Chapter 9.1. Sphere packings and packings of ellipsoids

For the characterization of many crystal structures, geometrical aspects have proved to be a useful tool. Among these, sphere-packing considerations stand out in particular.

A sphere packing in the most general sense is an infinite, three-periodic set of non-intersecting spheres (i.e. a set of non-intersecting spheres with space-group symmetry) with the property that any pair of spheres is connected by a chain of spheres with mutual contact. If all spheres are symmetry-equivalent, the sphere packing is called homogeneous, otherwise it is called heterogeneous.

A homogeneous sphere packing may be represented uniquely by the set of symmetry-equivalent points that are the centres of the spheres [point configuration, cf. ITA (2005, Section 14.1.2 )]. This point configuration is distinguished by equal shortest distances giving rise to a connected graph. As all spheres of a homogeneous sphere packing must be equal in size, their common radius can be calculated as half this shortest distance.

A heterogeneous sphere packing consists of at least two symmetry-distinct subsets of spheres, the centres of which form a respective number of point configurations. The radii of symmetrically distinct spheres can be either equal or different. In the first case, the heterogeneous sphere packing may be represented by its set of sphere centres, quite similar to a homogeneous one. In the case of different sphere radii, however, the knowledge of at least some of the radii is additionally necessary.

As there exists an infinite number of both homogeneous and heterogeneous sphere packings, it is convenient to classify the sphere packings into types: two sphere packings belong to the same type if there exists a biunique mapping that brings the spheres of one packing onto the spheres of the other packing and that preserves all contact relations between spheres.

The number of types of homogeneous sphere packings is finite whereas the number of types of heterogeneous sphere packings is infinite.

All definitions and properties mentioned so far may be transferred from sets of spheres in three-dimensional space to sets of circles in two-dimensional space, giving rise to heterogeneous and homogeneous packings of circles.

A characteristic property of types of homogeneous sphere (circle) packings is the number k of contacts per sphere (circle): for sphere packings and for packings of circles.

A sphere (circle) packing is called stable [close, cf. ITII (1972, Chapter 7.1)] if no sphere (circle) can be moved without moving neighbouring spheres (circles) at the same time. As a consequence, a stable sphere (circle) packing has at least four (three) contacts per sphere (circle), and not all these contacts must fall in one hemisphere (semicircle).

The density of a homogeneous sphere (circle) packing is defined as the fraction of volume (area) occupied by spheres (circles). It may be calculated as for sphere packings, and as for packings of circles. Here, r is the radius of the spheres (circles), n the number of spheres (circles) per unit cell, V the unit-cell volume, and A the unit-cell area.

Geometric properties of different sphere (circle) packings of the same type may be different. Such properties are, e.g., the density and the property of being a stable packing.

The homogeneous packings of circles in the plane may be classified into 11 types (cf. Niggli, 1927, 1928; Haag, 1929, 1937; Sinogowitz, 1939; Fischer, 1968; Koch & Fischer, 1978). These correspond to the 11 types of planar nets with equivalent vertices derived by Shubnikov (1916). If, in addition, symmetry is used for classification, the number of distinct cases becomes larger (31 cases according to Sinogowitz, 1939).

Table 9.1.1.1 gives a summary of the 11 types. In column 1, the type of circle packing is designated by a modified Schläfli symbol that characterizes the polygons meeting at one vertex of a corresponding Shubnikov net. The contact number k is given in column 2. The next column displays the highest possible symmetry for each type of circle packing. The corresponding parameter values are listed in column 4. The appropriate shortest distances d between circle centres and densities ρ are given in columns 5 and 6, respectively.

