Concomitant Ordering and Symmetry Lowering

Research: Science and Education

Concomitant Ordering and Symmetry Lowering William O. J. Boo and Daniell L. Mattern* Department of Chemistry and Biochemistry, University of Mississippi, University, MS 38677; *[email protected]

Concomitant ordering may be defined as subordinate ordering that is commensurate with a symmetrical structure, that is, the imposition of an additional patterning on a structure. The symmetrical structure may be a discrete shape or a periodically repeating motif. Consequently, many arguments regarding concomitant ordering apply equally well to point groups, plane groups, or space groups. Concomitant ordering is observed in nature when a symmetrical structure (e.g., a crystal) is cooled through a disorder– order transition. As a consequence of this ordering, the resulting symmetry is a subgroup of the original group. Our first goal is to demonstrate how symmetry-lowering methods can describe real structures that manifest concomitant ordering. Having established the principles of concomitant ordering, the second goal, for those unfamiliar with the concepts of crystallography, is to demystify space group symmetry and thereby stimulate interest in solid-state chemistry. Our third goal is to present simple hands-on exercises in which the student may create concomitant designs, emphasizing connections between art and science. Maximal Subgroups and Degree of Symmetry A subgroup of a point group, space group, or plane group is a maximal subgroup if no intermediate subgroup exists in a symmetry-lowering sequence. Maximal subgroups of space groups (or plane groups) are of two types (1a). Type I maximal subgroups, also called translationengleiche or t-type, have all translations retained during the symmetry-lowering step (no loss of lattice points). Type II maximal subgroups, also called klassengleiche or k-type, lose some translational symmetry (there is a thinning out of lattice points). Type II maximal subgroups are further divided into three categories: in Type IIa, the unit cell is decentered and its size is unchanged; in Type IIb, the unit cell increases in size; and in Type IIc the unit cell increases in size, but the space group (or plane group) notation is identical to (isomorphic with) the original group. (The unit cell of a Type I maximal subgroup may or may not be of a different size from that of the original group.) The maximal subgroups of the 17 plane groups and 230 space groups are given in the International Tables for Crystallography (1b). Megaw (2) stated “Structures may be said to belong to the same family if there is a one-to-one correspondence between all their atoms, and between their interatomic bonds.” The symmetries within a family of structures, however, may differ, for example, with changes in atom identities or bond lengths. Megaw denoted the ideal structure of a family as the aristotype, which necessarily has the highest symmetry. Those structures that deviate from the ideal are hettotypes. Hettotypes may be the consequence of concomitant ordering or, more simply, of small crystal distortions. Either of these may set in when a crystal is formed or may be the result of a solid-solid phase transition. The order of symmetry of a point group is an integer, which means the relative symmetries of point groups are quantitative.

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In contrast, the order of symmetry of a space (or plane) group is infinite. We therefore use the qualitative term degree of symmetry to describe the relative symmetries of space (or plane) groups. The two following rules are useful: (i) in a family of structures, the aristotype has the highest degree of symmetry and (ii) hettotypes are subgroups of the aristotype structure. Since the symmetry lowering associated with concomitant ordering usually involves the removal of only one or two symmetry elements, identifying maximal subgroups is useful in the assignment of space (or plane) groups. Concomitant Ordering on Crystals Some families of structures occur repeatedly both in nature and in man-made materials. Examples of these include perovskite (CaTiO3), hexagonal tungsten bronze (KxWO3, x = 0.18–0.32), and tetragonal tungsten bronze (KxWO3, x = 0.40–0.60). Members of these families may be qualitatively similar but not strictly isomorphic. A wide variety of ordering phenomena may occur in compounds having these structures, including magnetic ordering, Jahn–Teller cooperative ordering, electronic ordering, ordering of two or more different ions, and ordering of ions in partially-filled sites. In addition, some small crystal distortions may be the consequence of displacive transitions, in which no chemical bonds are broken. In each of these examples, symmetry elements are removed from the aristotype, making the resulting hettotype a subgroup. Magnetic ordering includes ferromagnetism, antiferromagnetism, and ferrimagnetism. Of these, antiferromagnetism is the most common. KVF3 is a classic antiferromagnet; at room temperature it has the ideal cubic perovskite structure (O1h–Pm3m). The V2+ ion has three unpaired 3d electrons and the allowed ms values are +3∙2, +1∙2, ‒1∙2, and ‒3∙2. Above 130 K there is random distribution among these states (paramagnetic). Below 130 K, antiferromagnetic ordering sets in (3, 4) and the spin vectors are oriented alternately up (+3∙2) and down (‒3∙2) (5). This ordering process is dramatized in Figure 1. The maximal subgroup sequences at the bottom of Figure 1 show the space 17–I4/mmm. There are two group of the ordered structure as D4h 1 17 pathways from Oh–Pm3m to D4h–I4/mmm, each containing a Type I and a Type IIb maximal subgrouping. Jahn–Teller cooperative ordering is a well-known effect. The theorem of Jahn and Teller (6) states that a cation with an orbitally-degenerate ground state may undergo distortion to a lower-symmetry system in a way that removes the ground-state degeneracy. Cr2+ and Cu2+ in octahedral environments fulfill these conditions and are often referred to as Jahn–Teller ions. For example, KCrF3 has the cubic perovskite structure at high temperatures, but below 370 °C the ground-state degeneracy is removed and Jahn–Teller cooperative ordering sets in (7). This 1 –P4/mmm, ordering, which lowers the crystal symmetry to D4h 1 is illustrated in Figure 1. D4h –P4/mmm is a Type I maximal subgroup of O1h–Pm3m.

