Possible space groups

The p6mm symmetric single layer of closely spaced spheres has the sixfold rotation points at the centers of the spheres, positioned at the corners of the unit cell at (x,y) = (0,0); and threefold rotation points at the two non-equivalent triangular voids between three neighboring spheres at the fractional coordinates x,y of 2/3, 1/3 and 1/3, 2/3 (Fig. 2a), referred to as sites A, B, and C. The sixfold rotation axis also includes threefold rotations as some of its operations. Each of these sites lies at the intersection of six (in-the-plane) reflection lines oriented at 30° to each other.

Fig. 2

a Symmetry elements (green) of the hexagonal layer of spheres (blue) with the planar space group p6mm. b Relevant symmetry elements, shown in green, except for the threefold axes (triangles) which are colored according to the A/B/C sites, of a multilayer arrangement of spheres in space group P3m1 (the glide planes have been omitted for clarity). The three non-equivalent (with respect to the hexagonal unit cell, black) positions of spheres in the consecutive layers are marked at x,y of 0,0 (site A) in blue, at 2/3, 1/3 (B) in red, and at 1/3, 2/3 (C) in purple. In both panels, an outline of four unit cells is marked in black

To start the close packing in 3D, we cover the first layer, positioned arbitrarily at the A sites, with another layer nesting at either the B or C positions. The difference in height (z coordinate) between these two layers is √6/3 times the hexagonal cell parameter a, or 2√6/3 times the sphere radius r. In all possible hexagonal arrangements, the cell parameter c is always a multiple of a√6/3.

A third layer can be positioned either exactly above the first one (where the second layer also has new depressions) or above the depressions not yet occupied by the previous layer (where the second layer has its second set of depressions), Fig. 2b. If the mutual disposition of the consecutive layers is repeated indefinitely as in the former case, the resulting packing is HCP with the hexagonal symmetry P63/mmc. If the second disposition of sequential layers is repeated, it leads to CCP with the cubic space group Fm3m.

There is an important difference between the HCP packing, which is uniaxial, i.e., grows symmetrically only in one direction (c), and the CCP packing, which is quadro-axial, meaning that the stack of layers grows (with identical sequence) in four directions (along cube body diagonals) simultaneously, even if we continue the stacking operation in just one of these directions. In fact, the Fm3m quadro-axial case is quite exceptional, as all other possibilities (hexagonal or trigonal) are uniaxial.

The repeating sequence of layers can be of any length N > 1. We will consider only such sequences that form primitive periods, i.e., are not repetitive themselves. Another distinction among those possibilities is between palindromic (i.e., reading the same both ways) and non-palindromic (or polar) sequences. Of course, palindromic sequences of layers will possess mirror symmetry or horizontal twofold axes, while for polar non-palindromic sequences such symmetry is forbidden. Another pattern, which we call “semipalindromic”, is characterized by the coincidence of the A sites and interchange of the B and C sites after the reversal of the reading direction. The symmetry of all these arrangements is restricted to several hexagonal and trigonal space groups, except for the CCP packing, which is cubic Fm3m. The description of HCP as “hexagonal close packing” is, therefore, somewhat imprecise, since almost all other closely packed arrangements are also hexagonal.

The prevalence of hexagonal symmetry results from the hexagonal arrangement of the spheres in each individual layer. It has already been mentioned that all three possible positions of spheres in the hexagonal unit cell of a layer coincide with a point of threefold symmetry and an intersection of at least three reflection lines. No matter how the consecutive layers are stacked upon each other, the stack will always possess the aforementioned threefold axes and vertical mirror planes, even if the sequence of layers is completely random and non-periodic. If the sequence is periodic, it will, therefore, form at least the trigonal space group P3m1 (Fig. 2). Depending on the particular sequence of layers, the resulting symmetry may be higher, but it will always be a supergroup of P3m1.

The possible space groups of periodic sequences were listed by Krishna and Pandey [18] as trigonal/hexagonal P3m1, P\(\overline\)m1, P\(\overline\)m2, P63mc, P63/mmc, R3m, and R\(\overline\)m. They also mistakenly mentioned F\(\overline\)3m as the symmetry of CCP, instead of the Fm3m space group. In fact, many other cubic space groups contain P3m1 as a subgroup, but this is irrelevant, as the only cubic possibility is Fm3m.

