration (p) while d = − .
25. In these figuresarrows denote the transition temperatures. In Fig.2(a), we give the case for h = 0 . , p = 0 .
50. In this case, the system represents re-entrance with twosequential first order phase transitions which take place at T t and T t . WhileFig.2(b) exhibits the reduced temperature dependence of the dynamical or-der parameter for h = 0 . , p = 0 .
75. For these values of the parameters,BC model with random single ion anisotropy undergoes a first and a secondphase transition sequentially. In Fig.2(c) we give an example of second or-der phase transition from ordered to disordered phase for h = 0 . , p = 0 . h = 0 . , p = 0 . d = − .
3) since then the critical temperature increases with increasingdiluteness for fixed h.On the other hand, it is well known fact that in the static limit ( ω = 0 . h − T planecollapses to a line with h = 0 and ending at T = T c , the static transitiontemperature of the unperturbed system [34]. Fig.3(b) shows the thermal vari-ations of M and for several values of static h while d = − .
5. In addition,Fig.3(c) gives the behaviors of M and as a function of static h for d = − . d = 0 .
25. The numberaccompanying each curve denotes the value of the crystal field concentration(p). The outermost curve corresponds to the pure BC model with no quenchedrandomness ( p = 0).As crystal field quenched randomness is introduced withdecreasing values of p, ordered phases and first-order phase transitions recede.This result is consistent with the RG theory predictions given in Ref.[17]. Fi-nally, we should stress that similar phase diagrams were also obtained in thekinetic of the mixed spin- and spin-1 Ising ferromagnetic system [29] as wellas the kinetic spin- Ising model [32]. The reason that the phase diagram issimilar to the one obtained for the kinetic spin- Ising model is due to thecompetition between
J, d and h . For positive crystal field values, the Hamilto-nian of the spin-1 model gives similar results to the Hamiltonian of the spin- Ising model.It has been given in Ref.[30] that pure kinetic BC model has four differentphase diagram topologies for negative d, which depend on d values. Now letus discuss the effect of randomness in the single ion isotropy on these phasediagrams:(1) For − . > d ≥ − . d = ∆ /zJ <
0) favors the annealedvacancies, namely the nonmagnetic states S i = 0. Fig.3(b) exhibits this fact:with increasing concentration of negative single ion isotropy ( d = − .
25) theordered phase recedes and the tricritical temperature moves to lower temper-atures. Whereas, the coexistence region (F+P) in the low temperature andfield region disappears with increasing vacancy in the single ion anisotropy(for p ≤ . − . < d ≤ − . d = 0 .
25 and d = − .
25 , whereas, the other DTCP occurs in the low hregion. In addition, the first-order phase transition lines exist at the low re-duced temperatures, and h values separate not only the P+F region from theF phase, but also from the P phase. When we introduce quench disorder inthe crystal field we found that this topology changes drastically with varyingcrystal field concentration ( p ). Our calculations has revealed that there arefour different phase diagram topologies which depend on p values:(2.a) Type 1 ( p ≥ . . > p ≥ . h = 0axis. Moreover, the P+F phase recedes and an ordered phase appears in theneighborhood of h = 0 , T = 0 (see Fig.3(d)).(2.c) Type 3 (0 . > p ≥ . . > p ≥ . − . > d ≥ − . − . < d ≤ − . . ≥ p > . . ≥ p > . . ≥ p > . . ≥ p ≥ . − . > d , the topology of the phase diagram is dramatically dif-ferent from the other intervals of the single ion anisotrpy amplitude : it doesnot include a P+F phase coexistence region at low temperature and low mag-netic field for pure spin-1 BC model. In Figs. 6(a) to (d) we illustrate the fourdifferent types of behavior depending on the the p:4(a) Type 1(1 ≥ p > . . ≥ p > . . ≥ p ≥ SUMMARY AND CONCLUSIONS
