Rigid ion

Pauling s calculated values for the ionic radii are not really the radii of the rigid ions, but the values which these ions would have if they formed a lattice with the NaGl structure. In another lattice the distance will be somewhat different, so that the distance in the... 

Now A8/A6 differs little from unity, and therefore n must have a large value (about 35) to satisfy the equation. Since n is actually much smaller than 35 for all kinds of ions, the CsCl lattice cannot be stable. The introduction of non-rigid ions therefore has a serious effect on Goldschmidt s treatment. In KF, where the ions are of equal size, a CsCl structure is no longer to be expected. It is the van der Waals forces, described in Section 46, tending towards high coordination numbers, which lead to the formation of CsCl lattices, but only in the compounds CsCl, CsBr and Csl which are composed of strongly polarizable ions. 

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. 

Since this approach does not account for long-range electrostatic potentials present in the extended material, the second approach chosen was the rigid-ion lattice energy minimization technique, widely used in solid-state chemistry. This method is based on the use of electrostatic potentials, as well as Born repulsion and bond-bending potentials parametrized such that computed atom—atom distances and angles and other material properties, such as, for instance, elastic constants, are well reproduced for related materials. In our case, parameters were chosen to fit a-quartz. 

Table IV. Calculated Rigid Ion Lattice Energies of Aluminum-Free Zeolite Lattices...

Scattering by Alkali Halides Melts A Comparison of Shell-Model and Rigid-Ion Computer Simulation Results. 

Most likely the blueprint for stereoselective transformations fulfiUing the three-functions-rule was one of the former rigid ions linked to a chiral environment and causing transformations in a substrate mixture... 

Ail interpretations based on the assumptions with respect to use of the Nernst-Planck relationships are, however, subject to sizable uncertainty because the constancy attributed to the diffusion coefficients used in these relationships is susceptible to sizable variability. Bead volume variations are ignored. This leads to variations in ion mobilities which dictate changes in the diffusion coefficients of the Nemst-Planck relationships. Sizable divergence of measurements firom prediction can be expected on this basis. Even in rigid ion exchangers such as zeolites the difference in size of the counterions exchanged usually affects their mobility, and so leads to variations of the diffusivities [3]. 

For metals and ceramics considered separately, reliable interaction potentials have been developed in the past. For many metals, embedded-atom-type potentials ° have proven successful. Alternatives and refinements have been developed.For ceramics, rigid-ion and shell model potentials and their refinements " have proven equally capable. 

Equation [15] is often referred as a rigid ion potential model. 

Both the rigid ion and shell potential models have been used in energy minimization studies of dense and microporous silica and molecular sieves. Kramer and coworkers reported parameters of the rigid ion potential model for silicates, aluminosilicates, and aluminophosphates. The model also includes parameters for extra-framework cations such as Na and Cl . Both the rigid ion and shell models were used by Catlow and coworkers in modeling silicate and zeolite structures. ... 

Kramer and co-workers used ab initio calculations of H4TO4 (T = Si, Al, P) clusters to derive parameters for the rigid ion potential model. The potential energy surface of the clusters was scanned along two modes of distortion, and the resulting potential curves were fitted using Eq. [15]. The set of parameters was refined by the use of experimental data on a-quartz. This procedure resulted in a parameterization that well reproduced both structure and elastic moduli of silicates, aluminosilicates, and aluminophosphates. Subsequently, this approach was extended to protonated forms of zeolites. ... 

De Boer and coworkers ° °" parameterized the shell model for silica polymorphs on the results of ab initio calculations of the potential energy surfaces, polarizabilities, and dipole moments of Si(OH)4 and (0H)3Si-0(H)-Si(OH)3 clusters. The structural characteristics and elastic moduli calculated with this set of parameters for three structures compared well with results computed with the use of both the rigid ion and the empirical shell models. ... 

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