Abstract. The geometric characteristics of the Voronoi-Dirichlet polyhedrons are determined for 79 crystallographic types of Zr(IV) atoms whose positions in crystal structures are surrounded by fluorine atoms. The volumes of the Voronoi-Dirichlet polyhedrons are found to be virtually independent of the coordination number (6, 7, or 8) of the zirconium atoms, and their average value is 8.0(2) µ³.

As primary information, we used the data on the crystal structure of 65 compounds that satisfy the following conditions: (a) their structures are determined with R0.10, and (b) Zr and/or F atoms are not statistically distributed over the positions of one or several regular systems of points. The sampled compounds involved 79 crystallographic types of the Zr(IV) atoms (17, 20, and 42 types with the C.N.s equal to 6, 7, and 8, respectively), and the geometric characteristics of their VD polyhedrons were calculated using a TOPOS set of structure-topological programs [4]. Note that 31 of the 65 chosen compounds contained the hydrogen atoms. Inasmuch as their positions were determined only for the structures of 15 compounds and the accuracy of determining the hydrogen coordinates, as a rule, is normally low, H atoms were disregarded when calculating the VD polyhedrons.

According to the results obtained, the C.N. of the majority of Zr atoms, if taken equal to the total number of faces in the corresponding VD polyhedrons (C.N._VDP), coincided with the values given in the original works. At the same time, the VD polyhedrons of eight Zr atoms (four with C.N. 7 and four with C.N. 8) have one (for seven atoms) or two {only for the Zr(l) atom in the structure of Cs₅Zr₄F₂₁Ç3?₂? [5]} additional small faces. All of these faces correspond to the small solid angles W_i [smaller than 1.5% of the total solid angle (471 sr) of the Zr atom] and abnormally long interatomic distances Zr–F (longer than 3 µ), so that the additional contacts can be considered, as in the case of the U(VI) compounds [1], as nonvalence interactions, the more so as, in most cases, they are related to the so-called indirect neighbors [6, 7], i.e., to those X, atoms (surrounding atom A) for which the A–X, sections do not intersect the corresponding faces of the VD polyhedron of atom A. As an example, Table 1 represents the main characteristics of the VD polyhedron of the Zr(l) atom in the structure of Cs₅Zr₄F₂₁Ç 3?₂?. Figure 1 shows the shape of the VD polyhedron. The atomic numbering in Table 1 and Fig. 1 is identical and coincides with that given in the original work [5].

As is known, each interatomic A-X_i contact with the distance of r_i(A–X) corresponds to a certain face of the VD polyhedron AX_n, with the area S_i of the face corresponding to a certain solid angle W_i [1]. Note that, similar to the uranium compounds [2], zirconium fluorides exhibit a clear-cut dependence between the lnW_i(A–X) and r_i(A–X) values (Fig. 2), for which the linear regression analysis gives equation

InW_i(Zr–F) = 6.67(5) – 1.95(2)r_i(Zr–F) (1)

Fig. 1. The Voronoi-Dirichlet polyhedron of the Zr(l) atom in the Cs₅Zr₄F_21Ç3?₂? structure [5].

where the correlation coefficient r = -0.963 for 581 faces of the VD polyhedrons of the zirconium atoms of 79 crystallographic types. In (1), the W_i values are expressed in percentages of the total solid angle (4p sr) of the zirconium atom, and the r_i(Zr–F) values are given in angstroms. Note that six faces with W_i(Zr–F) < 1% were disregarded when deducing equation (1). The data [2] and the results obtained suggest that the correlations of the type (1) with r > 0.9 are valid only for those VD polyhedrons of composition AX_n in which the strong chemical interaction between the atoms leads to the appearance of a short-range order in the mutual arrangement of the X atoms relative to the A atoms.

The total area (S_VDP) of the surface of the VD polyhedron of zirconium atom [23.1(8) µ³ in the compounds studied is independent, within 2s(S_VDP) of the number of the fluorine atoms coordinated to the zirconium atom.

The dimensionless second moment of inertia of the VD polyhedrons (G₃), which is an integral parameter characterizing uniformity of the fluorine arrangement around the Zr atoms [8], is constant, within 2s (G₃), and equal to 0.082(1) independently of the C.N. and shape of the coordination polyhedron.

Table 1. Main characteristics of the VD polyhedron of the Zr(1) atom in the Cs₅Zr₄F₂₁ Ç3?₂? structure [5]. The coordinates of the central atom Zr(1): x = 0.431, ? = 0.721, and z = 0.25, V_VDP = 8.203 µ³ S_VDP = 23.759 µ² G₃ = 0.0838256

** Q is the number of edges of the common face of the VD polyhedrons of Zr and of the F, atom whose number, coordinates, and Zr-F, distance aie given in the same row. The area (S_i) and the solid angle (W_i), for the same face of the VD polyhedron are expressed in percent of the total area of the surface of the VD polyhedron (S_VDP) and of the total solid angle (4p sr) of the zirconium atom, respectively.