Within the mean field approach, we have analyzed stationary states ofthe spin-1 Blume-Capel model with a random crystal field ∆ i under a time-dependent oscillating external magnetic field. The time evolution of the systemis described by a stochastic dynamics of the Glauber type. We have studied thetime dependence of the magnetization and the behavior of the dynamical or-der parameter as a function of reduced temperature for reduced magnetic fieldand different possibility (p) of the crystal field. We have also analyzed thermalvariations and temperature dependence of M for various values of crystal fieldconcentration (p) and for different static reduced magnetic field, respectively.Moreover, the behavior of M as function of static reduced magnetic field (h)for various values of the reduced temperature have been examined.The dynamic phase transition (DPT) points are found and the phase di-agrams are constructed in the reduced magnetic field and temperature plane.We have found that the behavior of the system strongly depends on the val-ues of random crystal field or random single-ion anisotropy. For all (p) andpositive values of reduced crystal field (d) the system behaves as the standardkinetic Ising model [32], and also kinetic mixed Ising spin-(1/2,1) model [29].We have observed that there exist F+P coexistence and first order dynamicalphase transitions in the low temperature and high field regime. Whereas, thedynamical phase transitions turns out to be second order with increasing tem-perature and decreasing magnetic field. Consequently, it shows that the systemexhibits dynamical tricritical point (DTCP). As we have mentioned in detailin the previous section, the introduction of random ion-anisotropy in kineticspin-1 BC model produces an effect which suppress the F+P coexistence region15n the low temperature low field and high field low temperature regimes. Weshould stress that this result in according with the previous results obtainedby Renormalization-Group theory [17]. Finally we should point out that someof the first-order phase lines and also the dynamic multicritical points (DTCPand DCEP) are very likely artifacts of the mean-field approach. The reasonof this artifact can be stated as follows: for field amplitudes less than thecoercive field and temperatures lower than the static ferromagnetic - param-agnetic transition temperature, the time dependent magnetization representsa nonsymmetric stationary solution even zero frequency limit. Meanwhile inthe absence of the fluctuations, the system is trapped in one well of the freeenergy and cannot go to other one [33]. On the other hand, this mean-fielddynamic study reveals that the random single ion anisotropy spin 1 Blume-Capel Model represents interesting dynamic phase diagram topologies. Sincethen, we hope that this work can stimulate further studies on kinetic featuresof kinetic random single ion anisotropy spin-1 Model systems theoretically andexperimentally. This research was supported by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK), including computational support through theTR-Grid e-Infrastructure Project hosted by ULAKBIM, and by the Academyof Sciences of Turkey. 16 eferences [1] Kirkpatrick T R and Thirumalai D 1987 Phys. Rev. B 36 5388.[2] Blume M, Emery V J and Griffiths R B 1971 Phys. Rev. A 4 1071.[3] Nicodemi M and Coniglio A 1997 J. Phys. A 30 L187.Arenzon J J, Nicodemi M and Sellitto M 1996 J. Phys. I 6 1143.[4] Coniglio A 1993 J. Phys. IV 3 C1-1 .[5] Schupper N and Shnerb N M 2005 Phys. Rev. E 72 046107.Angelini R, Ruocco G and Panfilis S De 2008 Phys. Rev. E 78 020502.[6] Blume M 1966 Phys. Rev. 141 517.Capel H W 1966 Physica 32, 966 .[7] Berker A N and Wortis M 1976 Phys. Rev. B 14 4946.McKay S R, Berker A N, and Kirkpatrick S 1982 Phys. Rev. Lett. 48 767.[8] Branco N S and Boechat B M 1997 Phys. Rev. B 56 11673.[9] Branco N S 1999 Phys. Rev. B 60 1033.[10] Snowman D P 2007 J. Magn. Magn. Mater. 314 69.Snowman D P 2008 J. Magn. Magn. Mater. 320 1622.Snowman D P 2008 Phys. Rev. E 77 041112.[11] McKay S R and Berker A N 1984 J. Appl. Phys. 55 1646.[12] Hoston W and Berker A N 1991 Phys. Rev. Lett. 67 1027.[13] Sellitto M, Nicodemi M, and Arenzon J J 1997 J. Phys. I 7 945.[14] Bouchiat H, Dartyge E, Monod P and Lambert M 1981 Phys. Rev. B 23 1375.[15] Katsumata K, Nire T and Tanimoto M 1982 Phys. Rev. B 25 428.
