W. Fischer & E. Koch (2001): 3-periodic minimal surfaces derivable from Laves nets. - Z. Anorg. Allg. Chem. 627, 2091 - 2094.
3-periodic minimal surfaces made up by catenoid-like surface patches may be related to Laves nets. Starting from these nets four new types of such surfaces with self-intersection exclusively along straight lines could be derived. They subdivide R3 into 2-periodic labyrinths and 1-periodic tubes.
E. Koch (2001): Self-intersecting three-periodic minimal surfaces forming one-periodic tubes or finite polyhedra. - Z. Kristallogr. 216, 430 - 437.
17 families of three-periodic minimal surfaces with straight self-intersections have been derived that subdivide R3 into an infinite number of one-periodic 'tubes' and/or finite 'polyhedra'. Their Euler characteristics vary between -3 and -18. Ten of these families show non-orientable minimal surfaces. They subdivide R3 either into congruent tubes (eight families), into congruent polyhedra, or into two different kinds of congruent polyhedra. The minimal surfaces of seven families are orientable. All of them cause two different kinds of spatial subunits: two kinds of tubes (one family), tubes and polyhedra (three families), or two different kinds of polyhedra (three families).
E. Koch (2000b): Self-intersecting three-periodic minimal surfaces forming two-periodic (flat) labyrinths. - Z. Kristallogr. 215, 386 - 392.
14 families of minimal surfaces with straight self-intersections have been derived which subdivide R3 into infinitely many congruent, two-periodic 'flat labyrinths'. For eleven families, all flat labyrinths are parallel to each other. Two sets of mutual perpendicular flat labyrinths have been found three times. All these minimal surfaces are non-orientable. Their Euler characteristics vary between -3 and -13.
E. Koch (2000a): Minimal surfaces with self-intersections along straight lines. II. Surfaces forming three-periodic labyrinths. - Acta Crystallogr. A56, 15 - 23.
47 families of minimal surfaces with straight self-intersections have been derived which subdivide R3 into a finite number of congruent, three-periodic labyrinths. In most of these cases, namely for 42 families, the number of labyrinths is two. Four congruent labyrinths have been found three times and eight congruent labyrinths twice. Minimal surfaces with three or six labyrinths seem not to exist. Most of these minimal surfaces are non-orientable. The surfaces of three families only are orientable ones.
E. Koch & W. Fischer (1999): Minimal surfaces with self-intersections along straight lines. I. Derivation and properties. - Acta Crystallogr. A55, 58 - 64.
A special kind of three-periodic minimal surface has been studied, namely surfaces that are generated from disc-like-spanned skew polygons and that intersect themselves exclusively along straight lines. A new procedure for their derivation is introduced in this paper. Several properties of each such surface may be deduced from its generating polygon: the full symmetry group of the surface, its orientability, the symmetry group of the oriented surface, the pattern of self-intersections, the branch points of the surface, the symmetry and periodicity of the spatial subunits demarcated by the surface, and the Euler characteristics both of the surface and of the spatial subunits. The corresponding procedures are described and illustrated by examples.
W. Fischer & E. Koch (1996b): Spanning minimal surfaces. - Phil. Trans. R. Soc. Lond. A354, 2105 - 2142.
Spanning minimal surfaces are 3-periodic minimal surfaces which contain straight lines, i.e. axes of 2-fold symmetry. We have used crystallographic knowledge of the space groups and of the corresponding arrangements of 2-fold axes involved for the derivation of new surfaces of this type and for their systematic description. Complete information on spanning minimal surfaces without self-intersections is given with respect to their symmetry and topology, together with some new results on spanning minimal surfaces with self-intersections along straight lines.
W. Fischer & E. Koch (1996a): Two 3-periodic self-intersecting minimal surfaces related to the Cr3Si structure type. - Z. Kristallogr. 211, 1 - 3.
Two new 3-periodic minimal surfaces with self-intersections have been derived that may be related to the Cr3Si (A15) structure type. One of these is onesided and wraps each atom of a Cr3Si structure separately, whereas the other one has two sides and separates each Cr chain from the isolated Si atoms.
E. Koch & W. Fischer (1993b): A crystallographic approach to 3-periodic minimal surfaces. In: Statistical Thermodynamics and Differential Geometry of Microstructured Materials, The IMA Volumes in Mathematics and its Applications, Vol. 51 - New York: Springer, 15 - 48.
Symmetry conditions are tabulated that must be fulfilled by all minimal balance surfaces, i.e. by all 3-periodic intersection-free minimal surfaces that subdivide R3 into two congruent regions. The 52 configurations of straight lines (2-fold rotation axes) that may be embedded within any 3-periodic intersection-free minimal surface are described in some detail and surface patches for minimal balance surfaces that span these line configurations are discussed. Crystallographic formulae for the calculation of the genera of 3-periodic intersection-free minimal surfaces are given. A list of all minimal balance surfaces known so far contains the respective inherent symmetries, the genera and in addition the orders, the site symmetries and the Wyckoff position of their flat points.
E. Koch & W. Fischer (1993a): Triply periodic minimal balance surfaces: a correction. - Acta Crystallogr. A49, 209 - 210.
In previous papers, 15 families of minimal balance surfaces that can be generated with the aid of disc-like surface patches were noted. Recently, it has become apparent that the inherent symmetry of two of these families, the C(S) surfaces and the Y surfaces, is higher than the symmetry used for their generation. As a consequence, C(S) surfaces are in fact P surfaces and Y surfaces are in fact D surfaces.
W. Fischer & E. Koch (1992): Symmetry aspects of 3-periodic minimal surfaces. In: Bifurcation and Symmetry. Internationale Schriftenreihe zur Numerischen Mathematik, Vol. 104 - Basel, Boston, Berlin: Birkhäuser, 123 - 133.
Symmetry properties of 3-periodic minimal surfaces subdividing R3 into two congruent regions are discussed. The relation between the order of a flat point and its site symmetry is established. Explicit formulae are given for the calculation of the genus of such a surface depending on the kind of surface patches that build up the surface. Making use of 2-fold axes that have to be embedded in a surface with given symmetry new families of minimal balance surfaces have been derived. Two examples of bifurcations related to minimal surfaces are mentioned.
W. Fischer & E. Koch (1990): Crystallographic aspects of minimal surfaces. - Coll. Phys. 51 - C7, 131 - 147.
Symmetry properties of 3-periodic minimal surfaces subdividing R3 into two congruent regions are discussed. The relation between the order of a flat point and its site symmetry is established. Explicit formulae are given for the calculation of the genus of such a surface, depending on the kind of surface patches that build up the surface. Making use of 2-fold axes that have to be embedded in a surface with given symmetry new families of minimal balance surfaces have been derived. Corresponding lists are referred to the different kinds of surface patches.
E. Koch & W. Fischer (1990): Flat points of minimal balance surfaces. - Acta Crystallogr. A46, 33 - 40.
The symmetry conditions for flat points of minimal surfaces have been studied in relation to the order b of points on such surfaces. Using symmetry aspects, a set of rules for the derivation of flat points have been developed. By means of these rules the flat points for the 45 families of minimal balance surfaces known so far have been determined. As a check for completeness the relation between the genus of a minimal surface and the orders of its flat points has been used.
