Stoner-Wohlfarth particles

The exponent m cannot be regarded as a fitting parameter but depends on the symmetry of the system. In most cases, m = 3/2 [16, 140, 158, 166, 167, 174, 175], but m = 2 for highly symmetric systems, such as aligned Stoner-Wohlfarth particles. In particular, the m = 3/2 law is realized for misaligned Stoner-Wohlfarth particles and for most domain-wall pinning mechanisms [5], Experimental values of m tend to vary between 1.5 to 2 [136, 158]. Linear laws, where m = 1, are sometimes used in simplified models, but so far it hasn t been possible to derive them from physically reasonable energy landscapes [5, 16, 176]. The same is true for dependences such as /H- l/H0 [177], where series expansion yields an m = 1 power law. 

Stancu A, Spinu L (1998) Temperature- and time-dependent Preisach model for a Stoner-Wohlfarth particle system. IEEE Trans Magnetics 34 3867-3875 Stein DL (1992) Spin glasses and biology. World Scientific, Singapore... 

This critical field called coercivity ff. or switching field Ff., is also equal to FF. If a field is applied in between 0 and 90° the coercivity varies from maximum to zero. In the case of this special example the applied field Ha = Hs = Hc = Hk. Based on the classical theory, Stoner-Wohlfarth (33) considered the rotation unison for noninteracted, randomly oriented, elongated particles. The anisotropic axis can be due to the shape anisotropy (depending on the size and shape of the particle) or to the crystalline anisotropy. In the prolate ellipsoids b is the short axis and a the longest axis. The demagnetizing factors are IV (in the easy direction) and The demagnetizing fields can then be calculated by Hda = — Na Ms, and Hdb = — Nb Ms. The shape anisotropy field is Hd = (Na — Nb)Ms. Then the switching field Hs = Hd = (Na — Nb)Ms. 

The Stoner-Wohlfarth approach works fairly well for very small particles, where VM = 0 is a good approximation. However, it has been known for decades that neither the Stoner-Wohlfarth theory nor the additional consideration of the curling mode account for the coercivity of real materials. For example, the coercivity of optimized permanent magnets is only 20-40% of the anisotropy field 2Kx nMs, and only a part of the discrepancy can be ascribed to the curling terms in Eq. (11). The reason is that real-structure imperfections make it impossible to consider the magnets as perfect ellipsoids of revolution. 

MODIFIED STONER-WOHLFARTH THEORY FOR HARD-MAGNETIC PARTICLE ARRAYS... 

For the above mentioned FePt particles, the particle diameter is clearly smaller than the critical particle size given by Eq. (8) for coherent rotation. Furthermore the strength of the magnetostatic interaction field acting on nearest neighbor particles is only about 2% of the anisotropy field for a particle distance of 2 nm. Thus the Stoner-Wohlfarth theory can be applied. 

However, the classical Stoner-Wohlfarth theory is modified, in order to take into account the agglomeration of particles. 

In Eq. (10) the sum is over all crystallites in the particle ensemble. V, is volume of particle i, K is the uniaxial anisotropy constant, u, is the anisotropy direction of particle i, and J, is the magnetic polarization vector of the particle. In addition to the classical energy terms of the Stoner Wohlfarth theory we have to consider the exchange energy between the crystallites that agglomerate to a particle. We assume that in average three crystallites agglomerate as schematically shown in Fig. 2. 

Figure 3 shows two hysteresis loops calculated with the modified Stoner-Wohlfarth model with high exchange interactions and low exchange interactions between the crystallites of a particle. Loop shape, remanence and coercive field are in good agreement with experimental results reported in [9],... 

Figure 3. Hysteresis loops for FePt particles calculated using the modified Stoner-Wohlfarth model for high exchange (A = 1.4 10"11 J/m) and low exchange (A = 0.4 10 11 J/m).

Fig. 4.53. Magnetisation behaviour of a single-domain particle assuming rotation-only mechanism (a) applied field at 90° from the easy direction (b) field collinear to the easy direction (Stoner Wohlfarth, 1948).

Fig. 4.54. Theoretical hysteresis loop of a polycrystalline sample made of single-domain particles with random orientation of easy axes (Stoner Wohlfarth, 1948).

Based on the use of the Stoner-Wohlfarth theory for single-domain partieles, the Neel-Brown relaxation model, and equilibrium functions, Medahuoi et al. have composed a model that allows for a direct comparison of theoretical simulations and experimental results from hyperthermia experiments carried out in iron nanoparticles with particles sizes ranging from 5.5 to 28 nm." In the low field region, the optimum particle volume Vopi) can be calculated from the Neel-Brown model ... 

To maintain thermal stability, hence a condition EB/kBT= In (for) needs to be fulfilled. For z = 10 years storage, 109-10u Hz [28] and ignoring dispersions, i.e. assuming monodisperse particles, this becomes Es/kBT= 40-45. Reversal for isolated, well-decoupled grains to first order can be described by coherent rotation over EB. This simple model, as first discussed by Stoner and Wohlfarth in 1948 [29], considers only intrinsic anisotropy and external field (Zeeman) energy terms. For perpendicular geometry one obtains the following expression ... 

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