Table 9.1.1.1| top | pdf | Types of circle packings in the plane Type k Symmetry Parameters Distance d Density 36 6 p6mm 1(a) 0, 0   a 0.9069 32434 5 p4gm 4(c) 0.8418 3342 5 c2mm 4(d) x, 0 b 0.8418 346 5 p6 6(d) x, y 0.7773 44 4 p4mm 1(a) 0, 0   a 0.7854 3464 4 p6mm 6(e) 0.7290 3636 4 p6mm 3(c)   0.6802 63 3 p6mm 2(b)   0.6046 482 3 p4mm 4(d) x, 0 0.5390 46.12 3 p6mm 12(f) x, y 0.4860 3.122 3 p6mm 6(c) 0.3907

With three exceptions (3⁶, 3⁴6, 46.12), all types include circle packings that are not similar in the mathematical sense and that differ, therefore, in their geometrical properties. The highest possible symmetry for a type of homogeneous circle packing corresponds necessarily to the lowest possible density ρ of that type. Therefore, homogeneous circle packings of type 3.12² with symmetry p6mm are the least dense. The highest possible density is achieved by the circle packings with contact number 6 referring to triangular nets with hexagonal symmetry.

All circle packings described in Table 9.1.1.1 are stable in the sense defined above. Only circle packings of types 3.12² and 48² may be unstable.

The number of homogeneous sphere-packing types is not known so far. Sinogowitz (1943) systematically derived sphere packings with non-cubic symmetry from planar sets of spheres, but he did not compare sphere packings with different symmetry and classify them into types. Fischer calculated the para­meter conditions for all cubic (Fischer, 1973, 1974) and all tetragonal (Fischer, 1991a,b, 1993) sphere packings. 199 types of homogeneous sphere packings with cubic symmetry and 394 types with tetragonal symmetry exist in all. 12 of these types are common to both systems. In a similar way, Zobetz (1983) calculated the sphere-packing conditions for Wyckoff position 6(c). Using a different approach, Koch & Fischer (1995) derived all types of homogeneous sphere packings with contact number k = 3. Because of the unique correspondence of each homogeneous sphere packing to a graph, studies on three-dimensional nets also give contributions to the knowledge on sphere-packing types. In particular, papers by Wells (1977, 1979, 1983), O'Keeffe (1991, 1992), O'Keeffe & Brese (1992) and Treacy, Randall, Rao, Perry & Chadi (1997) contain some information on sphere packings with k = 3 and k = 4.

Table 9.1.1.2 shows examples for sphere packings with high contact numbers and high densities in the upper part and with small contact numbers and low densities in the lower part. Column 1 gives reference numbers to designate the types in the following. Column 2 displays the contact numbers k. The highest possible symmetry for each type is described in column 3. Coordinates and metrical parameters referring to the most regular sphere packings of each type are listed in column 4; the respective shortest distances d between sphere centres are given in column 5. For a sphere packing that can be subdivided into plane nets of spheres with mutual contact, the direction and the type of these nets are shown in column 6. Column 7 contains stacking information: the contact numbers to the nets above and below, and the number of layers per translation period in the direction perpendicular to the layers. The last column displays the density with respect to the parameters of column 4. For all cases, this value gives the minimal density for that type of sphere packing.

Table 9.1.1.2| top | pdf | Examples for sphere packings with high contact numbers and high densities and with low contact numbers and low densities Type k Symmetry Parameters Distance d Net Stacking Density 1 12 P63/mmc 2(c) a (001) 36 3, 3 2 0.7405 2 12 4(a) 0, 0, 0 – {111} 36 3, 3 3 {001} 44 4, 4 2 3 11 Cmca 8(f) 0, y, z a (001) 36 3, 2 4 0.7187 4 11 P3121 6(c) x, y, z a (001) 36 3, 2 6 5 11 Fdd2 16(b) x, y, z c (010) 36 3, 2 8 6 11 P6522 12(c) x, y, z a (001) 36 3, 2 12 7 11 C2/m 4(i) x, 0, z b (001) 36 3, 2 12 8 11 P42/mnm 4(f) x, x, 0 c – – 0.7187 9 10 I4/mmm 2(a) 0, 0, 0 c {110} 36 2, 2 2 0.6981 10 10 P6222 3(c) a (001) 36 2, 2 3 11 10 Fddd 8(a) 0, 0, 0 a (001) 36 2, 2 4 12 10 Fddd 16(g) a (001) 36 2, 2 8 13 10 Cmcm 4(c) (001) 44 3, 3 2 0.6981 14 10 Pnma 4(c) c (010) 44 3, 3 2 15 10 P63/mmc 4(f) a (001) 36 3, 1 4 0.6657 16 10 6(c) 0, 0, z a (001) 36 3, 1 6 17 10 Cmcm 4(c) a (010) 44 4, 2 4 0.6657 18 10 I41/amd 8(e) 0, 0, z a (001) 44 4, 2 8 19 10 I4/m 8(h) x, y, 0 c – – 0.6619 20 10 18(f) x, y, z (001) 346 3, 2 3 0.6347   21 4 32(e) x, x, x – – 0.1235 22 4 48(j) 0, y, z – – 0.1033 23 4 I4132 48(i) x, y, z – – 0.0789 24 3 I4132 24(h) – – 0.0555