W. Fischer & E. Koch (1989c): Genera of minimal balance surfaces. - Acta Crystallogr. A45, 726 - 732.
The genus of a three-periodic intersection-free surface in R3 refers to a primitive unit cell of its symmetry group. Two procedures for the calculation of the genus are described: (1) by means of labyrinth graphs; (2) via the Euler characteristic derived from a tiling on the surface. In both cases new formulae based on crystallographic concepts are given. For all known minimal balance surfaces the genera and the labyrinth graphs are tabulated.
E. Koch & W. Fischer (1989b): New surface patches for minimal balance surfaces. IV. Catenoids with spout-like attachments. - Acta Crystallogr. A45, 558 - 563.
The way in which Schoen [Infinite Periodic Minimal Surfaces Without Self-intersections (1970), NASA Tech. Note No. D-5541] derived a simply connected surface patch for a C(H) surface cannot be generalized. One may, however, subdivide a C(H) surface into larger patches that are not simply connected. Surface patches of analogous shape give rise to five families of minimal balance surfaces unknown so far: tetragonally and orthorhombically distorted C(P) surfaces, surfaces complementary to Schoen's R2 and R3 surfaces with genus 25 and 37, respectively, and orthorhombic surfaces of a fifth family with genus 5 that are also complementary to oP surfaces.
W. Fischer & E. Koch (1989b): New surface patches for minimal balance surfaces. III. Infinite strips. - Acta Crystallogr. A45, 485 - 490.
Two new families of minimal balance surfaces are described. Their surface patches are not finite but have the shape of infinite strips. Such a strip is bounded by two congruent zigzag lines in one case or by a zigzag line and a meander line in the other case. In addition, certain minimal balance surfaces derived before with the aid of finite surface patches can also be generated from infinite strip-like surface patches.
E. Koch & W. Fischer (1989a): New surface patches for minimal balance surfaces. II. Multiple catenoids. - Acta Crystallogr. A45, 169 - 174.
Eight new families of minimal balance surfaces are described. Their surface patches belong to a new kind, called multiple catenoids. The generating circuits of such a minimal surface are two congruent concave polygons with one point of self-contact each. The new minimal balance surfaces are complementary to other minimal balance surfaces which are built up from catenoid-like surface patches and have been known before.
W. Fischer & E. Koch (1989a): New surface patches for minimal balance surfaces. I. Branched catenoids. - Acta Crystallogr. A45, 166 - 169.
Three new families of minimal balance surfaces have been derived. For this a new kind of surface patch, i.e. branched catenoid, has been used. A concave polygon with one point of self-contact and a convex polygon are the two generating circuits of such a minimal balance surface.
E. Koch & W. Fischer (1988): On 3-periodic minimal surfaces with non-cubic symmetry. - Z. Kristallogr. 183, 129 - 152.
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.
W. Fischer & E. Koch (1987): On 3-periodic minimal surfaces. - Z. Kristallogr. 179, 31 - 52.
A method is described to deduce all group-subgroup pairs of space groups or black-white space groups compatible with 3-dimensionally periodic surfaces subdividing space into two congruent labyrinths without self-intersection. Within the cubic crystal system 34 types of such group-subgroup pairs exist. For these, the symmetry conditions are tabulated that have to be fulfilled by all corresponding minimal surfaces. The resulting types of generating linear nets and generating circuits of minimal surfaces are discussed. They give rise to four new types of minimal surfaces. In addition a fifth new type has been found. The subgroup relations derived for the 34 types of group-subgroup pairs are given in a diagram.
All homogeneous sphere packings and all interpenetrating layers of spheres were derived that refer to the 22 orthorhombic bivariant lattice complexes. In total, sphere packings of 90 different types have been found. Only for 47 of these types the maximal inherent symmetry of their sphere packings is orthorhombic. Some examples demonstrate the usefulness of sphere packings for the comparison and description of crystal structures.
All homogeneous sphere packings and all interpenetrating sphere packings were derived that refer to the 6 invariant and the 11 univariant lattice complexes belonging to the orthorhombic crystal system. In total, sphere packings of 38 types have been found. Only for 17 types is the maximal inherent symmetry of their sphere packings orthorhombic. By means of a number of examples, the applicability of sphere packings for the comparison and description of simple crystal structures is demonstrated.
The 13 trivariant lattice complexes with trigonal symmetry are compatible with 218 types of homogeneous sphere packings, 7 types of interpenetrating sphere packings and one type of interpenetrating layers of spheres. Altogether, the lattice complexes with trigonal characteristic space group (with 0, 1, 2 or 3 degrees of freedom) give rise to 225 types of sphere packing. 110 of these types are compatible exclusively with one of the 13 trivariant lattice complexes, 31 in addition with some of the invariant, univariant or bivariant lattice complexes, whereas 6 types occur solely in such a lattice complex. 65 types encompass special sphere packings that can also be generated with hexagonal symmetry [Sowa, Koch & Fischer (2003). Acta Cryst. A59, 317–326; Sowa & Koch (2004). Acta Cryst. A60, 158–166; Sowa & Koch (2005). Acta Cryst. A61, 331–342]; cubic inherent symmetry occurs for certain sphere packings [Fischer (2004). Acta Cryst. A60, 246–249] belonging to 13 types. The maximal inherent symmetry is trigonal for 147 of the 225 types. The sphere packings of 61 types can be subdivided into connected layer-like subunits, those of 86 types into connected rod-like subunits. Such subunits may be used to construct some kind of ‘descriptive symbols’ that reflect certain properties of the sphere packings. Interpenetrating sphere packings with cubic inherent symmetry belong to one of the 7 types. All interpenetrating sphere layers that belong to the only type occurring in the trigonal crystal system show hexagonal inherent symmetry.
T. E. Dorozinski & W. Fischer (2006): A novel series of sphere packings with arbitrarily low density.- Z. Kristallogr. 221 , 563-566.
The search for a least dense packing of spheres makes sense only under suitable restrictions because otherwise sphere packings of arbitrarily low density may be constructed. As has been shown before, the condition that all spheres have to be equal as well in their size as in their number of contacts is not such an effective restriction. A sharper condition is the demand that all spheres must have congruent patterns of contacts, i.e.that they must coincide in their first neighbourhood. The example presented here proves that also this restriction is not sufficient to avoid sphere packings with densities approaching zero.
E. Koch, W. Fischer & H. Sowa (2006): Interpenetration of homogeneous sphere packings and of two-periodic layers of spheres. - Acta Crystallogr. A62, 152-167.
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 up into two or three subsets of parallel sphere layers with an angle of 90° or of 120°, 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.
W. Fischer (2005): Tetragonal sphere packings: minimal densities and subunits. - Acta Crystallogr. A61, 435 - 444.
For all 382 types of homogeneous sphere packings with tetragonal symmetry, the minimal sphere-packing densities have been calculated. The tabulated coordinates allow the graphic representation of a sample packing for each type. 1- and 2-periodic subunits of these sphere packings are listed in addition.
E. Koch, H. Sowa & W. Fischer (2005): On the density of homogeneous sphere packings. - Acta Crystallogr. A61, 426 - 434.
For some types of sphere packing with typical one- and two-dimensional parameter regions, the sphere-packing density as a function of the free parameters is discussed. In addition, some sphere-packing types with extraordinary density properties are presented. Until now, it was generally assumed that sphere packings with minimal density are also those of highest inherent symmetry. An example to prove the opposite is given.
