Systematics of the Spinel Structure Type

Phys. Chem. Minerals 4, 317 339 (1979) PHYSICS I] CHEMISTRY MIHERALS © by Springer-Verlag 1979

Systematics of the Spinel Structure Type

Roderick J. Hill 1, James R. Craig 2, and G.V. Gibbs 2

i CSIRO Division of Mineral Chemistry, P.O. Box 124, Port Melbourne, Victoria 3207, Australia 2 Department of Geological Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.

Abstract. Systematic trends in the geometry of 149 oxide and 80 sulfide binary and ternary spinels have been examined from the standpoint of ionic radius and electronegativity. The mean ionic radii of the octahedral and tetrahedral cations, taken together, account for 96.9 and 90.5% of the variation in the unit cell parameter, a, of the oxides and sulfides, respectively, with the octahedral cation exerting by far the dominant influence in sulfides. The mean electronegativity of the octahedral cation exerts an additional, but small, influence on the cell edge of the sulfides. The equation a = (8/3]f-d)dtet+(8/3)doct, where dtet and doct are the tetrahedral and octahedra] bond lengths obained from the sum of the ionic radii, accounts for 96.7 and 83.2% of the variation in a in the oxides and sulfides, respectively, again testifying to the applicability of the hard-sphere ionic model in the case of the spinel structure. Comparison of observed and calculated u values for 94 spinels indicates that up to 40% of the experimentally measured anion coordinates may be significantly in error. In addition to these compounds, u values are given for 52 spinels for which no data have previously been determined. Diagrams are presented for the rapid interpretation of the internal consistency of published data and the prediction of the structural parameters of hypothetical or partially studied spinels.

Introduction

The ideal spinel structure consists of a cubic close-packed array of anions in which one-eight of the tetrahedral and one-half of the octahedral interstices are occupied by cations. The arrangement of atoms is such that, perpendicular to each three-fold axis, layers occupied only by cations in octahedral coordination alternate with others in which the tetrahedral and octahedral sites are filled in the ratio two to one (Bragg, 1915; Nishikawa, 1915). In a binary spinel ABzX4, where X represents an anion and A and B are cations, two extreme distributions of the cations among the available sites are possible (Barth

0342-1791/79/0004/0317/$04.60

318 R.J. Hill et al.

and Posnjak, 1932): the 'normal' distribution A[B2]X¢, and the'inverse'distribution B[AB]X4, where the ions indicated by parentheses occupy octahedral sites. Intermediate cation distributions may be represented as (AI-xBx)- [AxBi-x]X4, where x is the so-called degree of inversion, equal to zero and unity for the normal and inverse arrangements, respectively.

When the cations are distributed randomly over each type of site, the unit cell contains eight formula units and is usually referred to space group Fd3m, although the symmetry can be lowered by Jahn-Teller and other effects (Goode- hough and Loeb, 1955; Dunitz and Orgel, 1957; Grimes and Collett, 1971). With long-range cation ordering over the tetrahedral and/or octahedral sites (particularly in the case of ternary and higher spinels) the symmetry may again be lowered and is often accompanied by the formation of complex superstruc- tures (Laves et al., 1963 ; White and Keramidas, 1972). In fact, the rationalization and prediction (and, indeed, the determination) of cation ordering schemes both within and between the tetrahedral and octahedral sublattices in spinels have been among the more interesting and persistent problems in crystal chemistry (for example, Goodenough and Loeb, 1955; Gorter, 1954; Dunitz and Orgel, 1957; Navrotsky and Kleppa, 1967). However, in spite of a huge number of structural studies spanning many years (Hafner, 1960; Wyckoff, 1965 ; Blasse, 1964; and references therein) the number of precise spinel refinements appearing in the literature is surprisingly small, especially considering their remarkable record of usefulness (Grimes, 1975) and the facility with which one may manipu- late the properties of a spinel material to meet the demands of a particular application.

Although several studies in the past have been devoted to the variation of cell parameters with cation radii and covalent forces in small groups of spinels, often with related compositions (for example, Mikheev, 1955; Vermaas and Schmidt, 1959; Raccah et al., 1966; Kugimiya and Steinfink, 1968; Reuter et al., 1969), the availability of a new set of widely applicable effective ionic radii (Shannon, 1976), has highlighted the need for a reappraisal of the structural systematics of this important group of compounds in a broader sense. The present study was, therefore, initiated with three major objectives in mind: (i) to examine the presence of systematic trends between spinel structural and electronic parameters; (ii) using the results of (i), to assess the accuracy and internal consistency of data already published for more than 200 members of the group; and (iii) to predict the details of partially studied and unstudied spinels.