The densest homogeneous sphere packings known so far may be derived from the densest packings of circles (3⁶ in Table 9.1.1.1). Such sphere packings can always be subdivided into parallel plane layers of spheres with six contacts per sphere within each layer and with three contacts to each of the neighbouring layers above and below (cf. Fig. 9.1.1.1 ). Consequently, the contact number k becomes 12. As there exist two stacking possibilities for each layer with respect to the previous layer, infinitely many stacking sequences can be derived in principle, but only two refer to homogeneous sphere packings. If for each layer the two neighbouring layers are stacked directly upon each other, a sphere packing of a two-layer type with hexagonal symmetry (type 1) results. It is called hexagonal closest packing (abbreviated h.c.p.). If for all layers the neighbouring layers are never stacked directly upon each other, a sphere packing of a three-layer type with cubic symmetry (type 2) is formed. It is designated cubic closest packing (c.c.p.). In spite of these terms, for a long time it was only known that the cubic closest packings are the densest ones that correspond to lattices (Minkowski, 1904). Only recently, Hsiang (1993) published a proof that there does not exist any packing of spheres of equal size with a higher density, but the completeness of this proof is still doubted (cf. e.g. Hales, 1994).

Figure 9.1.1.1| top | pdf | Two triangular nets representing two densest packed layers of spheres. The layers are stacked in such a way that each sphere is in contact with three spheres of the other layer.

Independently of the stacking sequences, closest packings of spheres contain ideal octahedral and ideal tetrahedral voids. The number of octahedra per unit cell equals the respective number of spheres, whereas the number of tetrahedral voids is twice as large. The distances between the centres and the vertices of these voids are and , respectively. Within a cubic closest packing, faces are shared only between octahedral and tetrahedral voids. Each edge is common to two octahedra and two tetrahedra. In contrast, piles of face-sharing octahedra are formed within a hexagonal closest packing, whereas the tetrahedra are arranged as pairs with one face in common. The other faces are shared between octahedra and tetrahedra. Again, each edge belongs to two octahedra and two tetrahedra.

Densest layers of spheres may also be stacked such that each sphere is in contact with two spheres of the previous layers (cf. Fig. 9.1.1.2 ). Such a stacking results in contact number 10. Again, infinitely many periodic stacking sequences are possible, but only four give rise to homogeneous sphere packings [types 9, 10, 11: cf. Hellner (1986); type 12: cf. O'Keeffe (1988)]. In the most symmetrical forms of these four cases, each sphere is located exactly above or below the middle of two neighbouring spheres of the adjacent layers. This kind of stacking gives rise to distorted tetrahedral voids only. The number of tetrahedra per unit cell is six times the number of spheres. Two kinds of differently distorted tetrahedra exist in the ratio 1:2. The two-layer type 9 corresponds to a tetragonal body-centred lattice with specialized axial ratio.

Figure 9.1.1.2| top | pdf | Two triangular nets representing two densest packed layers of spheres. The layers are stacked in such a way that each sphere is in contact with two spheres of the other layer.

