- Email: [email protected]
shell nanowire
Accepted Manuscript
An investigation of competition on the dynamic magnetic properties of the core/shell nanowire ¨ ut Um ¨ Temizer , Ersin Kantar PII: DOI: Reference:
S0577-9073(18)30506-9 https://doi.org/10.1016/j.cjph.2018.11.010 CJPH 703
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
5 April 2018 14 September 2018 6 November 2018
¨ ut Please cite this article as: Um ¨ Temizer , Ersin Kantar , An investigation of competition on the dynamic magnetic properties of the core/shell nanowire, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.11.010
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 proof before it is published in its final 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.
ACCEPTED MANUSCRIPT Highlights
The dynamics of Ising spin systems on hexagonal nanowire are compared.
The mean-field theory and the Glauber-type stochastic dynamics are used.
The effects of the Hamiltonian parameters on the system are examined.
The behavior of dynamic core, shell and total magnetizations are analyzed.
The dynamic hysteresis properties are presented.
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ACCEPTED MANUSCRIPT
An investigation of competition on the dynamic magnetic properties of the core/shell nanowire Ümüt Temizer1 and Ersin Kantar2* 1
Department of Physics, Bozok University, 66100 Yozgat, Turkey Sorgun Vocational School, Bozok University, Yozgat 66700, Turkey
2
Abstract
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In this study, we investigate the dynamic magnetic properties of Ising-type core/shell nanowires (NW) for different spin systems. The model of NW X(Spin-1/2)@Y with Y=Spin1/2, Spin-1 and Spin-3/2 are considered for discussing an effect of the nature of shell particle on the dynamic properties. The mean-field theory and Glauber-type stochastic dynamics have been successfully applied to this, and the results of the dynamic magnetic properties for the
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core/shell NW are obtained. Different shell spin states are employed to the analysis of dynamic magnetic behavior for core/shell NW. Results of numerical calculation for the magnetization and coercivity curves are discussed for the effect of shell particles, shell interaction and oscillating field frequency. All results present that dynamic magnetic
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properties of the NW strongly dependent on the shell particle and the shell interaction.
Keywords: Nanowire; Ising model; Core/shell structure; Dynamic magnetic properties;
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Hysteresis behavior.
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1. Introduction
Recently, composite nanostructures (core/shell and segmented nanostructures) have been
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attracted a great interest for their unique physical properties [1–3] and potential applications [4–9], ranging from high density recording media to drug delivery. Therefore, many
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experimental [11-14] and theoretical [15-20] works have been performed on the composite nanostructures. In the theoretical frame, the different types of core/shell nanostructures such as nanowire, nanotube, nanocube, nanoparticle and nanoribbon have been studied by Ising model. However, most of theoretical works on composite nanostructures are concerned with equilibrium behavior of these systems. On the other hand, the Ising model has been applied successfully to the investigation the dynamic magnetic properties of the core/shell *
Corresponding author Tel: + 90 (354) 502 00 55; Fax: + 90 (354) 502 00 54. E-mail address: [email protected] (E. Kantar)
ACCEPTED MANUSCRIPT nanostructures. Deviren et al. [21] have investigated the dynamic phase transitions in a cylindrical Ising nanowire for both ferromagnetic and antiferromagnetic interactions by using the effective-field theory with correlations and Glauber-type stochastic dynamics approach [22]. Ertaş and Kocakaplan [23] have studied the hexagonal Ising nanowire to analyze the nature of transitions and to obtain the dynamic phase transition points by utilizing the effective-field theory based on Glauber-type stochastic dynamic. Wang et al. [24] have been examined the dynamic properties of phase diagrams in cylindrical ferroelectric nanotubes with three different structure mechanisms by utilizing the effective-field theory with
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correlations. Yüksel et al. [25] have presented the dynamic phase transition features and stationary-state behavior of a ferrimagnetic small nanoparticle system with a core/shell structure by means of Monte Carlo simulations. Kantar and Ertaş [26] have studied the nonequilibrium magnetic properties of a cylindrical Ising nanowire system with core/shell by using a mean-field approach based on Glauber-type stochastic dynamics. Vatansever and
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Polat [27] have performed the nonequilibrium behavior of a single cubic core/shell ferrimagnetic nanoparticle system by making use of a classical Monte Carlo simulation technique with a standard Metropolis algorithm.
Despite the increasing interest different dynamic properties of the core/shell
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nanostructures by Ising model in the last years, to the best of our knowledge, the effect of the nature of shell particle has not been investigated so far. In this paper, we studied the thermal
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and hysteresis properties of the core/shell hexagonal nanowires, which contains core spin-1/2, and shell 1/2, 1 or 3/2 spins, by using mean-field theory as well as Glauber-type stochastic dynamic.