Geometry of the Spinel Structure

When referred to space group Fd3m (as the vast majority of spinals are) the tetrahedral cations, A, and octahedral cations, B, occupy fixed positions on equipoints 8 a and 16 d respectively. The separations between cations are, therefore, dependent only on the unit cell dimension, a, with the distances of closest approach being as given in Table 1. It should be noted that the B... B distance spans a shared edge between octahedra and is therefore much shorter than

Systematics of the Spinel Structure Type 319

Table 1. Interatomic distances as a function of the unit cell edge a, and the anion positional parameter u, * in a spinel A[B]zX,~ where A and B are the tetrahedral and octahedral cations respectively, and X is an anion

Atom pair Distance Comments

A ... A al~j~4 tet-tet cation separation A ... B a l / l l /8 tet-oct cation separation B . . . B a]/2/4 oct-oct cation separation

A - X a l / 3 ( u - 0.125 ) tet bond B - X a (3u2 - 2u+0.375) 1/2 oct bond

35... 32 a l /2 (2u-0 .25) tet edge X . . . X a]/2(0.75 2u) shared oct edge X ... X a(4uZ-2u+0.375) 1/2 unshared oct edge

*u value corresponding to unit cell origin at the centre of symmetry

the distance A. . .A between the insular tetrahedra, a situation which has often been exploited in attempts to deduce cation distributions and structural stability from electrostatic considerations (Verwey et al., 1948; Gorter, 1954; Blasse, 1964; Kugimiya and Steinfink, 1968; Thompson and Grimes, 1977).

The anions in the spinel structure are located on equipoint 32e with their detailed positions determined by one parameter u. Variations in u reflect the adjustment of the structure to accommodate differences in the relative effective radii of the cations in the tetrahedral and octahedral sites. For u=0.25 (or 0.375 if the alternative unit cell origin of 43m at -0.125, -0.125, -0.125 from the center of symmetry is used) the anions are arranged in ideal cubic closest-packing with the octahedral bond 1.155 times as long as the tetrahedral bond (the formulae in Table 1 illustrate the dependence of bond length on a and u). As expected, thi s particular situation is rarely realized and, in fact, of the 135 oxide and sulfide spinels considered in this study for which the anion coordinate has been determined, the u value for 129 of them lies in the range 0.25 to 0.27.

As u increases above 0.25 the anions move away from the nearest tetrahedral cation in a [111] direction, thereby increasing the size of the tetrahedron relative to the undistorted close-packed anion arrangement but without changing its overall 43m symmetry. At the same time the octahedra become smaller and assume 3-m (rather than rn3m) symmetry, and the length of the six edges shared with adjacent octahedra become shorter relative to the unshared edges (which remain essentially constant in length). If the ratio of the octahedral to tetrahedral bond lengths is R then it is possible to express the inverse relationship between u and R by the following algorithm:

R2/4-2/3 +(11 R2/48 - 1/18) 1/2 u - 2R 2 - 2 (1)

The octahedral and tetrahedral bond lengths are, therefore, identical at a u value of 0.2625. Note that since the bond lengths in a sulfide spinel are, in


general, longer than those in its oxide analogue (see below), the value of u for the pair of compounds will be different to the extent that R is different from unity.

For u less than 0.25, not only is the octahedron bond length more than 15% larger than the tetrahedron, but the shared edges between octahedra are actually longer than the unshared ones (Table 1). Noting this fact, and the preponderance of spinel structures with u>0.25, Kamb (1968) has suggested that in accordance with Pauling's (1960) Third Rule, the ability of the shared edges to shorten may be an important factor in the stabilization, at low pressure, of this structure type relative to the olivine polymorph. An upper limit for u can also be established by making use of the fact that shared edges between oxide octahedra will not, in general, fall below 2.4 ~ because of rapidly increasing nonbonded repulsion forces. Assuming a mean cell edge of 8.4 ~ for oxide spinels (see below) and using the appropriate expression from Table 1, the upper limit for u is 0.274, in relatively good agreement with the observed upper limit of 0.269 (the 2.4 ~ limit on anion-anion distance also applies to the tetrahe-

Octohedral cation rodius(~)for sulfides (rS =-= I.B2 ~)

