Rigid-ion model

Grille and Bechstedt [9] performed calculations of the phonon modes for zincblende AlGaN superlattices within a rigid ion model. In a three-parameter Keating model elastic forces are characterised including the electric forces. The authors predict a two-mode behaviour of the TO modes, but a one mode behaviour of the LO modes. Through symmetry arguments the wurtzite situation can be connected to the zincblende case and Cros et al inferred such a two-mode behaviour for E2 as well. In the above experiment, however, a two-mode behaviour could not be found for Ai(TO) but was predicted by theory. 

Computations of minimum-energy configurations for some off-centre systems were first carried out on the basis of polarizable rigid-ion models, mainly devoted to KChLi" " [95,167-169]. Van Winsum et al. [170] computed potential wells using a polarizable point-ion model and a simple shell model. Catlow et al. used a shell model with newly derived interionic potentials [171-174]. Hess used a deformation-dipole model with single-ion parameters [175]. At the best of our knowledge, only very limited ab initio calculations (mainly Hartree-Fock or pair potential) have been performed on these systems [176,177]. 

Ii3/2 multiplets. Frequencies and polarization vectors of phonons in the LiYp4 crystal were obtained at 8000 points in the irreducible part of the Brillouin zone using the rigid ion model of lattice dynamics derived on the basis of neutron scattering data. Matrix elements of electronic operators Vds) were calculated with the wave functions obtained from the crystal-field calculation. The inverse lifetimes of the crystal-field sublevels determine the widths of corresponding absorption lines. 

Figure 5.16. Comparison of experimental and rigid-ion model predicted stability regions for Na -containing leolited .

Several attempts have been undertaken to derive such MM force constants for modeling zeolite frameworks [39]. Typical examples are the rigid ion and the shell model which assume that the character of the bonds in the lattice is largely ionic. Within the rigid ion model developed by Jackson and Catlow [40], the potential energy is given by... 

An extension of the rigid ion model is the shell model taking additional ionic polarizabihties into account [46]. Whereas for silicon and aluminum atoms usually a low polarizability is assumed and they are, therefore, treated as rigid cations in the shell model, for oxygen anions and extra-framework cations an additional term of the form... 

The above potential is based on a rigid-ion-model (RIM), as no effect of atomic polarization is taken into account. A shell model (SM) was also developed, which considers a split core-shell structure for polarizable O atoms. As usual [23], core and shell are coupled by an elastic spring of force constant k, and are characterized by different electric charges zqc and ZQs- In addition to k, and zqs, also the core-shell displacement, is to be optimized, and contributes three positional parameters (unless reduced by symmetry) for eaeh O atom in the asymmetric unit. When an O atom is involved in the two-body interaction, the repulsive and, possibly, dispersive energy is eomputed by reference to the 0 shell position. All other atoms and interactions are treated as for the RIM case. 

As expected, the shell model proves to be superior for simulating the lattice dynamics of calcite, while static properties (cell geometry and elastic constants) are reprodueed better with the rigid ion model. However, the overall difference between RIM and SM results is eomparatevely modest, and that can be considered to be a good performanee for the simpler RIM... 

Figure 11. Density of vibrational frequencies of calcite, computed by the RIM (rigid-ion-model) potential (above), by the SM (shell model) potential and by Kieffer s model (below, thick and thin lines, respectively).

Fig. 4.1-62 AIN (wurtzite structure). Phonon dispersion curves (left panel) and phonon density of states (right panel), from a rigid-ion model calculation [1.58,59]...

Takagi R, Hutchinson F, Madden PA, Adya AK, Gaune-Escard M (1999) The structure of molten DyCls and DyNasClg simulated with polarizable- and rigid-ion models. J Phys Condens Matter ll(3) 645-658... 

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