Three-sublattice model

The present approach can be applied to the case of three-coupled subsystems (three-sublattice model). We consider quasi-binary compounds (Ri,R2)Co2, where Ri and R2 are different lanthanide atoms with concentrations xi and xi, respectively (xi +X2 = 1). In the expressions of the firee energy we change the notation slightly for the d-subsystem the coefficients are written as a and for the R, subsystem, the coefficients are With these notations, eq. (61) becomes... 

The three ordered stales of the Potts model correspond to a preferential occupation of one of the three sublattices a,b,c into which the triangular lattice is split in the (-/3x-v/3)R30° structure. In the order parameter plane (0x.0r), the minima of F occur at positions (1, 0)MS, (—1/2, i/3/2)yWs, (—1/2, -yf3/2)Ms, where Ms is the absolute value of the order parameter, i.e. they are rotated by an angle of 120° with respect to each other. The phase transition of the three-state Potts model hence can be interpreted as spontaneous breaking of the (discrete) Zj symmetry. While Landau s theory implies [fig. 13 and eqs. (20), (21)] that this transition must be of first order due to the third-order invariant present in eq. (34), it actually is of second order in d = 2 dimensions (Baxter, 1982, 1973) in agreement with experimental observations on monolayer ( /3x /3)R30o structures (Dash, 1978 Bretz, 1977). The reasons why Landau s theory fails in predicting the order of the transition and the critical behavior that results in this case will be discussed in the next section. 

To account for this, a full-profile fit of the XRD pattern was made for three different crystallographic models (1) pure B structure (2) pure B2 structure (3) cubic structure with the fee sublattice for the lead atoms and two different positions for the sulfur atoms. The last model allows sulfur atoms to occupy not only octahedral interstitials in the fee sublattice as it is common for the B structure, but also the occupation of tetrahedral interstitials, which is common for the 53 structure. The best fit to experimental XRD pattern in the case of the third model is shown in Fig. 1. 

Starting from the discrete logical structure of the reaction lat tice, let us describe a chemical reaction algebraically by employing continuous parameters. In the simplest case we shall investigate the dynamic sublattice of the three-dimensional reaction lattice (single parameter X-model). Thereafter, these basic ideas wiU be generalized in three ways ... 

Fig. 69. (a) Part of the body-centered cubic lattice ordered in the B2 structure (left part) and in the Dtp structure (right part). Left part shows assignment of four sublattices a, b, c and d, In the B2 structure (cf. also fig. 66a), the concentrations of A atoms are the same at the a and c sublatticcs, but differ from the concentrations of the b, d sublattices, while in the DOj structure the concentration of the b sublattice differs from that of the d sublatlice, but both differ from those of the a, c sublattices (which are still the same). In terms of an Ising spin model, these sublattice concentrations translate into sublattice magnetizations mu, mu, mc, m,i, which allow to define three order parameter components / = ma + mL- — mu — m,/, fa = m - mc + mu — m,j, and fa = -ma + m., + mu — nij. 

In order to establish the model of intergranular impedance for doped barium titanate, it is important to notice that miorostructure properties of BaTiOj based materials, expressed in their grain boundary contacts, are of basic importance for electric properties of these materials. The barrier character of the grain boundaries is especially pronounced for doped BaTiOs materials which are used as PTC resistors. Basically two types of dopants can be introduced into BaTiOs large ions of valence 3+ and higher, can be incorporated into Ba positions, while the small ions of valence 5+ and higher, can be incorporated into the Ti sublattice [9-11], Usually, the extent of the solid solution of a dopant ion in a host structure depends on the site where the dopant ion is incorporated into the host structure, the compensation mechanism and the solid solubility limit [12], For the rare-earth-ion incorporation into the BaTiOs lattice, the BaTiOs defect chemistry mainly depends on the lattice site where the ion is incorporated [13], It has been shown that the three-valent ions incorporated at the Ba -sites act as donors, which extra donor charge is compensated by ionized Ti vacancies (V -), the three-valent ions... 

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