Anisotropy random

Note that in the limit of isotropic spins (where Si 0), the results for coherent axes and for random anisotropy duly coincide and agree with ordinary high-temperature expansions. 

Note that the result for random anisotropy is identical to the result for isotropic dipoles. 

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

Random-anisotropy contributions, as epitomized by the a(r)-term in Eq. (34), are an important aspect of nanomagnetism, but their detailed treatment goes beyond the scope of this chapter. Some aspects of random-anisotropy magnetism will be treated in the chapter on soft magnets, and we also recommend reading or consulting the rich original and review literature [16, 48, 116, 124, 189, 193-197]. 

Following this introductory section, we will overview the development of random anisotropy models and discuss the origin of the magnetic softness in nanostructures. The nanostructural formation process and alloy development in the Fe-M-B-(Cu) alloys to which less attention has been addressed in the previous reviews, will be another focal point in this chapter. 

Figure 3 illustrates the expected grain-size dependence of the average random anisotropy for the material parameters of optimized nanocrystalline Fe-Si-B-Nb-Cu alloys [29]. We hereby have included the contribution of the random atomic scale anisotropy of the amorphous matrix as well as the case of a small uniform anisotropy Klt. 

The random anisotropy of the amorphous matrix becomes only visible for very small grains, resulting in a grain-size independent anisotropy. However, the related coercivity [Hc 0.001 A/m) is so small, that the situation shown for smallest grain sizes in Fig. 3 remains academic. 

Abstract The influence of randomly distributed impurities on liquid crystal (LC) orientational ordering is studied using a simple Lebwohl-Lasher t5q)e lattice model in two d=2) and three d=3) dimensions. The impurities of concentration p impose a random anisotropy field-type of disorder of strength w to the LC nematic phase. Orientational correlations can be well presented by a single coherence length for a weak enough w. We show that the Imry-Ma... 

In this article we address the so-called Random Anisotropy Nematic (RAN) ", in which interactions with arbitrarily oriented but quenched local spins can locally orient a nematic liquid crystal. We consider a slightly more generalized model than that discussed previously (see refs. ( ) and ( °)), which allows for the density of impurity sites to be changed. This system belongs to the family of continuously broken spin systems, and is much amenable to experimental test than some of the magnetic systems used in the 1970s. Our study is computational and is therefore complementary to the high-powered theoretical approaches discussed elsewhere. 

We suppose that in addition the LC ordering is perturbed by loeal site random anisotropy disorder of strength w. This type of interaetion was first introduced in magnets by Harris et al We have elsewhere labeled this model in a nematie eontext as the Random Anisotropy Nematie model (RAN)". In this study the RAN is modified so that only spins at a random fraetion p of sites are subject to random anisotropy, as discussed e.g. by Chakrabarti and Bellini et al... 

In eq. (1) each site / is subject to a random anisotropy with probability p. The quantity, is a random variable, taking the values 0 or 1, defined formally as follows ... 

A magnetic study of magnetron-sputtered amorphous U27Fe73 films was performed by Freitas et al. (1988). Arrott plots indicate magnetic ordering below 32 K. The freezing phenomena found at low temperatures were interpreted in terms of random-anisotropy ferromagnetism. 

For amorphous alloys described by the random anisotropy model (Cochrane et al. 1978) as well as for polycrystalline materials, the EMD is isotropically distributed. In this case, spin orientation and domain walls are randomly distributed. The magnetisation process therefore consists of two steps (i) the motion of 180°-domain walls leading to a magnetisation of MJ2 without any magnetostriction, and (ii) the rotation of spins into... 

To describe the magnetic properties of amorphous alloys containing rare earth elements with non-zero orbital moment (L 0) the Hamiltonian of eq. (25) is no longer suited. Harris et al. (1973) have proposed a model in which they assume that there is a local uniaxial field of random orientation at each of the rare earth atoms in an amorphous solid. This local uniaxial field of random orientation is closely associated with the presence of an equally random crystalline electric field. The Hamiltonian for this random anisotropy model (RAM) can be written as... 

It is interesting to note that in the amorphous alloys discussed above, where large random anisotropy fields far outweigh random exchange fields, the magnetization shows a temperature dependence similar to that observed in Gd-base spin glasses. 

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