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Fig. 1. Two-dimensional surface relating octahedral and tetrahedral cation radius (for both oxides and sulfides) to unit cell edge a, anion positional parameter u, and the ratio (dashed lines) of the octahedral to tetrahedral bond length, for all reasonable values of these properties in the spinel structure type


dron edge length for u values below 0.226). In fact, the condition that 0.25 < u < 0.274 for formation of the spinel structure places an upper and lower limit on the value of the octahedral to tetrahedral bond length ratio, namely that 1.155>R>0.886. In order to postulate the existence of a hypothetical spinel it is, therefore, necessary to know only the cation distribution (which can often be assessed from relative octahedral site preferences) and the ionic radii of the anion and cations (which are assumed to be spherical, rigid and in contact). Knowing R and the ionic radii it is then possible to completely characterize the spinel by estimating the u value from (1), and the cell dimension from the expressions for bond length in Table 1. Alternatively, a two-dimensional surface relating tetrahedral to octahedral cation radius (or bond length if the appropriate anion radius is included) can be constructed with contours in unit cell edge and anion positional parameter for all reasonable values of these properties in oxide and sulfide spinels (Fig. 1). Moreover, (dashed) lines corresponding to various values of the octahedral to tetrahedral bond length ratio can also be included. A diagram of this kind allows rapid interpretation of the internal consistency of published spinel structural data and also permits the prediction of parameters for hypothetical spinels of a given composition without the need to perform calculations.

As indicated by expression (1), variations in the ratio between octahedral and tetrahedral bond lengths are absorbed by changes in the u parameter. However, changes in the average bond lengths (radii of the cations) are accom- modated by expansion or contraction of the anion framework as a whole (i.e., the unit cell edge). If the lattice parameter is taken to be a weighted average of the projections of the octahedral and tetrahedral bonds on the unit cell edge (assuming u=0.25), then by taking due account of the relative numbers of bonds contributing to the repeat distance, the cell edge can be approximated by the following algorithm:

8(tet. bond) 8(oct. bond) acalc -- 3 ~ - ~ 3 (2)

Similar expressions have been devised by Kugimiya and Steinfink (1968) and Suchow and Ando (1970), although in these studies an extra term was included to give additional weight to the radius of the anion and/or less weight to the tetrahedral bond length: in both cases the predictive powers are poorer than those of expression (2) (see next Section). Note that the value of a calculated from (2) will be slightly different from that obtained via (1) to the extent that u differs from 0.25.

Analysis of Observed Spinel Structures

Table 2 is a list of 149 oxide and 80 sulfide binary and ternary spinels with cubic symmetry (i.e., the minimum requirements for inclusion are a chemical composition and a cell edge or u value). Each group is arranged alphabetically accordingly to the chemical symbol of the A-type cation(s) and then to the

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symbol of the B-type cation(s) (see columns at right). The crystallographic data of Blasse (1964) and Wyckoff (1965) form the basis of the compilation, but wherever possible the entries have been supplemented and amended by information from more recent and/or more precise studies.

In Fig. 2 the lattice parameter has been plotted against the anion u parameter for the 68 oxide and 65 sulfide spinels for which both values are available and the surface contoured (using the systematics outlined in Table 1) in terms of the limiting values of tetrahedral and octahedral bond length, d(A-X) and d(B-X), respectively. Since the sulfide ion is considerably larger the oxide ion (Shannon, 1976), there is a clear division between the oxides and sulfides in terms of cell dimension (at about 9.2 A) and, to a lesser extent, bond length (at about d(A-X)=2.15 A and d(B-J0=2.2 A). Also, since the radii of the cations are approximately the same for both groups, the range of octahedral to tetrahedral bond length ratios (and therefore u values) is observed to be roughly the same.

One notable difference between the oxides and sulfides, however, is the range of cell dimensions exhibited by each group. Whereas the oxides tend to cluster about their mean value (for all 149 species) of 8.39 A with a variance of 0.05, the 80 sulfides are spread much more loosely (variance=0.21) about their mean of 10.23 A. Moreover, while the oxides all have cell dimensions slightly expanded from the 7.81 A expected for cubic closest packing of 0 2. ions of radius 1.38 ~ (Shannon, 1976), nearly 60% of the sulfides have a lattice parameter smaller than 10.30 A, the value corresponding to closest packing of S 2- ions of radius 1.82 A. Indeed, in order to represent the minimum observed cell edge of 9.4 A in terms of a cubic closest packing of spheres, their radius has to be reduced to 1.66 A, approximately halfway between the S 2- radius