W. Fischer (2005): On sphere packings of arbitrarily low density. - Z. Kristallogr. 220, in the press.
The search for a least dense packing of spheres (circles) makes sense only under suitable restrictions because otherwise sphere packings of arbitrarily low density may be constructed. An effective restriction is that all spheres have to be symmetrically equivalent. It has been tried to weaken this condition such that the spheres (circles) have only to be equal in size as well as in their number of contacts. It turned out, however, that even then series of packings may be derived within which the density approaches zero. Examples for such series are presented.
H. Sowa & E. Koch (2005): Hexagonal and trigonal sphere packings. III. Trivariant lattice complexes of hexagonal space groups. - Acta Crystallogr. A61, 331 - 342.
All types of homogeneous sphere packings and interpenetrating sphere packings and layers were derived that correspond to point configurations of the 15 trivariant hexagonal lattice complexes. The respective sphere packings are assigned to 147 types. In total, sphere packings of 170 types can be realized with hexagonal symmetry. 103 types of sphere packing refer exclusively to trivariant hexagonal lattice complexes. For 23 of these types the corresponding sphere packings can be generated only in hexagonal lattice complexes with less than three degrees of freedom or with trigonal or lower symmetry. In addition, seven types of interpenetrating sphere packings and two types of interpenetrating sphere layers were found. Interpenetrating 4.82 nets of spheres with 120° angles between the nets were assumed to be not possible, so far. The sphere packings belonging to 85 of the 170 hexagonal types can be split up into parallel layers of spheres with mutual contact and can be characterized by symbols derived from those for the Shubnikov nets. The sphere packings of 135 hexagonal types may be subdivided into rod-like subsets of spheres with mutual contact. Such rods may be described by rolling up a plane net. Only 23 types of sphere packing cannot be symbolized on the basis of layers or rods of spheres with mutual contact. Examples for crystal structures are given that can be described by means of sphere packings.
W. Fischer (2004): Minimal densities of cubic sphere-packing types. - Acta Crystallogr. A60, 246 - 249.
The minimal sphere-packing densities have been calculated for all 199 types of homogeneous sphere packings with cubic symmetry. The tabulated coordinates allow the graphic representation of each type.
E. Koch & H. Sowa (2004): Exceptional properties of some sphere packings in the general position of P6222. - Acta Crystallogr. A60, 239 - 245.
Three types of sphere packing are described which show properties that have never been observed before: a certain set of generating symmetry operations corresponds to a parameter range in P6222 that is not simply connected, but disintegrates into two disjoint, non-congruent regions; the minimal sphere-packing density is different for these two regions; two sphere packings from different regions cannot be deformed into each other without opening sphere contacts although their sphere-packing graphs are isomorphic in the graph-theoretical sense. Two heterogeneous crystal nets with different symmetry described by Delgado-Friedrichs & O'Keeffe [Acta Cryst. (2003), A59, 351-360] show a similar behaviour. They are related to a type of tetragonal sphere packings with likewise unusual properties.
H. Sowa & E. Koch (2004): Quantification of the deviations from closest sphere packings. - Eur. J. Mineral. 16, 255 - 260.
In order to quantify the deviation of the anion arrangement in a crystal structure from a closest sphere packing two parameters are introduced: D drel is the relative difference between the longest and the shortest distance to neighbouring atoms within a distorted closest-packed arrangement and s is the standard deviation of the normalized distances to all neighbouring atoms. The applicability of these parameters is compared with that of the distortion parameter Ucp introduced by Thompson & Downs, (2001, Acta Cryst., B57, 119-127). Deviations from ideal sphere packings in pyroxenes, staurolite, kyanite, and olivine- and spinel-type structures are considered.
H. Sowa & E. Koch (2004): Hexagonal and trigonal sphere packings. II. Bivariant lattice complexes. - Acta Crystallogr. A60, 158 - 166.
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 packings 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 63 sphere layers. Such an arrangement was unknown so far. Some frequently occuring structure types that can be related to sphere packings are described as examples.
H. Sowa, E. Koch & W. Fischer (2003): Hexagonal and trigonal sphere packings. I. Invariant and univariant lattice complexes. - Acta Crystallogr. A59, 317 - 326.
All homogeneous sphere packings and all interpenetrating sphere packings were derived which 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.
W. Fischer & E. Koch (2002): Homogeneous sphere packings with triclinic symmetry. - Acta Crystallogr. A58, 509 - 513.
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 which 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.
H. Sowa & E. Koch (2002): Group-theoretical and geometrical considerations of the phase transition between the high-temperature polymorphs of quartz and tridymite. - Acta Crystallogr. A58, 327 - 333.
A model was derived for the temperature-dependent phase transition between the high-temperature polymorphs of quartz (P6422) and tridymite (P63/mmc). Only the Si framework is considered, and the transformation can be described as a deformation of a homogeneous sphere packing with three contacts per sphere (type 3/10/h1) in the common subgroup P6122 of P6422 and P63/mmc. The proposed model guarantees the three-dimensional connection of the crystal structure during the whole transformation process.
H. Sowa & E. Koch (2001): A proposal for a transition mechanism from the diamond to the lonsdaleite type. - Acta Crystallogr. A57, 406 - 413.
A phase transition between the diamond (Fd-3m) and the lonsdaleite types (P63/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 Fd-3m and P63/mmc. The frequently observed transition between the zinc-blende (F-43m) and the wurtzite types (P63mc) may be described in an anologous way as a deformation of a heterogeneous sphere packing in the subgroup Pna21. The proposed model guarantees the three-dimensional connection during the whole transformation process. By this property it is distinguished from other models.
H. Sowa & E. Koch (1999): Sphere configurations with the symmetry R-3m 18(h) .m. - Z. Kristallogr. 214, 316 - 323.
In deriving all sphere configurations which may occur with symmetry R-3m - .m, 36 topological different types were found. The sphere configurations of 19 types disintegrate into disconnected partial configurations: subunits with point-group symmetry occur four times, such with rod-group symmetry or layer-group symmetry eight times or five times, respectively. Seventeen types correspond to homogeneous sphere packings, and two types to interpenetrating sphere packings. In addition, the possible deformations of all homogeneous sphere packings in the subgroups R-3, R32, and R-3c of R-3m were investigated. Structural examples that can be described with the aid of sphere packings are given.
W. Fischer (1997): Comments on Novel regular quinquehedral packing obtained by the local approach by V.V.Manzhar. - Acta Crystallogr. A53, 402.
Comments are made on a paper by Manzhar [Acta Cryst. (1996), A52, 645-646] where the author gives the impression that there is no detailed information available on packings in R3.
E. Koch & W. Fischer (1995): Sphere packings with three contacts per sphere and the problem of the least dense sphere packing. - Z. Kristallogr. 210, 407 - 414.
Two procedures to derive all types of sphere packings with three contacts per sphere are described. 52 such types have been found. They are characterized by their shortest meshs and related to Wells' three-connected nets, if possible. It is proved that there exists no sphere packing with contact number three and a density lower than 5.5%, i.e. the density of the Heesch-Laves packing.
W. Fischer (1993): Tetragonal sphere packings. III. Lattice complexes with three degrees of freedom. - Z. Kristallogr. 205, 9 - 26.
