The Goldschmidt Tolerance Factor

From a crystallographic perspective, the ideal perovskite stmeture is inflexible, as the unit cell has no adjustable atomic position parameters, so that any compositional change must be accommodated by a change in lattice parameter. This is a 

The width of the cuboctahedral cage site, V2a, is equal to twice the A—X bond length  

This means that the ideal structure forms when the ratio of the bond lengths is 

This relationship was first exploited by Goldschmidt, in 1926, who suggested that it could be used to predict the likelihood that a pair of ions would form a perovskite stmcture phase. When this was initially proposed, very few crystal structures had been determined and so ionic radii were used as a substitute for measured bond lengths. For this purpose, it is assumed that for a stable structure to form the cations, just touch the surrounding anions (Goldschmidt s mle), then  

Note that it is necessary to use ionic radii appropriate to the coordination geometry of the ions. Thus r should be appropriate to 12 coordination, to octahedral coordination and r to linear coordination. Furthermore, it is best to use radii scales that mirror the X anion present, as radii appropriate to oxides, although a reasonable approximation for fluorides, are poor when apphed to chlorides and sulphides. 

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ABOs compounds containing lanthanum are closer to the ideal perovskite than those containing smaller rare earth ions. Compounds of the smaller rare earth ions appear to have a distorted perovskite structure of lower symmetry. When, however, the relationship between the radii is very far from being ideal (eq. 31), strong distortion may result giving an entirely different structure. The Goldschmidt tolerance factor, t, for the perovskite structure is related to the ionic radii by... 

A useful parameter by which the stability of perovskites can be judged is the Goldschmidt tolerance factor t (25). In a study of the thermochemistry of the lanthanide (IV) perovskites... 

For the ideal cubic lattice of ABO3 perovskite, A is the larger cation surrounded by eight BC>6 octahedra. In order to estimate the general possibility of perovskite lattice formation from ionic radii, the Goldschmidt tolerance factor (ts) can be used [i] ... 

As observed by TPR, the nature of the Ln affects the reducibility of Co in the perovskites LnCoO,. The Goldschmidt tolerance factor t = (r, + r )/[V 2 (r( +r(,)j obtained for the structures of LaCoO, PrCoOi, NdCoO, SmCoO,and GdCoO,were 0.899, 0.885, 0.878, 0.867 and 0.857, respectively. These tolerance factors indicate that considering solely geometric factors lanthanum, the largest ion in the series, forms the most stable perovskite structure. This trend is reflected in the TPR results where the perovskite LaCoO, the most stable structure, is reduced at the higher temperatures, 844 K (Fig. 5). 

The data obtained allowed us to verily that the Goldschmidt tolerance factor t, used to predict the oxide-based perovskites stability [11] could also be utilized to correlate the stability of fluoiinated ternary compounds. The t value is a lunction of the anionic and cationic radii, the closer the t factor is to unity, the greater is the perovskite stability. 

Similar reaction sequences can probably be established for compositions with other cation combinations as well. However, it is not clear whether a generalized reaction sequence can be established for complex Pb compositions. There have been other attempts to generalize the tendency to form the py-rochlore phase in complex Pb compositions through correlations in ionic size, ionic bonding, and so on [31,32]. Two parameters were used the Goldschmidt tolerance factor [33], defined as... 

Cf compounds have been acquired recently (Fuger et al. 1993). A linear relationship exists between the heat of formation of these perovskite-type oxide systems and the Goldschmidt tolerance factor, which permits estimations for the enthalpies of formation of a number of homologous compounds of this type. From data of such Cf compounds, the enthalpy of formation of CfOj was estimated to be - 854-1- lOkJ mol in good agreement with a value of — 858 kJ mol derived from interrelationships between the molar volumes of dioxides and their standard enthalpies of solution (Morss 1986). It has not been feasible to carry out such direct experimental measurements with CfOj and these indirect approaches have provided the only data of this type. Data for other actinide oxides can similarly be obtained by these indirect approaches. 

The structure stability of mixed conducting perovskite materials is usually defined in terms of the Goldschmidt tolerance factor t ... 

The perovskite stracture is illustrated in Figure 6.10(b). The ideal stoichiometry of a perovskite is ABX3, where X is an anion, B is a cation with octahedral coordination, and A is a large cation with cuboctahedral coordinatioa The two cation sites are cormnordy referred to as the B-site (coordination number = 6) and the A-site (coordination number = 12). In order for the ideal cubic stracture to be realized the size of the A- and B-site cations mnst be well matched. The size match or lack thereof between the two cations is given by the Goldschmidt tolerance factor, t ... 

The structural stability of perovskite can be evaluated by the Goldschmidt tolerance factor... 

It should be noted that the new compounds prepared by Berndt et al. (1975) only appear to be stable at lower temperatures at higher temperatures they decompose into one or more solid-solution phases, and it is therefore interesting to speculate on the possibility of the stable existence of other such compounds at even lower temperatures. The lower limit of 0.77-0.79 for the Goldschmidt tolerance factor is, of course, derived empirically from experimental observation... 

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