Journal of Chemical Education • Vol. 85 No. 5 May 2008 • www.JCE.DivCHED.org • © Division of Chemical Education

Research: Science and Education antiferromagnetic in KVF3

Jahn–Teller in KCrF3

electronic in K 0.50VF3

partially filled in Rb0.167VF3

> 130 K

> 370 °C

> 200 °C

> 300 °C

< 130 K

< 370 °C

< 200 °C

< 100 °C

high temp

low temp

high temp

random

V2+/ V3+ V2+

occupied

ordered

ordered

V3+

empty

1

IIb

–

1

Oh–Pm3m

1 D4h –P4/mmm

I

IIb

17 D4h–I4/mmm

5

D4h –P4/mbm

IIb

I

9 – Oh –Im3m

low temp

–

Oh –Pm3m

I

random

random

I

13 D4h –P42/mbc

I 1 D4h –P4/mmm

3

D6h–P63/mcm

2 C4v–P4bm

I 17 D2h–Cmcm

IIb

8 C4v–P42bc

IIa 12

D2h–Pnnm

Figure 1. Four examples of concomitant ordering: antiferromagnetic ordering in KVF3; Jahn–Teller cooperative ordering in KCrF3; electronic ordering in K0.50VF3; and ordering of partially filled sites in Rb0.167VF3.

Electronic ordering often occurs in mixed valence compounds (e.g., V2+/V3+). At elevated temperatures, the odd electron is delocalized and the cations occupy equivalent crystallographic sites. Below a critical temperature, the electrons become localized in the concomitant arrangement of lowest energy. This ordering process divides the ions into two valence states and the sites that they occupy become nonequivalent. For example, at low temperature the phase KxVF3 (KxVIIx VIII1−xF3, where x = 0.45–0.56) is a hettotype of the tetragonal tungsten bronze structure (8). At high temperatures KxVF3 is isomorphic with KxWO3 (9), but at temperatures below approximately 200 °C the odd electrons become localized (10). The associated symmetry lowering is a two-step maximal subgroup process from 5 –P4/mbm to C 8 –P4 bc, as shown in Figure 1. D4h 4v 2 The compound K0.50CrF3 has the same 2+/3+ ordered structure as that of K0.50VF3 shown in Figure 1. Above 400 °C, 5 –P4/mbm (10). Below K0.50CrF3 has the aristotype structure D4h 400 °C, cooperative Jahn–Teller ordering lowers its symmetry to 8 –Pba2 (11), which is a maximal subgroup of C 8 –P4 bc C 2v 4v 2 . Compounds of the general formula KxM IIx M III1−xF3 (where x = 0.40 to 0.60 and MII and MIII are not homonuclear) provide examples of ionic ordering (as opposed to electronic ordering). The single-crystal X-ray structure of K0.54Mn0.54Fe0.46F3 8 –P4 bc (12), which is the ordered was determined to be C 4v 2 structure in Figure 1. Furthermore, in several cases, magnetic properties have been reported to be consistent with this ordered