For tetragonal lattice complexes with three degrees of freedom the sphere-packing conditions, the generation classes and the (topological) types of sphere packings are tabulated. The use of the table is illustrated by means of two structural examples.
W. Fischer (1991b): Tetragonal sphere packings. II. Lattice complexes with two degrees of freedom. - Z. Kristallogr. 194, 87 - 110.
For tetragonal lattice complexes with two degrees of freedom the sphere-packing conditions, the generation classes and the (topological) types of sphere packings are tabulated. The exceptional situation for the lattice complex I41/amd 16h .m. is discussed and illustrated by diagrams.
W. Fischer (1991a): Tetragonal sphere packings. I. Lattice complexes with zero or one degree of freedom. - Z. Kristallogr. 194, 67 - 85.
The basic definitions connected with sphere packings are given. The way in which sphere packings with tetragonal symmetry have been derived and assigned to generation classes and (topological) types is described. The results for tetragonal lattice complexes with zero and one degree of freedom are listed and - in the latter case - the sphere-packing conditions are illustrated by diagrams.
E. Koch (1985): The geometrical characteristics of the a-ThSi2 structure type and of its parameter field. - Z. Kristallogr. 173, 205 - 224.
The 2-dimensional parameter field for the a-ThSi2-structure type is discussed with respect to limiting complexes, Dirichlet partitions and sphere configurations. On this basis privileged arrangements of the thorium atoms and of the silicon atoms are derived and compared with the crystal structure of a-ThSi2. The unusual geometrical properties of the idealized a-ThSi2 structure are elucidated. The deviations of the structures of EuSi2, CaSi2, SrSi2 and BaGe2 from the idealized ones are discussed.
E. Koch (1984): A geometrical classification of cubic point configurations. - Z. Kristallogr. 166, 23 - 52.
The point configurations of cubic lattice complexes with less than three degrees of freedom, i.e. of all special cubic Wyckoff positions, are classified with respect to three properties: (1) the symmetry of a point configuration regarded by itself (the occurrence of limiting complexes), (2) the type of Dirichlet partitions that corresponds to the point configuration, (3) the type of sphere configurations (sphere packings, interpenetrating sphere packings, rodlike or polyhedral sphere configurations) referring to the point configuration. Detailed information on the Dirichlet polyhedra and on some properties of the space tilings is given in addition.
E. Koch & W. Fischer (1978): Types of sphere packings for crystallographic point groups, rod groups and layer groups. - Z. Kristallogr. 148, 107 - 152.
All conceivable types of sphere packings for subperiodic crystallographic groups have been derived. To achieve this, it was possible to limit study to the characteristic classes of configuration-sets for the point complexes, row complexes and net complexes. The resulting 24 sphere-packing types of point groups, 60 sphere-packing types of rod groups and 75 sphere-packing types of layer groups are listed together with their generating symmetry operations. The sphere packings of point groups and rod groups can be classified uniquely by means of topological symbols. For this, modified Schläfli symbols have been used in the case of point groups. The sphere packings of rod groups can be described uniquely as plane nets rolled up according to a coincidence vector. Almost all sphere packings of layer groups may be assigned to plane nets or double nets, although a unique topological symbolism could not be developed in this case.
W. Fischer (1976): Eigenschaften der Heesch-Laves-Packung und ihres Kugelpackungstyps. - Z. Kristallogr. 143, 140 - 155.
The Heesch-Laves packing is the sphere packing of lowest density within the remarkable type 3/3/cl. It is proved, that sphere packings of this type only are built up from triangular units with three contacts per sphere. The sphere-packing condition for 3/3/cl as well in I213 as in P4132 covers a large range of permissible deformations and, in consequence, of density variation (0.0555 £ r < 0.5554). Sphere packings of this type may be deduced from the cubic closest packing, due to subgroup relations, by omitting either 1/4, 7/9 or 29/32 of the spheres. 2, 4 or 8 single sphere packings of type 3/3/cl may be joined to form eight different types of interpenetrating sphere packings without mutual contact.
W. Fischer & E. Koch (1976): Durchdringungen von Kugelpackungen mit kubischer Symmetrie. - Acta Crystallogr. A32 225 - 232.
An interpenetrating sphere packing is defined as a set of crystallographically equivalent spheres, which form two or more sphere packings without mutual contact. All types of interpenetrating sphere packings with cubic symmetry have been deduced, using the known symmetry regions for cubic space groups. These 39 types show 12 different kinds of interpenetration. Different types are called related if an interpenetrating sphere packing of one type can be generated from another one by omitting contacts or spheres. All such relations are tabulated.
W. Fischer (1974): Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit drei Freiheitsgraden. - Z. Kristallogr. 140, 50 - 74.
Sphere-packing conditions for all cubic lattice complexes with three degrees of freedom are given in form of tables. The corresponding sphere packings form 285 classes of symmetrically equivalent sphere packings, belonging to 182 different types. For the cubic system as a whole there exist 199 types of homogeneous sphere packings. A sphere packing with four contacts per sphere has been found, which shows the extremely low density of 0.0789.
W. Fischer (1973): Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. - Z. Kristallogr. 138, 129 - 146.
Sphere-packing conditions have been deduced for all cubic lattice complexes with less than three degrees of freedom. 129 classes of symmetrically equivalent sphere packings have been found, which form 78 types. The complete results are given in form of tables. Different types of sphere packings have been distinguished by the aid of two criteria, which are discussed here.
E. Koch & W. Fischer (1972): Wirkungsbereichstypen einer verzerrten Diamantkonfiguration mit Kugelpackungscharakter. - Z. Kristallogr. 135, 73 - 92.
The type of sphere packing corresponding to the cubic diamond configuration may be tetragonally distorted with symmetry P41. The Dirichlet domains have been studied for this case. 33 different types of these have been found. Each type is described by the conditions of its existence and by the number and kind of its faces. The various types are illustrated by drawings.
W. Fischer (1971): Existenzbedingungen homogener Kugelpackungen in Raumgruppen tetragonaler Symmetrie. - Z. Kristallogr. 133, 18 - 42.
The conditions for packing crystallographically equivalent spheres in three-dimensional space have been evaluated for the cases of tetragonal space groups. The concept of lattice complexes and the knowledge of subgroup relations have been used to facilitate the investigation. 1016 classes of symmetrically equivalent sphere packings have been found. These classes form 394 topologically defined types. Detailed results are given for space group P4/n as an example.
W. Fischer (1968): Kreispackungsbedingungen in der Ebene. - Acta Crystallogr. A24, 67 - 81.
The conditions for packing crystallographically equivalent spheres in three-dimensional space have not yet been evaluated systematically because of the great number of calculations involved. For the case of circles packed in the plane, this effort can be reduced by using the concept of lattice complexes (the metrical behaviour of 72 different sets of equivalent points is represented by 30 lattice complexes) and by taking into account the fact that - owing to symmetry relations- the 13 lattice complexes with 2 degrees of freedom contain all the others as special configurations. The results of these investigations are tabulated for all lattice complexes in two dimensions.
L. Bessais, C. Djéga-Mariadassou & E. Koch (2002): Structural and Mössbauer spectral study of the metastable phase Sm(Fe,Co,Ti)10. - J. Phys.: Condens. Matter 14, 1-10.
