Perovskite tolerance factor

Several authors have pointed out that the enthalpy for the formation of perovskite-type oxides from the individual oxides exhibits a correlation with the perovskite tolerance factor, t, which is defined as ... 

Here, R is the radius of the ions, and the subscripts A, B and O represent the corresponding ions in ABO3. This factor serves as a structure parameter to describe the extent of distortion of the perovskite structure from the ideal cubic prototype due to mismatch between the A-O and B-O bond lengths. Figure 2.10 shows the relationship of AHj n. the heat of formation of ABO3 from oxide precursors, to the perovskite tolerance factor, t. As indicated in Fig. 2.10, the stability of the perovskite structure increases as the tolerance factor increases towards 1, because... 

Figure 2.10 The enthalpy of formation of various perovskites from oxide precursors plotted as a function of perovskite tolerance factor [134].

The tolerance factor t for perovskites AMX3 is a value that allows us to estimate the degree of distortion. Its calculation is performed using ionic radii, i.e. purely ionic bonding is assumed ... 

In this equation rA is the radius of the cage site cation, rB is the radius of the octahedrally coordinated cation, and rx is the radius of the anion. The factor l is called the tolerance factor. Ideally, t should be equal to 1.0, and it has been found empirically that if t lies in the approximate range 0.9-1.0, a cubic perovskite structure is stable. However, some care must be exercised when using this simple concept. It is necessary to use ionic radii appropriate to the coordination geometry of the ions. Thus, rA should be appropriate to 12 coordination, rB to octahedral coordination, and rx to linear coordination. Within this limitation the tolerance factor has good predictive power. 

The enthalpies of formation of selected perovskite-type oxides are given as a function of the tolerance factor in Figure 7.17. Perovskites where the A atom is a Group 2 element and B is a d or / element that readily takes a tetravalent state [19, 20] show a regular variation with the tolerance factor. Empirically, it is suggested that the cations that give t close to 1 have the most exothermic enthalpies of formation. When t is reduced, the crystal structure becomes distorted from cubic symmetry and this also appears to reduce the thermodynamic stability of the... 

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),... 

The two mentioned ternary fluorides of cadmium with their tolerance factors of 1.00 and 0.88 resp. mark quite accurately the field of existence of cubic perovskites. As may be seen from the following Table 25 the tolerance factors of all cubic fluoroperovskites of the transition metals hitherto known lie within the range of these limits. 

A group of 8 ternary fluorides containing the transition metal ions Cr2+ and Cu + crystallizes in a tetragonedly distorted perovskite lattice. This distortion is caused by the Jahn-Teller effect displayed by the configurations d% d (Cr +) and d d (Cu2+) resp., rather than by geometrical reasons. As for their space requirements the ions Cr + and Cu + are very close in size to Mn2+ and Co + resp. and as a consequence the corresponding compounds do not differ in their tolerance factors. 

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. 

There exists quite a number of hexagonal oxidic perovskites 183, 332), but there seem to be only three types in the case of ternary fluorides. Their occurrence again clearly depends on the tolerance factor wich thus proves to be useful in classifying the hexagonal perovskites also. After having described their structures in detail they will be further discussed under a common point of view. 

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... 

The cubic AMO3 perovskite structure consists of an MO3 array of comer-shared MO6/2 octahedra with a large A cation at the body-center position. As is illustrated in tig. 1, this structure allows formation of the Ruddlesden-Popper (1957,1958) rock-salt/perovskite intergrowth structures MO (AMO3 ) . In all these structures, the mismatch between the equilibrium (A-O) and (M-O) bond lengths is given by the deviation from unity of the geometric tolerance factor... 

In contrast to dielectric losses permittivity is not, in general, sensitive to small amounts of impurities and for homogeneous dielectrics values can be calculated as described in Section 2.7.1, and the various mixture rules allow good estimates to be made for multiphase dielectrics. For Ba- and Sr-based dielectrics having the perovskite structure the variation of permittivity with temperature, which determines rf (see Eq. (5.37)), can be correlated with the tolerance factor t (see Section 2.7.3) [13] providing guidance for tailoring ceramics to have xf = 0. 

Stability of the K2NiF4 structure in oxides. Just as for perovskite oxides, a tolerance factor t may be defined for A2B04 oxides as... 

A tolerance factor [9,10] can be used to determine the phase transition in AB03 perovskite oxides, as given by t — (rA + > o )/V2(J b + ro), where rA, rB, and rQ are the ionic radii [11] of the A, B, and O ions, respectively. This indicates that the spatial margin relates to the type of phase transition. However, the atomistic explanation has not been given for the factor in order to distinguish between ferroelectric and antiferrodistortive phase transitions in AB03 perovskite oxides. 

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] ... 

One important thing to be kept in mind here is that the ionic radii are derived from the cation-anion bond distances measured experimentally at room temperature, not at the temperatures where the formation reactions take place. At present, there is no way to estimate exactly the tolerance factor at high (or low) temperatures. So, whether one can obtain a perovskite or related material from a given A-B-X combination or not depends mostly upon experimental tests. 

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