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SystematJcs of the Spinel Structure Type 331

of 1.82/~ [-representing Pauling's (1960) crystal radius slightly corrected for a decrease in coordination from six to four] and the so-called one-angle (or nonbonded) sulfur radius of 1.45/~ (Glidewell, 1975). Furthermore, if the 13 pairs of oxide and sulfide spinels having the same cation composition and ordering pattern listed in Table 2 are compared, it is found that a sulfide spinel, on the average, has a lattice parameter which is 1.57 A larger than its oxide analogue. Assuming closest packing of spheres this value is equivalent to a sphere size difference between the oxides and sulfides of 0.28 A, and, hence, to a sulfur radius of 1.66 A once again. Of course, as pointed out by Pauling (1960), the electron distribution function for an ion extends indefinitely, and, therefore, no single characteristic size can be assigned to it. Nevertheless, the agreement between the radii obtained from the minimum cell edge and from the difference between analogous oxide and sulfide spinel cell edges, indicates that the mean effective radius of sulfur is 1.66 ~ in this structure type. (Note that the cation radii of Shannon, 1976 must then be appropriately adjusted in order to reproduce the observed bond lengths.)

In the past it has been suggested that the smaller sulfide spinel cell dimensions are a function of the presence of appreciable covalency in the metal-sulfur bonds (Lotgering, 1956; Bouchard et al., 1965; Vaughan et al., 1971). To test this hypothesis a regression analysis of cell dimension, a, vs Xte t and Xoct, the mean electronegativities of the tetrahedral and octahedral cations (using the scale of Allred, 1961), was completed for all those oxides and sulfides for which a cell dimension and an inversion parameter has been determined. The results are summarized in Table 3, and indicate that while the regression may not be meaningful for the oxides (O), the sulfides (S) display a significant decrease in cell dimension with increasing electronegativity of the cations. In fact, however, a total of only 26.8% of the variation in thiospinel lattice parameter can be associated with a change in the covalency of the bonds, with the octahedral bonds making by far the major contribution (24.8%).

For those spinels in Table 2 for which a unique assignment of the valence state (and, where appropriate, the spin state) could be made, mean ionic radii were obtained for the tetrahedral and octahedral cations (using the data of Shannon, 1976), and the resultant values included in a regression analysis against cell edge. The least squares planes obtained (Table 3) for the oxides and sulfides account for 96.9 and 90.5%, respectively, of the variation in cell parameter. However, whereas for the oxides the tetrahedral and octahedral cation radii individually account for only 21 and 44%, respectively, of the variation in unit cell dimension, for the sulfides their corresponding contributions are 18 and 87%. It is, therefore, clear that, once again, the octahedral cation plays a far more dominant role in the structural systematics of the sulfide spinels than it does in the oxides. In fact, the inclusion of the tetrahedral cation radius with that of the octahedral cation in the regression with cell edge only reduces the residual sum of squares by a little over 3% (Table 3). Similarly, the inclusion of cation electronegativity with radius in the lattice parameter regression does not reduce the residual sum of squares to any extent, either for the sulfides or oxides, despite its importance when considered alone. Finally, although Mik- heev (1955) has presented a related equation for the determination of the oxide

332 R.J. HiE et a]

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spinel edge, the results are of limited use since he considered only a small number of compounds and did not have access to an accurate set of ionic radii.

Whereas empirical relationships of this kind, once established, can be used to predict cell dimensions for hypothetical and/or as yet unstudied spinels, they are founded on nothing more than the fact that the spinel structure contains two types of cation sites, and as a result must be continually updated as new data become available. Equation (2), on the other hand, is firmly based on a knowledge of the spinel structure and should be able to predict the cell dimension free of parameterization. Therefore, as a test of Eq. (2) and of its performance relative to the corresponding algorithms of Kugimiya and Stein- link (1968) and Suchow and Ando (1970) (hereafter KS and SA respectively), cell dimensions were calculated using all three relationships for the same set of spinels (radii of 0 2 and S 2 set equal to 1.38 and 1.82 A, respectively), and the resultant values regressed against the observed lattice parameters (Ta- ble 3). While expressions (2) and KS account for identical amounts of the residual sum of squares (as a result of the assignment of equal relative weights to the octahedral and tetrahedral bond lengths), unlike (2), the KS lines have slopes and intercepts which are significantly different from unity and zero respectively, and, therefore, do not predict the cell edges directly. The SA sulfide line suffers from the same disadvantage, although it is (fortuitously) able to account for a higher percentage of the cell edge variation as a result of the assignment of a larger weight to the octahedral bond length. This advantage does not apply in the case of the oxides (where the tetrahedral and octahedral bonds have a more equal influence on the lattice parameter), and as a result the SA oxide line accounts for less than 80% of the cell variation. It is, therefore, clear that expression (2) is superior in its ability to predict the lattice parameter of oxide and sulfide spinels alike (Fig. 3). Note, however, that the calculated cell edges for the sulfides are about 0.15 A larger than the corresponding observed values.