Furthermore, densest layers of spheres may be stacked in a mixed sequence with three contacts per sphere to one neighbouring layer and two contacts to the other layer. This kind of stacking results in five types of homogeneous sphere packings (3 to 7) with contact number 11.

Two other types of homogeneous sphere packings (15 and 16) with contact number k = 10 also refer to densest layers of spheres. In these cases, each sphere has three contacts to one neighbouring layer and one contact to the other layer that is stacked directly above or below the original layer.

Cubic closest packings may also be regarded as built up from square layers 4⁴ stacked in such a way that each sphere has four neighbouring spheres in the same layer and four neighbours each from the layers above and below (cf. Fig. 9.1.1.3 ). If square layers are stacked such that each sphere has contact to four spheres of one neighbouring layer and to two spheres of the other layer (cf. Fig. 9.1.1.4 ), sphere packings with contact number 10 result. In total, two types of homogeneous packings (17 and 18) with this kind of stacking exist. Sphere packings of type 9 may also be decomposed into 4⁴ layers parallel to (101) or (011) in a five-layer sequence. These nets are made up from parallel rhombi and stacked such that each sphere has contact with three other spheres from the layer above and from the layer below. If such layers are stacked in a two-layer sequence, sphere packings of type 13 with symmetry Cmcm result (O'Keeffe, 1998). Sphere packings of type 14 are also build up from 4⁴ layers, but here the rhombi occur in two different orientations (O'Keeffe, 1998). Sphere packings with high contact numbers may also be derived by stacking of other layers. Type 20, for example, refers to 3⁴6 layers where each sphere is in contact with three spheres of one neighbouring net and two spheres of the other one (Sowa & Koch, 1999). Such a sphere packing may alternatively be derived from the cubic closest packing by omitting systematically 1/7 of the spheres in each of the 3⁶ nets.

Figure 9.1.1.3| top | pdf | Two square nets representing two layers of spheres stacked in such a way that each sphere is in contact with four spheres of the other layer.

Figure 9.1.1.4| top | pdf | Two square nets representing two layers of spheres stacked in such a way that each sphere is in contact with two spheres of the other layer.

Sphere packings of types 8 and 19 (cf. Figs. 9.1.1.5 and 9.1.1.6) cannot be built up from plane layers of spheres in contact although their contact numbers are also high.

Figure 9.1.1.5| top | pdf | Sphere packing of type 8 (Table 9.1.1.2) represented by a graph: k = 11, P42/mnm, 4(f), xx0.

Figure 9.1.1.6| top | pdf | Sphere packing of type 19 (Table 9.1.1.2) represented by a graph: k = 10, I4/m, 8(h), xy0.

Table 9.1.1.2 contains complete information on homogeneous sphere packings with k = 10, 11, and 12 and with cubic or tetragonal symmetry.

The least dense (most open) homogeneous sphere packings known so far have already been described by Heesch & Laves (1933). Sphere packings of that type (24) cannot be stable because their contact number is 3 (cf. Fig. 9.1.1.7 ). As discussed by Fischer (1976), it is very probable that no homogeneous sphere packings with lower density exist; those discussed by Melmore (1942a,b) with ρ = 0.042 and ρ = 0.045 are heterogeneous ones. Recently, Koch & Fischer (1995) proved that the Heesch–Laves packing is the least dense homogeneous sphere packing with three contacts per sphere.

Figure 9.1.1.7| top | pdf | Least dense sphere packing known so far (type 24 of Table 9.1.1.2) represented by a graph: k = 3, I4132, 24(h), , y, − y. z coordinates given in multiples of 1/100.

The least dense sphere packings with contact number 4 derived so far are described as type 23. All sphere packings of this type are similar in the geometrical sense and are not stable. In contrast, the sphere packings of type 22 are stable. Sphere packings of type 21 (Heesch & Laves, 1933), which have been supposed to be the most open stable ones (cf. Hilbert & Cohn-Vossen, 1932, 1952), have a slightly higher density.