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The sections in the rest of the paper are arranged as follows: section 2 presents the model and formalism; section 3 contains the detailed the numerical results and discussions;
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section 4 contains a summary.
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2. The model and derivation of mean-field dynamical equations The Hamiltonian of the Ising-type NW with a core/shell structure, in which displayed in Fig. 1, is given by:
2 H =-J1 iS j -J C i i -JS S jSk - D S j - H i + S j , ij ii jk j i j
(1)
ACCEPTED MANUSCRIPT where the spin i located on the core takes the values ±1/2. Shell forms by spin-1/2, spin-1 or spin-3/2 particles with a spin Sj on the Hamiltonian. and indicate a summation over all pairs of nearest-neighboring sites. In the first three terms, namely J1, JC and JS indicate the core-shell (σ-S), core-core (σ - σ) and shell-shell (S-S) exchange interactions, respectively. The last two terms denote the crystal-field interaction D and time dependent external oscillating magnetic field H, respectively. In the case of spin-1/2, Hamiltonian does not
H(t) = H0 cos(wt),
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include the D term. H is given by
(2)
where H0 and w = 2πf are the amplitude and the angular frequency of the oscillating field, respectively. In this study, we define the core-shell interaction and shell interaction r = J1/JC
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and s = JS/JC, respectively.
The mean-field dynamic equations of motion can be obtained by means of Glaubertype stochastic dynamics. First, while the shell S- spins momentarily fixed, the master equation for - spins of the core can be written as
(3)
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d σ P σ1 ,σ 2 ,...,σ N ;t =- Wiσ (σ i σ i' ) P σ σ1 ,σ 2 ,...,σ i ,...σ N ;t dt i σ ¹σ' i i + Wiσ (σi' σi ) P σ σ1 ,σ 2 ,...,σ i' ,...σ N ;t , i σ ¹σ' i i
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where Wiσ σi σi' is the probability per unit time that the ith spin changes from the value
σ i to σ i' . Assuming that the system is in contact with a heat bath at a temperature T A, the
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factor,
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probability of change of state per unit time of each spin would be given by the Boltzmann
σ ' 1 exp -βΔE σi σi W σi σ = τ exp -βΔE σ σi σi' σ i
' i
where = 1/ k B TA , k B is the Boltzmann factor,
(4)
σi'
is the sum over the two possible values
σi'
of σ i' = ±1/2, and is a time constant that a transition from σ i and σ i' occurs on time scale .
ACCEPTED MANUSCRIPT ΔE σ σi σi' =- σi' -σi H+z CSJ1 S j +zCC J C σi j i
(5)
gives the change in the energy of the system when the -spin changes. The probabilities satisfy the detailed balance condition. Using Eqs. (2)-(4) with the mean-field approach, we obtain the mean-field dynamic equation for the -spins as d 1 mC =-mC + tanh[β x C +h 0 cos ξ ], dξ 2
(6)
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Ω
where h0 = H 0 / J C , x C =zCS J1 mS +z CC J C mC , mC , mS S , w t and = w . Moreover, z CS and zSS correspond to the number of nearest-neighbor pairs of spins σ and S, in which z CS = 6 and z CC = 2 . Similarly, the dynamic equations for the shell spins (spin-1/2,
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spin-1, and spin-3/2) have been obtained following as:
d 1 mS =-mS + tanh[β x S +h 0 cos ξ ], dξ 2
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2sinh[β x S +h 0 cos ξ ] d mS =-mS + , dξ 2cosh[β x S +h 0 cos ξ ]+exp[-βΔ]
(7)
(8)
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Ω
(9)
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3 1 3sinh[ β x S +h 0 cos ξ ]+exp[-2βΔ]sinh[ β x S +h 0 cos ξ ] d 2 2 Ω mS =-mS + , 3 1 dξ 2 cosh[ β x S +h 0 cos ξ ]+2exp[-2βΔ]cosh[ β x S +h 0 cos ξ ] 2 2
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where Δ=D/J C , xS =zSC J1 mC + zSS J S mS , zSC = 1,zSS = 4 . The total magnetization of per site can be defined as M T =
m +6m . 7 C
S
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In the next section, we investigate the effect of shell particles and shell interaction on
dynamic magnetic properties of NW.
