UDC 548.736

CRYSTALLOGRAPHY

E. A. KUZ’MIN, V. V. ILYUKHIN, Academician N. V. BELOV

CRYSTAL STRUCTURE OF POTASSIUM BICHROMATE K₂Cr₂O₇ (LOPEZITE)

Single crystals of chrompik—the triclinic modification of K₂Cr₂O₇—were isolated from solution and kindly placed at our disposal by E. N. Slavnova. The unit-cell parameters \((a = 7.52;\ b = 13.40;\ c = 7.40\ \text{Å};\ \alpha = 98^\circ,\ \beta = 90^\circ 50',\ \gamma = 96^\circ 10';\ Z = 4)\) are in agreement with those previously reported for natural lopezite \((^1)\). The good solubility of potassium bichromate in water made it possible without difficulty to obtain specimens of almost spherical shape. The experimental material for the X-ray structural analysis consisted of 2600 nonzero reflections \(hk0—hk8\) and \(0kl—3kl\) (MoK\(_\alpha\) radiation, \(\max \sin \vartheta/\lambda = 0.85\ \text{Å}^{-1}\)).

A detailed analysis of the three-dimensional Patterson synthesis (together with the well-established absence of a piezoelectric effect) made it possible to carry out the analysis under the assumption of a centrosymmetric variant of the structure. In interpreting the three-dimensional Patterson function, the most effective proved to be the “method of combining points of a vector system into \(n\) identical \(n\)-gons” \((^{2,8})\), which made it possible to localize the Cr and K atoms. Further refinement of the structure proceeded at the stage of electron-density syntheses \(\rho(xyz)\). At this stage (an isotropic thermal correction, averaged over groups of atoms, was introduced: \(B_{\mathrm{Cr}} \approx 1.1\ \text{Å}^2,\ B_{\mathrm{K}} \approx 1.4\ \text{Å}^2,\ B_{\mathrm{O}} \approx 1.7\ \text{Å}^2\)) the discrepancy factor \(R_{hkl}\) over the entire volume of the sphere of reflections is equal to 16.1%*. The coordinates of the basis atoms at the achieved value of the \(R\)-factor are given in Table 1. The centrosymmetric structure (space group \(P\bar{1}\)) is characterized by 66 parameters.

Table 1

Coordinates of the basis atoms in the structure of K₂Cr₂O₇

Each of the four crystallographically independent chromium atoms is surrounded by four oxygen atoms in a tetrahedron. The Cr tetrahedra are joined in pairs into diortho groups Cr₂O₇. Among the Cr—O distances in each diortho group, two elongated ones stand out (Cr₁—O₃ and Cr₂—O₃, Cr₃—O₈ and Cr₄—O₈, respectively) to the common atom O in the diortho group at

* The noncentrosymmetric variant of the structure, differing from the selected one only by a small displacement of the light atoms, did not give a substantial improvement of the \(R\)-factor (15.8%), but led to a sharp worsening of the interatomic distances.

against the background of the other six:

\[ \begin{gathered} \text{Dichromate group I }(\mathrm{Cr}_1-\mathrm{Cr}_2):\quad \mathrm{Cr}_1-\mathrm{O}_3 = 1.72_5\ \text{\AA} \\ \mathrm{Cr}_2-\mathrm{O}_3 = 1.85_6\ \text{\AA} \\ \text{with the remaining } \mathrm{Cr}_1,\ \mathrm{Cr}_2-\mathrm{O}=1.53\text{--}1.68\ \text{\AA}, \\ \text{Dichromate group II }(\mathrm{Cr}_3-\mathrm{Cr}_4):\quad \mathrm{Cr}_3-\mathrm{O}_8 = 1.84_2\ \text{\AA} \\ \mathrm{Cr}_4-\mathrm{O}_8 = 1.74_5\ \text{\AA} \\ \text{with the remaining } \mathrm{Cr}_3,\ \mathrm{Cr}_4-\mathrm{O}=1.51\text{--}1.63\ \text{\AA}, \\ \text{Angles: }(\mathrm{I})\ \mathrm{Cr}_1-\mathrm{O}_3-\mathrm{Cr}_2 = 127^\circ \\ (\mathrm{II})\ \mathrm{Cr}_3-\mathrm{O}_8-\mathrm{Cr}_4 = 122^\circ . \end{gathered} \]

The O—O distances in the tetrahedra are: 2.58–2.79 Å (Cr₁ tetrahedron), 2.48–2.87 Å (Cr₂ tetrahedron), 2.45–2.84 Å (Cr₃ tetrahedron), 2.67–2.90 Å (Cr₄ tetrahedron).

