Equivalent Crystal Theory

From a computational standpoint, the usefulness of the method relies on the simplicity of the calculations needed for the determination of the three equivalent crystals associated with each atom i. This is accomplished by building on the simple concepts of Equivalent Crystal Theory (ECT) [25,26], as will be discussed in detail below. The procedure involves the solution of one simple transcendental equation for the determination of the equilibrium Wigner-Seitz radius i WSE) of ch equivalent crystal. These equations are written in terms of a small number of parameters describing each element in its reference state, and a matrix of perturbative parameters Ay , which describe the changes in the electron density in the vicinity of atom / due to the presence of an atom j (of a different chemical species), in a neighboring site. The determination of parameters for each atom in... 

A.M. Rodriguez, G. Bozzolo, J. Ferrante, Structure and energetics of high-index Fe, Al, Cu and Ni surfaces using equivalent crystal theory. Surf. Sci. 307, 625-631 (1994)... 

ECT is based on the concept that there exists for each atom i, a certain perfect equivalent crystal with its lattice parameter fixed at a value so that the energy of atom i in the equivalent crystal is the BPS strain energy contribution ef. This equivalent crystal differs from the actual ground state crystal only in that its lattice constant may be different from the ground state value. We compute ef via perturbation theory, where the perturbation arises from the difference in the ion core electronic potentials of the actual defect solid and those of the effective bulk single crystal. 

The growth rate of many crystals is often observed to depend upon temperature in a manner consistent with nucleation theories. If measurements are made on growth from solutions of different concentrations then, at equivalent thicknesses, the dependence of growth rate upon concentration may be determined. Equations (3.16) and (3.17) can be used to predict the concentration dependence of this nucleation approach. 

FIGURE 16.24 In the crystal field theory of complexes, the lone pairs of electrons that serve as the Lewis base sites on the ligands (a) are treated as equivalent to point negative charges (b). 

A roughly equivalent valence-bond theory would result from allowing the 2s electron of each lithium atom to be involved in the formation of a covalent bond with one of the neighbouring atoms. The wave function for the crystal would be... 

The transitions of the coil-globule type were considered not only in the usual space, but also in the space of monomeric units orientation, where such transition is equivalent to the nematic liquid crystal ordering [60,61]. Such an approach using the formalism developed by I.M. Lifshitz has led to the creation of the theory of liquid crystal ordering in the solutions of semi-flexible macromolecules [62,63]. 

This stearolic acid has been thoroughly characterized 3 6 by the freezing-point curve, ultraviolet and infrared spectra, ozonization, and hydrogenation. It has been shown to be free both of positional isomers and of olefinic acids such as oleic and elaidic acids. Its properties include m.p. 46-46.5°, iodine number (Wijs titration, 30 minutes) 89.5, d 5 1.4510, d 5 1.4484, neutral equivalent 279.2-279.6 (theory 280.4), hydrogen uptake 95-100% of theory for a triple bond. The last trace of color is difficult to remove by recrystallization from petroleum ether. It can be removed, however, by crystallization from a 20-30% solution in acetone at —5 to —8°, or from an 8-10% solution at —20°, or by distillation (b.p. 189-190°/2mm.). 

Itinerant electrons occupy band states, which are molecular orbitals for an entire crystal they therefore belong equally to all like atoms on energetically equivalent sites and are described by band theory. 

The application of the theory of space-groups to crystal structure determination involves the assumption that equivalent sites in the crystal are occupied by identical atoms. Fortunately tins is true for... 

Mixed crystals are mentioned here chiefly as an introduction to the idea that sites equivalent according to space-group theory may in some circumstances be occupied by different atoms. As far as structure determination is concerned, we need not be detained by further consideration of mixed crystals nobody is likely to attempt to determine the structure of a mixed crystal without first knowing the structures of the pure constituents. The function of X-ray analysis here is to determine, as in the example of Cu3Au, whether a substance thought to be... 

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