Two-sublattice model

Definition of site fractions. The multiple sublattice model is an extension of earlier treatments of the two-sublattice models of Hillert and Steffansson (1970), Harvig (1971) and Hillert and Waldenstrom (1977). It allows for the use of many sublattices and concentration dependent interaction terms on these sublattices. To woiic with sublattice models it is first necessary to define what are known as site fractions, y. These are basically the fiactional site occupation of each of the components on the various sublattices where... 

To overcome this problem an extension of the sublattice model was proposed by Hillert et al. (1985) which is now known as the ionic two-sublattice model for liquids. As in the previous case it uses constituent fractions as composition variables, but it also considers that vacancies, with a charge corresponding to the charge of the cations, can be introduced on the anion sublattice so that the composition can move away from the ideal stoichiometry and approach an element with an electropositive character. The necessary neutral species of an electronegative element are added to the anion sublattice in order to allow the composition to approach a pure element. The sublattice formula for the model can then be written as... 

To complete this section it is interesting to show the equivalence between the ionic two-sublattice model and the associate model as demonstrated by Hillert et al. (1985). Equation (5.62) can be simplified for a system (A - )p(B+ , J3°)q... 

Condition Eq. 18 is discussed based on two approaches the two-sublattice model and the FE soft mode dressed anharmonically, giving the inequaUty ... 

There are further models which do not introduce new ideas but use the above frame for more complicated types of interactions. Bari and SivardiSre215) discussed a two sublattice model in analogy to the molecular field theory of antiferromagnetism. In this case there are two different interaction constants, viz. the intrasublattice and the intersublattice interaction. They also expand the one-sublattice model by an Heisenberg type magnetic interaction term between the HS states. Such an interaction may only become important for degenerate spin states. 

Transverse components of the magnetization and restrictions of the two-sublattice model [9,10]... 

The magnetic data of CoO(I) and CoO(II) have been correlated with a two sublattice model and exchange parameters valuated. 

We have also examined a two-sublattice model, where the displacement on one sublattice is opposite to that on the other, but this model shows only second-order spin-state transitions. In order to explain the occurrence of both first- and second-order spin-state transitions, we have explored a two-sublattice model where the spin states are coupled to the cube of the breathing mode displacement This model predicts first- or second-order transitions but only zero high-spin-state population at low temperatures. The most general model that predicts nonzero high-spin-state population at low temperatures, a first- or a second-order transition, and other features appears to be one where the coupling of the spin states to a breathing mode is linear and that to an ion-cage mode is quadratic. Nonetheless, spin-state transitions in extended solids need to be further explored to enable us to fully understand the mechanism of these transitions. 

In the case of rock salt, the cations form a face-centered-cubic (f.c.c.) array. Such an array is not compatible with a two-sublattice model, and there are four different types of magnetic order that can be considered (see Fig. 18). An f.c.c. structure is composed of four interpenetrating, simple-cubic (s.c.) sublattices. In ordering of the first kind, each s.c. sublattice is ferromagnetic and Mi = — M2, M3 = -M4 so that each cation has eight antiparallel and four parallel near neighbors. Then equation 101 becomes (all interactions assumed negative)... 

The difference between the compound energy model and the simple two-sublattice model can be illustrated with two ternary intermetallic phases from the Al-Mg-Zn system. One of these two phases is known to contain a constant composition of 54.5 atomic percent magnesium and an extended homogeneity range of aluminum and zinc, corresponding to the formula Mg6(Al,Zn)5. However, no crystallographic data is available for this phase. Therefore, it is appropriately thermodynamically modeled by two sublattices, in which one sublattice is exclusively occupied by Mg, while A1 and Zn are allowed to randomly mix on the second sublattice (Liang et al., 1998). 

Interstitial solid solutions are treated similarly. The structure is again approximated with a two-sublattice model, but where one sublattice is occupied by substimtional elements, and one by the smaller interstitial elements (e.g. C, N, H) and vacancies. It follows, from Eq. 11.27, that for a binary interstitial solution, the Gibbs energy is given by ... 

Within a mean-field approach the R-T exchange interaction is described by molecular fields that are aeting on the magnetic moments. In a two-sublattice model, the suitable effective molecular fields acting on the R and T moments can be written as... 

In the simplest approach, in which the two-sublattice model is used, the magnetic phenomena are still described by many parameters representing the anisotropy of... 

In a two-sublattice model, the phenomenological energy of a magnetic system in an external field Bq has been written as (Verhoef et al. 1989, 1990)... 

The final example for discussion is shown in Figure 14 that repeats the earlier results of calculations for the W-Re-C system at 2200 °C and 2300 °C as compared with experimental results at 2000 °C, Kaufinan. The important feature of these calculations was the unusual finding that the close packed hexagonal phase runs continuously from the well known W2C phase to the stable close packed hexagonal Re. The W2C phase is described by a two-sublattice model for the hep lattice of W atoms with an interstitial lattice containing C atoms... 

The rare earth perovskites are usually treated in terms of a two-sublattice model with a rare earth ion on the A site and a transition metal ion on the B site. Early neutron diffraction studies (Koehler et al., 1960) Indicated that there was little interaction between the two sublattices. This is also suggested by the respective ordering temperatures for example, in the orthoferrifes the iron lattice orders around 700 K and the rare earth lattice only at a few degrees kelvin. An advantage of the low ordering temperature of the rare earth sublattice is that studies can be made well above Tn without much effect from the thermal vibrations of the lattice. 

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