Dynamic hysteresis features in a two-dimensional mixed Ising system

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Dynamic hysteresis features in a two-dimensional mixed Ising system

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Department of Physics, Erciyes University, 38039 Kayseri, Turkey

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a r t i c l e

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Mehmet Ertas¸ 1 , Mustafa Keskin

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Article history: Received 6 March 2015 Accepted 11 April 2015 Available online xxxx Communicated by C.R. Doering Keywords: Dynamic phase transitions Dynamic hysteresis Glauber-type stochastic Effective-field theory

The dynamic hysteresis features in a two-dimensional mixed spin (1, 3/2) Ising system are studied by using the within the effective-field theory with correlations based on Glauber-type stochastic. The dynamic phase transition temperatures and dynamic hysteresis curves are obtained for both the ferromagnetic and antiferromagnetic interactions. It is observed that the dynamic hysteresis loop areas increase when the reduced temperatures increase, and the dynamic hysteresis loops disappear at certain reduced temperatures. The thermal behaviors of the coercivity and remanent magnetizations are also investigated. The results are compared with some theoretical and experimental works and found in a qualitatively good agreement. © 2015 Published by Elsevier B.V.

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1. Introduction

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Although the dynamic behaviors of Ising systems in the presence of an oscillating magnetic field have been studied for approximately two decades, the understanding of their dynamic properties is still much weaker than of the equilibrium properties. The dynamic phase transition (DPT) and dynamical hysteresis behavior are two interesting problems in dynamic systems. Theoretically, the DPT was first found in a study of the kinetic spin-1/2 Ising model under an oscillating magnetic field by using meanfield theory based on Glauber-type stochastic [1] that has been also called the dynamic mean-field theory (DMFT) and then the DPT has attracted much attention in recent years (see [2–14] and the references therein). Experimental evidences for the DPT have been found in many physical systems, such as ultrathin [Co/Pt]3 magnetic multilayers [15], cuprate superconductors [16], amorphous YBaCuO films [17], ultrathin Co/Cu(001) ferromagnetic films and highly anisotropic (Ising-like) [18,19] and ferroic systems with pinned domain walls [20]. On the other hand, hysteresis represents the intrinsic feature of dynamic phase transition in a wide class of ferroic materials [21] and dynamic hysteresis has been the subject of intensive research in the recent literature (see [22–27] and the references therein), particularly in the context of Ising ferromagnets, in which it has been shown that the system may undergo a first-order phase transition resulting in asymmetrical loops [28,29]. Moreover, the dynamic hysteresis properties (hysteresis area, coer-

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1

E-mail address: [email protected] (M. Ertas). ¸ Tel.: +90 352 207 66 66x33134.

http://dx.doi.org/10.1016/j.physleta.2015.04.017 0375-9601/© 2015 Published by Elsevier B.V.

civity and remanence) are very important in the magnetic recording media [30]. Real magnetic recording media quality tests and their relationship to the hysteresis based methods can be found in Ref. [31]. We should also mention that among various mixed spin systems, the mixed spin (1, 3/2) Ising system has gained a great deal of attention in the last years. The mixed spin (1, 3/2) Ising system was used to investigate iron nitride compounds, namely Fe4 N. Theoretical studies in the system of Fe4 N have been carried out in recent years in terms of its large saturation magnetization [32], chemical stability [33], and lower coercive force [34]. While the mixed spin (1, 3/2) Ising systems have been examined in detail within many different methods in equilibrium statistical physics (among them the mean-field theory (MFT) [35–37], the effectivefield theory (EFT) [38–42], the cluster variation method (CVM) [43,44], the Monte-Carlo (MC) simulation [45–51] etc.), the dynamics or nonequilibrium properties of mixed spin (1, 3/2) Ising systems have not been as thoroughly explored; there have been only few investigations, to our knowledge, about the dynamical aspects of the model. Hence, further efforts should be spent on investigating the time-dependent behavior of mixed spin-(1, 3/2) Ising systems. An early attempt to study the dynamics of the mixed spin-(1, 3/2) Ising model was made by Keskin et al. [52] using the DMFT, in particular they obtained the DPT and dynamic phase diagrams of system on two interpenetrating square lattices in detail (Fig. 1). They also studied [53] the existence of dynamic compensation temperatures and presented the dynamic phase diagram on a hexagonal lattice within the DMFT. Recently, Temizer [54] has examined the dynamic magnetic properties of the mixed spin-(1, 3/2) Ising system on a two-layer square lattice by using

