Structures with lattice-induced strain

Lattice-induced strains clearly cause the bonds to violate the network equations and their presence may be indicated by a large value of the bond strain index (BSI) defined in eqn (12.1) (Preiser et al. 1999, 73 in table 1)  

A second and complimentary measure of lattice strain is the global instability index (GII) defined by Salinas-Sanchez et al. (1992) using eqn (12.2)  

An interesting example of the use of the GII is provided by compounds with the formula Ln2BaCu05 (72165, Ln = rare earth) studied by Salinas-Sanchez 

Ba and CuOj ions. Only for Ln = Tm do all the pieces have the right size to fit together without strain. Replacement of Tm + with other rare earth ions causes the structure to become strained as indicated by the GII shown in Fig. 12.3 where it is plotted against the rare earth ionic radius. For these compounds the GII is closely approximated by eqn (12.3)  

Typically lattice-induced strain results in the bonds around one cation being stretched and the bonds around another cation being compressed as found in BaRuOs (10253) by Santoro et al. (1999, 2000). When this happens, the valence sum rule will be violated around the cations in question but the valence still distributes itself as uniformly as possible among the bonds, so that the experimental bond valences determined from the bond lengths remain as close as possible to the theoretical bond valences. For this reason the BSI is typically smaller than the GII for lattice-induced strains, though the opposite is true for compounds with electronically induced strain where the valence sum rule remains well obeyed. 

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Reconstructive phase transitions occur when major changes are made in the topology, i.e. when the bond graph is reorganized. The transitions usually observed in structures with lattice-induced strain are displacive and often second order (no latent heat). Reconstructive transitions arise when two quite different structures with the same composition have similar free energies. Unlike the displacive transitions they involve the dissolution of one structure and the recrystallization of a quite different structure. These phase transitions possess a latent heat and often display hysteresis. 

It is assumed that for an imstrained structure, GII is typically smaller than 0.1 valence units, whereas for structures with lattice-induced strain 0.1 < GII < 0.2. Crystal structures with GII > 0.2 are found to be unstable (Rao et al., 1998 Lufaso and Woodward, 2001). Vasylechko and Matkovskii (2004) calculated GII values for RAIO3 compoimds taking into account different possible CN of R-cations. It was shown that in terms of the GII concept, all RAIO3 perovskites can be considered as unstrained structures. High GII values 0.21 and 0.14 valence units, observed in CeA103 for CN 8 and LaA103 for CN 9, indicate that the chosen coordination numbers are not suitable for those structures. 

There are, obviously, no compounds to illustrate lattice-induced strains with GII 3> 0.2 vu. Such structures are unstable and cannot exist, but if it is possible to model structures of any arbitrary composition using the methods described in Chapter 11, it is possible to determine which compositions give rise to stable structures and which ones do not. A systematic exploration of different compositions occurring between a group of elements would then lead to an understanding of the phase diagram. For example, on the basis of a few simple rules, Skowron and Brown (1994) were able to predict most of the structures in the Pb-Sb-S phase diagram and their relative stabilities (Section 11.2.2.2). 

Periodic structures containing alternating layers of different ferroelectric and non-ferroelectric materials - superlattices (SLs) - have received increased attention. High-quality ferroelectric SLs with nearly atomically sharp interfaces in various perovskite systems have been synthesized and investigated [14, 24—27, 49-56], and also studied theoretically [44, 57-64]. Properties of such structures are not just a simple combination of the properties known for constituent bulk materials, as they are affected by both mechanical (lattice-mismatch-induced strain) and electrostatic boundary conditions at multiple interfaces located within close proximity to each other. Strain engineering is one of the most appealing ways to... 

A periodic arrangement of many epitaxially grown thin layers with lattice mismatch constitutes a strained-layer superlattice. An example of such a superlattice structure can be found in the vertical-cavity surface-emitting laser (VCSEL). As discussed by Choquette (2002) and Nurmikko and Han (2002), the control of layer thickness, elastic strain due to LAN to us mismatch, stress-driven crack formation and processing induced defects in the superlattice presents major scientific and technological challenges in the development of these devices. 

We will confine ourselves to those applications concerned with chemical analysis, although the Raman microprobe also enables the stress and strain imposed in a sample to be examined. Externally applied stress-induced changes in intramolecular distances of the lattice structures are reflected in changes in the Raman spectrum, so that the technique may be used, for example, to study the local stresses and strains in polymer fibre and ceramic fibre composite materials. 

As early as 1829, the observation of grain boundaries was reported. But it was more than one hundred years later that the structure of dislocations in crystals was understood. Early ideas on strain-figures that move in elastic bodies date back to the turn of this century. Although the mathematical theory of dislocations in an elastic continuum was summarized by [V. Volterra (1907)], it did not really influence the theory of crystal plasticity. X-ray intensity measurements [C.G. Darwin (1914)] with single crystals indicated their mosaic structure (j.e., subgrain boundaries) formed by dislocation arrays. Prandtl, Masing, and Polanyi, and in particular [U. Dehlinger (1929)] came close to the modern concept of line imperfections, which can move in a crystal lattice and induce plastic deformation. 

An intermediate case (Bazin et al., 2002) between bulk and surface diffraction is reached for nanoparticles when the contribution from surface atoms becomes significant and diffraction analysis in the limit of infinite periodic lattice models inadequately describes the diffraction data. A case study with diamond nanoparticles (Palosz et al., 2002) describes elegantly the possibilities and limitations of diffraction analysis of such samples there is a focus on the nonperiodic structure such as strain and disorder induced by the dominant presence of a nonideal surface termination. 

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