2. Numerical results and discussion The dynamic magnetic properties of the Ising-type core/shell NW for the three different shell spin states are studied by mean-field theory. In here, we use the fixed values of h0=0.1, JC=1.0 and =0.0. We start by studying the thermal effect on the dynamic magnetization curves of the NW systems by using the ferromagnetic and antiferromagnetic core-shell interactions,
ACCEPTED MANUSCRIPT such as r=1.0 and r=-1.0. In Figs. 2 and 3, we have presented the thermal magnetizations of the NW systems for high (s=1.0) and low (s=0.1) surface interactions. It is seen from the Fig. 2 that the magnetizations monotonically move from their maximum value at zero temperature. With increasing of the temperature all of them tend to zero continuously. In Fig. 2(a), the transition temperature values are different from each other for the spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems, which are obtained as 1.47, 3.19 and 5.63, respectively. It is worthwhile mentioning that the value of phase transition temperature is obtained as TC=2.98 in the equlibrium state of the hexagonal core/shell structure for spin- (1/2, 1) Ising system [28]. In
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this study, the value of h0 is 0.1 for spin-(1/2, 1) Ising system and this value is very close to the previous work value [28]. On the other hand, the obtained transition temperature values of the nanowire systems are the same for the both ferromagnetic and antiferromagnetic coreshell interactions. In Fig. 2(b), Q-type compensation behaviors are obtained for antiferromagnetic core-shell interactions. In Fig. 3, as similar to Fig. 2, the phase transition
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points occur at lower temperatures depending on s value, namely 0.99 for spin-(1/2, 1/2), 1.45 for spin-(1/2, 1), and 1.95 for spin-(1/2, 3/2). On the other hand, the total magnetization curves display the S-type compensation behavior as seen in Fig. 3(b). In this behavior, initially exhibits a steep decrease of the MT, and then it again shows a second steep decrease
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in the vicinity of critical temperature.
Fig. 4 shows the magnetizations versus the external magnetic field of the NW systems
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for r=1, s=1 and T=0.2. The system shows the ferromagnetic phase below the transition temperature, and the system illustrates the paramagnetic phase above the critical temperatures. Spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems display the first-order phase transition with
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hC=2.25, hC=4.20 and hC=6.05 respectively. We have presented the role of the shell particles and shell interaction on the dynamic
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hysteretic behavior, as seen in Figs. 5-7. Fig. 5 shows the hysteresis curves of the total, core and shell for shell spin = 1/2, 1 or 3/2 cases with r=1.0 and s=1.0. The system exhibits a soft
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magnetic characteristic with narrow dynamic hysteresis loops for the spin-(1/2, 1/2) system. With the raise of spin values of shell, the dynamic hysteresis loop areas tend to increase. In this case, the system displays a hard magnetic characteristic with widest hysteresis loops for the spin-(1/2, 3/2) system. In Figs. 6 and 7, the effect of the temperature and shell interaction on total coercive field (HC) curves is investigated for the spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems, respectively. In Fig. 6, as expected, we observe that the HC decreases with increasing of T. At zero temperature, HC is the smallest value for the spin-(1/2, 1/2) system. With the raise of spin values of shell, HC value is become higher. In Fig. 7, HC values are close to each
ACCEPTED MANUSCRIPT other in the lower shell interaction values for all systems. HC values increase with the increasing of the shell interaction for each spin systems. Finally, we have shown the effect of oscilating field frequency (w) on the dynamic magnetization curves and hysteresis behavior of core/shell NW in Figs. 8 and 9. In Figs. 8(a) and 8(b), dynamic magnetization curves have been obtained
for w=0.5 and
w=5respectively. This figure corresponds to the Fig. 2(a), except for w value. It is seen that from these figures, the dynamic transition temperature values are different from each other for
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the w values. We observe that the dynamic transition temperature increases with increasing of T. In Fig. 9, HC – w curve has been obtained to show the effect of w on hysteresis behavior of the NW systems. In this figure, firstly, the HC is constant as w values increase. Then, the HC starts to increase after the certain values of w.
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4. Summary and Conclusion
In this paper, we have studied the dynamic magnetic properties of a core/shell structured Ising-type nanowire by using the mean-field theory based on the Glauber-type stochastic dynamics. First, we have obtained the dynamic phase transition points for both ferromagnetic and antiferromagnetic cases. Then, we have studied the dynamic hysteresis properties and
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composed the dynamic magnetic and hysteretic properties of Ising-type nanowire. We have found that the nanowire systems show qualitatively similar dynamic thermal and hysteretic
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behaviors for the different spin states. The nanowire systems have the same dynamic magnetic properties by given transition temperatures for the ferromagnetic and
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antiferromagnetic core-shell interactions. From the magnetic hysteresis loops, we can deduced that the total system fit the shell hysteresis behavior. The coercive field decreases and
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increases with increasing the temperature and shell interaction, respectively.
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Acknowledgments
This work was supported by the Yozgat Bozok University Research funds (Grant no: 6602aSMYO17-130) References
[1] E.M. Palmero, C. Bran, R.P. del Real, C. Magén, M. Vázquez, J. Appl. Phys. 116 (2014) 033908.