We have performed a Mössbauer spectral analysis of nanocrystalline metastable P6/mmm SmTi(Fe1-xCox)9, correlated with structural transformation towards its equilibrium derivative I4/mmm SmTi(Fe1-xCox)9. The Rietveld analysis shows that the 3g site is fully occupied, while the 6l occupation is limited to hexagons surrounding the Fe-Fe dumb-bell pairs 2e. A specific programme for the Wigner-Seitz cell (WSC) calculation of the metastable disordered structure was used. The hyperfine parameter assignment based on the isomer shift correlation with the WSC volumes sequence leads to Co 3g preferential occupation, with Ti location in 6l sites. The mean hyperfine field increases with Co content in connection with the enhancement of the negative core electron polarization term upon additional Co electron filling. The same trend is observed for each individual site leading to the sequencc HHF{2e} > HHF{6l} > HHF{3g}.
E. Koch & W. Fischer (1996): DIDO95 and VOID95 - programs for the calculation of Dirichlet domains and coordination polyhedra. - Z. Kristallogr. 211, 251 - 253.
For the calculation and the drawing of Dirichlet domains and coordination polyhedra in crystal structures or point sets two programs - DIDO95 and VOID95 - have been derived. A third program - DEMODIDO - shows how the Dirichlet domain of a point changes if the metrical parameters or the coordinates vary along a one-dimensional path. All three programs are written in FORTRAN and may be used on an ordinarv PC under DOS.
W. Fischer (1986): Geometrical aspects of the patterns of conduction paths in fast ion conductors. - Cryst. Res. Techn. 21, 499 - 503.
A method for a straightforward calculation of geometrically privileged conduction paths in fast ion conductors is presented. It makes use of a special property of Dirichlet domains, namely that their vertices and edges define all polyhedral voids and all least narrow connecting passages within a rigid framework. The results of the procedure are exhaustive in the geometrical sense and may be used as a basis for physical investigations. The method is illustrated by two simple examples, a-AgI and sodium b-alumina. Restrictions are discussed.
E. Koch (1985): The geometrical characteristics of the a-ThSi2 structure type and of its parameter field. - Z. Kristallogr. 173, 205 - 224.
The 2-dimensional parameter field for the a-ThSi2-structure type is discussed with respect to limiting complexes, Dirichlet partitions and sphere configurations. On this basis privileged arrangements of the thorium atoms and of the silicon atoms are derived and compared with the crystal structure of a-ThSi2. The unusual geometrical properties of the idealized a-ThSi2 structure are elucidated. The deviations of the structures of EuSi2, CaSi2, SrSi2 and BaGe2 from the idealized ones are discussed.
E. Koch (1984): A geometrical classification of cubic point configurations. - Z. Kristallogr. 166, 23 - 52.
The point configurations of cubic lattice complexes with less than three degrees of freedom, i.e. of all special cubic Wyckoff positions, are classified with respect to three properties: (1) the symmetry of a point configuration regarded by itself (the occurrence of limiting complexes), (2) the type of Dirichlet partitions that corresponds to the point configuration, (3) the type of sphere configurations (sphere packings, interpenetrating sphere packings, rodlike or polyhedral sphere configurations) referring to the point configuration. Detailed information on the Dirichlet polyhedra and on some properties of the space tilings is given in addition.
W. Fischer (1980b): On a space-filling polyhedron with 26 faces. - match (communications in mathematical chemistry) 9, 103. [full text]
The number of faces of a space-filling polyhedron in Rn is bounded (Delone, 1961). For n£4, the estimate used in Delone's prove, however, seems to give no accessible value for this maximal number. According to a conjecture by Brunner & Laves (1977), the upper limit is about 26 for n=3. The highest number of faces actually found in the past for a space-filling polyhedron is 24 (Koch & Fischer, 1972). This refers to the Dirichlet domain (Voronoi polyhedron) of a slightly distorted diamond configuration. The distortion is connected with the group-subgroup relation between Fd3m and P41. In the present study, an attempt has been made to derive Dirichlet domains with even more faces in an analogous way. For this, the Dirichlet domains of cubic point configurations (Koch, 1972) with a fairly high number of faces have been used as a starting point. As a result, space-filling polyhedra with 25 and 26 faces have been derived in P4132 (general position) from a Dirichlet domain with 22 faces in I4132 (1/8,x,1/4-x with x=0.1835). The type of Dirichlet domains with 26 faces exists in a narrow, but well-established array around 0.1440, 0.1825, 0.0635.
W. Fischer (1980a): Normal homogeneous partitions of three-dimensional Euclidean space which are not partitions into fundamental regions of a space group. - match (communications in mathematical chemistry) 9, 101. [full text]
For n³3, it was not known so far whether there exists a normal homogeneous space partition of Rn into convex polyhedra, which does not form a partition into fundamental regions (asymmetric units) of a space group of Rn (Delone, 1961). In this connexion, normal means that adjacent polyhedra share entire faces, and homogeneous implies that the set of all polyhedra of the space partition forms an orbit under the action of some space group. For n=3 , a related problem has been solved in crystallography. There exist exactly five Wyckoff positions the point configurations (orbits of points) of which cannot be generated in the general position of a space group: I-43d (d) x¹1/8, Ia3d (f) x¹0, Fd3m (f) x¹1/4, F4132 (f) x¹l/4, Fd3 (f) x¹1/4 (cp. Fischer, 1973). All corresponding space partitions into Dirichlet domains (Koch, 1972) do not form partitions into fundamental regions of a space group in R3. These cases give rise to related solutions of the problem for n>3. It has been proved for Fd3m (f) x=1/8, that the corresponding Dirichlet domains (distorted octahedra) cannot be rearranged to another type of space partition. This example, therefore, is a new special type of answer to Hilbert's Problem 18 (1900).
E. Koch & W. Fischer (1980): Calculation of volume increments for organic compounds by means of Dirichlet domains. - Z. Kristallogr. 153, 255 - 263.
Volume increments of atoms in organic compounds have been calculated as average volumes of Dirichlet domains from 192 crystal structures. The results have been checked on the basis of 167 further compounds (R = 5.1 %) and are compared with volume increments derived with the aid of least-squares procedures by other authors.
W. Fischer & E. Koch (1979): Geometrical packing analysis of molecular compounds. - Z. Kristallogr. 150, 245 - 260.
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 hcp (32, 30, and 11, respectively).
E. Koch & W. Fischer (1974): Zur Bestimmung asymmetrischer Einheiten kubischer Raumgruppen mit Hilfe von Wirkungsbereichen. - Acta Crystallogr. A30, 490 - 496.
Two different methods for the deduction of asymmetric units are proposed and have been applied to cubic space groups. Both these methods are based on the knowledge of the Dirichlet domains (Wirkungsbereiche) for special sets of equivalent points: (1) The Dirichlet domains for points in general positions directly give rise to asymmetric units. For the limiting cases, where higher symmetry is simulated by relations between the coordinates, these Dirichlet domains are known as those of special positions in supergroups. (2) According to point symmetry the Dirichlet domains for special positions may be split into asymmetric units for the space group under consideration. Selection of the simplest asymmetric unit for each space group leads to 15 different polyhedra for all cubic space groups. The part of the border of the asymmetric unit that belongs to the asymmetric unit is specified for each space group.
E. Koch & W. Fischer (1973): Über den Einfluß der Kugelradien auf heterogene Wirkungsbereichsteilungen. - N. Jb. Min. Mh. 1973, 361 - 380.
