Stuffed Derivatives of Close-Packed Structures - Journal of Chemical

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Stuffed Derivatives of Close-Packed Structures

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Bodie E. Douglas Department of Chemistry, University of Pittsburgh, Pittsburgh, PA 15260; [email protected]

Buerger (1) described derivative structures as derived from simpler structures by generalization. Later he introduced the term stuffed derivatives of silica (2) for aluminosilicates (and some silicates with other metals partially replacing silicon atoms) based on a silica structure with metal ions in cavities of the framework. Palmer (3) reviewed the stuffed derivatives of tetrahedral silica polymorphs. β-Quartz, with small cavities, can only accommodate small ions such as Li+, for example, LiAlSiO4, β-eucryptite. Tridymite is hexagonal and the open channels along the c axis can accommodate large ions such as Na+, K+, and Ca2+. Nepheline (ideal formula, Na3KAl4Si4O16 ) is a common mineral based on the tridymite framework. β-Cristobalite is cubic without channels, but there is a cavity in the center of the unit cell. Carnegierite, NaAlSiO4, has Na+ ions in the cristobalite cavities. Stishovite,

a polymorph of silica from meteorites, is not in this group since its network of SiO6 octahedra is too dense without cavities for accommodating metal ions. Wells (4), Adams (5), Pearson (6), West (7), Galasso (8), Putnis (9), and others have discussed crystal structures in terms of octahedral and tetrahedral sites in close-packed structures, but they did not consider the total general pattern. The close-packing positions are A, B, and C for packing (P) atoms, PAPBPC for cubic close-packed (ccp) structures and PAPB for hexagonal close-packed (hcp) structures. There is a regular pattern of P, octahedral (O), and tetrahedral (T) layers of sites in close-packed structures; between two P layers there is an O layer and two T layers in the PTOT sequence (Figure 1) (10–12). Structures of metals and many other elements were described in the PTOT system (12, 13). Cubic Close-Packed Structures and Tetrahedral and Octahedral Sites

Figure 1. Spacing of PTOT layers in a close-packed structure.1

Figure 2. (A) The O and T sites between two layers. (B) A projection of OC, TA–, and TB+ sites between PA and PB layers.

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There is the same number of sites in each P, T, and O layer. The O layer is halfway between the P layers. Three atoms in adjacent P layers centered on an O site form opposite triangular faces of an octahedron. As seen in Figure 2 the position of the O site is different from those of the atoms of the adjacent A and B P layers. The A, B, or C position of a T site is the same as that of the apical atom from the more distant P layer. Three atoms of the closer P layer form the base of the tetrahedron. In the PATOTPB sequence sites of the lower T layer have an atom of the PB layer at the apex and point upward. They are designated as TB+. The sites of the next T layer have an atom of the PA layer at the apex, pointing downward. They are designated as TA−. For most considerations the + and − designations are not needed. The radius of the O site relative to that of the atoms in P layers is rO兾rP = 0.414. The T site is smaller, the ratio rT兾rP = 0.225. Figure 2 shows space-filling atoms to show the relative sizes of O and T sites. Ball and stick models are used in other figures to reveal internal features. Spacing for all layers between any adjacent close-packed P layers is PT0.25O0.5T0.75P relative to the distance between P layers, which is taken as 1.000. The spacing is determined by the geometry of the polyhedra formed by neighboring P atoms for the T and O sites. For metal alloys with C or H, the small atoms can occupy interstitial sites with little expansion. For inorganic compounds the atoms or ions in the T or O sites cause expansion of the P layers, but the relative spacing is unchanged and with filled O, TT, or TOT layers there is no loss of symmetry in the ccp structure. The P sites are in the ABC sequence of positions of atoms in layers in a ccp structure. The full sequence of positions is PATB OCTA PBTC OATB PCTAOBTC…. The unit cell of a ccp structure is face-centered cubic (fcc). The 12 nearest neighbors (P) around a central atom in a ccp structure form a cuboctahedron. Within the cuboctahedron eight T sites form a cube and six O sites form

