Born’s model

Born s model (Born, 1920) gives the solvation free energy of a spherical ion as ... 

Born-Model Calculations.- A very much more complex model is that due to Catlow and co-workers who have developed a methodology whereby structure prediction takes place based upon Born s model for ionic solids. The interactions which are considered, are, for the most part, non-bonded interactions, and thus can be considered as a potential model and not a force constant model. 

This equation contains the basic model used by M. Newton in his pioneering calculations of the solvated electron [7]. If the charge density is replaced by a classical unit charge at the origin of the sphere, the RF potential obtained after integration of Eq. (26) corresponds to Born s model for a metalized sphere immersed in an isotropic continuum. 

The first factor in the parenthesis is associated with m and the second with the short-range interactions. Since the ratio between p and Rq is generally of the order of 0.1, the largest contribution to the cohesion energy is m, which justifies a posteriori the hard-sphere model. Despite its simplicity, Born s model has been used with success in many different instances for example, it has helped in the interpretation of many bulk phonon dispersion curves (Bilz and Kress, 1979). 

In Born s model, in which the ionic charges have the same constant integer values in the bulk and at the surface, the reduction of the Madelung constants at the surface directly induces a reduction of the Madelung potential. However, when the electronic structure is more carefully considered, changes in the electronic distribution in the outer layers may also modify the Madelung potential. We will see in the following that, on most dense oxide surfaces, these modifications are small but, on more open surfaces, precise estimations have to be performed. 

Theoretically, the elasticity theory of continuous media may be used to study the long-wavelength modes. To determine the microscopic modes, numerical approaches are necessary. Most of them have used Born s model to estimate the inter-atomic forces. The semi-infinite crystals are modelled by thin films, whose thickness must be larger than the attenuation length of the surface modes. The complete MgO(OOl) phonon spectrum has been calculated, neglecting (Chen et al, 1977 Barnett and Bass, 1979) or taking into account (Lakshmi and de Wette, 1980) the surface relaxation. The same has been done for SrTiO3(001) (Prade et al, 1993). 

On SrTi03((X)l), Prade et al. (1993) have calculated the relaxation strength, the origin of the reconstruction and the vibration dynamics, using Born s model, with an account of the ionic polarization by the shell model. They have analyzed the manifestation of the anti-ferrodisplacive phase transition at the surface, a transition which involves rotations of TiOe octahedra and which induces a doubling of the lattice parameters. Introducing in their model force constants which vary with temperature. 

The bias observed between experimental measurements and Kieffer s model predictions is due to the relative paucity of experimental data concerning cutoff frequencies of acoustic branches, and also to the assumption that the frequencies of the lower optical branches are constant with K and equivalent to those detected by Raman and IR spectra (corresponding only to vibrational modes at K = 0). Indeed, several of these vibrational modes, and often the most important ones, are inactive under Raman and IR radiation (Gramaccioli, personal communication). The limits of the Kieffer model and other hybrid models with respect to nonempirical computational procedures based on the equation of motion of the Born-Von Karman approach have been discussed by Ghose et al. (1992). 

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

In Minero s Case 2, the concentration of the substrate is high or it is very hydrophobic, although it should be borne in mind that such a scenario may result in the oxidised substrate behaving as an extrinsic recombination centre. A direct consequence of this is that Minero s model is no longer applicable as his simplified kinetic scheme did not consider this reaction. Thus, the utility of the resultant expressions is limited to mainly very hydrophobic or poor water soluble compounds at low concentrations. Under these conditions, equation (9.59) simplifies to... 

Lin and Lee started the derivation from Kantor and Webman s two dimensional model of flexible chains that considers a vectorial Born-lattice model with a bending energy term between neighboring bonds [91]. As outlined in Fig. 11, the strain energy H of a chain composed of a set of N singly connected bonds, of length a under an applied force F at the two ends of the chain is ... 

A free energy of reduction AGred bears the relation, AGred = -nFE0, to the standard reduction potential <) > which is approximately identical to the first E /2- A combination of these relationships and Born s solvation model yields the following equation between the difference in electron affinity A a and the difference in the first half wave potential A i/2 for two species [33]. 

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