Journal of Structural Chemistry, Vol. 43, No. 4, pp. 615–621, 2002

Original Russian Text Copyright @ 2002 by S. V. Borisov, R. F. Klevtsova, S. A. Magarill,

N. V. Pervukhina, and N. V. Podberezskaya

EXPERIMENTAL CRYSTALLOGRAPHY

FROM ATOMIC TO SUPRAMOLECULAR SCALE

UDC 548.30.736S. V. Borisov, R. F. Klevtsova,

S. A. Magarill, N. V. Pervukhina, and N. V. Podberezskaya

Results of crystal structure analysis for various types of compounds are discussed in terms of anew concept of crystal formation. The effect of ordering “rigid” atomic groups ([Hg2]2+, [H2As×W18O60]7−, etc.) by families of crystallographic planes with interplanar distances comparable tothe sizes of the groups is considered. The wide occurrence of symmetrical packings of atoms and/orcenters of rigid atomic groups in structures is explained by a tendency toward maximum reductionin the number of degrees of freedom and, hence, a tendency toward minimum energy of the system.

Contemporary crystallography has evolved under the influence, or even pressure, of mathematical thinkingencouraged by advances in abstract mathematics. Thus, E. S. Fedorov’s titanic geometrical derivation of the spacesymmetry groups for crystals was almost simultaneously and independently duplicated as a special variety in generalgroup theory by mathematician A. Schoenflies. At the same time, one must bear in mind that crystallization is afundamental physical phenomenon, and reducing it to the final idealized image, i.e., a translation lattice, meansignoring its driving forces and, hence, inadmissible simplification of the sense and mechanism of the process.

The concept of the crystalline state was formulated a decade ago. According to this concept, the emergenceand maintenance of strict long-range order in the arrangement of atoms is explained by a consistent system ofstanding plane waves formed in a disordered ensemble of material particles bound by elastic (as a first approximation)forces during gradual loss of energy by this ensemble, i.e., cooling [1]. A diagram of such ordering is shownschematically in Fig. 1a and b. In the first stage, the centers of the atoms in a certain volume of this ensembleare on parallel equidistant planes and can move along these planes but cannot move in the perpendicular direction.The loss of one degree of freedom by the atoms leads to a decrease in the total energy of the atomic system. At thesecond stage of the ordering by a family of equidistant planes inclined to the first family, a two-dimensional set offamilies of ordering planes appears simultaneously. The orientation and interplanar distances for these planes aredefined by the relation

1/dhk0 = h1/d100 + k 1/d010,

where d100 and d010 are the interplanar distances of the first and second starting (basic) systems, i.e., those havingthe largest d; the integers h and k do not have a common divisor and the whole set of families of planes is describedby a linear combination of two vector parameters. Note that all planes of this set pass through all lines of intersectionof the basic families (100) and (010), which form a two-dimensional translation lattice in the plane of projectionnormal to both of them. The unit cell of this lattice is a parallelogram with minimal surface (Fig. 1c). The stateof the system is characterized by fixation of the atoms in two planes but the possibility of free shifts along thelines of intersection remains. This degree of freedom is lost by the atoms in ordering by a third family of parallelplanes intersecting the first two. (The splitting of the ordering process is, of course, conditional. In reality, these

Institute of Inorganic Chemistry, Siberian Branch, Russian Academy of Sciences, Novosibirsk. Translatedfrom Zhurnal Strukturnoi Khimii, Vol. 43, No. 4, pp. 664–670, July–August, 2002. Original article submittedFebruary 26, 2002.

0022-4766/02/4304-0615 $27.00 c© 2002 Plenum Publishing Corporation 615

Fig. 1. Diagram of stages in ordering of an atomic ensemble: (a) initial disordered disposi-tion; (b) one-dimensional ordering: the centers of the atom are on parallel equidistant planesnormal to the plane of the figure; (c) two-dimensional ordering: the centers of the atoms areat the points of intersection of the basic families of (100) and (010) planes (c). One of thederivative families of (110) planes and two planes of the (510) family are shown.