Vol. 84 No. 11 November 2007

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an octahedron (Figure 3). The O sites are in the centers of the square faces of the cuboctahedron; a P atom is added outside of the cuboctahedron to complete one octahedron in the figure. The patterns of occupancies of PTOT layers give detailed descriptions of over 300 different structures representing thousands of minerals and compounds in a recent book (12). This is not a formalism. The reality is demonstrated here by examination of the interstitial sites in the cuboctahedron. Stuffed Cuboctahedron for Simple Structures –

Sodium chloride, NaCl (Oh5, Fm3m, a = 5.6402 Å) has − Cl ions filling the P layers in a ccp sequence forming the cuboctahedron with Na+ ions filling O layers (PO, Figure 4A). Since P and O sites are equivalent for a ccp structure, roles of Cl− and Na+ ions can be reversed. The NaCl or halite structure is the most common structure for ionic or intermetallic MX compounds. The sphalerite (or zinc blende) ZnS mineral has a ccp arrangement of S atoms filling–P layers with Zn atoms filling one of two T layers (Td2, F 43m, a = 5.4093 Å) (PT, Figure 4B). This structure is common for MX compounds with small cations or high covalence. Fluorite, CaF2, has Ca2+ ions filling P layers in a ccp sequence forming a cuboctahedron with F– ions filling both T layers (PTT, Figure 4C). One T layer has tetrahedra pointing upward (T+) and one has tetrahedra pointing downward (T–). – With all T sites filled, the symmetry–is Oh5, Fm3m, a = 5.4628 Å. Lithium oxide, Li2O (Oh5, Fm3m, a = 4.873 Å), is described as the antifluorite structure since the roles of ions are reversed, with oxide ions filling P layers and Li+ ions filling both T layers. This is the more common situation since usually anions are larger than cations, but the radii are similar for Ca2+ and F−. BiLi3 is an intermetallic compound; Bi atoms form a fcc – cell with a ccp (ABC ) sequence of P layers (Oh5, Fm3m, a = 5.63 Å) and Li atoms fill O and both T layers (Figure 3). All sites of the cuboctahedron are filled for BiLi3. BiF3 has the same structure. NaCl, cubic ZnS, CaF2, Li2O, BiLi3, and BiF3 can be described as stuffed ccp structures or stuffed cuboctahedra. Thousands of compounds have these common structures. For these structures the atoms or ions in T and O sites are larger than the interstitial voids, but the relative spacing is retained. – The body-centered cubic (bcc) structure of Fe (Oh9, Im3m, a = 2.8664 Å) is represented by Figure 3 with all atoms the same. The cuboctahedron is formed by three P layers of Fe atoms in an ABC (ccp) sequence and Fe atoms also fill O and both T layers. Filling all of these sites with large Fe atoms opens the cuboctahedron, but the Oh symmetry elements are –identical for BiLi3 and bcc Fe. CsCl is cubic also (Oh1, Pm3m, a = 4.02 Å). Cs+ ions are at centers of the cube formed by Cl − ions. The symmetry elements of this stuffed cuboctahedron are the same as those for Fe and BiLi3. Cl− ions fill P and O layers with Cs+ ions filling both T layers. The P and O layers together are equivalent to the two T layers. The roles of Cs+ and Cl− can be reversed just as the unit cell can be described as a cube with Cs+ or Cl− at the center. The orientations and proportions of the cuboctahedron, the octahedron formed by O sites and the cube formed by T sites are determined by the geometry and symmetry of these www.JCE.DivCHED.org

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Figure 3. The T and O sites in the cuboctahedron for a ccp structure.

Figure 4. (A) The cuboctahedron for NaCl with Na+ ions filling O sites. Extra Cl– ions are added to complete octahedra. (B) The cuboctahedron for sphalerite, ZnS, Zn atoms fill one T layer. (C) The cuboctahedron for fluorite, CaF2, F– ions fill both layers of T sites.


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concentric polyhedra. These relationships are exactly the same for the PTOT sites for BiLi3, CsCl, and the bcc structure. They are well described as stuffed ccp structures represented by Figure 3.

is also 3⭈2PTOT. Some molecular compounds, such as SiF4, have bcc structures. These can be described as 3⭈2PTOT structures (Figure 3) with the molecules filling all P, T, and O layers.

