The Global Instability Index

11 Structure Analysis and Prediction 11.1 The Global Instability Index 

There are a number of tools that are useful in validating either a measured or a proposed structure. The most widely used is the valence sum rule (2) using bond valences calculated from the observed bond lengths. Experimental uncertainties will mean that the atomic valence and bond valence sum are rarely exactly the same, but for a well-determined structure the difference is usually around 0.05 vu. Larger differences are often found, indicating that some bonds are compressed or stretched by the steric constraints imposed on the structure (Sect. 8). In some cases the differences can be quite large, but for a strained structure to be in equilibrium both stretched and compressed bonds must be present, which is key to verifying the presence of steric strains. 

A more interesting index, but one more difficult to apply, is the bond strain index, B, which compares the observed and predicted bond valences by summing the squares of the deviation over all m bonds (24)  

This calculation requires ideal bond valences to be predicted using the network equations (14a) and (14b). B will not be zero if either electronic anisotropies (Sect. 7) or steric strains (Sect. 8) are present. It measures the deviation from the predictions of the network equations, but does not indicate the origin of these deviations, nor does it measure instability. B will always be large if hydrogen bonds are present, masking the possible presence of other effects. 

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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) ... 

Fig. 13.3. The phase diagram of Ao.33A o.67Mn03 (A = divalent cation, A = rare earth) as a function of temperature and the global instability index of the idealized perovskite structure. The points show the observed transition temperatures in various compounds. FMM = ferromagnetic metal, PMI = paramagnetic insulator, FMI = ferromagnetic insulator (from Rao et al. 1998).

Figure 3 The variation in the Global Instability Index of Ln2BaCu05 as a function of the ionic radius of Ln. The line is a fitted second order equation. Nd adopts a different structure. (Figure 12.3 from The Chemical Bond in Inorganic Chemistry by Brown, David (2001)). (Ref. 3. Reproduced by permission of Oxford University Press)...

For characterization of the overall structure stability of perovskites, the Global Instability Index (GII), has been proposed (Salinas-Sanchez et al., 1992 Lufaso and Woodward, 2001). GII is defined as ... 

In the aristotype ABX3 perovskite stmcture, with Pm3m space group symmetry, there is a single free parameter, the cubic lattice parameter, a. A simple method to model the crystal stmcture is to determine the stmcture parameter(s), a in this case, that minimize(s) the Global Instability Index, G (Eq. (21) in [8]). 

Fig. 10 Influence of b averaging on the global instability index G fm 128 compounds that contain both a chalcogenide and a halide anion. The solid line marks the ratio 1 1, the broken line a fourth order polynomial over all data...

To aid in analysis of these surface stmctures, a new metric was defined surface instability index (SII) [15]. The SII is calculated similar to the global instability index (G, Eq. 2.21 of Brown [16]), except that only the atoms in the surface structure and the first bulk layer are included. Enterkin and coworkers demonstrated... 

VaList requires an input list of bond lengths in CIF, GSAS, ICSD or FULPROF formats. It calculates bond valences and their sums around individual atoms as well as the global instability index and occupation numbers where a site is occupied by two different atoms. 

Fig. 2 Bond valence sums of the ions and global instability index versus lattice parameter for the cubic perovskite SrTiOs...

The bond valence parameters, calculations of bond valences, and global instability index are more fully described in [8, sections 3, 8 and 11]. 

The observed and predicted global instability index versus tolerance factor for untilted cubic ABX3 perovskites is shown in Fig. 4. As expected the G is smallest for compounds with a tolerance factor near 1. For unstrained structures the normal maximum value of G is 0.2 v.u., but here we see a number of compounds where G > 0.2. This can be explained in part by realizing that when t is significantly different than 1 the compounds can be classified as strained. The largest G observed (G = 0.47) is for KTa03 t = 1.08) which has shown on multiple occasions to be a cubic perovskite [13, 30, 31]. These studies used samples prepared with several synthetic techniques and both polycrystalline powders and single crystals have been examined, which support the accuracy of the crystal structure. 

F. 14 Contour plot of the SPuDS calculated global instability index with the lattice parameter a and fractional coordinate x for the La2Su207 pyrochlore. The white diamond with green outline represents the SPuDS predicted lattice parameter and Ifactiraial coradinate, respectively. The yellow square with red outline represents the observed lattice parameter and fractiraial coordinate, respectively [71], Contour lines are at 0.29, 0.5, 0.75, and 1 valence units... 

The predicted structures obtained by optimizing the bond valences of the ions have BVSs close to their formal valences. The degrees of freedom in the spinel structure enable the simultaneous optimization of the BVS for each of the A-, B-, and X-site ions. Within calculation limitations, a zero global instability index (G) is fotmd for each of the predicted structures. The experimentally determined structures have a larger G, ranging from about 0.04 to 0.27 v.u. with an average of 0.15 v.u. for the examined structures. Unlike the cubic perovskite and pyrochlore structure, the spinel is not a strained structure. Regardless of the sizes of the A and B... 

Adams S, Moretzki O, Canadell E (2004) Global instability index optimizations for the localization of mobile protons. Solid State Ionics 168 281-290... 

Fig. 17 Left-hand side variation of global instability index G of 48 ab initio structure models of Li ion conducting oxides versus the relative difference between the volume of the ab initio structure models and the experimentally determined unit cell volume for the same phase. Right-hand side linear correlation between the (unsealed) activatirai energy barriers derived by the B VSE method for experimental and for ab initio calculated structure models. In both graphs the six cases for which the ab initio model overestimates the unit cell volume by more than 8% are marked as open symbols. These less reliable data points are excluded liom the calculation of the r.h.s. linear correlation...

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