Elke Koch

Geometrical packing analysis of molecular compounds

From a geometrical point of view the packing of molecules in crystal structures has been studied for 192 organic compounds. For this, Dirichlet domains of the atoms have been calculated by means of radical planes. These domains have been joined to packing polyhedra of the molecules. Two molecules are called adjacent if their packing polyhedra share faces. Utilizing this concept, coordination numbers have been calculated and a classification of structures into packing types has been performed with the aid of packing graphs. As a result, coordination number 14 was calculated more frequently (94 structures) than 12 (46 structures). In total, 51 packing types have been found, 9 of which stand out because of their frequency. It should be noted that two packing types with coordination number 14 show frequencies (18 and 17) comparable to those of the types ccp, bcc, and hep (32, 30, and 11, respectively).

Hexagonal and trigonal sphere packings. I. Invariant and univariant lattice complexes

All homogeneous sphere packings and all interpenetrating sphere packings have been derived that refer to the seven invariant and the 23 univariant lattice complexes belonging to the hexagonal crystal family. The respective sphere packings may be assigned to 66 types. In addition, one case of interpenetrating sphere packings was found. For five types, the inherent symmetry of some sphere packings with specialized metrical and coordinate parameters may become cubic. For two further types, namely 8/4/c1 (body-centered cubic lattice) and 12/3/c1 (face-centered cubic lattice), the inherent symmetry is cubic for all corresponding sphere packings. By means of a large number of examples, the applicability of sphere packings for the comparison and description of simple crystal structures is demonstrated.

Interpenetration of homogeneous sphere packings and of two-periodic layers of spheres.

All systems of interpenetrating sphere packings that occur with highest symmetry in the cubic, hexagonal or tetragonal crystal family are tabulated. Homogeneous sphere packings belonging to 49 different types may be intertwined to systems of interpenetrating sphere packings belonging to 74 types. For all compatible lattice complexes, the coordinate and lattice parameters are given. The corresponding patterns of interpenetration are analysed. For the interpenetration of two, three, four, five and eight sphere packings, eleven, three, five, one and two different patterns, respectively, are distinguished. In addition, four types of interpenetrating layers of spheres were found. Each such sphere configuration splits into two or three subsets of parallel sphere layers with an angle of 90 degrees or of 120 degrees , respectively, between the directions of the normals of the layers. A single sphere layer corresponds either to a honeycomb net or to a net built up from quadrangles and octagons.

Normalizers of space groups : A useful tool in crystal-structure description, comparison and determination

Abstract After an illustrative example for the ‘symmetry of symmetry’ the group-theoretical concept of normalizers is introduced by a series of definitions. Subsequently, this concept is applied to space groups for which Euclidean and affine normalizers are discussed. Their implications on point configurations, Wyckoff positions and coordinate descriptions of crystal structures are explained. The derivation of all equivalent descriptions of a crystal structure with the aid of the tables of normalizers given in the International Tables for Crystallography, Vol. A is elucidated by several examples.

The implications of normalizers on group–subgroup relations between space groups

A hierarchy of classifications for subgroups of space groups by means of Euclidean and affine normalizers is introduced. The different levels of this classification scheme are illustrated in detail with examples and its usefulness for various problems is demonstrated. The Euclidean (or affine) normalizers of a space group G and of one of its subgroups U may either coincide [N(G)=N(U)], or form a group-subgroup pair [N(G) ⊃ N(U) or N(G) ⊂ N(U)], or share only a common subgroup [N(G) ⊇N(U) and N(G) ¢ N(U)]. The different implications of these cases on the equivalence classes of subgroups (or supergroups) are discussed. A procedure is given to calculate the normalizers. The same concept may be applied to number of equivalent subgroups or supergroups, other crystallographic groups without problems.

A proposal for a transition mechanism from the diamond to the lonsdaleite type

A phase transition between the diamond (Fd3;m) and the lonsdaleite types (P6(3)/mmc) may be described as a deformation of a homogeneous sphere packing with three contacts per sphere (type 3/10/o1) in the common subgroup Pnna of Fd3;m and P6(3)/mmc. The frequently observed transition between the zinc-blende (F4;3m) and the wurtzite types (P6(3)mc) may be described in an analogous way as a deformation of a heterogeneous sphere packing in the subgroup Pna2(1). The proposed model guarantees the three-dimensional connection during the whole transformation process. By this property it is distinguished from other models.

Hexagonal and trigonal sphere packings. II. Bivariant lattice complexes.

All homogeneous sphere packings were derived which correspond to point configurations of the 26 bivariant lattice complexes belonging to the hexagonal crystal family. They may be assigned to 109 sphere-packing types. Among these, there is a type of sphere packing with contact number 10 that was not described before. For seven of the 109 types, the inherent symmetry of the sphere packings with minimal density is cubic. In addition, three types of interpenetrating sphere packings were found and one type of interpenetrating 6(3) sphere layers. Such an arrangement was unknown so far. Some frequently occurring structure types that can be related to sphere packings are described as examples.

Homogeneous sphere packings with triclinic symmetry

All homogeneous sphere packings with triclinic symmetry have been derived by studying the characteristic Wyckoff positions P -1 1a and P -1 2i of the two triclinic lattice complexes. These sphere packings belong to 30 different types. Only one type exists that has exclusively triclinic sphere packings and no higher-symmetry ones. The inherent symmetry of part of the sphere packings is triclinic for 18 types. Sphere packings of all but six of the 30 types may be realized as stackings of parallel planar nets.

On 3-periodic minimal surfaces with non-cubic symmetry

A list of those 547 group-subgroup pairs of space groups is given which are not incompatible with balance surfaces for symmetry reasons. The symmetry conditions that have to be fulfilled by all balance surfaces are tabulated in addition. Two kinds of non-cubic minimal balance surfaces have been derived completely: (1) 7 families of minimal balance surfaces which may be generated by skew circuits of 2-fold axes that are disk-like spanned, (2) 7 families of minimal balance surfaces which may be generated by pairs of parallel flat circuits of 2-fold axes that are catenoid-like spanned.

Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden

Dirichlet domains have been studied for all cubic lattice complexes with less t han three degrees of freedom. The method of their derivation is described here. 117 types of polyhedra and 143 types of space part i t ions have been found. Detailed results are given for two lattice complexes as examples. I t could be shown t h a t a method proposed by NOWACKI (1935) will not yield all possible homogeneous space partit ions into Dirichlet domains.