The large K cations also occupy 4 independent crystallographic positions and, as usual, have an ill-defined environment: K₁ and K₂ in seven-vertex polyhedra, which occur more frequently for moderately large

Fig. 1. K bichromate $\mathrm{K}_2\mathrm{Cr}_2\mathrm{O}_7$. Arrangement of K cations in the $yz$ projection. A quadrupole of 4 seven-vertex polyhedra around K is highlighted. Common edges are marked by triple lines

($r_i = 0.97$–0.99 Å) Na and Ca cations: this is a combination of a trigonal prism and a semioctahedron “attached” to one of the rectangular faces of the prism. If the K₁ polyhedron is sufficiently compact: 6 distances K₁—O are 2.67–2.78 Å (with an average of 2.73 Å) and one anion is removed to 2.97 Å, then the K₂ polyhedron may be characterized as very loose: 6 neighbors are removed to distances of 2.78–3.00 Å (average 2.91 Å), and only one distance is much shorter and almost equal to the sum of the ionic radii: 2.69 Å.

The coordination of K₃ is sixfold, which for such a large cation is a rather rare case. The coordination polyhedron is a flattened octahedron, noted earlier for the still larger Rb ($^4,^5$). The distances K₃—O = 2.73–2.93 Å, with an average of 2.84 Å (anions more than 3.10 Å away were not included in the coordination sphere). Finally, K₄ is located in the eight-vertex polyhedron already described by us ($^6$), encountered in compounds of the type $\mathrm{M}_n\mathrm{BX}_4$, where $\mathrm{M}=\mathrm{K}, \mathrm{Rb}, \mathrm{Cs}, \mathrm{NH}_4, \mathrm{Ba}$; $\mathrm{B}=\mathrm{S}, \mathrm{Be}, \mathrm{Cr}, \mathrm{P}$ and $\mathrm{X}=\mathrm{F}, \mathrm{O}$. Seven distances K₄—O vary within the narrow interval 2.82–2.95 Å around an average of 2.88 Å, and again one (the eighth) is substantially shorter: 2.69 Å.

In the structural motif of $K_2Cr_2O_7$ (lopezite) one can distinguish two quadrupoles of K-polyhedra. The first quadrupole is composed of two pairs of hemivertex polyhedra: $K_1$ and $K_2^{**}$, $K_2$ and $K_1^{**}$*, situated (each pair) at approximately the same height (along the $a$ axis):

$$ K_1^{**}=0.20,\qquad K_2=0.14,\qquad K_2^{**}=0.46,\qquad K_1=0.40 \qquad (\text{Fig. 1}). $$

The hemivertex polyhedra are joined along a common edge into pairs, which condense into foursomes also along edges. At the midpoint of the edge common to $K_1$ and

Fig. 2. K-bichromate $K_2Cr_2O_7$. The second quadrupole of K-polyhedra: two octads and two octahedra in the same $yz$ projection. Common edges are marked by triple lines.

$K_1^{**}$ there is a center of symmetry; $K_1$ has common edges with $K_2$ and $K_2^{**}$; analogously $K_1^{**}$ has common edges with $K_2$ and $K_2^{**}$, but $K_2$ and $K_2^{**}$ are not directly connected to one another. The planar quadrupole (the plane of the quadrupole $\sim(\bar{1}12)$) of $K_1—K_2$ hemivertex polyhedra is connected with translationally identical ones not directly, but through Cr-tetrahedra: (the $Cr_1$ tetrahedron in the direction $a[100]$ and $Cr_2$ in the direction $c[001]$).