ACCEPTED MANUSCRIPT [2] J. García, V.M. Prida, L.G. Vivas, B. Hernando, E.D. Barriga-Castro, R. MendozaReséndez, Vázquez, M. (2015). Journal of Materials Chemistry C, 3 (2015), 4688. [3] S. Ishrat, K. Maaz, K.J. Lee, M.H. Jung, G.H. Kim, Journal of Solid State Chemistry, 199, (2013) 160. [4] N. Sounderya, Y. Zhang, Recent Patents on Biomedical Engineering 1, (2008), 34. [5] G.V. Kurlyandskaya, M.L. Sanchez, B. Hernando, V.M. Prida, P. Gorria, M. Tejedor, Appl. Phys. Lett., 82, (2003) 3053. [6] S. Nie, S.R. Emory, Science, 275, (1997) 1102.
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[7] H. Zeng, J. Li, J.P. Liu, Z.L. Wang, S. Sun, Nature, 420, (2002) 395. [8] D.W. Elliott, W.-X. Zhang, Environ. Sci. Tech., 35, (2001) 4922.
[9] J.E. Wegrowe, D. Kelly, Y. Jaccard, Ph. Guittienne, J.Ph. Ansermet, EPL, 45, (1999) 626. [10] H.L. Su, G.B. Ji, S.L. Tang, Z. Li, B.X. Gu, Y.W. Du, Nanotechnology 16, (2005) 429 [11] K-H. Seong, J-Y. kim, J-J. Kim, S-C. Lee, S-R. Kim, U. Kim, T-E. Park, H-J. Choi,
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Nano Letters 7, (2007) 3366.
[12] N. Ahmad, J.Y. Chen, J. Iqbal, W.X. Wang, W.P. Zhou, X.F. Han, J. Appl. Phys. 109, (2011) 07A331.
[13] J.P. Xu, Z.Z. Zhang, B. Ma, Q.Y. Jin, J. Appl. Phys. 109, (2011) 07B704.
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[14] Z. H. Yang, Z. W. Li, L. Liu, L. B. Kong, J. Magn. Magn. Mater. 323, (2011) 2674. [15] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 322 (2010) 3410.
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[16] A. Zaim, M. Kerouad, and M. Boughrara, J. Magn. Magn. Mater. 331, (2013) 37 [17] W. Jiang, F. Zhang, X.-X. Li, H.-Y. Guan, A.-B. Guo, Z. Wang, Physica E 47 (2013) 95102.
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[18] O. Iglesias, X. Batlle, A. Labarta, Physical Review B 72 (2005) 212401. [19] E. Kantar, J. Alloys Compd. 676, (2016) 337
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[20] E. Kantar, Philosophical Magazine, 97, (2017) 431–450 [21] B. Deviren, E. Kantar, M. Keskin, J. Magn. Magn. Mater 324, (2012) 2163
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[22] R. J. Glauber J Math Phys. 4 (1963) 294–307 [23] M. Ertaş and Y. Kocakaplan, Phys. Lett. A 378, (2014) 845 [24] C. Wang, Z.Z. Lu, W.X. Yuan, S.Y. Kwok, B.H. Teng, Phys. Lett. A 375, (2011) 3405 [25] Y. Yüksel, E. Vatansever, and H. Polat, J. Phys.: Condens. Matter 24, (2012) 436004 [26] E. Kantar, M. Ertaş, Superlattices and Microstructures 75 (2014) 831. [27] E. Vatansever, H. Polat, Physica A 394 (2014) 82. [28] E. Kantar and Y. Kocakaplan, Solid State Communications 177 (2014) 1–6.
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List of the Figure Captions
Figure 1. (Color online) Schematic representation of Ising-type core/shell NW. The one
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center and six surface magnetic atoms are located on the hexagonal nanostructure.
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The red and blue spheres indicate core and shell magnetic atoms, respectively.
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ACCEPTED MANUSCRIPT Figure 2. (Color online) The thermal variation of the total and partial magnetizations for ferromagnetic and antiferromagnetic C-S interactions. (a) Fixed values of r = 1.0, h0=0.1, =0.0, w=2.0 and s= 1.0.
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(b) Fixed values of r = -1.0, h0=0.1, =0.0, w=2.0 and s= 1.0.
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Figure 3. (Color online) The thermal variation of the total and partial magnetizations for ferromagnetic and antiferromagnetic C-S interactions.
(a) Fixed values of r = 1.0, h0=0.1, =0.0, w=2.0 and s= 0.1.
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(b) Fixed values of r = -1.0, h0=0.1, =0.0, w=2.0 and s= 0.1.
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Figure 4. (Color online) The first–order phase transitions of the total and partial
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magnetizations for T=0.2 with r = 1.0, =0.0, w=2.0 and s= 1.0. (a) The core and shell magnetization curves.
