Lattice, local defects

We can anticipate that the highly defective lattice and heterogeneities within which the transformations are nucleated and grow will play a dominant role. We expect that nucleation will occur at localized defect sites. If the nucleation site density is high (which we expect) the bulk sample will transform rapidly. Furthermore, as Dremin and Breusov have pointed out [68D01], the relative material motion of lattice defects and nucleation sites provides an environment in which material is mechanically forced to the nucleus at high velocity. Such behavior was termed a roller model and is depicted in Fig. 2.14. In these catastrophic shock situations, the transformation kinetics and perhaps structure must be controlled by the defective solid considerations. In this case perhaps the best published succinct statement... 

Conventional physical descriptions of materials in the solid state are concerned with solids in which properties are controlled or substantially influenced by the crystal lattice. When defects are treated in typical solid state studies, they are considered to modify and cause local perturbations to bonding controlled by lattice properties. In these cases, defect concentrations are typically low and usually characterized as either point, linear, or higher-order defects, which are seldom encountered together. 

The diSuse scatter arises because dislocations are defects which rotate the lattice locally in either direction. This gives rise to scatter, from near-core regions, which is not travelling in quite the same direction as the diffraction from the bulk of the crystal. This adds kinematically (i.e. in intensity not amplitude) and gives a broad, shallow peak that mnst be centred on the Bragg peak of the dislocated layer or substrate since all the local rotations are centred on the lattice itself. We can model the diffuse scatter quite well by a Gaussian or a Lorentzian function of the form ... 

Let us analyze these results one step further and ask about a quantitative measure of the Kirkendall effect. This effect had been detected by placing inert markers in the interdiffusion zone. Thus, the lattice shift was believed to be observable for an external observer. If we assume that Vm does not depend on concentration and local defect equilibrium is established, the lattice site number density remains constant during interdiffusion. Let us designate rv as the production (annihilation) rate of the vacancies. We can derive from cA+cB+cv = l/Vm and jA +/ B +./v = 0 that... 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

A domain wall under an external electric field moves in a statistical potential generated by their interaction with the lattice, point defects, dislocations, and neighboring walls. Reversible movement of the wall is regarded as a small displacement around a local minimum. When the driven field is high enough, irreversible jumps above the potential barrier into a neighboring local minimum occur (see Figure 1.23). 

Vacancies are missing atoms and are the simplest defects. Because higher temperatures increase vibrational motion and expand a crystal, more vacancies are formed at higher temperatures. However, even near the melting point, the number of vacancies is small relative to the total number of atoms, about 1 in 10,000. The effect of a vacancy on the rest of the lattice is small, because it is a localized defect and the rest of the lattice remains unaffected. Self-interstitials are atoms displaced from their normal location and appear in one of the interstices in the lattice. Here, the distortion spreads at least a few layers in the crystal because the atoms are much larger than the available space. In most cases, the number of these defects is much smaller than the number of vacancies. 

The analysis of the solutions of equations for the motion of a crystal lattice with defects showed [10, 89] that there occurs a kind of displacement from the reaction zone of the vibrations corresponding to the dense part of the phonon spectrum, and the main contribution to the rate constant is made by local vibration. 

We performed a calculation of the relaxation rates using the phonon Green s functions of the perfect (CsCdBr3) and locally perturbed (impurity dimer centers in CsCdBr3 Pr ) crystal lattices obtained in Ref. [8]. The formation of a dimer leads to a strong perturbation of the crystal lattice (mass defects in the three adjacent Cd sites and large changes of force constants). As it has been shown in Ref. [8], the local spectral density of phonon states essentially redistributes and several localized modes appear near the boundary of the continuous phonon spectrum of the... 

In addition, potential local defects including gem OH groups and silanol nests may be easily detected. For example in the case of Boralite (B-MFI), water molecules held at trigonal Boron sites, gem and OH groups were identified (23) which indicate either lattice defect or dynamic equilibria involving adsorbed water molecules and dessociated water. Similar defects were also thought to be present in the case of Cr-MFI (24). 

Defects in solids are ubiquitous and can be found both in the bulk and at the surface of materials.Two classes can be distinguished point defects and extended defects. The former, also called local defects, produce a modification of the site environment of an otherwise perfect lattice for instance, the absence of an atom in a lattice position (vacancy), the presence of an atom in an interstitial position (interstitial defect), or the substitution of an atom for another atom of a different chemical species at a regular lattice site (substitutional defect). Figure 45 shows typical examples of local defects in an ionic solid. 

Figure 47 Schematic example of a local defect in a two-dimensional lattice as modeled by the supetceU approach.

When a local defect is created in the lattice, it is accompanied by a displacement of the surrounding atoms, thereby removing at least some of the translational and point group symmetry that is a property of the perfect lattice. Even if the effect of the defect is so localized that it is imperceptible on the structure a few tens of A from the defect center (thereby permitting the application of the lattice-defect method described in Section 2.5), relaxing symmetry restrictions effectively introduces additional degrees of freedom into the permitted atomic displacements. 

The width of the reflections depends on both the size of crystallites and local lattice fluctuations (defects). The smaller the crystallites, the... 

Krenzer and Ruland applied a Fourier analysis to the meridional reflections of PpPTA [121]. They concluded that an interpretation of the line broadening by lattice defects in terms of a one-dimensional paracrystalline disorder along the c-axis is not justified, but that the disorder is the result of local defects such as chain ends incorporated in the crystalline lattice. 

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