Vector magnetization of a distribution of cubic particles

A model for the vector magnetization of a distribution of particles with cubic anisotropy is presented. Recent work by the authors modeled the vector magnetization of a distribution of uniaxial particles by decomposing the total magnetization into reversible and irreversible components. In this paper, using an energy approach applicable to a generic plane, the model is extended to include cubic anisotropy projected to the (100) plane. The magnitude of the irreversible component is modeled using a Preisach differential-equation approach; however, other valid models can be used. The direction of the reversible component is modeled using the minimum energy approach of the classical Stoner–Wohlfarth model and taking into account the anisotropy ﬁeld. The formulation of the generalized model is derived and its results are discussed considering (i) oscillation and rotational modes, (ii) lag angle, and (iii) magnetization trajectories. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4974892]


I. INTRODUCTION
Modifications and extensions of the Stoner-Wohlfarth model 1 have been widely used to model vector magnetization in different anisotropic dispersions. 2, 3 We have recently introduced a new vector model for the magnetization of a distribution of uniaxial particles. 4 The total magnetization was modeled as the vector sum of the irreversible magnetization, M I , and the reversible magnetization, M R . In the case of uniaxial media, the direction of M I is perpendicular to the single hard axis (i.e. parallel with the easy axis) but in the case of cubic anisotropy projected to the (100) plane, M I has two components perpendicular to each of the hard axes. The magnitude of each component of M I is determined by a Preisach differential-equation approach; 5 however, other valid models can be used. The magnitude of M R is computed by a single-valued function of the applied field 6 and its direction is determined by the minimum energy approach of the classical SW model. The susceptibility of M R along the easy axis is zero and the total magnetization is parallel to one of the easy axes when a sufficiently large external field is applied along it. In this paper, we generalized the idea to cubic anisotropy projected to the (100) plane where the easy axes lay along the x and y axes and the hard axes lay along the y = x and y = x straight lines in regular Cartesian coordinate system.

II. MINIMUM ENERGY APPROACH
SW particles are single domain planar structures with uniaxial anisotropy, immersed in a uniform magnetic field H. Two components of the energy are the cubic anisotropy energy and the Zeeman where α, β, and γ are the directional cosines with respect to the x, y, and z axes. Define θ R as the angle between M R and the x-axis, and θ H as the angle between the applied field and the x-axis a general expression for the normalized Gibb's free energy can be derived. We will evaluate the threedimensional expansion of the cubic anisotropy energy density on a generic plane given by z = ax + by, where a and b are arbitrary constants, which passes through the origin of the Cartesian system of reference. This particular case does not lead to a loss of generality due to the cubic symmetry of the energy density function.
In order to formulate the in-plane magnetic anisotropy, the relationship between the three directional cosines must be evaluated. The unitary magnetization vector is a straight line passing through the origin and is given in parametric form by: x = αt, y = βt, and z = γt. The orthogonal projection of the unitary magnetization vector on the generic z plane yields: The obtained directional cosines are no longer normalized. Equations 3 and 4 show the steps to obtain the normalized directional cosines on the generic plane, α , β , and γ .
Adding the Zeeman energy term and assuming that K 1 >>K 2 , the normalized Gibb's free energy is given by: where c is an arbitrary constant and h x and h y are the applied field components in the x and y axis directions normalized to the anisotropy field (H k ), respectively. Equation 5 shows the general expression from which special cases of interest can be derived. For example, if the plane of interest is z = 0 (the xy plane), setting γ = 0 (α 2 + β 2 = 1) will lead to the normalized Gibb's free energy equation of the biaxial anisotropy ((100) cubic anisotropy) case which is given by: By taking derivative of Equation 6 and using h x = h cos(θ H ) and h y = h sin(θ H ), the relationship between θ R , θ H , and the normalized applied field, h, for cubic anisotropy projected to the (100) plane is given by: The corresponding relationship for a distribution of uniaxial particles as derived in Ref. 4 is given by:

