J Supercond Nov Magn DOI 10.1007/s10948-009-0503-8
O R I G I N A L PA P E R
Anisotropic and Particle-Particle Interaction Effect in a One-Dimensional System of Magnetic Particles A.A. Obeidat · M.A. Gharaibeh · D. Al-Safadi · D.H. Al Samarh · M.K.H. Qaseer · N.Y. Ayoub
Received: 24 November 2008 / Accepted: 15 June 2009 © Springer Science+Business Media, LLC 2009
Abstract We have studied the effect of the shape anisotropy in a system that consists of a chain of N identical spherical particles each of magnetic dipole moment μ and that has an easy axis. By considering two particle interactions (Dimer Model) we have investigated two different distinct cases depending on the direction of the applied field H and the orientation (ξ ) of the easy axis relative to H . We found that for the randomly oriented easy axis (ξ ) and for H parallel or perpendicular to the chain the anisotropy has no effect on the ferromagnetic state. For fixed orientation (ξ ) an interplay between ferromagnetic-like and anti-ferromagnetic-like behavior exists. The existence of each behavior is strongly dependent on the anisotropy K and the direction of H relative to the chain. Keywords Magnetic anisotropy · Ferromagnetism · Antiferromagnetism · Ferrofluid 1 Introduction Fine magnetic particles systems and in particular magnetic fluids (ferrofluids) have many industrial, medical, agricultural, and physical applications, to mention only a few of
N.Y. Ayoub on leave from Department of Physics, Yarmouk University, Irbid, Jordan. A.A. Obeidat () · M.A. Gharaibeh · D. Al-Safadi · D.H. Al Samarh · M.K.H. Qaseer Department of Physics, Jordan University of Science & Technology, Irbid, 22110, Jordan e-mail:
[email protected] N.Y. Ayoub School of Applied Natural Sciences, German Jordanian University, Amman, Jordan
the many fields that these systems could be applied to [1– 10]. In media storage devices that depend on magnetic fine particles it is important to understand the magnetic behavior of these systems especially when they form chains and study the effect of the external magnetic field on the magnetic initial susceptibility of the chain. As a physical application, recently Palm et al. [11] have used a ferrofluid-based neural network in order to design an analogue associative memory. Different models have been proposed to study the magnetic behavior of fine particle systems (ferrofluids) [3, 5, 10, 12–17]. Chantrell et al. [12] studied the effect of particle size on the magnetic behavior of such fluids. Ayoub et al. [13] suggested a dipole-dipole interaction to explain the Curie-Weiss behavior for a three-dimensional model of fine particles. Luo et al. [18] have studied the glassy behavior of a quenched ferrofluid and found features of random anisotropy similar to what is found in amorphous ferromagnets. Rosenbaum et al. [19] have studied the effect of an interparticle interaction on the transition from glassy behavior to the random field limit. Eberbeck [20] has studied the glassy behavior of four ferrofluids in the frozen state and found that the resulting ordering temperature T0 approximately agrees with the energy interaction between particle moments as calculated from the structure of particle aggregates. Kruse et al. [21] have studied the field-induced anisotropy in ferrofluids using two-dimensional small angle X-ray scattering. Wagner et al. [22] have studied complex magnetic liquids and found that, due to the competition of the magnetic dipole-dipole interaction with a soft Yukawa potential, as a result these colloids were made anisotropic even in the presence of weak magnetic fields. Berkov et al. [23] have numerically studied through simulations the effect of a magnetocrystalline anisotropy on the magnetic proper-
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ties in the cases of the rigid model and a model in which the particles have internal degrees of freedom. In this work we will apply a simple model to study the magnetic behavior of a magnetic fluid by considering a onedimensional chain of N identical spherical fine particles. We will introduce the magnetic anisotropy of particles in our calculations. We believe that it plays a major role in determining the magnetic state of the assembly. As we shall see, when the magnetic anisotropy dominates, the ordering temperature T0 , reveals a phase transition from ferromagneticlike behavior to antiferromagnetic-like behavior. This transition will depend on the direction of the magnetic anisotropy (i.e. the easy magnetic axes of the particles). We will calculate the initial susceptibility χ when the magnetic field H is applied parallel and perpendicular to the chain. We will adopt the statistical approach and introduce the magnetic anisotropy (i.e., the magnetic easy axes will be considered at fixed orientation relative to H ) in the Hamiltonian of the system. We will calculate the partition function (Z) of our assembly and from this we get the magnetic initial susceptibility of the assembly.