If the number of layers in the primitive period is a multiple of three and the three subsections are related by the 31 (or 32) axes, or in other words are circular permutations of the letters (sites) A → B → C → A (or the reverse relation), as, for example, in the nine-layered sequence ABC-BCA-CAB, then the lattice becomes rhombohedral. The symmetry remains the same as in the hexagonal description (axes), but the cell is three times smaller; therefore, the rhombohedral interpretation is preferred.

Figure 3 illustrates the hexagonal/rhombohedral interpretation of the 15-layered stacking sequence ABCBA-CABAC-BCACB, representing the 15R polytype from the example of Krishna and Verma [39]. This sequence consists of three blocks related by the A → C → B → A shift of the sites. In the 15-layer interpretation, it has the P\(\overline\)m1 hexagonal symmetry, while in the rhombohedral axes the space group is R\(\overline\)m, with the same symmetry elements. In our nomenclature, this case is, of course, listed as ABACBCACBABCBAC. In the hexagonal cell there are 15 spheres in the unit cell, while in the rhombohedral unit cell there are only 5 spheres.

Fig. 3

Illustration of the packing within the 15R example from Krishna and Verma [39]. Shown is a projection of nine hexagonal unit cells (black) corresponding to the P\(\overline\)m1 space group, with positions of spheres in sites A, B, and C marked in blue, red and purple, respectively, and with their z-coordinates expressed as multiples of 1/15. One rhombohedral unit cell of the R\(\overline\)m space group is shown in gray. Selected relevant symmetry elements are marked in green

The three centrosymmetric space groups, P63/mmc, P\(\overline\)m1, and R\(\overline\)m, have centers of symmetry located at z = 0 and z = 1/2. In various arrangements of sphere layers, these sites can be either occupied by spheres or empty, inside octahedral voids (vide infra). If both locations, at (0,0,0) and (0,0,1/2), are occupied, the space group can be designated as S-type. If both sites are empty, the designation is of O-type. If only one of these two sites is occupied by a sphere, the space group is SO-type. This classification of centrosymmetric space groups is correlated with our (site-specific) ABC nomenclature, see Sect. “A site-dependent A/B/C notation”. If the (0,0,0) site (of type A) is occupied, the layer sequence begins with the letter A and corresponds to space group types S or SO. If it is empty, then the occupied site with the lowest z-coordinate is site B, and the layer sequence begins with B. Let us consider three examples from Table 1. (i) For entry 1 with the BC sequence, in the HCP packing, both centers of symmetry (at (0,0,0) and (0,0,1/2)) are empty; therefore the symmetry is P63/mmc (O) type. (ii) For entry 3, with the sequence ABAC, the symmetry is P63/mmc (S) type, since both centers of symmetry are occupied by a sphere. Because of the mirror plane at z = 1/4, in space group P63/mmc both centers of symmetry are equivalent, and the SO case is, therefore, not possible. (iii) Entry 4, with the sequence ABCBC, belongs to the P\(\overline\)m1 (SO) space group, since the site at z = 0 is occupied by a sphere, while the site at z = 1/2 is empty. The special case of CCP, with the ABC sequence and true Fm3m symmetry, may be expressed in the primitive rhombohedral unit cell with symmetry R\(\overline\)m (SO), since then the site at (0,0,0) is occupied by a sphere and the site at (0,0,1/2) is empty.

Identification of the space groups

Certain characteristics of the periodic layer sequences allow us to identify the symmetry of the packing arrangements. A systematic approach to identifying the space group symmetry of periodic packings of hexagonal layers was presented by Patterson and Kasper [15]. Here, we propose a different algorithm to identify the symmetry from the layer sequences, based on characteristic properties of the seven trigonal/hexagonal space groups in question, as listed in Table 2.

Table 2 Selected characteristics of the seven trigonal/hexagonal space groups of the close-packed layers of spheres

The following five tests for the presence of symmetry elements will lead to an unambiguous identification of the space group of the whole arrangement, as summarized in Fig. 4 and Table 2. The tests check the presence of (1) twofold axes parallel to the hexagonal cell edges a [100] and b [010], (2) twofold axes parallel to the diagonal (i.e., rotated 30° from the previous ones) directions [210] (or the reciprocal axis a*) and [120] (or b*), (3) a 21 axis (included in 63) along the hexagonal c direction, (4) a 31 (or 32) axis along the hexagonal c direction, and (5) the F cell.