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w = 2π f are the amplitude and angular frequency of the oscillating field, respectively. The system is in contact with an isothermal heat bath at absolute temperature, T A . Now, we use the EFT with correlations [63,64] to obtain the EFT equations. Within the framework of the EFT with correlations, one can easily find the magnetizations (m A ), quadrupole moment (q A ) on A sublattice and the magnetizations (m B ), quadrupole moment (q B ), octupolar (r B ) on B sublattices as coupled equations as follows

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Fig. 1. The sketch of the spin arrangement on the two interpenetrating square lattice. The lattice is formed by lattices of A (open circles) and B (solid circles) spins.

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the DMFT for both ferromagnetic/ferromagnetic and antiferromagnetic/ferromagnetic interactions. Very recently, Shi et al. [55] studied the dynamical response of the mixed spin-1 and spin-3/2 Ising model with Fe4 N structure in the presence of a sinusoidal oscillating magnetic field by using DMFT. They presented dynamic phase diagrams of the Fe4 N compound in the T /| J |–h0 /| J | plane. Moreover, they also studied [56] the dynamic compensation behavior of the mixed spin-1 and spin-3/2 ferrimagnetic Ising model by using DMFT on a layered honeycomb structure in which two kinds of spins (spin-1 and spin-3/2) occupy sites alternately. Since the DMFT neglects nontrivial thermal fluctuations to study the dynamic properties, the results given by DMFT need further investigation by using more reliable techniques. The EFT considers partially the spin–spin correlations and results in an improvement over the MFT; hence, in recent literature, the dynamic or nonequilibrium behaviors of Ising systems have been studied by using the effective-field theory with the Glauber-types stochastic dynamics, which is also called the dynamic effectivefield theory (DEFT) (see [56–62] and references therein). Moreover, to our knowledge, no works have been directed to study dynamic hysteresis features, which is very important technological implications such as for high frequency devices applications, in a twodimensional mixed spin-1 and spin-3/2 Ising system. Thus, the purpose of the work is to investigate dynamic hysteresis behaviors in a two-dimensional mixed spin-1 and spin-3/2 Ising system in detail within the DEFT. For this aim, the paper was organized as follows: In Section 2, we are briefly present the model and formulations. The results and discussions are summarized in Section 3, and finally Section 4 contains our conclusions.

mA





F m (x)|x=0 = [a0 + a1m B + a2 q B + a3 r B ]4 qA F q (x)|x=0 ⎧ ⎫ m B ⎨ ⎬   4 = 1 + m A sinh( J ∇) + q A cosh( J ∇) − 1 qB ⎩ ⎭ rB ⎫⎤ ⎡⎧ ⎨ G m (x)|x=0 ⎬ × ⎣ G q (x)|x=0 ⎦ , ⎭ ⎩ G r (x)|x=0

F m (x) = F q (x) =

2 sinh[β(x + h)] 2 cosh[β(x + h)] + exp(−β D ) 2 cosh[β(x + h)] 2 cosh[β(x + h)] + exp(−β D )

2. Model and formulations

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We considered the two-dimensional Ising Hamiltonian with crystal-field parameter under the influence of time varying external magnetic field i.e.

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H=−J



σi S j − D



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2 i

σ +

   σi + Sj , − h(t ) i



 S 2j

j

(1)

j

where σi = ±1.0, 0.0 and S j = ±3/2, ±1/2. This model describes the mixed spin system consisting of two sublattices A and B, namely two interpenetrating square lattices; one is occupied by a spin-1 particle with spin moment σi at site i, while the other is occupied by spin-3/2 with spin moment S j at site j. In the Hamiltonian, i j  represents a summation over all pairs of nearest-neighboring sites. J , D, h(t ) represent the bilinear nearestneighbor exchange interaction, single-ion anisotropy or crystalfield interaction, a time-dependent external oscillating magnetic field, respectively. h(t ) is given by h(t ) = h0 sin( wt ), where h0 and