stages can occur simultaneously.) The points of intersection of the three basic families of planes form a three-dimensional lattice in space, which, in the ideal case (simple compounds, for example), will be translational; i.e.,the minimal parallelepiped cut out by these basic planes is a unit cell, and the edges of the parallelepiped are a,b, and c translations with α, β, and γ angles between them. Again, simultaneously, a three-dimensional set ofplanes appears which passes through all the points of intersection of the three basic families (100), (010), and (001)according to the equation 1/dhkl = h1/d100 +k 1/d010 + l 1/d001 [2]. Note, however, that the formation of a familyof equidistant parallel planes does not mean, generally speaking, the occurrence of translation symmetry betweenthe atoms from adjacent planes. Therefore, searching in a concrete structure for the basic planes (100), (010), and(001) which determine the entire spatial system of families of planes (and, therefore, cover all reflections of thediffraction pattern), we can choose a triplet whose “unit cell” will be smaller than the true translation cell of thecrystal structure. Thus, very often, some sorts of atoms (for example, heavy cations) can be ordered by systems ofplanes with periodicity smaller than the true one. Then, for these atoms only, there is strict pseudoperiodicity ifthey are located exactly at the points of intersection of these planes or there is approximate pseudoperiodicity ifthe atoms are located near the points of intersection of these planes, as suggested by intensification of some X-rayreflections [3]. In many structures of heavy metal sulfides [4, 5], whose elements of symmetry include parallel mirrorplanes ∼2 A apart (so that the centers of all atoms are on the mirror planes; i.e., they are ideally ordered by onecoordinate) an interesting phenomenon is observed: the anions and cations of the structure are ordered by differentsystems of planes normal to the mirror planes (Fig. 2). The translation lattice shared by all atoms can be likenedto “the least common multiple” of the cation and anion sublattices. In a more general case, it can be assumedthat the atomic positions are determined by several linearly independent systems of planes [6]. Furthermore, it wasascertained that the ordering procedure depends to a greater extent on the effective size of the atom, i.e., the sizeof minimal contacts with objects comparable in size, rather than on the chemical nature of the atom. An exampleis known [7] where the Cl anion in the La2TaO4Cl3 structure lie in the cation sublattice rather than the “oxygenplanes” because the chlorine–cation and cation–cation distances are of the same order. On the other hand, in theCsBi3S5 structure, the large Cs cation is ordered together with the S anions, again because they are similar insize [6].

For the sulfide structures, the effect of “splitting” was revealed and described in detail. In general, thismeans that the points of intersection of the basic crystal-forming planes can be partly vacant, i.e., not all occupiedby atoms, and the arrangement of atoms and vacancies obeys a certain law (see Fig. 2). In this case, the interplanar

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Fig. 2. Projection xz of the Tm5S7 structure (C2/m, a = 12.455 A, b = 3.740 A, c = 11.268 A, β = 105.4◦,Z = 2); the Tm atoms in the net nodes that are points of intersection of the (202), (203) and (020) planesare shown by large circles (filled for y = 0 and open for y = 1/2). The S atoms at the points of intersectionof (203), (204), and (020) are shown by small circles (solid for y = 0 and empty for y = 1/2). Taking intoaccount the two (020) planes coinciding with the mirror plane of symmetry, we obtain 20 nodes of the Tmsublattice and 28 nodes of the S sublattice (in each sublattice, half the nodes are vacant).

Fig. 3 Fig. 4

Fig. 3. Cation matrix of the β-K2UF6 structure (P 62m, a = 6.55 A, c = 3.75 A, and Z = 1); U are largecircles, K are small circles, and the vacancies of the K net are shown by asterisks.

Fig. 4. Unit cell of the β-W cubic structure (A15, Pm3n, a = 5.04 A, and Z = 8): two sorts of the W atoms(filled and open circles) are at the points of intersection of the {400} planes.

distances for the basic crystal-forming planes can be substantially reduced compared with the occupied planes.An example is the widespread β-K2UF6-type cation framework, in whose hexagonal structure the crystal-formingplanes are the (002) basic planes, and one of the translationally independent planes contains the hexagonal K netwith two nodes per unit cell and the adjacent plane contains the U net (one node per unit cell) centering thehexagons in both the under- and overlying K nets [3] (Fig. 3). This distribution of atoms and vacancies in thetrigonal 36 nets of the total cation framework is caused by the high coordination numbers for both U (12K + 2U)and K (6U + 5K), which indicates that this common packing is favorable for high symmetry of the structure. Itis not merely incidental that about 25% of the structures of complex fluorides and oxides of heavy elements, as arule with the 1 : 2 stoichiometry of the two types of cations, have β-K2UF6 cation frameworks [7]. The splittingof a close-packed plane into two, in which the nodes occupied by atoms in one plane are vacant in the adjacentone and vice versa and the total number of vacancies is equal to the number of atoms (see Fig. 2), is not a singlemode of transformation of close-packed planes. In the β-W(A15) structure with high cubic symmetry (Pm3n), the(004), (040), and (400) planes cut out a cubic subcell with parameter a/4, in which only one-eighth of the nodes areoccupied by atoms (Fig. 4). It is significant that this structure involves a very ingenious variant of the conjugation

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TABLE 1Relative Densities of Occupation by Hg Atoms (ρHg)

and Centers of Dumbbells [Hg2] (ρc) for the Most Closely Packed Planes (hkl)

of Eglestonite Hg6Cl3O2H = [Hg2]3Cl3O2H

hkl dhkl ρHg hkl dhkl ρc

004 4.01 0.51 004 4.01 0.90

224 3.27 0.48 044 2.83 0.95

026 2.54 0.45 444 2.31 0.94

444 2.31 0.35 008 2.00 0.92

Fig. 5 Fig. 6

Fig. 5. Section of the eglestonite structure cut by the (444) plane: (a) arrangement of Hg atoms; (b) arrange-ment of the centers of [Hg2]2+ dumb-balls.