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(b) The total magnetization curves.
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Figure 5. (Color online) The total and partial hysteresis loops for the ferromagnetic C-S interactions r = 1.0 and fixed values of T=0.1, h0=0.1, =0.0, w=0.01 , and s= 1.0. Red, blue and black lines denotes the spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems, respectively. (a) The total hysteresis curves.
ACCEPTED MANUSCRIPT (b) The core hysteresis curves.
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(c) The shell hysteresis curves.
Figure 6. (Color online) The temperature dependence of the coercive field for the fixed
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values of r= 1.0, s = 1.0, h0=0.1, =0.0 and w=0.01
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Figure 7. (Color online) The effect of the surface interaction on coercive field for the fixed
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values of T= 0.1, r = 1.0, h0=0.1, =0.0 and w=0.01
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Figure 8. (Color online) The thermal variation of the total and partial magnetizations for w=0.5and w=5. (a) Fixed values of r = 1.0, h0=0.1, =0.0, w=0.5 and s= 1.0. (b) Fixed values of r = 1.0, h0=0.1, =0.0, w=5.0 and s= 1.0.
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Figure 9. (Color online) The effect of the oscilating field frequency on coercive field for the fixed values of T= 0.1, r = 1.0, h0=0.1, =0.0 and s=1.0
An investigation of competition on the dynamic magnetic properties of the core/shell nanowire ¨ ut Um ¨ Temizer , Ersin Kantar PII: DOI: Reference:
S0577-9073(18)30506-9 https://doi.org/10.1016/j.cjph.2018.11.010 CJPH 703
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
5 April 2018 14 September 2018 6 November 2018
¨ ut Please cite this article as: Um ¨ Temizer , Ersin Kantar , An investigation of competition on the dynamic magnetic properties of the core/shell nanowire, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.11.010
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 proof before it is published in its final 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.
ACCEPTED MANUSCRIPT Highlights
The dynamics of Ising spin systems on hexagonal nanowire are compared.
The mean-field theory and the Glauber-type stochastic dynamics are used.
The effects of the Hamiltonian parameters on the system are examined.
The behavior of dynamic core, shell and total magnetizations are analyzed.
The dynamic hysteresis properties are presented.
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An investigation of competition on the dynamic magnetic properties of the core/shell nanowire Ümüt Temizer1 and Ersin Kantar2* 1
Department of Physics, Bozok University, 66100 Yozgat, Turkey Sorgun Vocational School, Bozok University, Yozgat 66700, Turkey
2
Abstract
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In this study, we investigate the dynamic magnetic properties of Ising-type core/shell nanowires (NW) for different spin systems. The model of NW X(Spin-1/2)@Y with Y=Spin1/2, Spin-1 and Spin-3/2 are considered for discussing an effect of the nature of shell particle on the dynamic properties. The mean-field theory and Glauber-type stochastic dynamics have been successfully applied to this, and the results of the dynamic magnetic properties for the
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core/shell NW are obtained. Different shell spin states are employed to the analysis of dynamic magnetic behavior for core/shell NW. Results of numerical calculation for the magnetization and coercivity curves are discussed for the effect of shell particles, shell interaction and oscillating field frequency. All results present that dynamic magnetic
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properties of the NW strongly dependent on the shell particle and the shell interaction.
Keywords: Nanowire; Ising model; Core/shell structure; Dynamic magnetic properties;
ED
Hysteresis behavior.
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1. Introduction
Recently, composite nanostructures (core/shell and segmented nanostructures) have been
CE
attracted a great interest for their unique physical properties [1–3] and potential applications [4–9], ranging from high density recording media to drug delivery. Therefore, many
AC
experimental [11-14] and theoretical [15-20] works have been performed on the composite nanostructures. In the theoretical frame, the different types of core/shell nanostructures such as nanowire, nanotube, nanocube, nanoparticle and nanoribbon have been studied by Ising model. However, most of theoretical works on composite nanostructures are concerned with equilibrium behavior of these systems. On the other hand, the Ising model has been applied successfully to the investigation the dynamic magnetic properties of the core/shell *
Corresponding author Tel: + 90 (354) 502 00 55; Fax: + 90 (354) 502 00 54. E-mail address: [email protected] (E. Kantar)
ACCEPTED MANUSCRIPT nanostructures. Deviren et al. [21] have investigated the dynamic phase transitions in a cylindrical Ising nanowire for both ferromagnetic and antiferromagnetic interactions by using the effective-field theory with correlations and Glauber-type stochastic dynamics approach [22]. Ertaş and Kocakaplan [23] have studied the hexagonal Ising nanowire to analyze the nature of transitions and to obtain the dynamic phase transition points by utilizing the effective-field theory based on Glauber-type stochastic dynamic. Wang et al. [24] have been examined the dynamic properties of phase diagrams in cylindrical ferroelectric nanotubes with three different structure mechanisms by utilizing the effective-field theory with
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correlations. Yüksel et al. [25] have presented the dynamic phase transition features and stationary-state behavior of a ferrimagnetic small nanoparticle system with a core/shell structure by means of Monte Carlo simulations. Kantar and Ertaş [26] have studied the nonequilibrium magnetic properties of a cylindrical Ising nanowire system with core/shell by using a mean-field approach based on Glauber-type stochastic dynamics. Vatansever and
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Polat [27] have performed the nonequilibrium behavior of a single cubic core/shell ferrimagnetic nanoparticle system by making use of a classical Monte Carlo simulation technique with a standard Metropolis algorithm.