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(2a)

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(2b)

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where ∇ = ∂/∂ x is the differential operator. Here, we give ai coefficients in Appendix A and define the F (x) and G (x) functions as follows:

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(3b)

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G m (x + h ) 1 3 sinh[3β(x + h)/2] + sinh[β(x + h)/2] exp(−2β D )

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2 cosh[3β(x + h)/2] + cosh[β(x + h)/2] exp(−2β D )

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(4a) G q (x + h )

=

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G r (x + h ) 1 27 sinh[3β(x + h)/2] + sinh[β(x + h)/2] exp(−2β D )

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2 4 cosh[3β(x + h)/2] + 4 cosh[β(x + h)/2] exp(−2β D )

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τ τ

dt dm B dt

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where β = 1/k B T A , k B is the Boltzmann constant which k B is taken 1.0 throughout the paper and T A is the absolute temperature. As one can see, in our treatment besides m A and m B parameters, such as quadrupole (q) and octupolar (r ) order parameters naturally appear, which one is able to evaluate. We only study behavior of m A and m B . Reason that the behavior of m B and r B is similar, and since we did not include the biquadratic interaction in the Hamiltonian one does not have to study behavior of q B [56–58, 60,61]. However, we need to use the equations for q B and r B to determine the behaviors of m A and m B . In this point, we can calculate the set of the dynamical effective-field equations by applying Glauber-type stochastic dynamics. We employ Glauber transition rates, which the system evolves according to Glauber-type stochastic process at a rate of 1/τ transitions per unit time. Here, τ = wt and w = 2π f . Hence, the frequency of spin flipping, f , is 1/τ . After some manipulations the set of dynamic equations of motion for the magnetizations are obtained as:

dm A

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(4b)

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= −m A + f (m B , q B , r B ),

(5a)

= −m B + g (m A , q A ).

(5b)

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Fig. 2. (Color online.) The reduced temperature dependence of magnetizations (M A , M B , M T ) for the ferromagnetic interaction, J = 1.0, w = 0.05π , h0 /| J | = 0.5 and D /| J | = −1.4, −1.0, 0.1, 1.0, 4.0, 8.0: (a) for the M A and M B , (b) for the M T .

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Here, f and g functions came from the expanding right-hand side of Eqs. (2a) and (2b), respectively. These functions consist long coefficients that and can be easily calculated by employing differential operator technique, namely exp(α ∇) f (x) = f (x + α ). But, these coefficients will not be expressed here because of complicate and long expressions.

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3. Numerical results and discussions

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In this section, the dynamic phase transition temperatures and dynamic hysteresis curves are obtained for both the ferromagnetic and antiferromagnetic interactions.

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3.1. Dynamic Phase Transitions (DPT) temperatures: thermal variations of the dynamic magnetization

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The dynamic order parameters were defined as the time averaged magnetization over a full period of the oscillating magnetic field.

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M A,B =

1

2π m A , B (ξ )dξ.

2π

(6)

0

Total magnetization is defined as M T = ( M A + M B )/2. Eq. (6) has been solved by combining the numerical methods of the Adams–Moulton predictor–corrector with Romberg integration and examined the thermal behavior of M A , B , T for both ferromagnetic and antiferromagnetic cases, for fixed J , h0 /| J |, w val-

Fig. 3. (Color online.) The reduced temperature dependence of magnetizations (M A , M B , M T ) for the antiferromagnetic interaction, J = −1.0, w = 0.1π , h0 /| J | = 0.1 and D /| J | = −1.5, −1.0, 0.1, 1.5, 5.0: (a) for the M A and M B , (b) for the M T .