On the basis of the material known at that time, Slack (1983) tried to develop empirical formulae for the minimal and the maximal density of circle packings and sphere packings depending on the contact number. A paper by O'Keeffe (1991) on four-connected nets pays special attention to the densest and the least dense sphere packings with four contacts per sphere.

Sphere packings have been used for the description of inorganic crystal structures in different ways and by several authors (e.g. Brunner, 1971; Figueiredo & Lima-de-Faria, 1978; Frank & Kasper, 1958; Hellner, 1965; Hellner, Koch & Reinhardt, 1981; Koch, 1984, 1985; Laves, 1930, 1932; Lima-de-Faria, 1965; Lima-de-Faria & Figueiredo, 1969a,b; Loeb, 1958; Morris & Loeb, 1960; Niggli, 1927; Smirnova, 1956a,b, 1958a,b, 1959a,b,c, 1964; Sowa, 1988, 1997).

In the simplest case, the structure of an element may be described as a sphere packing if all atoms are interrelated by equal or almost equal shortest distances. This does not imply that the atoms really have to be considered as hard spheres of that size. Often such sphere packings are homogeneous ones with a high contact number k (e.g. Cu, Mg with k = 12; Pa with k = 10; W with k = 8). Low values of k (e.g. diamond with k = 4, white tin with k = 6) and heterogeneous sphere packings (La with k = 12) have also been observed for structures of elements.

Crystal structures consisting of different atoms may be related to sphere packings in different ways:

Because of their importance for problems in digital communication (error-correcting codes) and in number theory (solving of diophantine equations), densest sphere packings in higher dimensions are of mathematical interest (cf. Conway & Sloane, 1988).

Special homogeneous or heterogeneous sets of spheres may be subdivided into a small number i of subsets such that each subset, regarded by itself, forms a sphere packing and that spheres of different subsets do not have mutual contact. Sets of spheres with these properties are called interpenetrating sphere packings.

The cubic Laves phases are a well known example for heterogeneous interpenetrating sphere packings. The Mg atoms in MgCu₂ [, 8(a)], for example, correspond to a sphere packing with shortest distances and contact number k = 4 whereas the copper atoms [16(d)] refer to another sphere packing with shortest distances and k = 6. The shortest distances between centres of different spheres are .

The crystal structure of Cu₂O gives an example of a different kind. If one takes into account the size of the atoms, sphere contacts can only be expected between different spheres. As a consequence, the heterogeneous set of spheres disintegrates into two heterogeneous but congruent subsets with no mutual contact.

In the case of homogeneous interpenetrating sphere packings, all i subsets have to be symmetry-equivalent. Then the symmetry of each subset is a subgroup of index i of the original space group. Homogeneous interpenetrating sphere packings with cubic symmetry have been derived completely by Fischer & Koch (1976). They may be classified into 39 types. For 33 of the 39 types, the number i of subsets is 2; i is 3, 4, and 8 for 1, 3, and 2 types, respectively.

Remarkable are those homogeneous interpenetrating sphere packings that are built up from sphere packings of type 24 (Table 9.1.1.2), i.e. that type with the least dense sphere packing. Combinations of 2, 4, or 8 such sphere packings result in altogether 8 different types of interpenetrating sphere packings (Fischer, 1976). The P atoms in the crystal structure of Th₃P₄ give an example for such interpenetrating sphere packings built up from two congruent subsets (Koch, 1984).

Complete results for other crystal systems are not available. With tetragonal symmetry, interpenetrating sphere packings are known, built up from 2, 3, or 5 congruent subsets (Fischer, 1970). Analogous interpenetration patterns are formed by hydrogen bonds within certain molecular structures (Ermer, 1988; Ermer & Eling, 1988).

Interpenetrating sphere packings may be brought in relation to interpenetrating labyrinths as formed by periodic minimal surfaces or by periodic zero-potential surfaces without self-intersection (cf. e.g. Andersson, Hyde & von Schnering, 1984; Fischer & Koch, 1987, 1996; von Schnering & Nesper, 1987).