2 Results and Discussion Our system of magnetic particles consists of N identical spherical particles. The chain of the particles is constrained
to one dimension. We will consider two particle interactions with N/2 pairs confined to the length L0 . Therefore, the total Hamiltonian HT of our system can be described as HT = Hp–p + Ha + Hp–H
The first term Hp–p represents the particle-particle magnetic interaction given by Hp–p =
2 3(μ 2 · r) μ 1 · μ 1 · r)(μ − 3 3 r r
(2)
where μ 1 and μ 2 are the dipole moments of the two particles under consideration, respectively. The interparticle separation is r. The second term of the Hamiltonian, Ha , represents the anisotropic energy, and its contribution is given by Ha = KV sin2 β
(3)
where K is the uniaxial anisotropy constant, V is the particle volume and β is the angle between the magnetic moment and the easy direction (see Figs. 1–2). The last term in HT is the external field-dipole interaction and is given by Hp–H = −
2 i
Fig. 1 (a) System of two particles each of magnetic dipole moment μ oriented at angle θ relative to the chain axis (z-axis). The figure shows the polar angles and the azimuthal angles of each particle. Eˆ is the easy axis, taken to be randomly oriented. The applied field H is parallel to the z-axis. (b) Same as Fig. 1a, but the chain is parallel to the y-axis and H is parallel to the z-axis (or perpendicular to the chain direction). The chain is oriented along the y-axis
a
b
(1)
μ i · H
(4)
J Supercond Nov Magn Fig. 2 (a) Same as Fig. 1a, but Eˆ is at fixed angle (ξ ) relative to H . (b) Same as Fig. 1b but the Eˆ is at fixed angle (ξ ) relative to the H
a
b For a dilute system of particles we will apply the MaxwellBoltzmann distribution law, where the partition function is given by Z=
HT d Exp − kB T
(5)
where kB is the Boltzmann constant and is the phase space. In order to calculate the magnetization M and the initial susceptibility χ of the assembly, we will consider two distinct cases, depending on the easy axis orientation ξ relative to H . In each case H is to be taken parallel and perpendicular to the chain. In the first case we will consider randomly oriented easy axes. Figure 1a shows this configuration with applied magnetic field H parallel to the assembly. The figure shows H applied at different angles relative to the easy direction of each particle (i.e., ξ and ξ are the angles of particle 1 and particle 2 relative to H , respectively). We will refer to this as a dimer model with randomized easy axes. For an assembly of N/2 pairs the total partition function is given by (Z)N/2 ZT = (N/2)!
The total magnetization M of the assembly will be calculated using the relation M = kB T
∂ ln(ZT ) ∂H
The initial susceptibility χ can be calculated from the relation ∂M χ = lim (8) H →0 ∂H Now we will consider the two different cases: Case 1: Randomly Oriented Easy Axes Consider H is parallel to the chain as shown in Fig. 1a. In the limit of kμH 1, (1) reduces to BT μH − 2KV k T Z=e B (cos θ + cos θ ) Exp kB T 2 μ KV g + × 1+ J kT kT z3 2 2 1 μ KV d (9) J + g+ 2 kT z3 kT where g(θ, θ , φ, φ ) = 2 cos θ cos θ − sin θ sin θ cos(φ − φ )
(6)
(7)
J (θ, θ , α, α , φ, φ , ψ, ψ ) = cos2 β + cos2 β
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cos β = cos θ cos α + sin θ sin α cos(φ − ψ)
We found for χ : N μ2
cos β = cos θ cos α + sin θ sin α cos(φ − ψ ) χ=
and
T−
μ2 (z0 +zi ) 3kB z2 z2 0 i
3kB − 8KV 3kB
(1 − 3 cos2 ξ )
d = dz sin θ dθ sin θ dθ sin ξ dξ sin ξ dξ dφdφ dψdψ
This is a Curie-Weiss form with T0 equal to
(see Fig. 1a). Performing the necessary calculation we get the following result for the initial susceptibility χ :
T0 =
χ=
T
N μ2 3kB 2 − μ3k zi 2+z20 z0 zi
(10)
where zi and z0 are the minimum and maximum separations between the two particles. This is a Curie-Weiss form of χ with a positive Curie constant C = Nμ2 /3kB and ordering temperature T0 = μ2 zi +z0 3k 2 2 . z0 zi
This result shows that T0 does not depend on the anisotropic constant K. Therefore, we conclude that K has no effect on χ when H is parallel to the chain and the particles have randomly oriented easy axes. The behavior is isotropic with ferromagnetic behavior. Now let us consider H perpendicular to the chain. Fig 1 we ure 1b shows this configuration. In the limit of kμH BT perform the same calculations as above and we get for χ N μ2 χ=