Notation for Close-Packed and Related Structures (14, 15)

Symmetry

The ccp and hcp structures involve close-packed (P) layers with each atom surrounded by a hexagonal planar arrangement of six neighbors in the central P layer. For all close-packed structures the first two P layers are identical with those in Figure 1. They differ in the positions of atoms for the third layer. For hcp the third layer is identical to the first layer, AB AB…. For ccp the positions of atoms of the third layer are shifted from those of the first and second layers, giving ABC ABC…. The notation is 2P for hcp since there are two layers repeating and for ccp the notation is 3P for the repeating three layers. The sodium chloride structure has a ccp arrangement of chloride ions with sodium ions filling layers of octahedral sites (O). This is a PO structure with three P layers (ABC ) and six P and O layers repeating giving the notation 3⭈2PO. LiFeO2 has a disordered structure with Li and Fe filling each O layer, 3⭈2PO1兾2 1兾2. The annealed form is ordered with Li and Fe filling alternate O layers, 3⭈4POFePOLi. CdCl2 has a ccp sequence of P layers filled by Cl− ions with Cd2+ ions filling alternate O layers (POP, for PAOPB PC OPA PB OPC). Nine layers repeat, 3⭈3POP, a layer or sand– wich structure, D3d5, R3m, a = 3.85, and c = 17.46 Å. There is only weak van der Waals attraction between adjacent P layers of Cl− ions without Cd2+ ions between them. The two structures of zinc sulfide, ZnS, sphalerite (or zinc blende) and wurtzite, have sulfur atoms in P layers with zinc atoms filling one T layer between P layers. Sphalerite is cubic with P layers in an ABC sequence, giving the notation – 3⭈2PT. Chalcopyrite, CuFeS2, has this structure (D2d12, I4d, a = 5.24, and c =10.30 Å) with Cu and Fe filling one T layer, 3⭈2PT1兾2 1兾2. Calcium fluoride has both T layers filled by F− ions with P layers filled by Ca2+ ions in an ABC sequence giving 9 layers repeating, 3⭈3PTT. Wurtzite, ZnS, is hexagonal with a hcp (AB) sequence for sulfide ions filling P layers and zinc ions filling one T layer between P layers. Four layers (PATPBT) repeat giving the notation 2⭈2PT. BiLi3 has a ccp arrangement of Bi atoms (ABC sequence) with Li atoms filling the O layer and both T layers between P layers in the sequence PTOT shown in the cuboctahedron (Figure 3). The three P layers and a total of twelve layers repeat giving the notation 3⭈4PTOT. The repeating sequence for cubic CsCl is PTOT with Cl− ions filling P and O layers with Cs+ ions filling both T layers. For a ccp structure P and O sites are interchangeable and all T sites are equivalent. Each P and O site is at the center of a cube formed by T sites. Each T site is at the center of a cube formed by four P and four O sites. The unit cell can have either ion at the center of the cube formed by the other ion since both T layers together are interchangeable with the combined P and O layers. Consequently, only six layers repeat, giving the notation 3⭈2PTOT. The bodycentered cubic structure of a metal such as Fe is also a PTOT structure with Fe atoms filling all four layers. The notation 1848


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The symmetry of the cuboctahedron in a ccp structure is Oh; the symmetry is the same for NaCl with O sites filled by Na+ ions, CaF2 with all T sites filled by F–, and BiLi3 with all O and T sites filled by Li. The symmetry of bcc Fe is also Oh with Fe in all P, T, and O sites, and the symmetry of CsCl is Oh with Cs in P and O sites and Cl in all T sites. Stuffing the cuboctahedron causes no distortion for these structures. The symmetry of sphalerite (ZnS) is lowered (Td2) because only one T layer is filled. Stuffed silica derivatives commonly have lower symmetry than the parent silica polymorph because of distortion caused by stuffed atoms or ions. The fully stuffed ccp structures cited have no distortion. The cavities of a fcc cell (for ccp) are the T and O sites falling on symmetry axes or planes. More Complex Stuffed Cubic Close-Packed Structures Many compounds and minerals treated in close-packed layered PTOT structures (12) can be described as having stuffed close-packed structures, –represented by Figure 3. Spinel, MgAl2O4 (cubic, Oh7, Fd 3m, a = 8.08 Å), has an interesting combination of occupancies of T and O layers. Oxygen atoms fill P layers in an ABC sequence. Between two P layers Al atoms occupy one-fourth of O sites and both T layers are one-fourth occupied by Mg. Between the next two P layers three-fourths of O sites occupied by Al with no T sites occupied, Figure 5. This complex pattern is clear from the notation 3⭈6PT1兾4O1兾4T1兾4PO3兾4. The index 3 designates the ABC sequence of P layers and the product indicates that 18 layers repeat. The versatility of the PTOT system is illustrated by – K2PtCl6. Its structure (cubic, Oh5, Fm3m, a = 9.76 Å) can be described as an M2X compound, an antifluorite compound with [PtCl6]2− ions filling P layers in an ABC sequence and with K+ ions filling all T layers (3⭈3PTT). Also the structure can be described as a ccp structure with Cl and K filling