structure (13). Ionic ordering in these compounds occurs when the crystals are formed. Ordering of ions in partially filled sites was reported to occur in the phase RbxVF3 (x = 0.18–0.32) (14). These compounds are analogs of hexagonal tungsten bronze 3 –P6 /mcm) (15), which is the aristotype structure. Above (D6h 3 300 °C the RbxVF3 compounds exist as a single phase with the aristotype structure. Ordering occurs slowly as the crystals are cooled over the range 300 to 100 °C. Superstructures form with 1∙2 filled sites (x = 0.167), 2/3 filled sites (x = 0.222), and 3∙4 filled sites (x = 0.250) belonging to space groups D12 2h–Pnnm, iso17–Cmcm, and D5 –Pmma, respectively. The ordering morphic D2h 2v of half-filled sites in Rb0.167VF3 is illustrated in Figure 1. There is 3 –P6 /mcm to D12–Pnnm. just one pathway from D6h 3 2h FeF3 provides an example of a crystal structure that exhibits a small displacive transition. At high temperatures FeF3 has the cubic ReO3 structure (space group O1h–Pm3m). As it is cooled, a displacive transition occurs at 410 °C, changing it to rhom6 –R3c) (16). The symmetry route is bohedral (space group D3d 1 5 6 –R3c. Oh–Pm3m; Type I to D3d –R3m; Type IIb to D3d In each of the above examples, concomitant ordering lowers the symmetry of the crystal. Disorder–order phenomena and small displacive transitions demonstrate that the degree of symmetry of a crystal is temperature dependent. The form existing at the higher temperature always has the higher symmetry.

© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 5 May 2008 • Journal of Chemical Education

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Concomitant Ordering on Molecules, Ions, and Clusters

Concomitant Ordering as an Educational Aid

Although concomitant ordering is usually considered to be a process associated with crystals, concomitant coordination is becoming increasingly useful in describing the structures and properties of complex molecules, ions, and clusters. The wide variety of fullerene molecules (17), for example, provides an almost unlimited source of possible derivatives. Since many of these structures emulate icosahedral symmetry, the maximalsubgroup minimal-supergroup plot that includes the icosahedral (Ih) aristotype, given in the International Tables for Crystallography (1c), is quite useful. Structures displaying concomitant coordination often occur as the predominant product in elaborate preparation schemes. The deuterated molecule C60D36, which has tetrahedral (T) symmetry, is one example (18). One of the most dramatic examples exists in metal carbon clusters called met–cars (metallocarbohedrenes). The first member of the class discovered was Ti8C12 (19). The proposed cagelike structure rivals buckminsterfullerene in beauty. The twenty atoms are located at the vertices of a pentagonal dodecahedron (Figure 2A). The eight titanium atoms are concomitantly ordered such that they lie on the vertices of a cube (Figure 2B). The resulting structure of Ti8C12 is shown in Figure 2C. It is noteworthy that the proposed structure of Ti8C12 involves two aristotype structures: the dodecahedron, which has icosahedral symmetry (Ih, order of symmetry = 120) and the cube, which has octahedral symmetry (Oh, order of symmetry = 48). Concomitant coordination lowers the symmetry of Ti8C12 to tetrahedral (Th, order of symmetry = 24), which is a subgroup of both Ih and Oh.

Connections made between art and science are particularly helpful in teaching crystallography and symmetry. There are abundant examples of one-dimensional (linear) and twodimensional (plane) space groups found in designs from past cultures. These provide valuable stepping stones to the more formidable three-dimensional case. Hargittai and Lengyel, for example, illustrated the 7 linear groups (21) and 17 plane groups (22) with ornamental examples of Hungarian needlepoint. Other authors have compiled patterns found on ancient pottery, wallpaper, and mosaics (23, 24). These decorative patterns help to clarify the abstract concept of infinite translations and the ways that point group symmetry may be combined with translational symmetry. It is possible to generate purely geometrical configurations of the 17 plane groups by applying the symmetry elements to an appropriate asymmetric shape. Patterns generated with scalene triangles by Buerger (25) are more-or-less the standard and have been adopted by numerous authors (26, 27). The student, however, may still feel overwhelmed by the mind-boggling number of three-dimensional space groups (230) and may wonder what relationships exist among them. Here again, it is appropriate to utilize the plane groups. Concomitant ordering applied to twodimensional geometrical designs can vindicate the existence of the 17 plane groups and can make the relationships between them clear. von Fedorov, who was the first to discover that there are 230 three-dimensional crystallographic space groups, said, “All crystals are either cubic or hexagonal, at least approximately” (28). von Fedorov realized that the cubic and hexagonal lattices contain all of the symmetry elements found in the 230 space groups and that all of the other space groups are subgroups of the cubic system or hexagonal system, or both. This means that it is possible to create all of the other space groups by systematically removing symmetry elements from the space groups of the cubic or hexagonal lattices. For two-dimensional systems, we may modify von Fedorov’s statement to read, “All two-dimensional crystals are either square or hexagonal, at least approximately.” We will demonstrate what von Fedorov meant, first with the 10 two-dimensional crystallographic point groups and then with the 17 plane groups. For continuity, we will use only the aristotype structures (square and hexagonal) of Buerger’s patterns. From these, we will generate all of the other point groups and plane groups by applying subordinate patterning, that is, through concomitant ordering.