As pointed out in the previous Section, the cell edge may also be calculated from the rat io of the octahedral to tetrahedral bond lengths by means of Eq. (1) together with a rearrangement of one or other of the bond length expressions in Table 1 (as coarsely graphed in Fig. 1). Although this method has the apparent advantage that it does not need to assume ideal cubic closest packing of the anion lattice, a regression analysis of the cell parameter calculated via equation (1), vs the observed dimension, does not produce a significant reduction of the residual sum of squares (Table 3) over that given by Eq. (2). However, the fact that the intermediate step involves an estimation of the u parameter for the particular spinel under consideration does represent a significant advantage. At first sight the accuracy of the u parameter calculated from (1) does not appear to be high (Table 3 and Fig. 4). However, when a regression analysis is carried out only on the 20 oxide spinels (marked with an asterisk in Table 2 and as filled circles in Fig. 4) for which a u parameter has been determined with a precision of 0.001 or better, Eq. (1) is seen to account for no less than 98 percent of the variation in the observed values (Table 3). A similar analysis for the sulfides was not possible due to a surprising paucity of precise


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de te rmina t ions , but in the l ight o f the oxide results it can be deduced tha t i f the ca lcu la ted u value for a spinel differs f rom the observed value by more than 0.0018 (i.e., two s t anda rd er rors o f the es t imate) , then there is at least a 95% chance tha t the observed u value a n d / o r the ca t ion d i s t r ibu t ion is incor- rect. Since the cell d imens ion ca lcu la ted via express ion (1) is able to account for 97 and 85% of the observed lat t ice p a r a m e t e r va r i a t ion in the oxides and sulfides respect ively (Table 3), it can be conc luded tha t it is more of ten than not the u value which is in er ror , ra ther than the invers ion pa ramete r . In fact, based on the above cr i ter ion, up to 40% of the observed an ion coord ina te s l isted for the oxides in Table 2, a long with a somewha t smal ler f rac t ion for the sulfides, can be cons idered to have been p o o r l y charac ter ized . Therefore , in add i t i on to the 94 oxide and sulfide spinels for which a u value has been


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determined both experimentally and by calculation, Table 2 presents u parameters for a further 52 c o m p o u n d s for which no measurements have as yet been made.

Whereas a few isolated studies in the past have addressed themselves at least in part to the prediction o f u parameters for individual spinels (studies of the impor tan t mantle phase SiMg20~ by Kamb, 1968 and Yagi et al., 1974, are two examples), in all cases the est imation has come f rom a compar i son of the measured cell edge with observed trends of a v s u in a series o f c o m p o u n d s with related composit ions. The success of our method rests, o f course, on the systematics contained within Table 1 and the fact that what happens on the B site does not, in general, affect the A site, and v i c e v e r s a . A n illustration o f the case in point is clearly visible in the lower left hand por t ion o f Fig. 2


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Fig. 5. Plot of observed values of the lattice parameter and anion positional parameter u, for oxide spinels of composition AV20~ (solid circles), MgB204 (open circles) and MnCr204, contoured for various values of the octahedral and tetrahedral bond lengths

Fig. 6. Plot of observed values of the lattice parameter and anion positional parameter u, for sulfide spinels of composition ACr2S4 (solid circles') and CuB2S4 (open circles), contoured for various values of the octahedral and tetrahedral bond lengths

where the three silicate spinels SiB2O4, with B=Ni, Co, and Fe, are closely parallel to a contour of constant tetrahedral bond length. In fact, plots of this kind have occasionally appeared in the literature (Raccah etal., 1966; Reuter et al., 1969; Yagi et al., 1974) and can be very stringent tests of the accuracy of observational data.