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50 100 150 200 250-1.0-0.50.00.51.0 m( ) (a) m( ) (b) m( ) (c)Fig. 1. Time variance of the magnetization ( m ( ξ )) while p=0.9 : (a) Correspondingto a paramagnetic phase (P) for d=-0.25, h=0.5, and T=0.7; (b) Exhibiting a ferro-magnetic phase (F) for d=-0.25, h=0.25, and T=0.5; (c) Representing a coexistenceregion (F+P) d=-0.25, h=0.75, and T =0.1. .0 0.2 0.40.00.20.40.60.81.0 T t2 d=-0.25h=0.70p=0.50 M T T t1 (a) T c T t d=-0.25h=0.70p=0.75 M T (b) d=-0.25h=0.4p=0.75
M T Tc (c) T t d=-0.25p=0.75h=0.75 M T (d)Fig. 2. Dynamical order parameter as a function of reduced temperature. T c and T t indicate second and first order phase transition temperatures respectively. (a)The system under goes two successive first order phase transitions, there existsre-entrance. (b) Two successive phase transitions: the first one is a first-order andthe second one a continuous phase transition and there is re-entrance. (c) Thesystem under goes a second order phase transition. (d) The system shows a firstorder phase transition. .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.2 d=-0.3h=0.0 M T (a)
M T (b)
M h (c)Fig. 3. (a) Thermal variations of M for various values of crystal field concentration(p) for vanishing external field. The number accompanying each curve illustratesthe value of p. (b) Temperature dependence of M for several values of static externalfield amplitudes (h) while p=0.5. The number accompanying each curve denotes thevalue of h. (c) The behavior of M as function of static h for d=-0.5. The numberaccompanying each curve denotes the value of the reduced temperature (T). .0 0.2 0.4 0.6 0.80.00.20.40.60.81.0 TCP d=0.25 P F+P F h T (a) TCP TCP
P+F PF h T TCP (b)
TCP
F+P
F+P PF d=-0.525p=0.94 h T TCP (c) P P+F
P+F FF d=-0.525p=0.85 h T
TCP (d) F F+P
F+P P d=-0.525p=0.75 h T TCP (e)
F+P PF d=-0.525p=0.25 h T TCP (f)Fig. 4. Dynamic phase diagrams of the Blume-Capel model with crystal field ran-domness in the (T,h) plane for various values of the single ion anisotropy concen-tration (p). Dotted and solid lines represent the first-order and second-order phasetransitions, respectively. (a) d=0.25, the number accompanying each curve denotesthe value of p. (b) d=-0.25 the number accompanying each curve illustrates thevalue of p. (c) d=-0.525 and p=0.94. (d) d=-0.525 and p=0.85. (e) d=-0.525 andp=0.75. (f) d=-0.525 and p=0.25. .0 0.1 0.20.00.40.8 TCPP+FP+F
F PP+F d=-0.625p=0.95 h T TCP (a)
F+P
F+P
F P d=-0.625p=0.50 h T TCP (b)
F+P
F+P PF d=-0.625p=0.25 h T CEPTCP (c)
PF+P F d=-0.625p=0.0 h T TCP (d)Fig. 5. Dynamic phase diagrams of the Blume-Capel model with crystal field ran-domness in the (T,h) plane for various values of the single ion anisotropy con-centration (p) while d=-0.625. Dotted and solid lines represent the first-order andsecond-order phase transitions, respectively. (a) p=0.95, (b) p=0.50, (c) p=0.25, (d)p=0.0. .00 0.06 0.121.01.21.4 TCP
F+P
F+P PF d=-1.0p=1.0 h T TCP (a)
TCPCEPF+P
F+PF+P
F FP d=-1.0p=0.75 h T TCP (b)
TCP
F+P F F+P PF d=-1.0p=0.5 h T CEP
TCP (c) PF F+P d=-1.0p=0.0 h T TCP (d)Fig. 6. Dynamic phase diagrams of the Blume-Capel model with crystal field ran-domness in the (T,h) plane for different values of the single ion anisotropy con-centration (p) while d=-1.0. Dotted and solid lines represent the first-order andsecond-order phase transitions, respectively. (a) p=1.0, (b) p=0.75, (c) p=0.5, (d)p=0.0.(d)Fig. 6. Dynamic phase diagrams of the Blume-Capel model with crystal field ran-domness in the (T,h) plane for different values of the single ion anisotropy con-centration (p) while d=-1.0. Dotted and solid lines represent the first-order andsecond-order phase transitions, respectively. (a) p=1.0, (b) p=0.75, (c) p=0.5, (d)p=0.0.