III. DIRECTION OF THE FIELD
The anisotropy field, H k , is an equivalent field with a magnitude of H k = 2K 1 M S (where M S is the saturation magnetization) as defined in the literature. [7][8][9] Figure 1 shows the generalized graphical realization of the vector relationship between H, H k , and M R in terms of the three arbitrary constants a, b, and c. In Ref. 10 we assumed that for a distribution of uniaxial particles the anisotropy field remains parallel to the easy axis. Here, with the purpose of deriving the direction of the anisotropy field, we write Equation 8 as: Using the law of sines and assigning the values of 2, 1, and 0.5 to a, b, and c, respectively, reduces the vector relationship in Fig. 1 to the special case of uniaxial anisotropy as formulated in Equation 9. It can be determined from Fig. 1 that a and c are the only independent parameters and that bθ R + θ R = aθ R which shows that the angle that the anisotropy field H k makes with the x-axis is −θ R . In other words, when the reversible component of the magnetization rotates counterclockwise the anisotropy field rotates clockwise with the same angular frequency. The reversible component of the magnetization and the anisotropy field lie in the same direction only when they are along the easy axis and are in opposite directions when they are along the hard axis (in this case the y-axis). Note that as the magnetization rotates by 90 o , the anisotropy field rotates -90 o and will be located in the opposite direction of the reversible component of the magnetization.
For the case of cubic anisotropy projected to the (100) plane, Equation 7 can be written as: Assigning the values of 4, 3, and 0.5 to a, b, and c, respectively, reduces the vector relationship in Fig. 1 to the special case of cubic anisotropy projected to the (100) plane as formulated in Equation 10.
The angle between the anisotropy field, H k and the x-axis is −3θ R . By inspection, it can be verified that the anisotropy field and the reversible component of the magnetization are in the same direction only along the easy axis and they are in the opposite directions along the hard axis. For example, noting that y=x is a hard axis, as the reversible component of the magnetization rotates 45 o , the anisotropy field rotates -135 o and is located in the opposite direction of the reversible component of the magnetization. This method is similar to the image theory in electromagnetics and shows that anisotropy field plays a significant role in analyzing and modeling the total magnetization vector. It can be extended for biaxial anisotropy with non-perpendicular easy axes and further generalized to three-dimensional axes.

A. Total magnetization vector
The reversible component of the magnetization has been discussed in detail. Consideration is now given to the irreversible component. In the case of uniaxial particles, we proposed that the irreversible component is along the easy axis and its magnitude is computed by a Preisach differential-equation approach. 5 In the case of cubic anisotropy, the irreversible component can be further decomposed into two components, M Ix and M Iy , each perpendicular to the hard axes (y=x and y=-x). This is consistent with the uniaxial anisotropy case where laying along the easy axis is equivalent to being perpendicular to the hard axis. Having modeled the magnitude and direction of M I and M R , the total magnetization vector M can now be calculated as their vector sum. The graphical realization of the total magnetization vector M of the proposed model for the cubic anisotropy projected to the (100) plane is shown in Fig. 2.

B. Lag angle curves
Lag angle is defined as the difference between the applied field angle θ H and the total magnetization angle θ M . A computer simulation is performed based on the formulation of the proposed model and the result for the simulated lag angle curve is shown in Figure 3 for two different applied fields. The direction of M I is shown in Fig. 2 and its magnitude is computed using a Preisach differentialequation approach. The direction of M R is given by Equation 7 and its magnitude is computed by a single-valued magnetization function of the applied field. The simulation inputs are the magnitude of the normalized rotating field (h) and the angle step size. The output is the angle of the total magnetization vector (θ M ). As the field increases, the maximum lag angle decreases. The applied field angle where the maximum lag angle occurs approaches 45 degrees (i.e. hard axis) as the field increases. The lag angle curve has a period of 180 degrees for a distribution of uniaxial particles and a period of 90 degrees for a distribution of cubic particles.

C. Magnetization trajectory
The normalized total magnetization trajectory, which is the polar plot of θ M as θ H completes a full revolution, traced for a normalized rotating applied field of magnitude h=1.25 (corresponding to the solid line plot in Fig. 3) is shown in Fig. 4. Maximum magnitudes of the magnetization occur along the easy axes and minimum magnitudes occur along the hard axes. Note that the trajectory for a distribution of uniaxial particles is elliptical with the major axis laying along the single easy axis.

D. Oscillation, rotation, and bifurcation curve
Similar to the case of a distribution of uniaxial particles, if the applied field is smaller than a threshold, then as the field rotates the magnetization will oscillate around the easy axis (oscillation mode). This threshold was shown to be h = 0.5 since the bifurcation curve is an astroid. 4 For a distribution of cubic particles, the bifurcation curve is a wind rose and the rotation-oscillation threshold is found to be h = 0.25 as shown in Equations 11 and 12. For a detailed study of the wind rose bifurcation curve, refer to Ref. 11. For h < 0.25, the applied field remains inside the bifurcation curve and does not change its state which refers to oscillation mode.

V. CONCLUSION
A generalization of the vector magnetization model that was previously developed for a distribution of uniaxial particles is presented and a generalized Gibbs free energy function for cubic anisotropy projected to the (100) plane is derived. The proposed model decomposes the magnetization into reversible and irreversible components and takes into account the anisotropy field. The lag angle simulation shows that the total magnetization for a distribution of cubic particles projected to the (100) plane has a period of 90 degrees which is consistent with the existence of two in-plane easy axes. The magnetization trajectory is discussed as well as the oscillation and rotation modes of magnetization. Finally, it was shown that the proposed model reduces to the simpler case of a distribution of uniaxial particles as well as the classical Stoner-Wohlfarth case of a single ellipsoidal particle.