Symmetry and Order Rosen noted that “disorder stands in positive correlation with symmetry” and “organization implies and is implied by reduction of degree of symmetry” (20). Concomitant ordering is consistent with these statements. It sets in when the temperature of a crystal is lowered, which decreases the degree of symmetry of the crystal. This relationship may seem counterintuitive: degree of symmetry and randomness increase together, whereas ordering lowers the degree of symmetry. The most stable state of a structure at high temperature is that approaching maximum symmetry. In contrast, the most stable state of a structure at low temperature is that of lowest energy and is the most ordered. A

B

C

The Ten Two-Dimensional Crystallographic Point Groups

indeterminant atom

titanium atom

carbon atom

Figure 2. (A) The 20 atoms of Ti8C12 at the vertices of a pentagonal dodecahedron, with no distinction made between Ti and C (point group Ih). (B) The 8 titanium atoms are concomitantly ordered such that they lie on the vertices of a cube (point group Oh). (C) The structure of Ti8C12 resulting from this concomitant ordering (point group Th).

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The two-dimensional crystallographic point groups have two aristotypes: C4v (square) and C6v (hexagonal). Buerger employed 8 scalene triangles to represent the C4v aristotype (order of symmetry = 8) and 12 scalene triangles for the C6v aristotype (order of symmetry = 12). These motifs are shown as the top patterns of Figures 3A and 3B, respectively. We obtain hettotype structures (subgroups) in Figure 3 by systematically painting some of the scalene triangles white. Such a concomitant ordering on the C4v aristotype of Figure 3A can remove the mirror planes and lower the symmetry to C4 (order of symmetry = 4). Additional white paint “partially quenches” the rotational symmetry, reducing it to a hettotype with two-fold rotational symmetry,


Research: Science and Education A

B

C

12—

order of symmetry

8—

4—

2—

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C4

C2

Cs

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4— C 2v 3—

C3v

C3

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2—

C 3v 3m

C2v 2mm C3 3

3—

Cs

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C6 6

6— 4—

2—

C2

C4v 4mm

8—

C4v 6—

C 6v 6mm

12—

C6v

Cs m C1 1

1—

C1

Figure 3. (A) Concomitant ordering on Buerger’s C4v (4mm) motif (the top pattern). (B) Concomitant ordering on Buerger’s C6v (6mm) motif (the top pattern). (C) Maximal subgroups and minimal supergroups of the 10 two-dimensional crystallographic point groups.

point group C2 (order of symmetry = 2). Further subordinate ordering results in the asymmetric group C1 (order of symmetry = 1). White paint can also be applied to the aristotype C4v in a different way, giving the symmetry-lowering sequence C4v → C2v → Cs → C1. There is yet a third sequence (which does not contribute additional point groups) by following the arrows downward from C4v → C2v → C2 → C1. By adding white paint to the second aristotype motif, C6v (Figure 3B), we obtain the hettotype structures with point group symmetries C6, C3v, C3, C2v, C2, Cs, and C1 (all of the subgroups of C6v). Figure 3B shows six unique pathways from C6v to C1. By combining Figures 3A and 3B, we obtain the maximalsubgroup and minimal-supergroup plot of the two-dimensional crystallographic point groups (designated by both Schoenflies and International symbols) illustrated in Figure 3C. The maximal subgroups of any point group are indicated by the connecting arrows pointing downwards. The maximal subgroups of C6v (6mm), for example, are C2v (2mm), C6 (6), and C3v (3m). A similar plot and one for the 32 three-dimensional crystallographic point groups are given in the International Tables for Crystallography (1d). The Square and Hexagonal Lattices and Their Subgroups In keeping with von Fedorov’s statement, we will use only the square and hexagonal lattices to generate all 17 plane groups. The square and hexagonal lattices are illustrated in Figures 4A and 4B, respectively. Two possible unit cells are shown for each. The unit cells defined by a and b are primitive (p), meaning there is one lattice point per unit cell. Those defined by a' and b' are centered (c), and have two lattice points per unit cell. Note that the larger unit cells can be made primitive by decentering (for example, by making the motifs in the centers different from those on the corners). A crystallographic point group (Figure 3C) is defined as a point group that maps a point lattice onto itself (1e). A square lattice, for example, appears unchanged after operation of the