Fig. 6. Section of the (NH4)7[H2AsW18O60] · 16H2O structure by the (040) plane: W atoms (black circles)occupy the sites of the cation sublattice formed by intersection of the (040), (822), and (406) planes and thederivative (424) planes (for illustration, four 2a× 2c cells are shown).

of hexagonal and square nets (a hexagon is inscribed in a square [3]) with retention of high coordination numbersof atoms and high symmetry.

Let us now turn from analysis of the ordering of individual atoms to the ordering of whole atomic groups.It is reasonable to introduce the notion of a “rigid” atomic group, implying atoms linked by strong chemicalbonds so that the interatomic distances and bond angles in this group are kept almost constant for structuresof different composition. Examples of rigid groups are [PO4]3−, [SO4]2−, and [SiO4]4− complex anions; clustergroups, for example, [Hg2]2+ and [Hg3]4+; etc. As we have shown [8], for a crystal-chemical description too, it isreasonable to consider isolated [Hg2]2+ and [Hg3]4+ groups as a single cation. Its surrounding by anions is similarto the surrounding of a large (for example, Cs) cation, and the coordination polyhedra are convex polyhedra withuniform distribution of the edge lengths and nearly equal distances from the vertices to the geometrical center ofthe cluster group. An objective criterion for the ordering of the centers of atomic groups is the calculation of therelative densities of occupation of crystallographic planes by these centers using the KAP PLATS software [9]. Thisprocedure was carried out for a number of typical structures with [Hg2]2+ and [Hg3]4+ “polycations.” As a rule,there is a large difference between the densities of occupation of crystallographic planes by mercury atoms takenseparately and by the centers of [Hg2]2+ or [Hg3]4+ groups (Table 1). Figure 5 shows the arrangement of the Hgatoms in the close-packed (444) plane of the mineral eglestonite [Hg2]3Cl3O2H (Ia3d, a = 16.03 A, Z = 16 [10])

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TABLE 2Relative Densities of Packing (ρW)

of Crystallographic Planes by W Atomsfor (NH4)7[H2AsW18O60] · 16H2O

hkl dhkl ρW

040 3.56 0.60424 3.25 0.51406 3.23 0.65822 2.54 0.75337 2.39 0.59

Note. Planes with ρW > 0.5 are given.

and, for comparison, the arrangement of the centers of [Hg2] dumbbells in the same plane. In the former case, theaveraged deviation of Hg atoms from the (444) plane exceeds 0.5 A and the arrangement of these atoms in the planeis far from regular. In the latter case, the deviation is not more than 0.15 A with an ideal kagome net on the plane.This, in turn, means that in the other planes [(400), (040), (004)] having nearly maximum densities of occupationby the centers of [Hg2] dumbbells, the arrangement is also regular. A similar situation is also encountered in otherstructures, and this suggests that the ordering of the centers of gravity for such groups by families of planes is themain crystal-forming factor.

As regards larger atomic groups, the above statement is also true. An analysis of crystal structures with theKeggin heteropolyanions [PW12O40]3−, [H2W12O42]10− paratungstate ion, and the Dowson anion [H2AsW18O60]7−

revealed [11, 12] that these rigid atomic groups with sizes of 10–15 A are, as a rule, packed in structures in the samemanner as the atoms. As early as 1971, Evans offered a classification of the then known heteropolycompounds ofMo and W, distinguishing four packing modes for Keggin-type anions [13]. This work was continued and extendedto include other types of polyanions [11, 12]. Using the KAP-PLATS software package for analysis of the positionof the centers of gravity of rigid groups, it is possible to assess the regularity of packing, single out a set of basiccrystal-forming planes, and establish the symmetry of their relationships in the set. As a result, with allowance forthe data of [13], it was found that the crystallographic planes most closely packed by the centers of rigid groups(interplanar distances within 8–12 A) cut out a tetragonal cell and a nearly cubic I cell in 25–30% of compounds,a primitive cubic cell in ∼20%, and a nearly cubic F cell in ∼40%, and the arrangement of the centers on theclose-packed planes is the higher, the higher the intrinsic symmetry of the atomic group.