Despite the increasing interest different dynamic properties of the core/shell
M
nanostructures by Ising model in the last years, to the best of our knowledge, the effect of the nature of shell particle has not been investigated so far. In this paper, we studied the thermal
ED
and hysteresis properties of the core/shell hexagonal nanowires, which contains core spin-1/2, and shell 1/2, 1 or 3/2 spins, by using mean-field theory as well as Glauber-type stochastic dynamic.
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The sections in the rest of the paper are arranged as follows: section 2 presents the model and formalism; section 3 contains the detailed the numerical results and discussions;
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section 4 contains a summary.
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2. The model and derivation of mean-field dynamical equations The Hamiltonian of the Ising-type NW with a core/shell structure, in which displayed in Fig. 1, is given by:
2 H =-J1 iS j -J C i i -JS S jSk - D S j - H i + S j , ij ii jk j i j
(1)
ACCEPTED MANUSCRIPT where the spin i located on the core takes the values ±1/2. Shell forms by spin-1/2, spin-1 or spin-3/2 particles with a spin Sj on the Hamiltonian.
H(t) = H0 cos(wt),
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include the D term. H is given by
(2)
where H0 and w = 2πf are the amplitude and the angular frequency of the oscillating field, respectively. In this study, we define the core-shell interaction and shell interaction r = J1/JC
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and s = JS/JC, respectively.
The mean-field dynamic equations of motion can be obtained by means of Glaubertype stochastic dynamics. First, while the shell S- spins momentarily fixed, the master equation for - spins of the core can be written as
(3)
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d σ P σ1 ,σ 2 ,...,σ N ;t =- Wiσ (σ i σ i' ) P σ σ1 ,σ 2 ,...,σ i ,...σ N ;t dt i σ ¹σ' i i + Wiσ (σi' σi ) P σ σ1 ,σ 2 ,...,σ i' ,...σ N ;t , i σ ¹σ' i i
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where Wiσ σi σi' is the probability per unit time that the ith spin changes from the value
σ i to σ i' . Assuming that the system is in contact with a heat bath at a temperature T A, the
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factor,
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probability of change of state per unit time of each spin would be given by the Boltzmann
σ ' 1 exp -βΔE σi σi W σi σ = τ exp -βΔE σ σi σi' σ i
' i
where = 1/ k B TA , k B is the Boltzmann factor,
(4)
σi'
is the sum over the two possible values
σi'
of σ i' = ±1/2, and is a time constant that a transition from σ i and σ i' occurs on time scale .
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gives the change in the energy of the system when the -spin changes. The probabilities satisfy the detailed balance condition. Using Eqs. (2)-(4) with the mean-field approach, we obtain the mean-field dynamic equation for the -spins as d 1 mC =-mC + tanh[β x C +h 0 cos ξ ], dξ 2
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where h0 = H 0 / J C , x C =zCS J1 mS +z CC J C mC , mC , mS S , w t and = w . Moreover, z CS and zSS correspond to the number of nearest-neighbor pairs of spins σ and S, in which z CS = 6 and z CC = 2 . Similarly, the dynamic equations for the shell spins (spin-1/2,
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spin-1, and spin-3/2) have been obtained following as:
d 1 mS =-mS + tanh[β x S +h 0 cos ξ ], dξ 2
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2sinh[β x S +h 0 cos ξ ] d mS =-mS + , dξ 2cosh[β x S +h 0 cos ξ ]+exp[-βΔ]
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3 1 3sinh[ β x S +h 0 cos ξ ]+exp[-2βΔ]sinh[ β x S +h 0 cos ξ ] d 2 2 Ω mS =-mS + , 3 1 dξ 2 cosh[ β x S +h 0 cos ξ ]+2exp[-2βΔ]cosh[ β x S +h 0 cos ξ ] 2 2
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where Δ=D/J C , xS =zSC J1 mC + zSS J S mS , zSC = 1,zSS = 4 . The total magnetization of per site can be defined as M T =
m +6m . 7 C
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In the next section, we investigate the effect of shell particles and shell interaction on
dynamic magnetic properties of NW.
