Goldschmidt tolerance factor, perovskite

Goldschmidt tolerance factor perovskites Gokhshtein, Yakov Peysachovich... 

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. 

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... 

Before dealing with these structure t es in detail, the clear-cut dependence of the occurence of these distorted perovskites on the radius ratio of the ions in question should be mentioned. The tolerance factor defined by Goldschmidt [115),... 

This type of orthorhombic perovskite structure appears, if the tolerance factor of Goldschmidt is smaller than t — 0.88. The example of the compound NaMnFs [t = 0.78), showing doubled lattice constants a and h (287), is likely to mark the lower limit of the field in which orthorhombic fluoro-perovskits of the GdFe03-t3q>e may occur. Fluoroperovskites which have a smaller tolerance factor than t = 0.78 never have been observed so far, nor do fluoride structures of the ilmenite type seem to exist, which might be expected for ya = Me, corresponding to 1=1/1/2=0.71. 

Likewise, the TPO experiments showed that reoxidation of cobalt to form the perovskite structure is more favorable for larger lanthanides. Figure 5 shows a good correlation between the oxidation temperature obtained from the TPO profiles with the Goldschmidt s tolerance factor. Katsura et al. [18] studied the thermodynamics between 1473 and 1673 K of the oxidation of iron to the rare earth perovskites according to the reaction ... 

Figure 5. Goldschmidt s tolerance factor t versus (a) reduction and (b) oxidation temperatures obtained by TPR and TPO experiments for the perovskites LnCoO,.

Fig. 7.5 Behaviour of L11C0O3 perovskites (a) H2 ( ) and CO (O) yields from partial oxidation of methane as a function of the lanthanide ionic radii and (b)Goldschmidt s tolerance factor t and the oxidation temperature obtained from temperature-programmed oxidation experiments (after Ref. 108).

The lower limits for cationic radii are rA > 0.09 nm, and rB > 0.051 nm in the case of oxides. Goldschmidt (79), on the basis of geometric considerations, defined the tolerance limits of the size of ions through a tolerance factor t = (rA + rx)l 2(rB + rx), where rA,rB, and rx are the radii of the respective ions t would be equal to one for the ideal cubic structure (Fig. 2). In fact, the perovskite structure exists in oxides only between the limits 0.75 < t < 1.0 with t between 0.8 and 0.9 in most cases. For t > 1 the calcite and aragonite structures are prevalent, whereas for t < 0.75 the stable structure is ilmenite. Roth (20) has classified the limits of the existence of these competing structures according to the ionic radii values. 

The tolerance factor (t), introduced by Goldschmidt (1926) is used to define the range of ionic radii of the A, B and X elements allowed by the perovskite structure ... 

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