Following some general remarks on heterogeneous sphere packings the influence of atomic or ionic radii on heterogeneous Dirichlet partitions is discussed using two examples: perowskite-type structures under sphere-packing conditions (depending on two radius ratios) and fluorite-type structures (depending on two radii). For the second example the changes of some properties of the Dirichlet domains with varying radii are shown.
W. Fischer & E. Koch (1973): Über heterogene Wirkungsbereichsteilungen in Abhängigkeit von zwei Parametern. - N. Jb. Min. Mh. 1973, 252 - 273.
Some general ideas on types of heterogeneous "Dirichlet partitions" (space partitions into Dirichlet domains) are given. Two examples illustrate the dependence of these types on two parameters, one of which is a radius ratio of two spheres. In connection with these the structure types of CsCl, NaCl, CaF2 and Pyrite are discussed.
E. Koch (1973): Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. - Z. Kristallogr. 138, 196 - 215.
Dirichlet domains have been studied for all cubic lattice complexes with less than three degrees of freedom. The method of their derivation is described here. 117 types of polyhedra and 143 types of space partitions have been found. Detailed results are given for two lattice complexes as examples. It could be shown that a method proposed by NOWACKI (1935, Homogene Raumteilung und Kristallstruktur. Diss. Zürich ) will not yield all possible homogeneous space partitions into Dirichlet domains.
E. Koch & W. Fischer (1972): Wirkungsbereichstypen einer verzerrten Diamantkonfiguration mit Kugelpackungscharakter. - Z. Kristallogr. 135, 73 - 92.
The type of sphere packing corresponding to the cubic diamond configuration may be tetragonally distorted with symmetry P41. The Dirichlet domains have been studied for this case. 33 different types of these have been found. Each type is described by the conditions of its existence and by the number and kind of its faces. The various types are illustrated by drawings.
W. Fischer, E. Koch & E. Hellner (1971): Zur Berechnung von Wirkungsbereichen in Strukturen anorganischer Verbindungen. - N. Jb. Min. Mh. 1971, 227 - 237.
In order to extend the concept of Dirichlet domains ("Wirkungsbereiche") from sets of points to sets of spheres with different radii some possible constructions are proposed. Three of these are of special interest, the ones which use (1) perpendicular bisectors, (2) "Verhältnisebenen" (planes perpendicular to the connecting line of the centers of two spheres, subdividing this line according to the radius ratio) and (3) radical planes. Advantages and disadvantages of these possibilities are discussed and illustrated by the example of CaCl2.
E. Koch & W. Fischer (2005): Normalizers of space groups: A useful tool in crystal-structure description, comparison and determination. - Z. Kristallogr. 220, in the press.
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.
E. Koch & U. Müller (1990): Euklidische Normalisatoren für trikline und monokline Raumgruppen bei spezieller Metrik des Translationengitters. - Acta Crystallogr. A46, 826 - 831.
A listing of the Euclidean normalizers for triclinic and monoclinic space groups having translation lattices with specialized metric is given. These normalizers have not been included in previous tabulations. For convenience, in the case of monoclinic space groups, only the second setting (c-axis unique) is considered and the metric of the cells is restricted within certain limits which warrant that all specialized cases and all cell choices according to International Tables for Crystallography [(1987) Dordrecht: Kluwer] are included.
E. Koch (1986): Implications of the Euclidean normalizers of space groups in reciprocal space. - Cryst. Res. Techn. 21, 1213 - 1219.
Each crystal structure with symmetry G exactly corresponds to i different but equivalent coordinate descriptions of the atomic arrangement and to i different but equivalent structure-factor lists. The number i equals the index of G in its Euclidean normalizer NE(G). The isometries of one coset of G in NE(G) transform an original coordinate description and an original structure-factor list into a certain other equivalent description and equivalent structure-factor list. Two intermediate groups are uniquely defined: G Í K(G) Í L(G) Í NE(G). G and K(G) are class-equivalent whereas K(G) and NE(G) are translation-equivalent. L(G) is the most comprehensive intermediate group from the same Laue class as G. In course of a structure determination with direct methods all information needed to "fix the origin" by appropriate restriction of some phases may be derived from the additional translations of K(G). If no anomalous scattering has been observed and if NE(G) is centrosymmetric but G is not, the positions of the inversion centers in L(G) determine tho appropriate phase restriction to "fix the enantiomorph".
E. Koch (1984): The implications of normalizers on group-subgroup relations between space groups. - Acta Crystallogr. A40, 593 - 600.
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) É N(U) or N(G) Ì N(U)], or share only a common subgroup [N(G) Ë N(U) and N(U) Ë N(G)]. The different implications of these cases on the equivalence classes of subgroups (or supergroups) are discussed. A procedure is given to calculate the number of equivalent subgroups or supergroups.
W. Fischer & E. Koch (1983): On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. - Acta Crystallogr. A39, 907 - 915.
The Euclidean normalizers of space groups form the appropriate mathematical tool for several problems treated independently by crystallographers in the past, e.g. the comparison, the classification and the standardized description of crystal structures. Explicit tables are presented that enable the user to handle Euclidean normalizers in an easy way and, especially, to calculate all descriptions of a crystal structure compatible with a chosen space-group setting. The use of the tables is illustrated by different examples, and the role of Euclidean normalizers for crystal-structure determination is discussed.
E. Koch & W. Fischer (1975): Automorphismengruppen von Raumgruppen und die Zuordnung von Punktlagen zu Konfigurationslagen. - Acta Crystallogr. A31, 88 - 95.
For all space groups the groups of inner automorphisms are given. They are isomorphic with groups of motions, but fall into four sets according to their dimension. In the triclinic and monoclinic systems, however, the corresponding groups of all automorphisms cannot be represented by groups of motions. These groups of automorphisms therefore are only tabulated for the other cases. The relation between groups of automorphisms and Cheshire groups is discussed. By means of automorphisms sets of equivalent points are combined to 'Konfigurationslagen'. For these complete tables are included.
E. Koch & H. Sowa: The cubic limiting complexes of lattice complexes with trigonal characteristic symmetry. - Z. Kristallogr. 220, accepted.
Point configurations with cubic inherent symmetry may be generated within certain trigonal space groups. The corresponding limiting-complex relationships were studied. They are caused by a specialization of the axial ratio c/a and require also special coordinate parameters in most cases. The limiting complexes were derived with the aid of group-subgroup relationships. The material was checked making use of the occurrence of cubic sphere packings within trigonal space groups. The results are presented in a table referring to all lattice complexes with rhombohedral characteristic space groups (two invariant, five univariant, four bivariant, four trivariant) and to one univariant, one bivariant and one trivariant lattice complex with trigonal characteristic space groups.
E. Koch & W. Fischer (2003): The cubic limiting complexes of the tetragonal lattice complexes. - Z. Kristallogr. 218, 597 - 603.
For all tetragonal lattice complexes the limiting complexes with cubic characteristic space-group type have been derived. These limiting complexes are caused by a specialization of the axial ratio c/a. However, they may require in addition special coordinate parameters. The limiting complexes have been determined with the aid of group-subgroup relationships. In order to check the material, the occurrence of cubic sphere packings within tetragonal lattice complexes was used. The results are presented in four separate tables referring to the 4 invariant, the 24 univariant, the 38 bivariant and the 48 trivariant tetragonal lattice complexes.
