Exchange stiffness

Figure 3. Exchange energy as a function of the wave vector of the magnetization inhomogenity Lindhard function (solid line) and exchange-stiffness or continuum approximation (dashed line). For late 3d elements (Cu), k/2kf = 1 corresponds to modulation wavelength of 0.23 nm...

Figure 3 compares the Lindhard function (solid line) with the exchange-stiffness approximation (dashed line). We see that the exchange-stiffness approximation works well unless k is comparable to kf. 

In noncubic materials, A must be replaced by the 3x3 exchange-stiffness tensor A v, and the energy is X J A v dM/dxy dM/dxv dV. Here the indices ju and v denote the spatial coordinates x, y, and z of the bonds. The energy is anisotropic with respect to the nabla operator = d/d (bond anisotropy) but isotropic with respect to the magnetization M. By contrast, the relativistic anisotropic exchange Xa i Ha(i VMa VMP dV is isotropic with respect to V but anisotropic with respect to M. 

Here Ms(r) is the spontaneous magnetization, A)(r) is the first uniaxial anisotropy constant, A(r) denotes the exchange stiffness, and n(r) is the unit vector of the local anisotropy direction. H is the external magnetic field, and Hd is the magnetostatic self-interaction field. The latter can be written as... 

In the past, nucleation fields such as Eq. (15) have been obtained for several cases spherical particles in an infinitely hard matrix [110], small inclusions in a matrix of arbitrary anisotropy and exchange stiffness [105] [111], various types of multilayers [111, 113], and some core-shell and nanowire configurations [105, 114, 115], For a discussion of the unphysical limit of very small inclusions, L = 0, see e.g. [5],... 

The term V(yfVm) reflects due to local character of exchange stiffness A(r) [111]. For sharp phase boundaries, the exchange term reduces to the boundary condition... 

The micromagnetic exchange constant (exchange stiffness) A [J/m] is proportional to the exchange integral J. For a cubic lattice having one nonequivalent atom, A is given by [36]... 

Chaudari et al. (1974) have prepared GdCoCu amorphous materials by rf sputtering. Their films are richer in Co, because there is a Gd resputtering effect. They have observed that a drive field for bubble domain propagation in T-bar devices is proportional to the magnetization. They have studied particularly the magnetic quality factor and the exchange stiffness versus the temperature. 

A = exchange stiffness constant. A schematic representation of this model is given in Fig. 4.3-27. The basic effect of decreasing the grain size consists of local averaging of the magnetocrystalline anisotropy energy Ki for D = 10-15 nm at Lq = 30-50 nm (about equal to... 

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