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

Abstract

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.

On the density of homogeneous sphere packings

“…Asymmetric parameter regions. The one-dimensional parameter regions of the hexagonal sphere-packing type 4/4/h5 in P6 3 /mmc 12j xy 1 4 and in P6 3 22 12i xyz (Sowa & Koch, 2005a) do not show any symmetry. The ends of the parameter region at y = 0 and at y = 1 9 correspond to sphere packings of types 5/4/h5 with = 0.40307 and 6/3/h28 with = 0.35828, respectively.…”

“…The parameter region of sphere-packing type 3/6/h3 in P6 3 22 12i xyz (cf. Sowa & Koch, 2005a) represents an analogous two-dimensional example (Fig. 4).…”

“…Sphere-packing type 10/3/h2 exhibits an unusual onedimensional parameter field in the general position of P6 1 22 (cf. Sowa & Koch, 2005a Sphere-packing density as a function of the coordinate parameter y for sphere-packing type 4/4/h5 in P6 3 /mmc 12j xy 1 4 or P6 3 22 12i xyz.…”

“…The one-dimensional parameter region 0 < y < 0.06274 in P6 1 22 12c xyz of the hexagonal sphere-packing type 4/6/h14 (cf. Sowa & Koch, 2005a) is a typical example (Fig. 6).…”

Orthorhombic sphere packings. II. Bivariant lattice complexes

“…The derivation of the sphere packings closely follows the procedure used before for trigonal and hexagonal (Sowa et al, 2003;Sowa & Koch, 2004, 2005 and for orthorhombic sphere packings . The derivation of the sphere packings closely follows the procedure used before for trigonal and hexagonal (Sowa et al, 2003;Sowa & Koch, 2004, 2005 and for orthorhombic sphere packings .…”