ANNNI model

Selke, W. (1989) Long period structures in alloys-statistical mechanics in the ANNNI model and related concepts. In Alloy Phase Stability, eds. Stocks, G.M. and Gonis, A., NATO ASI Series, Series E Applied Sciences (Kluwer Academic Publishers, Boston, USA), Vol. 163, p. 205. 

A particular complex problem has been the modelling of Si/W(l 10) Amar et have included pairwise interactions up to the sixth nearest neighbor shell, as estimated experimentally from field-ion microscopic studies The predicted phase diagram (Fig. 30) exhibits (5 x 1), (6 x 1) and p(2 x 1) commensurate phases, as well as a broad regime of an incommensurate phase. In contrast to the ANNNI model the present model does seem to have a finite-temperature Lifshitz point, where the incommensurate, commensurate... 

The systems exhibiting the 1x2 structure are then also expected to show commensurate and incommensurate phases, quite similar to those observed for the ANNNI model [75]. In the lattice gas systems the presence of incommensurate phases is restricted to the situations in which the substrate lattice can be divided into a certain number of equivalent interpenetrating sublattices and the ordered state corresponds to the preferential occupation of one of those sublattices. Incommensurability is manifested by the presence of regions with different occupied sublattices and the formation of walls between the domains of commensurate phase. In the case of the discussed here systems exhibiting 1x2 ordered phase we have two sublattices, since particles occupy alternate rows. Figure 6 shows examples of equilibrium configurations demonstrating the formation of incommensurate structure when the ordered 1x2 phase is heated up. 

Basic models Ising model, Potts model, clock model, ANNNI model, etc.. 184... 

While the standard lattice gas and Ising models (including the ANNNI model) start out from a two-state description of each lattice site (S = 1), we have already mentioned the Potts model (Potts, 1952 Wu, 1982) where... 

Figure 9.24. Phase diagram obtained by the ANNNI model.

Measurements of spin correlations (Lander et al. 1978) and collective excitations (Lander etal. 1979a, b, Haig and Furrer 1984) revealed large anisotropies - the interactions are strong within the (100) planes and weak between the planes, in accordance with the ANNNI model. Optical measurements (Schoenes and Vogt 1978) suggest a non-integer number of 5f electrons and f-d hybridization. 

The emergence of the types of polytypes described above may be rationalized on the basis of a variety of different ideas. Rather than exploring the phenomenon of polytypism in detail, we give a schematic indication of one approach that has been used and which is nearly identical in spirit to the effective Hamiltonian already introduced in the context of stacking fault energies (see eqn (9.18)). In particular, we make reference to the so-called ANNNI (axial next-nearest-neighbor Ising) model which has a Hamiltonian of the form... 

The latter situation in fact is predicted for the so-called ANNNI (axial next-nearest neighbor Ising) model (Selke, 1988,1992). This Ising model has a competing interaction Jj < 0 in one lattice direction only, and thus the Hamiltonian is... 

Yamashita and Miyazima [58] and Yamashita [59], [60] adopted the modified ANNNI (axial next-nearest-neighbor Ising) model with the third-neighboring interaction (ANNNI + Jj, model) and wrote the Hamiltonian... 

Some other subphases were also predicted at finite temperatures. Our treatment and ANNNI -I- model give essentially the same structures for the major subphases, SmCy and AF, as displayed in Figure 9.23, although the corresponding rational numbers are different [49]. 

Predicting these magnetic structures is extremely difficult since in each case there is a fine balance between different interactions. Probably the considerations discussed earlier will be necessary, together with some features of the anisnttrcrpic next-nearest-neighbor interaction model (ANNNI), as expounded by Bak and van Boehm (1980), Villain and Gordon (1980), and Fisher and Selke (1980). 

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