Figure 5 is an expansion of a more densely populated portion of Fig. 2 contoured according to the expressions given in Table 1 for various constant octahedral and tetrahedral bond lengths. The six vanadium (solid circles) and three magnesium (open circles) spinels lie close to lines of bond length 1.96 and 2.025 ~, respectively, in good agreement with the values expected from the ionic radii of Shannon (1976). Both of these trends demonstrate the quite remarkable facility with which the spinel structure is able to accommodate cations of widely different size in one site (through large changes in u and a), while leaving the remaining site essentially unchanged. An additional species of composition MnCr204 completes a parallelogram of compounds (indicated by double symbols) each common to two bond length contours. In contrast, spinel itself, MgAI204, lies well away from the intersection of the lines MgB204 and AA1204 since the cations are partially disordered over the A and B sites (Fischer, 1967).


In Fig. 6 two series of spinels of composition Acr2a+s4 (solid circles) and Cul+BzS4 (open circles) have been plotted as in Fig. 5 with lines of constant bond length added so as to give the best fit to each set of points. While displaying considerably more scatter than the oxide data in Fig. 5, the best-fit octahedral bond length of 2.41/~ is surprisingly close to the value expected for an ionic bond between Cr 3+ and sulfur (Shannon, 1976). However, whereas the best-fit tetrahedral contour lies well within the range 2.24 to 2.35 A observed for a number of sulfides in which the copper atom is unambiguously monovalent (Lotgering and van Stapele, 1968), it is about 0.15 A shorter than the value expected from the Shannon radii. Although this is certainly in agreement with the observed contraction of the anion array in thiospinels, we are at a loss, as in the case of the cell dimension dependence on ionic radius and electronegativity, to explain why the octahedral and tetrahedral cations tend to behave so differently.

In previous studies of the thiochromite spinels it has been noted (Raccah et al., 1966; von Philipsborn, 1971) that the u parameter for metallic CuCr2S~ is anomalously high relative to the other semiconducting species, and that this discrepancy is evidence for the formation of aA*-band states at a Cu 2+ ion (Goodenough, 1967). While the assignment of a formal valence to atoms in significantly covalent compounds is, at the very least, conjectural, it is interesting to note that the u and a data for CuCrzS4 (plotted as a plus sign in an open circle in Fig. 6) place it very close to the intersection of the Cul+B2S~ line with one obtained by subtracting 0.032 A (the difference between the radius of a Cr 3+ ion and that of a 1 : 1 mixture of Cr 3+ and Cr 4+) from the best-fit ACr3+S4 contour. Therefore, rather than being 'anomalously ' high relative to the other thiochromates, the u value for CuCr2S 4 is consistent with the presence of the valence state Cul+[Cr3+Cr4+]S4, as suggested by Lotgering and van Stapele (1968), instead of Cu 2 +Cr23+ $4. In fact, were the u parameters for all the ACrzS4 spinels in Fig. 6 equally reliable, then an attempt could be made to discern other variations in cation valence states from the detailed scatter of points.

Conclusions

The lattice constants of binary and ternary oxide spinels are a simple and approximately equal function of the effective radii of the octahedral and tetrahedral cations, but are essentially independent of their electronegativities. For thiospinels the cell edge, though less well characterized, is primarily a function of the radius of the octahedral cation alone, with small additional contributions from the tetrahedral cation radius and octahedral cation electronegativity. This result attests to the applicability of the hard-sphere ionic model in the case of the spinel structure. Comparison of observed and calculated anion positional parameters indicates that up to 40% of the published experimentally determined coordinates may be in error by more than 0.002a. Diagrams of cell edge v s

anion coordinate are a rapid means of ascertaining the internal consistency of spinel structural data, and of predicting the parameters for hypothetical


c o m p o u n d s , a n d c a n r e v e a l i m p o r t a n t i n f o r m a t i o n a b o u t t h e v a l e n c e s t a t e s

a n d e f fec t ive r a d i i o f t h e c o n s t i t u e n t c a t i o n s .

Acknowledgements. All calculations presented in this study were performed with a local modification of the program GENSTAT (Department of Statistics, Rothamsted Experimental Station, Harpenden, Herts, England), using the CSIRO Cyber 76 computer. It is a pleasure to acknowledge the support of a Queen Elizabeth II Fellowship (awarded to RJH) and grant DMR75-03879 from the Materials Research Division of the National Science Foundation. Ms Beryl Coe is kindly thanked for typing the manuscript.

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Received January 12, 1979

Systematics of the Spinel Structure Type

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