W. Fischer & E. Koch (1995): Lattice complexes with at most one comprehensive complex. - Acta Crystallogr. A51, 586 - 587.
Most lattice complexes with less than three degrees of freedom are contained as limiting complexes within several comprehensive complexes corresponding to general Wyckoff positions. There exist, however, 21 exceptional cases: 3 lattice complexes without any and 18 lattice complexes with exactly one such comprehensive complex each. They are listed together with their comprehensive complexes. This information may be used to restrict geometrical investigations of all point configurations to general positions of space groups.
E. Koch & W. Fischer (1985): Lattice complexes and limiting complexes versus orbit types and non-characteristic orbits: a comparative discussion. - Acta Crystallogr. A41, 421 - 426.
The concept of lattice complexes is compared with the concept of orbit types, and the correspondences and differences are worked out in some detail. For this, it turned out to be necessary to distinguish whether a set of symmetrically equivalent points is regarded as attached to or as detached from its generating space group. The terms 'crystallographic orbit' and 'point configuration' used as synonyms so far are applied here to discriminate between these distinct meanings. On this basis, the terminology within both concepts is redefined stressing the aspect of equivalence classes. After that, the relations between non-characteristic orbits and limiting complexes are carefully discussed for the first time.
W. Fischer & E. Koch (1978): Limiting forms and comprehensive complexes for crystallographic point groups, rod groups and layer groups. - Z. Kristallogr. 147, 255 - 273.
Different complexes in R3 may share point configurations. Such point configurations in common necessarily give rise to limiting-form relations between complexes: a complex A, the comprehensive complex, contains all point configurations of a second complex B, which is a limiting form of A. Complete tables of the comprehensive complexes and of the limiting forms are given for all point complexes, row complexes and net complexes in R3. For these, the limiting forms have been derived by means of group-subgroup relations. The corresponding two-coloured groups have been used to facilitate this procedure.
E. Koch & W. Fischer (1978): Complexes for crystallographic point groups, rod groups and layer groups. - Z. Kristallogr. 147, 21 - 38.
Point complexes, row complexes and net complexes in three-dimensional space are defined - by analogy to lattice complexes - as sets of point configurations in crystallographic point groups, rod groups and layer groups, respectively. For this the point configurations are united stepwise via point-positions and configuration-sets. With subperiodic groups, however, affine isomorphisms and affine automorphisms rather than isomorphisms and automorphisms of groups have to be used as equivalence relations. The assignment of all the configuration-sets of crystallographic point groups, rod groups and layer groups to point complexes, row complexes and net complexes, respectively, is given explicitly by tables.
W. Fischer & E. Koch (1974b): Kubische Strukturtypen mit festen Koordinaten. - Z. Kristallogr. 140, 324 - 330.
The concept of "cubic structure types with fixed coordinates" is defined as an extension of A. NIGGLI'S (1971, Z. Kristallogr. 133, 473-490) "nonvariant cubic structure types". For this, configurations of invariant cubic lattice complexes are used additionally when occurring in cubic site sets with degrees of freedom. For each cubic space group these configurations are tabulated with their relative positions. Sample structures are given for some structure types, which are covered by the extended definition only.
E. Koch (1974): Die Grenzformen der kubischen Gitterkomplexe. - Z. Kristallogr. 140, 75 - 86.
Special point configurations of a lattice complex may show a symmetry higher than the symmetry of the lattice complex. Using subgroup relations of space groups all these special point configurations of cubic lattice complexes have been deduced as "limiting forms".
W. Fischer & E. Koch (1974a): Eine Definition des Begriffs "Gitterkomplex". - Z. Kristallogr. 139, 268 - 278.
Using some concepts of set theory and of group theory a lattice complex (Gitterkomplex) is defined as a set of point configurations. For this the difference between space groups and classes of space groups has to be considered. In order to define lattice complexes point configurations are combined stepwise to form sets of equivalent points, "Konfigurationslagen", and classes of these. Some properties of lattice complexes are explained and possibilities of using this concept of lattice complexes are discussed.
H. Burzlaff, W. Fischer, E. Hellner & A. Niggli (1974): Zur Entwicklung des Begriffs "Gitterkomplex". - Z. Kristallogr. 139, 246 - 251.
W. Fischer, H. Burzlaff, E. Hellner & J.D.H. Donnay (1973): Space Groups and Lattice Complexes. - National Bureau of Standards Monograph 134, Washington.
The lattice complex is to the space group what the site set is to the point group - an assemblage of symmmetry-related equivalent points. The symbolism introduced by Carl Hermann has been revised and extended. A total of 402 lattice complexes are derived from 67 Weissenberg complexes. The Tables list site sets and lattice complexes in standard and alternate representations. They answer the following questions: What are the co-ordinates of the points in a given lattice complex? In which space groups can a given lattice complex occur? What are the lattice complexes that can occur in a given space group? The higher the symmetry of the crystal structures is, the more useful the lattice-complex approach should be on the road to the ultimate goal of their classification.
E. Koch & E. Hellner (1971): Die Pattersonkomplexe der Gitterkomplexe. - Z. Krist. 133, 242 - 259.
Sets of Patterson vectors have been evaluated for all lattice complexes. By simple transformations the vector sets for all sets of equivalent points within the 230 space groups can be derived. Only one pair of lattice complexes turned out to be completely homometric.
H. Burzlaff, W. Fischer & E. Hellner (1969): Bemerkungen zu "Die Gitterkomplexe der Ebenengruppen" (Acta Cryst. A24, 57) (1968) und "Kreispackungsbedingungen in der Ebene" (Acta Cryst. A24, 67) (1968). - Acta Crystallogr. A25, 710.
The authors' work on the lattice complexes of plane groups is related to the earlier publications of N. L. Smirnova and her co-workers.
H. Burzlaff, W. Fischer & E. Hellner (1968): Die Gitterkomplexe der Ebenengruppen. - Acta Crystallogr. A24, 57 - 67.
The concept of lattice complexes due to C. Hermann is applied to the sets of equivalent points in planar groups. Based on bis nomenclature for invariant lattice complexes a symbolism is derived with the use of symbols for point complexes and their arrangement in addition. It is possible to construct each lattice complex from its symbol. The concept can be applied to space groups. All of these lattice complexes are tabulated together with their sets of equivalent points. Some applications are discussed. The efficiency of the conception is demonstrated by tables containing the relations between sets of equivalent points in different planar groups.
E. Hellner & E. Koch (1982): The garnet-like cyanide framework of ammonium ferrocyanide hydrate with a channel system for ionic conductivity. - Acta Crystallogr. B38, 376 - 379.
The CN dumb-bells in the cyanide framework of ammonium ferrocyanide hydrate occupy almost exactly the O positions in the garnet structure. In contrast to other crystal structures with a garnet-like framework all large voids are filled up. Two interpenetrating systems of channels are formed by the voids with centers at 16(b) 1/8,1/8,1/8, i.e. a Y** configuration, and at 24(c) 0,1/4,1/8, i.e. a V* configuration. These voids are occupied by one quarter of the NH4+ ions and by the H2O molecules, respectively.
E. Hellner, E. Koch & A. Reinhardt (1981): The homogeneous frameworks of the cubic crystal structures. - Physik Daten - Physics Data 16-2, 1 - 67.