Fig. 4

Selection of the space group of a stack of layers of spheres via a series of yes/no tests for the presence of symmetry axes or lattice type

The presence of the 63 axis (Fig. 5a) requires that the number of layers in the repeating unit be even and that the A positions in both halves of the sequence period are the same, but positions B and C swap places, as in, e.g., ABAC. In the HCP arrangement the A positions are not occupied at all, but are nevertheless formally the same in both halves.

Fig. 5

The effect of actions of characteristic symmetry operations on the sequence of layers located at positions A, B, and C. a The 63 (or 21) screw axis parallel to c (down the view direction). b The twofold axes parallel to the diagonal a* and b* directions. c The twofold axes parallel to the a and b directions. d The mutual orientation of the hexagonal (black) and rhombohedral (gray) unit cells. Blue circles correspond to the A positions at (x,y) = (0, 0), red to B positions (2/3, 1/3), and purple to C positions (1/3, 2/3). The fractions are the z-coordinates of these positions. All views of the unit cells are down the crystallographic hexagonal c axis. The rhombohedral unit cell is drawn in the obverse setting

The presence of the twofold axes parallel to a* (Fig. 5b) requires that the sequence of layers be palindromic, i.e., the backward sequence is the same as the original one, e.g., ABCBCB.

The presence of the twofold axes parallel to a (Fig. 5c) requires that the sequence of layers be “semipalindromic”, i.e., the backward sequence with the B and C sites interchanged should be the same as the original sequence, e.g., ABCBC.

The rhombohedral lattice requires that the number of layers be a multiple of three, and each of the three blocks must differ from the previous one by the A → B → C → A permutation or its inverse direction. The relation between the hexagonal and rhombohedral unit cells is presented in Fig. 5d.

Symmetry of periodic N-layered sequences

Table 3 presents the frequency of the possible space groups in relation to the number of layers N (in three-symbol notation). The space group Fm3m occurs for only one singular case, at the beginning of the table, for N = 3 with the ABC sequence. One must wait until N = 9 to find the first rhombohedral lattice, which in that case has the R\(\overline\)m symmetry (although the Fm3m case can also be presented as rhombohedral). The first R3m symmetry is encountered much later, at N = 21. Indeed, R3m is the rarest space group after Fm3m. The most common case is by all means the P3m1 space group with the lowest symmetry. It appears with rapidly increasing frequency, starting from N = 9.

Table 3 Numbers of arrangements with different symmetry in all combinations of layers up to N = 27 layers

Table 1 presents the systematic generation of all possible A/B/C sequences up to N = 10. The sequences are also annotated using the numerical and Pauling, Hägg, and Zhdanov systems. Each case is presented with full crystallographic description, including space group and sphere coordinates. Table 1 shows that some of the sequences are polar, i.e., are direction dependent, while some are non-polar or palindromic, i.e., read the same both ways. Within the polar category, we also distinguish semipalindromes, which become palindromic with the simple substitution rules (A → A, B ↔ C) applied during inverse reading, i.e., transforming one of the sites (letters) onto themselves and exchanging the remaining two. The palindromic and polar character of the layer sequences is presented in Table S1, containing all Pauling h/c sequences up to N = 10 with the corresponding numerical and letter nomenclature.

Generating the sequences in Table 1 (and counting them in Table 3) one has to observe a number of rules to prune redundancy, i.e., to eliminate sequences that appear unique but are in fact equivalent to other entries. Those rules include checking (i) cyclic permutations of the sequences, (ii) inversion of the reading direction, and (iii) permutation of layer labels. For example, there is no ABACB sequence at N = 5. The reason is as follows: ABACB = BABAC (cyclic permutation) = CABAB (backwards) which becomes ABCBC as in Table 1 after the following relabeling of the layers C → A; A → B; B → C.

Expansion of the period

An extra layer may be added to any primitive sequence of length N not only at the end but indeed at any position. The new layer has always Pauling type c because it must be the third possibility between two different layers. The new primitive sequence is not necessarily of length N + 1, because after the insertion the sequence could become degenerate and would be reduced to a shorter primitive sequence, e.g., (N + 1)/2, as in the following example (the inserted layer is in bold): ABCBC → ABCABC = ABC. If we add another layer every other layer of the starting sequence, the new sequence becomes twice as long but it preserves the original symmetry, for example: in BC → ABAC → ACBCABCB = ABCBACBC → ACBACABCABCABACB → …; all these sequences have the same P63/mmc symmetry.