2. Numerical results and discussion The dynamic magnetic properties of the Ising-type core/shell NW for the three different shell spin states are studied by mean-field theory. In here, we use the fixed values of h0=0.1, JC=1.0 and =0.0. We start by studying the thermal effect on the dynamic magnetization curves of the NW systems by using the ferromagnetic and antiferromagnetic core-shell interactions,
ACCEPTED MANUSCRIPT such as r=1.0 and r=-1.0. In Figs. 2 and 3, we have presented the thermal magnetizations of the NW systems for high (s=1.0) and low (s=0.1) surface interactions. It is seen from the Fig. 2 that the magnetizations monotonically move from their maximum value at zero temperature. With increasing of the temperature all of them tend to zero continuously. In Fig. 2(a), the transition temperature values are different from each other for the spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems, which are obtained as 1.47, 3.19 and 5.63, respectively. It is worthwhile mentioning that the value of phase transition temperature is obtained as TC=2.98 in the equlibrium state of the hexagonal core/shell structure for spin- (1/2, 1) Ising system [28]. In
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this study, the value of h0 is 0.1 for spin-(1/2, 1) Ising system and this value is very close to the previous work value [28]. On the other hand, the obtained transition temperature values of the nanowire systems are the same for the both ferromagnetic and antiferromagnetic coreshell interactions. In Fig. 2(b), Q-type compensation behaviors are obtained for antiferromagnetic core-shell interactions. In Fig. 3, as similar to Fig. 2, the phase transition
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points occur at lower temperatures depending on s value, namely 0.99 for spin-(1/2, 1/2), 1.45 for spin-(1/2, 1), and 1.95 for spin-(1/2, 3/2). On the other hand, the total magnetization curves display the S-type compensation behavior as seen in Fig. 3(b). In this behavior, initially exhibits a steep decrease of the MT, and then it again shows a second steep decrease
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Fig. 4 shows the magnetizations versus the external magnetic field of the NW systems
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for r=1, s=1 and T=0.2. The system shows the ferromagnetic phase below the transition temperature, and the system illustrates the paramagnetic phase above the critical temperatures. Spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems display the first-order phase transition with
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hC=2.25, hC=4.20 and hC=6.05 respectively. We have presented the role of the shell particles and shell interaction on the dynamic
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hysteretic behavior, as seen in Figs. 5-7. Fig. 5 shows the hysteresis curves of the total, core and shell for shell spin = 1/2, 1 or 3/2 cases with r=1.0 and s=1.0. The system exhibits a soft
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magnetic characteristic with narrow dynamic hysteresis loops for the spin-(1/2, 1/2) system. With the raise of spin values of shell, the dynamic hysteresis loop areas tend to increase. In this case, the system displays a hard magnetic characteristic with widest hysteresis loops for the spin-(1/2, 3/2) system. In Figs. 6 and 7, the effect of the temperature and shell interaction on total coercive field (HC) curves is investigated for the spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems, respectively. In Fig. 6, as expected, we observe that the HC decreases with increasing of T. At zero temperature, HC is the smallest value for the spin-(1/2, 1/2) system. With the raise of spin values of shell, HC value is become higher. In Fig. 7, HC values are close to each
ACCEPTED MANUSCRIPT other in the lower shell interaction values for all systems. HC values increase with the increasing of the shell interaction for each spin systems. Finally, we have shown the effect of oscilating field frequency (w) on the dynamic magnetization curves and hysteresis behavior of core/shell NW in Figs. 8 and 9. In Figs. 8(a) and 8(b), dynamic magnetization curves have been obtained
for w=0.5 and
w=5respectively. This figure corresponds to the Fig. 2(a), except for w value. It is seen that from these figures, the dynamic transition temperature values are different from each other for
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the w values. We observe that the dynamic transition temperature increases with increasing of T. In Fig. 9, HC – w curve has been obtained to show the effect of w on hysteresis behavior of the NW systems. In this figure, firstly, the HC is constant as w values increase. Then, the HC starts to increase after the certain values of w.
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4. Summary and Conclusion
In this paper, we have studied the dynamic magnetic properties of a core/shell structured Ising-type nanowire by using the mean-field theory based on the Glauber-type stochastic dynamics. First, we have obtained the dynamic phase transition points for both ferromagnetic and antiferromagnetic cases. Then, we have studied the dynamic hysteresis properties and
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composed the dynamic magnetic and hysteretic properties of Ising-type nanowire. We have found that the nanowire systems show qualitatively similar dynamic thermal and hysteretic
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behaviors for the different spin states. The nanowire systems have the same dynamic magnetic properties by given transition temperatures for the ferromagnetic and
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antiferromagnetic core-shell interactions. From the magnetic hysteresis loops, we can deduced that the total system fit the shell hysteresis behavior. The coercive field decreases and
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increases with increasing the temperature and shell interaction, respectively.