point group C4v (4mm). Each of the subgroups of 4mm (4, 2mm, 2, m, and 1) can also map a square lattice onto itself. Since a crystallographic point group can map a twodimensional lattice onto itself, it may combine with the lattice to form a plane group. Groups obtained by the combination of point operations with translations are called symmorphic (26). Each of the ten crystallographic point groups may combine with an appropriate primitive lattice (3m can combine in two ways). The eleven resulting plane groups are p6mm, p6, p4mm, p4, p3m1, p31m, p3, p2mm, p2, pm, and p1. A centered lattice yields two additional symmorphic plane groups: c2mm and cm. The remaining four plane groups, which are nonsymmorphic, employ a new symmetry operation called a glide plane ( g). A glide plane combines mirror reflection and translational symmetry into a single operation, which can also map a lattice onto itself. If, for example, the glide plane is parallel to a, the operation is reflection across the plane plus translation of a/2. The nonsymmorphic plane groups are p4gm, p2mg, p2gg, and pg. The full crystallographic notation for a plane group consists of four symbols. The first identifies the unit cell as primitive ( p) or centered (c). The second identifies the highest order of rotation (6, 4, 3, 2, or 1). The third denotes a symmetry of reflection orthogonal to the a direction; m indicates a mirror reflection, g

A

B

ab

ab

bb

bb a

a b

b

Figure 4. (A) The square lattice, plane group p4mm. (B) The hexagonal lattice, plane group p6mm.


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p4mm

p4

p2mm

c2mm

p4gm

isomorphic

p2

pm

cm

p2mg

p2gg

p1

pg

Figure 5. A maximal-subgroup symmetry-lowering schematic, demonstrating all of the possible subgroups of a square lattice. A unit cell for each structure is shown in the lower left of each pattern. The unit cells show the elements of symmetry: double lines = mirror planes; broken lines = glide planes; diads and squares = two- and four-fold rotation axes, respectively.

a glide reflection, and 1 no reflection. The final symbol denotes a symmetry of reflection that is not orthogonal to the a direction, also with symbols m, g, or 1. Consider, for example, the plane groups p3m1 and p31m. The mirror planes in p3m1 are orthogonal to a, but in p31m they are not. We have chosen to use the internationally accepted short forms; the short form of p211, for example, is simply p2. Concomitant Ordering on p 4mm and p 6mm Configurations of Scalene Triangles Buerger (25) illustrated the plane groups with 17 unique configurations of scalene triangles. We will use only two of Buerger’s configurations (p4mm and p6mm), which will serve as aristotype structures, and we will obtain all of the hettotypes by systematically applying concomitant patterns on them. As was the case with the point groups in Figure 4, the plane group hettotypes are obtained by removing rotational symmetry and 714

mirror planes. In addition, removal of lattice points is a symmetry-lowering event (in the conversion of a mirror plane to a glide plane, for example, half of the lattice points are removed). There are 12 plane groups possible on a square lattice: p4mm, p4, p2mm, c2mm, p2, pm, cm, p1, p4gm, p2mg, p2gg, and pg. The first 8 are symmorphic and the last 4 are nonsymmorphic. Buerger’s configuration for p4mm is shown at the top of Figure 5. In this figure, maximal subgroups of any plane group are designated by arrows pointing downwards. The 5 maximal subgroups of p4mm ( p4, p2mm, c2mm, p4gm, and isomorphic p4mm) appear in the second row. Second-generation maximal subgroups ( p2, pm, cm, p2mg, and p2gg) appear in the third row, and third-generation maximal subgroups (p1 and pg) in the fourth row. Only one structure is shown for each plane group (except for the isomorphic subgroup p4mm). A unit cell containing the elements of symmetry is shown at the lower left of each structure. Structures were chosen such that their unit cells would have the dimensions of one of those shown



