Homogeneous Sphere Packings Research Articles

Monoclinic sphere packings. III. Trivariant lattice complexes of P2/c and P21/c.

All homogeneous sphere packings were derived that refer to the trivariant lattice complexes of monoclinic space-group types P2/c and P21/c. In total, sphere packings of 55 types have been found. The maximal inherent symmetry is monoclinic for 17 types while the other types comprise at least one sphere packing with cubic (four cases), hexagonal (six cases), tetragonal (eight cases) or orthorhombic (20 cases) symmetry.

Monoclinic sphere packings. II. Trivariant lattice complexes with mirror symmetry.

All homogeneous sphere packings were derived that refer to the three trivariant monoclinic lattice complexes with mirror symmetry. In total, 29 types of sphere packings have been found. Only for three types is the maximal inherent symmetry of their sphere packings monoclinic whereas the inherent symmetry is orthorhombic for 13 types, tetragonal for eight types, hexagonal for four types and cubic for one type.

Sorption Kinetics of CH4 and CO2 Diffusion in Coal: Theoretical and Experimental Study

Experimental and theoretical analyses with empirical correlations were framed for diffusion of gas species CH4 and CO2 in coal samples from Jharia coal fields, India, considering the intrinsic pore parameters. Coefficient of diffusion (D) and diffusivity (Deff) for a single and binary component coal–gas system were estimated by adopting unipore gas kinetic models for gas flow on the integration of Fick’s law and Langmuir relation. The rigorous study was carried out in estimating crossover pressure, which is dominant in distinguishing the flow regime for two primary types of diffusion: Knudsen and molecular as well as the transition between two regimes. Investigation reveals that experimental values of coefficients of diffusion of CH4 and CO2 in random homogeneous isotropic sphere packing of coal samples are in good agreement with the results of theoretical calculations. For the pressure range investigated, variation of coefficient of diffusion was found to follow a dual nature with a stable trend at press...

Monoclinic sphere packings. I. Invariant, univariant and bivariant lattice complexes.

All homogeneous sphere packings were derived that refer to the two invariant, the four univariant and the three bivariant lattice complexes belonging to the monoclinic crystal system. In total, sphere packings of 29 types have been found. Only for five types is the maximal inherent symmetry of their sphere packings monoclinic whereas the inherent symmetry is orthorhombic for nine types, tetragonal for five types, hexagonal for six types and cubic for four types.

Orthorhombic sphere packings. V. Trivariant lattice complexes of space groups belonging to crystal class 222

This paper completes the derivation of all types of homogeneous sphere packing with orthorhombic symmetry. The nine orthorhombic trivariant lattice complexes belonging to the space groups of crystal class 222 were examined in regard to the existence of homogeneous sphere packings and of interpenetrating sets of layers of spheres. Altogether, sphere packings of 84 different types have been found; the maximal inherent symmetry is orthorhombic for only 36 of these types. In addition, interpenetrating sets of 63nets occur once. All lattice complexes with orthorhombic characteristic space group give rise to 260 different types of sphere packing in total. The maximal inherent symmetry is orthorhombic for 160 of these types. Sphere packings of 13 types can also be generated with cubic, those of seven types with hexagonal and those of 80 types with tetragonal symmetry. In addition, ten types of interpenetrating sphere packing and two types of sets of interpenetrating sphere layers are obtained. Most of the sphere packings can be subdivided into layer-like subunits perpendicular to one of the orthorhombic main axes.

Orthorhombic sphere packings. IV. Trivariant lattice complexes of space groups without mirror planes belonging to crystal classmmm

All homogeneous sphere packings and all interpenetrating layers of spheres were derived that can be realized in the ten orthorhombic trivariant lattice complexes belonging to the space groups of crystal class mmm without mirror symmetry. Altogether, sphere packings of 186 different types have been found; the maximal inherent symmetry is orthorhombic for 124 of these types. In addition, ten types of interpenetrating sphere packings were detected, and in three lattice complexes interpenetrating 6(3) nets occur.

Orthorhombic sphere packings. III. Trivariant lattice complexes with mirror symmetry

All homogeneous sphere packings and all interpenetrating layers of spheres were derived that refer to the 18 orthorhombic trivariant lattice complexes with mirror symmetry. In total, sphere packings of 51 different types have been found. Only for 28 of these types is the maximal inherent symmetry of their sphere packings orthorhombic. Some crystal structures that can be described by means of sphere packings are listed.

Orthorhombic sphere packings. II. Bivariant lattice complexes

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 is the maximal inherent symmetry of their sphere packings orthorhombic. Some examples demonstrate the usefulness of sphere packings for the comparison and description of crystal structures.

Orthorhombic sphere packings. I. Invariant and univariant lattice complexes

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.

Hexagonal and trigonal sphere packings. IV. Trivariant lattice complexes of trigonal space groups

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. Some examples depict crystal structures that can be described by means of sphere packings.

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.

Tetragonal sphere packings: minimal densities and subunits

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.

On the density of homogeneous sphere packings

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.

Hexagonal and trigonal sphere packings. III. Trivariant lattice complexes of hexagonal space groups

All types of homogeneous sphere packing 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 4.8(2) nets of spheres with 120 degrees 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 are given for crystal structures that can be described by means of sphere packings.

Minimal densities of cubic sphere-packing types.

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.

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.

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.

Relations between the zinc-blende and the NaCl structure type.

A new model for the transformation from the zinc-blende to the NaCl type is developed on the basis of the deformation of a 4-connected heterogeneous sphere packing in space group P3(2). The geometrical features of this transition are discussed by investigating the analogous deformations of a 4-connected homogeneous sphere packing in the supergroup P3(2)21 of P3(2). In this space group, the atomic arrangement corresponds to a diamond configuration. During the phase transition, it is deformed under preservation of all sphere contacts until a cubic primitive lattice is reached, the analogue of the NaCl arrangement formed by like atoms. The intermediate structure that occurs between the zinc-blende and the NaCl type is compared with the cinnabar type.

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.

Group-theoretical and geometrical considerations of the phase transition between the high-temperature polymorphs of quartz and tridymite

A model was derived for the temperature-dependent phase transition between the high-temperature polymorphs of quartz (P6(4)22) and tridymite (P6(3)/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 P6(1)22 of P6(4)22 and P6(3)/mmc. The proposed model guarantees the three-dimensional connection of the crystal structure during the whole transformation process.