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ues and variations values of D /| J |. Its behavior gives the DPT point and the type of the dynamic phase transition, namely firstor second-order phase transition lines. Figs. 2(a) and 2(b) illustrate the temperature variations of dynamic magnetizations (M A and M B ) and total magnetization (M T ), respectively, for ferromagnetic interaction with J = 1.0, w = 0.05π , h0 /| J | = 0.5 and D /| J | = −1.4, −1.0, 0.1, 1.0, 4.0, 8.0. It is observed from Fig. 2 for D /| J | = −1.4 and −1.0 that the system undergoes a first-order phase transition from the ferrimagnetic phase to the paramagnetic phase. Because, dynamic magnetizations go to zero discontinuously as the temperature increases and first-order phase transitions occurs at 2.25 and 2.75 values. Hence, 2.25 and 2.75 values are the first-order phase transition temperature where the jump or discontinuity occurs. On the other hand, for D /| J | = 0.1, 1.0, 4.0 and 8.0 that the system undergoes a second-order phase transition from the ferrimagnetic phase to the paramagnetic phase. Because, M A , M B , and M T arrive to zero continuously as the temperature increases and second-order phase transitions occurs at 3.64, 4.1, 4.95 and 5.42 values. Hence, 3.64, 4.1, 4.95 and 5.42 values are the second-order phase transition temperature. Critical value of crystal field, namely D C /| J |, for transition to second-order from first-order transition is −0.33. On the other hand, Fig. 3 is similar to Fig. 2 but Fig. 3 is obtained for antiferromagnetic case and J = −1.0, w = 0.1π , h0 /| J | = 0.1, D /| J | = −1.5, −1.0, 0.1, 1.5, 5.0 values. In Fig. 3, we can see that all the phase transitions are the second-order phase transition and the system does not show a dynamic first-order phase transition. We also see that the secondorder phase transition temperatures increase if the values of D /| J | increases continuously.

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Fig. 4. The dynamic hysteresis loops at different reduced temperatures which are around the dynamic critical temperature T t = 3.05 for the ferromagnetic interaction and J = 1.0, w = 0.05π , D /| J | = 0.5: (a) for the M A , (b) for the M B , (c) for the M T .

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tion decrease with increasing the temperature, except the behavior of coercive for the sublattice A that is initially increase then decrease. Fig. 6 shows dynamic magnetic hysteresis loops of dynamic magnetizations (M A , M B , M T ) for antiferromagnetic interaction and J = −0.03, w = 0.05π , D /| J | = 0.1 at T | J | = 0.15, 0.22, 0.4, 0.6 and 1.2. It is observed that hysteresis loops for M A and M T show multiple hysteresis loops at T | J | = 0.15, 0.22, 0.4, but the system illustrates single and double hysteresis loops for M B at T | J | = 0.15, 0.22, 0.4. Moreover, at T | J | = 0.6 and 1.2 values, the system shows single loop for M B and M T while the system displays double loops for M A . We should also mention that the phenomenon of multiple hysteresis loops have also been observed for different systems, such as in single chain magnets with antiferromagnetic interchain coupling [70] and in molecular-based magnetic materials [71]. More than multiple hysteresis loop behaviors have been also seen experimentally in CoNiP/Cu, CoFeB/Cu, FeGa/CoFeB and FeGa/Py multilayered nanowires [72]. Based on the above results, the dynamic coercivity and remanent magnetization behavior are depicted for the antiferromagnetic case in Fig. 7. The dynamic behavior of remanent magnetization for the antiferromagnetic case has a similar to ferromagnetic case, but the behavior of sublattice B is different which is initially decrease then increase. Variation of the coercivity magnetization are very different. We also examined the effects of the frequency and crystal field on the hysteresis behaviors of the system as seen Figs. 8(a) and 8(b). Fig. 8(a) is obtained for four selected typical frequency values, namely w = 0.05π , 0.06π , 0.08π , 5.0π in the case of J = 1.0, T /| J | = 3.1 and D /| J | = 0.5. With the increase of the frequency value, the hysteresis loop becomes narrower. On the other hand, Fig. 8(b) is calculated for five selected typical crystal field values, namely D /| J | = 1.0, 0.0, −1.0, −2.0, −4.2 in the case of J = 1.0, T /| J | = 3.1 and w = 0.05π . From the figure it is observed that with decreasing of the crystal field, hysteresis loop area is narrowing.

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Fig. 5. Reduced temperature-dependent coercivity and remanent magnetizations acquired from the dynamic hysteresis curves of J = 1.0, w = 0.05π , D /| J | = 0.5: (a) coercivity, (b) remanent magnetizations.

