Crystallization model, simplification

According to two simultaneous hypotheses15,16, the principal component of the 7 of simple metals (from lithium to aluminum) originates in the shift in the zero-point energy of the plasma models of the system when one perfect crystal is broken into two separate crystals. Some simplifications and arbitrary assumptions lead finally to the relation... 

In the previous sections, we briefly introduced a number of different specific models for crystal growth. In this section we will make some further simplifications to treat some generic behavior of growth problems in the simplest possible form. This usually leads to some nonlinear partial differential equations, known under names like Burgers, Kardar-Parisi-Zhang (KPZ), Kuramoto-Sivashinsky, Edwards-Anderson, complex Ginzburg-Landau equation and others. 

We have assumed so far, implicitly, that the interactions are strictly local between neighboring atoms and that long-ranged forces are unimportant. Of course the atom-atom interaction is based on quantum mechanics and is mediated by the electron as a Fermi particle. Therefore the assumption of short-range interaction is in principle a simplification. For many relevant questions on crystal growth it turns out to be a good and reasonable approximation but nevertheless it is not always permissible. For example, the surface of a crystal shows a superstructure which cannot be explained with our simple lattice models. 

Applying MD to systems of biochemical interest, such as proteins or DNA in solution, one has to deal with several thousands of atoms. Models for systems with long spatial correlations, such as liquid crystals, micelles, or any system near a phase transition or critical point, also must involve a large number of atoms. Some of these systems, including synthetic polymers, obey certain scaling laws that allow the estimation of the behaviour of a large system by extrapolation. Unfortunately, proteins are very precise structures that evade such simplifications. So let us take 10,000 atoms as a reasonable size for a realistic complex system. 

A promising simplification has been proposed by Bader (1990) who has shown that the electron density in a molecule can be uniquely partitioned into atomic fragments that behave as open quantum systems. Using a topological analysis of the electron density, he has been able to trace the paths of chemical bonds. This approach has recently been applied to the electron density in inorganic crystals by Pendas et al. (1997, 1998) and Luana et al. (1997). While this analysis holds great promise, the bond paths of the electron density in inorganic solids are not the same as the more traditional chemical bonds and, for reasons discussed in Section 14.8, the electron density model is difficult to compare with the traditional chemical bond models. 

In the development of the set of intermolecular potentials for the nitramine crystals Sorescu, Rice, and Thompson [112-115] have considered as the starting point the general principles of atom-atom potentials, proven to be successful in modeling a large number of organic crystals [120,123]. Particularly, it was assumed that intermolecular interactions can be separated into dispersive-repulsive interactions of van der Waals and electrostatic interactions. An additional simplification has been made by assuming that the intermolecular interactions depend only on the interatomic distances and that the same type of van der Waals potential parameters can be used for the same type of atoms, independent of their valence state. The non-electric interactions between molecules have been represented by Buckingham exp-6 functions,... 

The experimental values of hcogp lead to the observed electron concentrations no (again referred to one formula unit), when using average values for Acc and the information about volumes taken from the crystal stmctures. (65) The expected values n are calculated on the basis of the bond model (Table 7). The experimentally derived values Uq are 20% smaller than the expected values Ug for the pure elements. In spite of the simplifications in the calculations, e.g. taking m instead of m, the determined electron deficiency seems to be real, as a decreased electron concentration is also calculated from the surface plasmon energies measured for Rb and Cs hy Kunz. 87)... 

The quasi-molecular Hamiltonian (4.11) has had an immensely rich past as a model for point impurities in crystals. For reasons of symmetry and also of the wish for simplification only a few modes were normally included in the second sum in (4.11). These modes have been named interaction- , cluster- or configurational- modes. Although as we have remarked, the range of application is very wide we have made a very narrow selection of those instances in which there has been significant experimental information on the character of the interaction mode. 

When the crystal size approaches a, the pair becomes confined and the effects are observable. This model is a simplification, since other correlation functions (with more complex solutions) are possible. However, several experimental works have demonstrated the validity of the expression for near-spherical nanoparticles [65-79]. More complex solutions may be necessary for anisotropic shapes, as experimentally demonstrated by Buhro and Covin [80] for InAs. In general, the third term of (18) is neglected and the second term is only significant when < Rp [79]. hi these particles, the mode is known as strong confinement, whereas for larger, albeit still small particles, the mode is known as weak confinement [64]. 

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