It is of interest to trace the further stages of ordering at the atomic level. Since the distances betweenthe heavy atoms (W and Mo) in heteropolyanions are 3.5–4.0 A, the effect of their consistent ordering within theunit cell should be expected for crystallographic planes with dhkl ∼ 2.5–3.5 A, and just this is observed in reality(Table 2). Figure 6 shows the section of the (NH4)7[H2AsW18O60] · 16H2O structure by the (040) plane. This planeintersects a number of [H2AsW18O60] polyanions, but the position of heavy atoms from different polyanions areclose to the points of intersection of the (040), (822), and (406) basic planes of the cation sublattice, or, in otherwords, we have a consistent (i.e., mutually correlated) arrangement of heavy atoms within the whole crystal [12].Naturally, a sizable part of sublattice points turns out to be vacant because the packing of polyanions has voidsfilled by other atoms of the structure. The full number of intersection sites of the basic sublattice planes within theunit cell is determined by the value of the determinant of the indices of the planes forming the cation framework: 160(Det [/040/822/406/] = 160). Of them, the heavy cations of W whose disposition orders these planes occupy 62points. The vacant sites of the cation sublattice that are between polyanions are a convenient place for the remainingatoms of the structure.

Of course, the structures with the rather unusual atomic groups considered above do not exhaust all typesof ordering. It is merely easier to detect ordering in simple cases, where the centers of groups form, for example,an I-cell or an F -cell. At the same time, it is possible to identify more complicated situations. Thus, Kalinina etal. [14] showed that in the structures of the cubane cluster [Zn(NH3)4]2+

3 [Mo4Te4(CN12)]6− and its Cd analog, thepacking of anions and cations proceeds according to the β-W scheme (A15 type) and the same type of packing ofthe [NaHg12] and [KHg12] groups is found in the compound NaK29Hg48 = [NaHg12] · [KHg12]3K26 [15].

In the present paper, we do not seek to consider all possible packing modes for rigid atomic groups. Theyare characterized by a tendency to high symmetry with retention of maximal density, which is associated with the

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Fig. 7. (a) Set of planes for an octahedron [eight (111) planesincluding those with negative indices]; the points of intersec-tion of the planes are the nodes of a face-centered lattice(black circles); (b) the orientation of the three basic planes ofa rhombic dodecahedron (a total of 12 planes including thoserelated centrosymmetrically); the points of intersection arethe nodes of a body-centered lattice.

presence of simple or modified planar 36 nets. The nets are formed in the basic crystallographic planes, and thetendency toward high symmetry implies that such planes of different orientation tend to become symmetricallyequivalent. This provides for additional means of reducing the number of degrees of freedom for the structure and,hence, a means of minimizing the energy of the system (for example, of four independent planes, one independentplane remains after they are related by a fourfold axis). The most widely occurring set of basic structure-formingplanes are the set of an F lattice, i.e., any three linearly independent planes of the four equivalent planes (111),(111), (111), and (111) related by symmetry Fm3m (set of an octahedron), and the set of an I lattice, i.e., threeindependent planes of the six planes (110), (101), (011), (110), (101), and (011) related by symmetry Im3m (setof a rhombic dodecahedron) (Fig. 7). Earlier it was shown that conversion from one set to another is possible bylinear deformation [16] and, indeed, intermediate situations are often encountered in practice.

Real structures exhibit a certain degree of approach to the ideal symmetry of corresponding sets but, actually,this may be sufficient for structure stabilization. Thus, it can be stated that the stability of a particular structureis provided for by both its true symmetry and additional pseudosymmetric relations between the individual rigidatomic groups and/or individual sorts of atoms formed within the true symmetry. In other words, if the formationof a structure with a small unit cell is impossible (for example, because of complex composition), small subcells forall atoms or for part of the atoms are formed.

The aforesaid leads to the following conclusions concerning practical crystallography on both atomic andsupramolecular scales.

1. The crystal state, i.e., the presence of long-range order (translation lattice) in the arrangement of atomsor “rigid” atomic groups is achieved through their ordering by a consistent system of families of parallel equidistantplanes. The consistency implies that

1/dhkl = h1/d100 + k 1/d010 + l 1/d001.

2. The major contribution to the ordering is made by those families of equidistant parallel planes whoseinterplanar distances (dhkl) are comparable to the sizes of the objects being ordered. The sizes of the objects areestimated from the shortest distances between them in a condensed medium of given composition.

3. If the atomic configurations in a certain group hardly change in structures with different combinationsof this group with other components, such a group is considered “rigid” and treated as a structural unit whoseposition is determined by the center of mass of the group.

4. The system of families of ordering planes tends to maximal symmetry. Symmetry reduces the number ofdegrees of freedom for the system. Approaching a symmetric spatial arrangement lowers the system energy and isthus favorable even if ideal symmetry is not achieved. This statement is supported by the wide natural occurrenceof symmetrically related sets of ordering planes, whose points of intersection allow realization of translation latticesand sublattices of the F and I types and other symmetric cases.

This work was supported by the Russian Foundation for Basic Research (Grant No. 01-05-65104).

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