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Acknowledgments
This work was supported by the Yozgat Bozok University Research funds (Grant no: 6602aSMYO17-130) References
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ACCEPTED MANUSCRIPT [2] J. García, V.M. Prida, L.G. Vivas, B. Hernando, E.D. Barriga-Castro, R. MendozaReséndez, Vázquez, M. (2015). Journal of Materials Chemistry C, 3 (2015), 4688. [3] S. Ishrat, K. Maaz, K.J. Lee, M.H. Jung, G.H. Kim, Journal of Solid State Chemistry, 199, (2013) 160. [4] N. Sounderya, Y. Zhang, Recent Patents on Biomedical Engineering 1, (2008), 34. [5] G.V. Kurlyandskaya, M.L. Sanchez, B. Hernando, V.M. Prida, P. Gorria, M. Tejedor, Appl. Phys. Lett., 82, (2003) 3053. [6] S. Nie, S.R. Emory, Science, 275, (1997) 1102.
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[16] A. Zaim, M. Kerouad, and M. Boughrara, J. Magn. Magn. Mater. 331, (2013) 37 [17] W. Jiang, F. Zhang, X.-X. Li, H.-Y. Guan, A.-B. Guo, Z. Wang, Physica E 47 (2013) 95102.
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[18] O. Iglesias, X. Batlle, A. Labarta, Physical Review B 72 (2005) 212401. [19] E. Kantar, J. Alloys Compd. 676, (2016) 337
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[20] E. Kantar, Philosophical Magazine, 97, (2017) 431–450 [21] B. Deviren, E. Kantar, M. Keskin, J. Magn. Magn. Mater 324, (2012) 2163
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[22] R. J. Glauber J Math Phys. 4 (1963) 294–307 [23] M. Ertaş and Y. Kocakaplan, Phys. Lett. A 378, (2014) 845 [24] C. Wang, Z.Z. Lu, W.X. Yuan, S.Y. Kwok, B.H. Teng, Phys. Lett. A 375, (2011) 3405 [25] Y. Yüksel, E. Vatansever, and H. Polat, J. Phys.: Condens. Matter 24, (2012) 436004 [26] E. Kantar, M. Ertaş, Superlattices and Microstructures 75 (2014) 831. [27] E. Vatansever, H. Polat, Physica A 394 (2014) 82. [28] E. Kantar and Y. Kocakaplan, Solid State Communications 177 (2014) 1–6.
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List of the Figure Captions
Figure 1. (Color online) Schematic representation of Ising-type core/shell NW. The one
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center and six surface magnetic atoms are located on the hexagonal nanostructure.
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The red and blue spheres indicate core and shell magnetic atoms, respectively.
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(b) Fixed values of r = -1.0, h0=0.1, =0.0, w=2.0 and s= 1.0.
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Figure 3. (Color online) The thermal variation of the total and partial magnetizations for ferromagnetic and antiferromagnetic C-S interactions.
(a) Fixed values of r = 1.0, h0=0.1, =0.0, w=2.0 and s= 0.1.
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(b) Fixed values of r = -1.0, h0=0.1, =0.0, w=2.0 and s= 0.1.
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Figure 4. (Color online) The first–order phase transitions of the total and partial
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magnetizations for T=0.2 with r = 1.0, =0.0, w=2.0 and s= 1.0. (a) The core and shell magnetization curves.
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(b) The total magnetization curves.
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Figure 5. (Color online) The total and partial hysteresis loops for the ferromagnetic C-S interactions r = 1.0 and fixed values of T=0.1, h0=0.1, =0.0, w=0.01 , and s= 1.0. Red, blue and black lines denotes the spin-(1/2, 1/2), (1/2, 1) and (1/2, 3/2) systems, respectively. (a) The total hysteresis curves.
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(c) The shell hysteresis curves.
Figure 6. (Color online) The temperature dependence of the coercive field for the fixed
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values of r= 1.0, s = 1.0, h0=0.1, =0.0 and w=0.01
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Figure 7. (Color online) The effect of the surface interaction on coercive field for the fixed
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Figure 8. (Color online) The thermal variation of the total and partial magnetizations for w=0.5and w=5. (a) Fixed values of r = 1.0, h0=0.1, =0.0, w=0.5 and s= 1.0. (b) Fixed values of r = 1.0, h0=0.1, =0.0, w=5.0 and s= 1.0.
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Figure 9. (Color online) The effect of the oscilating field frequency on coercive field for the fixed values of T= 0.1, r = 1.0, h0=0.1, =0.0 and s=1.0