Figure 5. The cuboctahedron for spinel, MgAl2O4, with Al3+ ions in O layers and Mg2+ ions in T layers. One Mg2+ is added outside of the cuboctahedron to show that both T layers are partially filled between PA and PB layers.


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P layers with Pt occupying one-fourth of alternate O layers, 3⭈3P1兾4 3兾4P1兾4 3兾4O1兾4. Pt atoms only occupy O sites surrounded by six Cl atoms. This description of K2PtCl6 is similar to that of the perovskite structure, BaTiO3, tetragonal, C4v1, P4mm, a = 3.9998, and c = 4.018 Å. The cell is nearly cubic. CaTiO3, another member of the perovskite group, is more distorted (orthorhombic, Pbnm, a = 5.37, b = 5.44, and c = 7.64 Å). The space group is an elegant description of the symmetry of a crystal. The space groups for BaTiO3 and CaTiO3 show no similarities, but they are alike chemically. For both structures oxide ions occupy three-fourth of P layers with Ba or Ca in one-fourth of P sites. Ti occupies one-fourth of O layers, only the O sites surrounded by oxide ions, 3⭈2P1兾4 3兾4O1兾4, Figure 6A shows the unit cell of BaTiO3, a cube of Ba atoms with the TiO6 octahedron in the center. This is the CsCl structure, 3⭈2PTOT, with TiO6 octahedra filling P and O layers and Ba filling both T layers. The Ti is at the center of the cell and oxide ions are in the centers of faces. Figure 6B shows a cuboctahedron with Ba at the center. The TiO6 octahedra are outside of the cuboctahedron, the O sites in the centers of square faces of the cuboctahedron are vacant because there are two Ba neighbors, one at the center and one outside of the cuboctahedron. The structure of ReO3 is similar to that of perovskite, but only oxygen occupies three-fourth of P lay-

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ers and there is no distortion (cubic, Oh1, Pm3m, a = 3.734 Å), 3⭈2P3兾4O1兾4. For complex structures, extended layers need to be considered since an isolated polyhedron might not reveal the pattern for partial filling. For ZnI 2 (16) all T layers are one-eighth or three-eighth filled by Zn atoms (Figure 7). The CrystalMaker computer program allows layers to be extended to determine that the pattern is 3⭈6PT3兾8T3兾8PT1兾8T1兾8. This notation describes the complex structure simply. This unusual pattern for the ZnI2 structure results from the stacking of Zn4I10 units. Each pyramidal Zn4I10 unit (lighter color in the figure) consists of a ZnI4 tetrahedron above three ZnI4 tetrahedra. The apical iodine atoms of these three tetrahedra form the base of the fourth ZnI4 tetrahedron. Each iodine atom is bonded to two Zn atoms. Stuffed Hexagonal Close-Packed Structures and Severe Limitations The polyhedron of twelve neighbors about an atom in a hcp structure is also a tetraekadecahedron, but the central atom and its six neighbors (B positions) are in a symmetry plane (Figure 8). The polyhedron, called a twinned cuboctahedron, is produced by reflection of the upper (or lower) half of the cuboctahedron. For the central atom in a

Figure 7. The unit cell for ZnI2 with Zn4I10 units in lighter color.

Figure 6. (A) The unit cell of BaTiO3 (perovskite) with the TiO6 at the center. (B) The cuboctahedron for BaTiO3 with Ba at the center.