T+
3kB μ2 (z0 +zi ) 6kB z2 z2
(11)
0 i
The ordering temperature is given by T0 = −
μ2 (z0 + zi ) 6kB z02 zi2
(12)
Again this is a Curie-Weiss behavior with negative ordering temperature. This suggests antiferromagnetic behavior with no effect of the magnetic anisotropy. Therefore, for randomly oriented axes, whether H is parallel or perpendicular to the chain, there is no anisotropic effect. The field direction specified the magnetic state where for H parallel the ferromagnetic state exists and for H perpendicular antiferromagnetic state exists. Case 2: Fixed Easy Axes Relative to H We adopt two configurations. In the first configuration we will consider H parallel to the chain. This configuration is shown in Fig. 2a. We performed the same calculation of the partition function ZT , where the phase space in this case is d = dz sin θ dθ sin θ dθ dφdφ dψdψ
μ2 (z0 + zi ) 8KV − (1 − 3 cos2 ξ ) 3kB z02 zi2 3kB
(13)
(14)
It is clear in this case that T0 depends on the anisotropy constant K. Therefore, K is playing a role in determining the magnetic state of the assembly. To see the magnetic behavior for this configuration we will consider two orientations of easy axis ξ . In the case of ξ = 0, i.e., the easy axis is aligned with H , the ordering temperature reduces to 2 μ (z0 + zi ) 16KV T0 = (15) + 3kB z02 zi2 3kB In this case T0 is always positive and ferromagnetic behavior exists. This is true because μ is parallel to H . For the case of ξ = π/2, i.e., the easy axis is perpendicular to H , T0 is given by 2 μ (z0 + zi ) 8KV T0 = (16) − 3kB z02 zi2 3kB In this limit the magnetic state depends strongly on the anisotropic constant K and the magnetic dipole μ of each particle. If the anisotropy dominates (K > μ), antiferromagnetic behavior exists; otherwise ferromagnetic behavior exists. Therefore, an interplay between ferromagnetic and antiferromagnetic states exists. The magnetic state is determined by K and μ. Now, for H perpendicular to the chain, our result for χ is given by N μ2 χ=
μ2 T − − 6k B
(z0 +zi ) z02 zi2
3kB − 8KV 3kB
(1 − 3 cos2 ξ )
(17)
In this case T0 is given by T0 = −
μ2 (z0 + zi ) 8KV − (1 − 3 cos2 ξ ) 6kB z02 zi2 3kB
For ξ = 0, T0 reduces to −μ2 (z0 + zi ) 16KV T0 = + 6kB z02 zi2 3kB
(18)
(19)
It is clear that T0 depends on μ and K. For weak anisotropy T0 is negative and antiferromagnetic behavior exists; otherwise T0 is positive and ferromagnetic state dominates.
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Now for ξ = π/2, T0 reduces to μ2 (z0 + zi ) 8KV T0 = − − 6kB z02 zi2 3kB
References (20)
In this case T0 is always negative and an antiferromagnetic interaction exists. Therefore, in the perpendicular configuration and for ξ = 0 the anisotropy plays a major role in determining the magnetic state. For strong magnetic anisotropy a ferromagnetic state exists, while for ξ = π/2 antiferromagnetism exists.
3 Conclusion In this work we have investigated the effect of the anisotropy K on a magnetic fluid using a particle-particle interaction for a chain consisting of N identical particles of N/2 1, two distinct cases were systems. In the limit kμH BT adopted. The two cases are random easy axes and fixed easy axes relative to the applied field H . For both cases H is applied parallel or perpendicular to the chain. For the random orientation case we found that the anisotropy has no physical significance and the system has an isotropic response to both directions of H . For fixed orientation ξ of the easy axis relative to H we found for H parallel or perpendicular to the chain that the anisotropy plays a major role in determining the magnetic state. For H parallel to the chain we found that for ξ = 0 (easy axis aligned with H ) the ordering temperature is always positive and ferromagnetic like-behavior exists as one expects. For ξ = π/2 and if K dominates, the system goes into antiferromagnetic-like behavior. In the case of H is perpendicular to the chain and for ξ = 0, and for weak anisotropy antiferromagnetic-like behavior dominates. For ξ = π/2, T0 is always negative and antiferromagneticlike behavior exists, as expected. Acknowledgement We would like to thank the Jordan University of Science and Technology, Irbid, Jordan for financial support grant # 150/2005.
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