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3.2. The dynamic hysteric (hysteresis area, coercivity field, remanences magnetization) behaviors

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4. Summary and conclusion

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We solved Eq. (5) by combining the numerical methods of Adams–Moulton-predictor–corrector for a given set of system parameters to find the dynamic histeric features of the mixed spin-1 and spin-3/2 Ising and presented in Figs. 4–8. Fig. 4 displays dynamic magnetic hysteresis loops of dynamic magnetizations (M A , M B , M T ) for ferromagnetic interaction and J = 1.0, w = 0.05π , D /| J | = 0.5 at T | J | = 0.1, 2.9, 3.1, 4.0, 6.0 and 8.5. From this figure, one can see that the dynamic magnetic hysteresis loop areas increase as the temperature increases and at a certain temperature loop areas decrease with increasing the temperature that is in a good, quantitatively, agreement with some theoretical [24,65–67] and experimental [68,69] results. We also calculate the thermal behaviors of the dynamic coercivity magnetic field which is defined as the field at which the sign of the magnetization changes and remanent magnetizations that is the magnetization when the magnetic field is dropping to zero for J = 1.0, w = 0.05, D /| J | = 0.5, and different reduced temperatures from 0.1 to 8.5 and also plotted in Figs. 5(a) and 5(b), respectively. It is observed that the coercive field and the remanent magnetiza-

In this study, we investigated the dynamic hysteresis features and dynamic phase transition temperatures of a two-dimensional mixed spin-1 and spin-3/2 Ising system in the existence of oscillating magnetic field by means of the effective-field theory with correlations based on Glauber-type stochastic. We obtained the dynamic phase transition temperatures and dynamic hysteresis curves for both the ferromagnetic and antiferromagnetic cases. It is observed that the dynamic hysteresis loop areas increase when the reduced temperatures increase, and the dynamic hysteresis loops disappear at certain reduced temperatures for both the ferromagnetic and antiferromagnetic interactions. Depending on the Hamiltonian parameters, some characteristic hysteresis behaviors are found, such as the existence of single, double, triple, quartet and quintet hysteresis loops for antiferromagnetic system and single hysteresis loops for ferromagnetic system. The thermal behaviors of the coercivity and remanent magnetizations are also studied. The results are compared with some recently published theoretical and experimental works and a qualitatively good agreement are observed. Finally, we hope that our detailed theoretical investigation may stimulate further researches to study magnetic properties of mixed spin systems, and also will motivate experimentalists to investigate the dynamic histeric behavior and dynamic phase transitions in real material systems.

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Appendix A. The Van der Waerden coefficients

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The Van der Waerden coefficients a0 , a1 , a2 , and a3 for the spin-3/2 in Eq. (2a) are given as follows:

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Fig. 6. The dynamic hysteresis loops at different reduced temperatures which are around the dynamic critical temperature T C = 0.3 for the antiferromagnetic interaction and J = −0.03, w = 0.05π , D /| J | = 0.1: (a) for the M A , (b) for the M B , (c) for the M T .

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Fig. 7. Reduced temperature-dependent coercivity and remanent magnetizations acquired from the dynamic hysteresis curves of J = −0.03, w = 0.05π , D /| J | = 0.1: (a) coercivity, (b) remanent magnetizations.

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a0 =

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a1 =

1



8 1 12

 9 cosh







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J ∇ − cosh

2







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J∇

2

,

 1 3 27 sinh J ∇ − sinh J∇ ,

     1 3 − cosh a2 = J ∇ + cosh J∇ , 2

2

     1 1 3 −3 sinh a3 = J ∇ + sinh J∇ . 3

2

2

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[16] [17] [18] [19] [20] [21]

References

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[11] [12] [13] [14] [15]

2

1 2

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Fig. 8. (Color online.) Dynamic hysteresis behaviors for the ferromagnetic interaction and (a) J = 1.0, T /| J | = 3.1 and at w = 0.05π , 0.06π , 0.08π , 0.5π , (b) J = 1.0, w = 0.05π and at D /| J | = 1.0, 0.0, −1.0, −2.0, −4.2.

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