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Figure 8. The tetraekadecahedron (twinned cuboctahedron) for 12 neighbors of a central atom in a hcp structure with O and T sites.


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PB layer the triangles of three neighbors of each adjacent PA layer form a trigonal prism. The O sites in centers of square faces occupy C positions aligned in channels and form a trigonal prism. Each T site slightly below and above the central PB layer has three atoms in this PB layer as their base, these tetrahedra share bases. These pairs of T sites are very close together with no screening. T sites in B positions also occur as close pairs above and below PA layers. Two of these T sites are added above and below the twinned cuboctahedron in

Figure 9 (A) The hcp CN (coordination number) 12 polyhedron for NiAs with Ni atoms in O sites aligned at C positions. (B) The hcp CN 12 polyhedron for ReO2 with one-half filled and staggered O sites. Three oxygen atoms are added to complete the ReO6 octahedra. (C) The hcp CN12 polyhedron for wurtzite, ZnS, Zn atoms fill one T layer.

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Figure 8. The eight T sites within the twinned cuboctahedron form a bicapped trigonal prism. This figure does not represent the structure of a known compound, it was created in the computer program CrystalMaker for this publication. Nickel arsenide, NiAs, has a hexagonal structure (D6h4, P63兾mmc, a = 3.618, and c = 5.034 Å) with As atoms filling P layers in an AB sequence and Ni atoms filling O sites, 2⭈2PO, Figure 9A. The structure of NiAs is the hcp equivalent of NaCl, but the O sites occupied by Ni are aligned in C positions without screening. As a result of Ni–Ni bonding NiAs crystals have high electrical conduction along the direction of alignment. This structure is rare while the NaCl structure is most common for MX compounds. The NiAs structure is limited to compounds with polarizing metals and polarizable nonmetals because of poor screening of the cations. Six Ni atoms form a trigonal prism around As. Compounds with ratios of atoms other than 1:1 often have partial occupancies of layers. ReO2 (orthorhombic, D2h14, Pbcn, a = 4.8094, b = 5.6433, and c = 4.6007 Å) has Re in one-half of O sites with oxygen in P layers with A and B positions, 2⭈2PO1兾2. The Re atoms are staggered to avoid Re– Re close neighbors in C positions (Figure 9B). Sanmartinite, ZnWO4 (C2h1, P2兾c, a = 4.72, b = 5.70, c = 4.95 Å, and β = 90.15⬚) has a structure similar to that of ReO2 with W and Zn each occupying one-half of alternate O layers, 2⭈2PO1兾2PO1兾2, and they are also staggered. Cadmium iodide, CdI2, has a hexagonal structure, C3v1, P3m1, a = 4.24, and c = 6.84 Å, with I– ions in PA and PB layers. The Cd2+ ions occupy alternate O layers, avoiding interaction of close Cd2+ ions without shielding. The notation is 2⭈3兾2POP, the first index indicates that the P layers have an AB sequence, but only three layers repeat. The layer structure is similar to that of CdCl2 (based on an ABC sequence). There are many polytypes of CdI2, including the CdCl2 structure and structures with more complex sequences of P layers. Wurtzite, ZnS (C6v4, P63mc, a = 3.8230, and c = 6.2565 Å) is the hcp counterpart of cubic sphalerite (Figure 9C). Both structures are common. Wurtzite is a good example of a stuffed hcp structure. One set of T sites is occupied by Zn atoms, 2⭈2PT, with no unfavorable interaction because only one of the two T layers is filled. There is no counterpart of the fluorite, 3⭈3PTT, structure for a hcp framework. Pairs of close T sites prevents full occupancy of T sites in a hcp structure without bonding. Molybdenum sulfide, molybedenite, MoS2 (D6h4, P63兾m mc, a = 3.1604, and c = 12.295 Å) has a layer structure, 2⭈3TPT. The Mo atoms are in AB P layers with S in T layers, TBPATB TAPBTA, Figure 10. The distance (3.162 Å) between layers of S atoms bonded to Mo is close enough for S⫺S bonding. MoS2 is a solid lubricant; the S⫺Mo⫺S sandwiches slide easily over one another because of weak non-directional attraction between nonbonded S layers. Cotton et al. (17) said the S atoms are close-packed with Mo in trigonal prismatic interstices. The Mo atoms are in trigonal prismatic sites because the S atoms are in the same A or B positions. The S atoms are in T layers with Mo atoms in P layers. No compound is know for a hcp structure with all T and O layers filled. There are some cases with these layers partially filled. Chrysoberyl, BeAl2O4, has an orthorhombic structure, D2h16, Pnma, a = 9.407, b = 5.4781, and c = 4.4285 Å. Oxygen atoms are close-packed with an AB sequence. Al