In the quadrupole of $(2K_3+2K_4)$-polyhedra the four K cations: $K_3$, $K_3^{**}$, $K_4$ and $K_4^{**}$ are situated almost in the plane $(1\bar{1}1)$. The polyhedra are combined into a quadrupole as follows: $K_4$ has common edges with $K_3$ and $K_3^{**}$, and likewise $K_4^{**}$ with $K_3$ and $K_3^{**}$; however, if $K_3$ and $K_3^{**}$ have no common elements, then $K_4$ and $K_4^{**}$ have an additional bond—a common edge passing through the center of symmetry (Fig. 2). As in the case of the first quadrupole, the second quartet of K-polyhedra is not connected with translationally identical ones directly. Here too the Cr-tetrahedra ($Cr_3$ and $Cr_4$ tetrahedra) act as the linking elements.

When the quadrupoles of both kinds are joined into a three-dimensional framework, the $K_3$ octahedron from the second quartet has only one common edge with the $K_2$ hemivertex polyhedron of the first quadrupole, whereas the $K_4$ octad is connected (also by one edge) with the $K_1$ hemivertex polyhedron, but already of a quadrupole translationally identical to the first.

In the space filled with bulky polyhedra, the remaining voids are occupied by $Cr_2O_7$ diortho groups (encountered in chromates for the first time). It may be noted that in the structure of lopezite (K-bichromate) the relationship between K and Cr $(CrO_4)$ repeats that which is characteristic of Ca(Na) and Si$(SiO_4)$

* Here and below, $K_n^{**}$ denotes an atom related to the basic $K_n$ by reflection in a center of symmetry.

in silicates: the relative sizes of the K- and Cr-polyhedra are such that not a single Cr-tetrahedron, but the diortho group \(\mathrm{Cr_2O_7}\) (Fig. 3), is stretched over the edge of a large seven-vertex polyhedron around K, and K bichromate may be regarded as the first deciphered analogue of the diorthosilicates of Chapter II of crystal chemistry, not only of silicates,* but also of chromates, and in general of compounds with the complex anion \(\mathrm{BX_4}\).

Fig. 3. \(\mathrm{K_2Cr_2O_7}\). \(yz\) projection with separated diortho groups \([\mathrm{Cr_2O_7}]\)

In a number of works the “morphological” absence of a center of symmetry in \(\mathrm{K_2Cr_2O_7}\) is emphasized, in contradiction to the purely dry result of the X-ray investigation. Apparently, K bichromate is yet another example of the manifestation in a macrocrystal of lower symmetry in comparison with the microsymmetry established by X-ray structural analysis (hypomorphy, hyposymmetry according to Kleber \((^{9,10})\)).

Institute of Crystallography
Academy of Sciences of the USSR

Received
6 I 1967

REFERENCES

\(^1\) H. Strunz, Mineralogische Tabellen, 1962.
\(^2\) M. J. Buerger, Acta Crystallogr., 3, 87, 243 (1950).
\(^3\) M. Buerger, Crystal Structure Analysis in Vector Space, IL, 1961.
\(^4\) V. V. Ilyukhin, N. V. Belov, Crystallography, 6, 6 (1961).
\(^5\) N. Mustafaev, V. V. Ilyukhin, N. V. Belov, Dokl. Akad. Nauk SSSR, 159, No. 6 (1964).
\(^6\) N. Mustafaev, V. V. Ilyukhin, N. V. Belov, Dokl. Akad. Nauk SSSR, 162, No. 5 (1965); Crystallography, 10, 6 (1965).
\(^7\) N. V. Belov, Essays on Structural Mineralogy, XII, Mineralogical Collection of the Lvov Geological Society, No. 15, 1961.
\(^8\) Kh. S. Mamedov, N. V. Belov, Geochemistry, 11, 1087 (1964).
\(^9\) N. V. Belov, Essays on Structural Mineralogy, VIII, Mineralogical Collection of the Lvov Geological Society, No. 11 (1957).
\(^10\) W. Kleber, Wissenschaftliche Zs. d. Humboldt Universität zu Berlin, 5, No. 1 (1956).

* Cf. also the crystal-chemical similarity and analogy with silicates of Chapter II of phosphates (\(^7\)), borates (\(^8\)), fluoroberyllates (\(^{3,6}\)).