By adding new layers repeatedly, one at a time, one would create a progression (family) of ascending sequences. However, the relations between different family members are not unique in the sense that there may be multiple ways of arriving at a given member from its predecessors, as illustrated by ABCBC = CABAB → CABABA = ABACAB → ABCACAB = ABABCAC ← BABCAC = BCACBA ← BCACA = ABCBC. In this way, all sequences can be stepwise derived from the simplest BC progenitor.

Periodic layer sequences as bracelets of two or three bead colors

Periodic repeats of uniquely arranged N layers A/B/C can be represented by circular bracelets of N beads formed using beads of three colors and a strict rule that neighbors of the same color are forbidden. The periodicity of the layer arrangement is modeled by the circular topology of the bracelet, whose period (sequence of beads) is read over and over again as we go around the chain. Since the N-layer period of the stack must be primitive, we require that the sequence of beads in our bracelet is not formed by repetition of a shorter subsequence.

Figure 6 shows the column and bracelet representations of selected layer sequences for the six space groups possible for N = 12. The columns are analogous to the black-and-white illustrations used by Pauling [31, 32]. They are viewed along the diagonal [110] direction of the hexagonal unit cell, with N colored balls (beads) representing layers arranged along the c-axis of the cell. The blue balls correspond to layers at the A sites at 0, 0, z, the red balls on the right correspond to layers at the B sites at 2/3, 1/3, z, and the purple balls on the left represent the C sites at 1/3, 2/3, z. The green marks depict the symmetry elements characteristic for each of the space groups. Note that if the first layer is of type A, the blue ball has z = 0, but if the first layer is of type B, the corresponding red ball lies at z = 1/2N.

Fig. 6

Column and bracelet representations of selected examples from the six space groups occurring for periodic sequences of N = 12 closely packed layers (Table 3). Spheres in A/B/C layers are colored blue/red/purple. Black lines mark the bottom and top of the unit cell along z. Green color marks the relevant symmetry axes in each space group; full lines designate palindromic twofold operations, which preserve the color of the sites (spheres); dashed lines and empty axis symbols designate semipalindromic operations, transforming the A (blue) sites onto themselves and exchanging the B (red) and C (purple) sites. The curved (circle arcs) screw axes drawn inside the bracelets represent operations on infinite linear stacks and not on the closed, circular bracelets, of course

Essentially the same information is also included in the bracelet representation in the form of a ring of N beads with colors corresponding to the three A/B/C sites. The z = 0 coordinate corresponds to the top point of the circle. This point can only be filled by the sphere of A type at the 0, 0, 0 cell origin. If this site is empty, at the top of the bracelet there are two circles at the two sides of that point, a red one standing for site B and a purple one corresponding to site C. It is assumed that the z-coordinate increases with the clockwise rotation on the bracelet ring.

It is easy to visualize the symmetry elements acting on each column or bracelet. Note, however, that Fig. 6 illustrates only those symmetry elements that are present in addition to the basic symmetry of the P3m1 space group. The green twofold axes that transform all sites onto sites of the same type are marked with full lines and occur in palindromic sequences. The dashed or empty twofold axes transform the blue A sites onto themselves, but interchange the B and C sites. The empty symbol of the 31 screw axis is characteristic for the R cell and corresponds to the following shift of sites: A → B → C → A (or backwards).

There is a mathematical theory that allows calculation of the number of possible N-bead bracelets of k colors. To the best of our knowledge, the first paper presenting the count of such sequences on N elements was published by McLarnan [27], and later elaborated by several authors. We will address this issue from the mathematical point of view in a subsequent paper.