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atoms occupy one-half of each O layer and each T layer is one-eight occupied by Be. Staggering avoids unfavorable interaction for Be and Al. The notation is 2⭈4PT1兾8O1兾2T1兾8. Structures of Metals Bonded atoms are embedded in the electron cloud of a central atom. The stronger the bonds, the more deeply the atoms are embedded. For a ccp structure each atom has 12 close neighbors forming a cuboctahedron. A metal atom has too few electrons for ordinary bonding to 12 neighbors and M⫺M bonds are weak compared to bonding for ionic or covalent compounds. A cuboctahedron should be a good description of the shape of an atom in a ccp structure. A bodycentered cubic metal atom can be represented by a cuboctahedron containing a cube formed by eight atoms deeply embedded in the triangular faces and an octahedron formed by six atoms in centers of the square faces of the cuboctahedron. These eight atoms in T sites are much closer and six atoms in O sites are slightly closer to the central atom than those at the corners of the cuboctahedron. This is a stuffed cuboctahedron maintaining the Oh symmetry. The embedded atoms push apart the atoms of the “empty” cuboctahedron of a ccp structure, but in the stuffed cuboctahedron there is a total of 12 + 6 + 8 neighbors for the central atom. The prediction of a lower density for a bcc

structure compared to a ccp or hcp structure is limited to hard spheres, it does not apply to metals since metal atoms are highly compressible (18). The ccp, hcp, and bcc structures are the structures of almost all true metals. There are comparable numbers of metals with ccp and hcp structures. Since the bcc structure is also very common for metals, why are there no examples of a comparable structure based on a hexagonal cell with all T and O sites filled? The consideration of the bcc structure as a stuffed ccp derivative provides the answer, the T and O sites in a bcc structure are staggered in A, B, C positions and well screened. For the hcp structure all O sites are aligned along the c axis at C positions without shielding and T sites just above and below each P layer are at the same positions and very close without shielding, two tetrahedra share the base. Full occupancy of all T and O sites is not possible, even with expansion of the hcp framework. Summary Many publications have treated simple compounds in terms of occupancy of T and O layers between close-packed layers, but few have recognized the overall pattern, PTOT. The ball and stick cuboctahedron for the ABC sequence of P layers including all T and O sites makes the many patterns obvious that are obscured by space-filling models. The similar twinned cuboctahedron for the AB sequence of P layers including all T and O sites shows that only the PT (wurtzite) structure avoids serious problems for filled layers. The serious problems for filled O layers (NiAs) and both T layers filled (MoS 2 ) are expected from the twinned cuboctahedron with all sites shown. There are many examples of compounds with staggered O layers not more than halffilled for ABC and AB sequences for P layers. There are no examples of PTOT structures with full layers for an AB sequence for P layers. The ccp, hcp, and bcc structures are common for metals. The bcc structure can be described as a cuboctahedron with all P, T, and O layers (PTOT) layers filled by the metal atoms. Notation, based on the PTOT layers, gives simple descriptions of structures, even for complex patterns (Table 1). Note 1. Figures were prepared using the CrystalMaker computer program and exported as TIFF files. A free demo is available for Windows and Macintosh at http://www.crystalmaker.com/ (accessed Aug 2007). WSupplemental

Material

Interactive computer graphics of some molecules are available in this issue of JCE Online. Literature Cited

Figure 10. The hcp CN 12 polyhedron for the molybdenite, MoS2, layer structure with S atoms filling both T layers.

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1. 2. 3. 4.