p6mm

p6

p3

p3m1

p2

p31m

cm

p1

c2mm

p2mm

pm

isomorphic

p2mg

p2gg

pg

Figure 6. A maximal-subgroup symmetry lowering schematic, demonstrating all of the possible subgroups of a hexagonal lattice. A unit cell for each structure is shown in the lower left of each pattern. The unit cells show the elements of symmetry: double lines = mirror planes; broken lines = glide planes; diads, triangles, and hexagons = two-, three-, and six-fold rotation axes, respectively.

in Figure 4A. Unlike the crystallographic point groups, which can be arranged in a unique maximal-subgroup and minimalsupergroup schematic, there is no single arrangement for the plane groups. The schematic shown in Figure 5 is only one of several possible. Buerger’s configuration for p6mm appears at the top of Figure 6. There are 14 subgroups of p6mm, 3 of which are nonsymmorphic. Following the same procedure used in Figure 5, the maximal subgroups of p6mm (p6, p3m1, p31m, c2mm, and isomorphic p6mm) appear in the first row below p6mm. The second generation of maximal subgroups (p3, p2, cm, p2mm, p2mg, and p2gg) and the third generation of maximal subgroups ( pl, pm, and pg) appear in the third and fourth rows, respectively. Except for the isomorphic subgroup p6mm, all of the structures have unit cells with the dimensions of those shown in Figure 4B. Of the configurations chosen in Figures 5 and 6, Type I maximal subgroups are the most common. In the sequence p4mm → p4 → p2 → p1, for example, all transitions are Type I.

All of the lattice points are preserved; consequently, the unit cell remains the same size. Type II transitions are fewer in number; however, there are several examples of Type IIa transitions in Figures 5 and 6. In Type IIa transitions, lattice points are lost by decentering the unit cell. This is clearly illustrated in two of the transitions in Figure 6: c2mm → p2mm, and cm → pm. The only example of a Type IIb transition occurs in Figure 5: p4mm → p4gm. In this transition, the diagonal mirror planes of p4mm are replaced by glide planes. Finally, p4mm → p4mm (isomorphic) and p6mm → p6mm (isomorphic) are examples of Type IIc transitions. These transitions preserve the plane group symmetry, but lattice points are lost owing to the increase in size of the unit cell. By applying concomitant ordering and imposing the conditions of maximal subgroups for symmetry lowering, we generated all 17 plane groups from p4mm and p6mm. This twodimensional demonstration is consistent with von Fedorov’s declaration that all crystals are either cubic or hexagonal, at least


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A

B

C

D

E

F

1

2

3

Figure 7. All 17 plane groups are represented by imposing concomitant patterns on the three regular tilings (trigonal, tetragonal, and hexagonal). Each plane group appears only once. It is left as an exercise for the student to correctly identify each group.

approximately. In the next section, for the serious student, we suggest a hands-on exercise for creating two-dimensional designs that have the 17 plane group symmetries. Concomitant Designs on the Regular Tilings In a recent article, we described the generation of highsymmetry three-dimensional shapes (which can describe the topologies of molecules, ions, and clusters) by the perturbation and fusing of regular two-dimensional tilings (29). Here, we will exploit the perturbation of two-dimensional patterns by using concomitant ordering to remove symmetry elements. The trigonal, tetragonal, and hexagonal tilings (the regular tilings) belong to plane groups p6mm, p4mm, and p6mm, respectively. This suggests that designs having the symmetries of the 12 subgroups of p4mm may be realized by superimposing different concomitant patterns on a tetragonal tiling and that designs of the 14 possible subgroups of p6mm may be created by concomitant ordering on the trigonal or hexagonal tilings. In fact, it can be done with tiles of two colors (e.g., black and white). In this manner, all 17 plane groups are represented in Figure 7. The simple patterns in this figure include examples from each of the three tilings. Identifying which patterns belong to each of the plane groups is a good exercise for the beginner. The key to the analysis of Figure 7 is found in the online supplement. Making two-color patterns on the regular tilings is something anyone can do and finding ways to produce the 17 crystallographic plane groups is an excellent learning experience. All that is needed is multiple copies of the three regular tilings to serve as aristotypes. It is wise to begin with concomitant order716

