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) ... 

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. 

If possible, the lattice-induced strain will relax in such a way as to minimize both strain indices as illustrated in the following sections by the structure of... 

The relaxation of La2Ni04 to La2Ni04,i8 illustrates a couple of important points. Firstly, the defect and electronic modes of relaxation necessarily work together since the change in oxidation state of NP+ is directly related to the amount of interstitial present. This simultaneous relaxation of both the stretched and the compressed layers is a feature found in many, if not all, of the observed mechanisms for relaxing lattice-induced strain. Secondly, the lattice-induced strain is directly responsible for the crystallization of a stable compound with a fixed, but irrational, composition, involving a fixed, but nonintegral, oxidation state for nickel. 

The second difference between the nickel and copper compounds is that La2Cu04 is a superconductor, being the first of the Cu02 layer compounds in which superconductivity was observed. The lattice-induced strain is a necessary condition for superconductivity since it stabilizes the higher oxidation state needed to provide the superconducting carriers as discussed in Section 13.3.2. 

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. 

The examples discussed in this chapter show that there are many different ways in which lattice-induced strain can be relaxed or aeeommodated, the partieular mode depending on the properties of the elements and the struetures involved. Many of these compounds have unusual properties resulting from non-integral stoichiometry, the presence of non-integral oxidation states, or the spontaneous breaking of symmetry, all of which are the direct consequence of lattice-induced strain. 

Lattice-induced strains are characterized by large values of the GII because the environments around some atoms are stretched and around other atoms are compressed but, since the valence is still distributed as uniformly as possible among the bonds, the BSI remains small. This contrast with the electronically driven distortions discussed in Chapter 8 where the GII is small (the valence sum rule is obeyed) but the BSI is necessarily large. 

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). 

In Chapter 12 the layered perovskite, La2Ni04 (65917), was used as an example of a structure which displays lattice-induced strain. This compound is typical of the large class of perovskite-related structures. All show some degree of lattice-induced strain and, because the mechanism of relaxation depends on the details... 

Whatever composition and structure is chosen, a perfect match between the lattice spacings of the different layers is virtually impossible to achieve. Thus all perovskites show some evidence of lattice-induced strain. Section 13.3 describes some of the ways in which these strains can be relaxed, and the more effective the method of relaxation, the greater the amount of mismatch that can be accommodated. Since many relaxation mechanisms have already... 

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. 

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