Goldschmidt factor

The majority of new materials for SOFCs are perovskite stmctured oxides of general form ABO3.5 [23]. The ideal perovskite structure is a cubic close-packed ABO3 structure where the B-site cation sits within the octahedral interstices. Fig. 3.5. This stmcture is very flexible toward cation composition and tolerates large substitution fractions on either cation site. The Goldschmidt factor, a ratio of A, B, and O ionic radii, is often utilized to predict if a metal oxide will crystallize into the perovskite structure [24]. The A site of the commonly utilized perovskites is typically occupied by La, Ca, Sr, or Ba. The B site is typically a transition metal. Other stmctures investigated include double perovskites, apatites, and fluorites. 

This theoretical result is completely substantiated by experiment. Goldschmidt,31 from a study of crystal structure data, observed that the radius ratio is large for fluorite type crystals, and small for those of the rutile type, and concluded as an empirical rule that this ratio is the determining factor in the choice between these structures. Using Wasastjerna s radii he decided on 0.67 as the transition ratio. He also stated that this can be explained as due to anion contact for a radius ratio smaller than about 0.74. With our radii we are able to show an even more satisfactory verification of the theoretical limit. In Table XVII are given values of the radius ratio for a large number of compounds. It is seen that the max-... 

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. 

Goldschmidt.48 emphasized the necessity of making corrections of this type for the effect of ligancy, and suggested the factor 1.03 for changing from ligancy 6 to 8, and 0.93 to 0.95 for changing from 6 to 4. 

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... 

The factor t varies from 0.8 to 1. For an ideal cubic structure t should be > 0.89 (based on the Goldschmidt radii). 

Since the vek computation was only rough, as Fersman himself admitted, the hardness values calculated from the formulae (3.3) and (3.4) were not sufficiently accurate and failed to consider all crystallochemical factors. Next, Sobolev in 1946 established a relationship between hardness and coordination number, attempting to extend the applications of Goldschmidt s formula to complex minerals, including silicate minerals. It was discovered by that time that crystal hardness with both ionic and covalent bonds, depends on ... 

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... 

In the secondary distribution of the elements in the Upper Mantle and Crust, Goldschmidt believed that during the formation of a crystalline phase the principal factor controlling the behaviour of an element is the size of its ions. From tables of ionic radii one could predict the crystal structures in which a given ion was most likely to occur. This concept has immense qualitative value in crystal chemistry, and has been remarkably successful in view of the assumptions... 

To measure the departure from the ideal condition, Goldschmidt introduced a tolerance factor, t, defined by the equation ... 

Goldschmidt tolerance factor perovskites Gokhshtein, Yakov Peysachovich... 

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] ... 

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