Sphere coordination in 3D close packing

Regardless of the layer sequence, each sphere in a close packing environment is surrounded by 12 close (touching, kissing) neighbors: six in its own layer, plus three below and three above in the adjacent layers. The coordination number is therefore 12. To envision the coordination polyhedron, let us start from the basal hexagon, which divides the coordination polyhedron into two parts, called triangular cupolas. A triangular cupola has a triangular face (defined by the three spheres from an adjacent layer), joined to the basal hexagon by three squares at the edges and three equilateral triangles at the vertices. All the edges of a cupola have equal length. If the central layer is in Pauling’s h environment, the squares and the triangles (around the hexagon) of the two cupolas match and the resulting polyhedron is called twisted cuboctahedron or anticuboctahedron. It has 12 vertices, 6 square + 8 triangular = 14 faces, and 24 edges. If the central layer is in Pauling’s c environment, the adjacent layers are rotated by 60° relative to each other, and the cupolas are matched square-to-triangle. The resulting polyhedron is twisted with respect to the previous one and it is just the cuboctahedron (double twist of the cuboctahedron restores the cuboctahedron). It also has 12 vertices, 14 faces, and 24 edges.

Coordination polyhedra can be constructed around each sphere in the close packing environment. Such polyhedra (cuboctahedra and anticuboctahedra) would of course interpenetrate. However, one could also imagine constructing the same polyhedra on a twice smaller scale, with vertices placed at the touching points of the spheres, to create a lattice of closely adjoining polyhedra centered on each sphere. By this construction one would completely fill the space with these polyhedra. In the ABC arrangement the space would be filled contiguously with cuboctahedra, Fig. 7a; in the BC mode—with anticuboctahedra, Fig. 7b. In any other sequence of layers we would have filled the space closely with a mixture of cuboctahedra (around spheres in Pauling c environment) and anticuboctahedra (around spheres in Pauling h environment). Obviously, the symmetry of the arrangement of the coordination polyhedra in the 3D space is the same as that of the spheres themselves.

Fig. 7

Cuboctahedron (a) and anticuboctahedron (b) as coordination polyhedra for closely packed spheres in Pauling c and h environments, respectively. Rhombic dodecahedron (c) and its “antipolyhedron” (d) as the Dirichlet domains of spheres in Pauling environments c and h, respectively. The small red circles mark the places where the central sphere (placed at the red site) touches its 12 neighbors

The 3D space filled closely with equal spheres can be divided not only into the coordination polyhedra around each sphere, as above, but also into fundamental regions [33, Sect. 2.2.8], also known as Dirichlet domains or Voronoi polyhedra. In the arrangement of closely packed spheres, these are the polyhedra circumscribed around each sphere by faces tangent to its surface at all the points where it kisses its neighbors. These domains are dual with respect to the coordination polyhedra, i.e., have the sites of the vertices and faces interchanged. The Dirichlet domain around a sphere coordinated by a cuboctahedron in the Pauling c environment has the shape of the rhombic dodecahedron, Fig. 7c, while the Dirichlet domain around a sphere coordinated by anticuboctahedron in the Pauling h environment has the shape of “anti rhombic dodecahedron”, Fig. 7d. Again, all Dirichlet domains fill the 3D space without any gaps or overlaps, and the symmetry of their arrangement is the same as the space group of the underlying stack of hexagonal layers of spheres.

The voids in the close packing of spheres

A stack of closely packed hexagonal layers of equal spheres also has voids. There are two types of voids enclosed by spheres from adjacent layers in all possible modes of closest packing. Figure 8 shows a fragment of two consecutive hexagonal layers of spheres. As mentioned earlier, the spheres of each upper layer always lie above the centers of the equilateral triangles formed by three touching spheres in the lower layer, and in each hexagonal layer there are always twice as many triangles as spheres. As a consequence, half of the triangles in the lower layer are overlapped by triangles from the upper layer, while the other half are covered by spheres. As seen in Fig. 8, the former situation creates octahedral voids, while the latter one leads to tetrahedral voids. Overall, there are twice as many tetrahedral as octahedral voids.

Fig. 8

Fragments of two stacked close-packed hexagonal layers of spheres. The lower layer is in blue, the upper one in green. Each fragment is built from six identical equilateral triangles. The vertical line segments emphasize the two types of overlap “experienced” by the upper triangles: a triangle-upon-sphere case (black line, left), and triangle-upon-triangle (orange line, right). In the former case, centers of the four closest spheres from the two layers define a tetrahedron and a tetrahedral void. In the latter case, there are six close equidistant spheres on the plan of an octahedron, defining the octahedral void. The voids are shown in red outline