Nearest-neighbor distance theory

Sokel carried out the variational calculations necessary to calculate C44 for a few materials. In a similar way Chadi and Martin used the special points method to calculate C44 with the artificial restriction that all nearest-neighbor distances remain fixed. They then did a valence force field calculation, such as will be discussed in the next section, to correct for internal displacements. The results of these calculations are listed in Table 8-3. In spite of the extra complications, the theory seems to be about as accurate as it was for the calculation of c, — c,2. 

Let us now turn to the structure factors of Eq. (16-5), to determine them first for the perfect crystal. What we do here is formulate the diffraction theory for crystal lattices, since the interaction of the electron waves with the crystal is a diffraction phenomenon. A perfect crystal is characterized by a set of lattice translations T that, if applied to the crystal, take every ion (except those near the surface) to a position previously occupied by an equivalent ion. The three shortest such translations that arc not coplanar are called pihuitive translations, t, Tj, and Tj, as indicated in Section 3-C. For the face-centered cubic structure, described also in Section 3-A, such a set is [011]a/2, [101]u/2, [ll0]a/2. The nearest-neighbor distance is d = n 2/2. Replacing one of these by, for example, [0lT]a/2, would give an equivalent set. For a body-centered cubic lattice, such a set is [Tll]u/2, [lTl]a/2, and [11 l]ti/2, and the nearest-neighhor distance is For each of these struc-... 

This is no longer true at temperatures far below Tc where the correlation length is small, of the order of the nearest-neighbor distance, and where the lattice structure of the Ising model, which seems negligible near T plays a crucial role. Now the interfacial tension depends on the direction with respect to the lattice axes, an effect important if one would make precise nucleation theories [16]. In... 

The FvdM as well as the BMVW model neglects thermal fluctuation effects both are T = 0 K theories. Pokrovsky and Talapov (PT) have studied the C-SI transition including thermal effects. They found that, for T 0 K the domain walls can meander and collide, giving rise to an entropy-mediated repulsive force of the form F where I is the distance between nearest neighbor walls. Because of this inverse square behavior, the inverse wall separation, i.e. the misfit m, in the weakly incommensurate phase should follow a power law of the form... 

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. 

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