Most of the sixteen invariant cubic lattice complexes correspond to anion frameworks in ionic compounds or to basic frameworks in intermetallic compounds. In addttion 22 further homogeneous anion frameworks have been found in cubic crystal structures. This number may be compared with the number of 198 different types of homogeneous sphere packings which exist in total with cubic symmetry. For the sixteen invariant lattice complexes as well as for the 22 other homogeneous frameworks informations on their sphere-packing properties and on their voids are tabulated and discussed in some detail. Structural examples are given, if possible.
E. Hellner & E. Koch (1981): Cluster or framework considerations for the structures of Tl7Sb2, alpha-Mn, Cu5Zn8 and their variants Li22Si5, Cu41Sn11, Sm11Cd45, Mg6Pd and Na6Tl with octuple unit cells. -Acta Crystallogr. A37, 1 - 6.
Eight intermetallic compounds described so far by clusters are compared. It is shown that a common framework exists for all these crystal structures. The voids within this common part are filled in different ways. Moreover, for each of these crystal structures all atoms together form a three-dimensional connected framework. Therefore, the framework description seems to be more adequate than the cluster description.
E. Koch & E. Hellner (1981): The frameworks of sodalite-like structures and of tetrahedrite-like structures. - Z. Kristallogr. 154, 95 - 114.
Crystal structures containing atoms at the mirror planes of I-43m or in related positions of subgroups of I-43m are discussed. The frameworks of these crystal structures are either heterogeneous or they are homogeneous. In the latter case their frameworks are built up from nearly ideal tetrahedra with common vertices around a point configuration of the cubic invariant lattice complex W*. The geometrical properties of such frameworks are discussed in dependence with the position of the framework atoms. Two types of frameworks, W*(4tc) and [F2''-I(4t)], referring to sodalite-like structures and tetrahedrite-like structures, respectively, are sufficient to describe all these crystal structures with homogeneous frameworks.
E. Hellner & E. Koch (1980): A comparison of the crystal structures of Sb2Tl7, Cu5Zn8 ( gamma-brass) and Ir3Ge7. - Can. J. Chem. 58, 708 - 713.
The crystal structures of Sb2Tl7, Cu5Zn8, and Ir3Ge7 are discussed with the aid of shortest distances of Dirichlet domains and of voids in their frameworks. It will be pointed out that all three frameworks must be regarded as different though relations between the atomic parameters are obvious. The Sb2Tl7 structure represents a superstructure of an I lattice. The frameworks of Sb2Tl7 and of Cu5Zn8 contain tetrahedral voids only, while in Ir3Ge7 voids with a larger coordination number also exist.
E. Hellner & E. Koch (1979): The oxygen framework of leucite and analcime. - Miner. Petrogr. Acta 23, 303 - 311.
In leucite and analcime the Si and Al atoms together form a distorted [P'4-Y**]xx configuration. The corresponding idealized configuration [P'4-Y**] together with the configuration Y** of the potassium ions in leucite or the H2O molecules in analcime may be regarded as a cubic primitive lattice P'4 which is shifted against the origin by (1/8,1/8,1/8) and has a translation period of only one quarter the edge length of the unit cell. The Si and Al atoms form a nearly ideal sphere packing with four contacts per sphere. The oxygen atoms build up corner-connected tetrahedra around [P'4-Y**]xx and correspond to a sphere packing with six contacts per sphere. The framework of the oxygen atoms contains ten different kinds of voids if one regards the first coordination shells only. Three of them can be added to the octahedra around Y** resulting in icosahedra. These icosahedra overlap in very flat tetrahedra around V*. In this way two interpenetrating systems of channels are built up.
E. Hellner, R. Gerlich, E. Koch & W. Fischer (1979): The oxygen framework in garnet and its occurence in the structures of Na3Al2Li3F12, Ca3Al2(OH)12, RhBi4 and Hg3TeO6. - Physik Daten - Physics Data 16-1, 1 - 31.
The oxygen framework I222[6o] of garnet with symmetry Ia3d is derived from the boron framework P[6o] of CaB6 with symmetry Pm3m. Two frameworks P[6o] and P'[6o] which are shifted by the vector (1/2 1/2 1/2) against each other are united to I<6o> with symmetry Im3m. I<6o> is considered as an idealization of the garnet framework in a unit cell with a' = 1/2 agarnet. The behaviour of the voids within the framework during the transition from I<6o> to the garnet framework I222[6o] is studied. It is pointed out that in addition cryolithionite, hydrogarnet, a-RhBi4 and mercury tellurate crystallize with the garnet framework. The self-coordination number and the connection of coordination polyhedra are discussed for this framework. Relations between the oxygen parameters, the lattice constants and the radii of the cations in the structures with the garnet framework are shown.
P.M. de Wolff, Y. Billiet, J.D.H. Donnay, W. Fischer, R.B. Galiulin, A.M. Glazer, Th. Hahn, M. Senechal, D.P. Shoemaker, H. Wondratschek, A.J.C. Wilson & S.C. Abrahams (1992): Symbols for symmetry elements and symmetry operations. Final report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry. - Acta Crystallogr. A48, 727 - 732.
New or redefined printed symbols are proposed in the light of the recently accepted redefinition of symmetry elements [de Wolff et al. (1989). Acta Cryst. A45, 494-499]. In particular, the letter e covers certain glide planes which hitherto had no unique symbol, such as those called 'either a or b'. The use of e in the Hermann-Mauguin symbol of five different space groups is recommended. For e planes projected in a direction parallel to the plane, a graphical symbol is proposed which removes the ambiguity of their present designation. The letter k is proposed for a newly defined class of glide planes which until now were without specific symbol. The symbols for symmetry operations introduced in the space-group descriptions of International Tables for Crvstallography (1989), Vol. A (Dordrecht: Kluwer Academic Publishers) are recommended for general use, with modifications only for glide reflection operations.
P.M. de Wolff, Y. Billiet, J.D.H. Donnay, W. Fischer, R.B. Galiulin, A.M. Glazer, M. Senechal, D.P. Shoemaker, H. Wondratschek, Th. Hahn, A.J.C. Wilson & S.C. Abrahams (1989): Definition of symmetry elements in space groups and point groups. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry. - Acta Crystallogr. A45, 494 - 499.
A 'geometric element' is defined, for any given symmetry operation, as a geometric item that allows the operation (after removing any intrinsic translation) to be located and oriented. In the case of an inversion, a (screw-) rotation or a (glide-) reflection, it is respectively a point, line or plane. In the case of a rotoinversion, the geometric element consists of the axis of the rotation part and the center of the inversion part. As a general concept, the geometric element may be justified by a mathematical definition (as given in the Appendix). A 'symmetry element' (of a given crystal structure or object) is defined as a concept with a double meaning, namely the combination of a geometric element with the set of symmetry operations having this geometric element in common ('element set'). There is no overlap between element sets of a given structure. Together with the identity and the translations, for which a geometric element is not defined, the element sets cover all symmetry operations.
P.M. de Wolff, N.V. Belov, E.F. Bertaut, M.J. Buerger, J.D.H. Donnay, W. Fischer, Th. Hahn, V.A. Koptsik, A.L. Mackay, H. Wondratschek, A.J.C. Wilson & S.C. Abrahams (1985): Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry. - Acta Crystallogr. A41, 278 - 280.
Last update: July 2007