ing that gives the simplest periodic patterns possible (such as those shown in Figure 7). With practice and planning, more ornamental patterns become possible; however, serendipity often provides pleasant surprises. Since a tiling is theoretically of infinite dimensions, the number of possible patterns for each plane group is also infinite. Making original designs is an active exercise that develops creativity and proves a strong connection between art and science. Literature Cited 1. International Tables for Crystallography, Vol. A; Hahn, T., Ed.; D. Reidel Publishing: Dordrecht, Holland, 1989; (a) p 727, (b) pp 102–707, (c) p 787, (d) pp 781–782, (e) p 752. 2. Megaw, H. D. Crystal Structures: A Working Approach; Saunders: Philadelphia, 1973; p 282. 3. Cros, C.; Feurer, R.; Grenier, J. C.; Pouchard, M. Mater. Res. Bull. 1976, 11, 539. 4. Williamson, R. F.; Boo, W. O. J. Inorg. Chem. 1977, 16, 649. 5. Hong, Y. S.; Williamson, R. F.; Boo, W. O. J. J. Chem. Educ. 1980, 57, 583–587. 6. Jahn, H. A.; Teller, E. Proc. R. Soc. London 1937, A161, 220–235. 7. Cousseins, J. C.; deKozak, A. C. R. Acad. Sci. 1966, 263, 1533. 8. Magneli, A. Ark. Kemi 1949, 1, 213. 9. Hong, Y. S.; Williamson, R. F.; Boo, W. O. J. Inorg. Chem. 1980, 19, 2229. 10. Yeh, Y. K.; Hong, Y. S.; Boo, W. O. J.; Mattern, D. L. J. Solid State Chem. 2005, 178, 2191. 11. Hong, Y. S.; Baker, K. N.; Shah, A. V.; Williamson, R. F.; Boo, W. O. J. Inorg. Chem. 1990, 29, 3037.


Research: Science and Education 12. Banks, E.; Nakajima, S.; Williams, G. J. B. Acta Crystallogr., Sect. B 1979, B35, 46. 13. Hong, Y. S.; Williamson, R. F.; Baker, K. N.; Du, T. Y.; Seyedahmadian, S. M.; Boo, W. O. J. Inorg. Chem. 1992, 31, 1040. 14. Hong, Y. S.; Williamson, R. F.; Boo, W. O. J. Inorg. Chem. 1981, 20, 403. 15. Magneli, A. Acta Chem. Scand. 1953, 7, 315. 16. Tressaud, A.; Dance, J. M.; Menil, F.; Portier, J.; Hagenmuller, P. Z. Anorg. Allg. Chem. 1973, 399, 231. 17. Boo, W. O. J. J. Chem. Educ. 1992, 69, 605–609. 18. Hall, L. E.; McKenzie, D. R.; Davis, R. L.; Attalla, M. I.; Vassallo, A. M. Acta Cryst. 1998, B54, 345–350. 19. Guo, B. C.; Kerns, K. P.; Castleman, A. W., Jr. Science 1992, 255, 1411. 20. Rosen, J. A Symmetry Primer for Scientists; Wiley-Interscience: New York, 1983; p 166. 21. Hargittai, I.; Lengyel, G. J. Chem. Educ. 1984, 61, 1033– 1034. 22. Hargittai, I.; Lengyel, G. J. Chem. Educ. 1985, 62, 35–36. 23. Washburn, D. K.; Crowe, D. W. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis; University of Washington

Press: Seattle, 1988. 24. Stevens, P. Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions; The M.I.T. Press: Cambridge, MA, 1999. 25. Buerger, M. J. Elementary Crystallography; Wiley: New York, 1967. 26. O’Keeffe, M.; Hyde, B. G. Crystal Structures; Mineralogical Society of America: Washington, DC, 1996; pp 14–20. 27. Hargittai, I.; Hargittai, M. Symmetry Through the Eyes of a Chemist; VCH Publishers Inc.: New York, 1987; pp 345–349. 28. von Fedorov, E. Z. Krist. 1895, 24, 209–252. 29. Boo, W. O. J.; Mattern, D. L. J. Chem. Educ. 2002, 79, 1017– 1023.

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Key to Figure 7


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Concomitant Ordering and Symmetry Lowering

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