Buerger, Martin J. J. Chem. Phys. 1947, 15, 1–16. Buerger, Martin J. Am. Mineral. 1954, 39, 600–614. Palmer, David. Rev. Mineral. 1994, 29, 83–120. Wells, A. F. Structural Inorganic Chemistry, 5th ed.; Oxford University Press: Oxford, 1984.


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Table 1. Inorganic Structures for Combinations of Occupancies of P, T, and O Layers in ccp and hcp Systems a ccp (ABC)

PO

PT

PTT

PTOT

NaCl

ZnS (sphalerite)

CaF2

BiLi3, BiF3

CaC2

Ice (Ic)

Li2O

K3C60

FeS2 (pyrite)

MgAgAs

CsCl

UB12

K2[PtCl6]

SiF4 (molecular)

CaCO3 (calcite)

Fe (bcc)

Partial filling

Partial filling

Partial filling

Partial filling

LiFeO2 (PO1/2 1/2) (POPO; annealed)

γ-Ga2S3 (PT2/3)

Tl2O3 (PT3/4T3/4 )

MgAl2O4 (spinel) (PT1/4O1/4T1/4PO3/4)

BiF5 (PO1/5)

CuFeS2 (PT1/2 1/2)

Cu2O (cuprite) (PT1/4T1/4) SnI4 (PT1/8T1/8)

NbO (P3/4O3/4) b

ZnI2 (PT3/8T3/8PT1/8T1/8)

BaTiO3 (perovskite) (P1/4 3/4O1/4) ReO3b (P3/4O1/4) hcp (AB)

PO

PT c

TPT or PTT

ZnS (wurtzite)

NiAs

PTOT

d 2

MoS (molybdenite)

None

Ice (Ih) Partial filling

Partial filling

Partial filling

ZnWO4 (PO1/2PO1/2)

GeS2 (PT1/2)

BeAl2O4(chrysoberyl) (PT1/8O1/2T1/8)

ReO2 (PO1/2)

Cu3AsS4 (enargite) (PT3/4 1/4)

TiO2 (rutile) (PO1/2) Cs3O (PO1/3) Al2O3 (corundum) (PO2/3) a

For detailed discussions of these structures, see ref 12. bFor perovskite P layers are 1/4 occupied by Ba and 3/4 occupied by O. For ReO3 the P c d layers are only 3/4 filled. O sites are aligned at C positions without screening for NiAs. The MoS2 structure has pairs of S atoms close enough for bonding.

5. Adams, D. M. Inorganic Solids; Wiley: New York, 1974. 6. Pearson, W. P. The Crystal Chemistry and Physics of Metals and Alloys; Wiley: New York, 1972. 7. West, A. R. Basic Solid State Chemistry, 2nd ed.; Wiley: New York, 1999. 8. Galasso, F. S. Structure and Properties of Inorganic Solids; Pergamon: Oxford, 1970. 9. Putnis, A. Introduction to Mineral Sciences; Cambridge University Press: Cambridge, 1992. 10. Ho, S.-M.; Douglas, B. E. J. Chem. Educ. 1968, 45, 474– 476. 11. Ho, S.-M.; Douglas, B. E. J. Chem. Educ. 1969, 46, 207–216. 12. Douglas, B. E.; Ho, S.-M. Structure and Chemistry of Crystalline Solids; Springer: New York, 2006.

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13. Ho, S.-M.; Douglas, B. E. J. Chem. Educ. 1972, 49, 74– 80. 14. Douglas, B. E.; McDaniel, D. H.; Alexander, J. J. Concepts and Models of Inorganic Chemistry, 3rd ed.; Wiley: New York, 1994; pp 220–221. 15. Douglas, B. E.; Ho, S.-M. Structure and Chemistry of Crystalline Solids; Springer: New York, 2006; pp 25–26. 16. Douglas, B. E.; Ho, S.-M. Structure and Chemistry of Crystalline Solids; Springer: New York, 2006; p 139. 17. Cotton, F. A.; Wilkinson, G.; Murillo, C. A.; Bochmann, M. Advanced Inorganic Chemistry, 6th ed.; Wiley: New York, 1999; p 925. 18. Douglas, B. E.; Ho, S.-M. Structure and Chemistry of Crystalline Solids; Springer: New York, 2006; p 46.


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