Magnetic anisotropy solve

Resuming the main line of our consideration, let us show how to consistently take into account the effect of the particle magnetic anisotropy by solving the Brown equation (4.90). Taking n as the polar axis of the coordinate framework, we recover the situation considered as an illustration in Section II.B. Namely, the dimensionless particle magnetization is expressed as Eq. (4.54) and the particle energy as Eq. (4.55). Then the nonstationary solution of the kinetic equation (4.90), which is equivalent to Eq. (4.27), is presented in the form of expansion (4.56) whose amplitudes satisfy Eqs. (4.60) and (4.61). 

Subsequent developments [16,18] have focused on g-factor and hfc calculations from two-component DFT with SO coupling included variationally. In this case, it is not necessary to solve LR equations. Instead, the EPR parameters are calculated in an expectation-value fashion with the magnetic-field derivatives of OZ and SZ [as seen in equations (12.14b), (12.14e)] for g-factors, and the nuclear magnetic moment derivatives of OP and FCh-SD [see equations (12.14c), (12.14f)] for hfc, respectively. Two forms of SO DFT EPR calculations were developed. One approach allows for spin-polarization and was dubbed the magnetic anisotropy (MA) route, inspired by DFT studies of magnetic anisotropy in van Wiillen [29] and Schmitt et al. [30], Related HF and DFT methods have been put forward by Hrobarik et al. [31], Jayatilaka [32], and Malkin et al. [33,34]. The other approach follows works by van Lenthe, Wormer, and van der Avoird (LWA) [35,36] which does not treat spin polarization but has some advantages due to its simplicity and computational efficiency and tends to perform well for g-factors. 

The development of the bubble domain memory has been remarkable in that since the discovery of the growth induced anisotropy in garnets, problems connected with materials have been relatively few and not too difficult to solve. A major reason is that the different sizes and magnetic properties of the rare earths offer a wide range of choices for the materials designer. 

It is interesting that in the RBigSis system, the Gd phase has a transition at 3.2 K with no transition above 1.8 K for the Tb phase, while the RB25 phase shows opposite behavior with a transition only observed for the Tb phase. This is an indication of the effect of the anisotropy of the lanthanide ions (in the Gd case, a lack of anisotropy) on the magnetic interaction and should be useful toward solving the explicit mechanism. 

Ignoring the damping term in Eq. (17), the resonance is described by dM/dt = KMxHeff). For homogeneously magnetized ellipsoids of revolution, the effective field is equal to the applied field H = H ez plus the anisotropy field Ha, and the resonance problem is solved by the diagonalization of a 2x2 matrix. This uniform or ferromagnetic resonance (FMR) yields resonance frequencies determined by [17]... 

Consider a configuration where the undisturbed optic axis of the liquid crystal is aligned parallel to the surface of the electrodes. The dielectric anisotropy Ae is assumed to be positive. The analogous magnetic case was treated by Saupe [49]. The problem of the deformation by an electric field, which is more difficult because of the large dielectric anisotropy, was solved by several authors [50, 51]. 

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