# 1.10 Close packed structures cubic and hexagonal close packing

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1 1.9 Description of crystal structures The most common way for describing crystal structure is to refer the structure to the unit cell. The structure is given by the size and shape of the cell and the position of atoms (i.e. atomic coordinates). However, this is insufficient to visualize the whole crystal structures in 3D. Referring to a larger part of the structure (comprising perhaps several unit cells) and considering the arrangement relative to each other, their coordination numbers, interatomic distances, types of bonding, etc are more important. Two of the most useful ways of describing structures are based on close packing and space-filling polyhedra. They provide greater insight into the crystal chemistry than is obtained using unit cells alone Close packed structures cubic and hexagonal close packing Many crystal structures can be described using the concept of close packing. Structures are usually arranged to have the maximum density. Consider the most efficient way of packing equal-sized spheres in 3D. Fig shows the most efficient way to pack spheres in 2D. Each sphere, A is surrounded by, and is in contact with, six others; i.e. A has six nearest neighbors. The coordination number of six is the maximum possible for a planar arrangement of contacting, equal-sized spheres. 1

2 2

3 Regular repetition Close packed layer (cp layer) The cp layer has three close packed directions: XX, YY, ZZ (spheres are in contact in the directions). A non-cp layer is shown in the inset of Fig. 1.12; a lower coordination number is observed (4 for A). The most efficient way to pack spheres in 3D is to stack cp layers on top of each other: [hexagonal close packed] and [cubic close packed]. - The most efficient way for two cp layers A and B to be in contact Each sphere of one layer to rest in a hollow between three spheres in the other layer, e.g. at P or R in Fig Fig shows such a position for 2 layers. Atoms in second layer may occupy either P or R (Fig. 1.12), not both, nor a mixture of the two. Hexagonal Close Packing (hcp) - Fig. 1.13, suppose that A lies underneath the B layer. A third layer on top of B: positions S or T For S position, the third layer directly over the A layer. As subsequent layers are added, the following sequence arises: ABABAB This is known as hexagonal close packing (hcp) 3

4 Cubic Close Packing (ccp) - In Fig. 1.13, the third layer is placed at T, then all three layers are staggered relative to each other and it is not until the fourth layer is positioned (at A) that the sequence is repeated.. The third layer is called C, and it gives ABCABCABC i.e. cubic close packing (ccp) Fig hcp and ccp are two simplest stacking sequences and are by far the most important in structure chemistry. Other more complex sequences with larger repeat units, e.g. ABCACB or ABAC are related to the phenomenon of polytypism ( 多樣型 ). In 3D cp structures, each sphere is in contact with twelve others and this is the maximum coordination number possible for contacting and equal-sized spheres. (A common non-cp structure is body centered cube, with CN = 8, Fig. 1.9c for α Fe) 6 of these neighbors are coplanar (Fig. 1.12) Two groups of 3 spheres One group in the plane above One group in the plane below Fig and 1.14 Fig. 1.15: hcp and ccp differ only in the relative orientations of these two groups of 3 spheres. 4

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6 1.11 Relationship between cubic close packed and face centered cubic The unit cell of a ccp arrangement is the face centered (fcc) unit cell (Fig. 1.9(a) with spheres at the corners and face centers). - The relationship between ccp and fcc is not obvious since the faces of the fcc unit cell do not correspond to cp layers. - The cp layers are parallel to the {111} planes of the fcc unit cell (Fig. 1.16, the spheres labeled 2 7 in (a) and (b) form part of a cp layer, revealed by removing sphere 1) - In an fcc structure, cp layers occur in four orientations and are perpendicular to the body diagonals. (the fcc cube has four body diagonals, originating from 8 corners) - Fig. 1.16(c), the cp layers in one orientation are seen edge-on. 從側邊看 - Fig. 1.16(d), the cp layers are seen perpendicular to the layers. ( 對角 corner 的 atoms 剛剛好對到, 均為 A layers) 6

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8 1.12 Hexagonal unit cell and close packing An hcp arrangement of spheres has a hexagonal unit cell. - The basal plane of the cell coincides with a cp layer of spheres (Fig, 1.17b) - The unit cell centains two spheres One at the origin (and hence at all corners) 1 One inside the cell at position 3, 2 3, 1 (dashed circle in b). 2 - cp Layers occur in only one orientation in a hcp structure (hexagonal unit cell). - Fig shows that the two axes in the basal plane are of equal length (a = 2r, if the spheres of radius r touch) - The angle γ is 120, c-axis is a sixfold rotation axis. See further from the auxiliary materials 8

9 1.13 Density of close packed structures In cp structures, 74.05% of the total volume is occupied by spheres. This is the maximum density possible if structures constructed of spheres of only one size. Ex: the fcc unit cell (4 spheres in one unit cell). One at a corner and three at face centers The cp directions (XX, YY, ZZ in Fig. 1.12) occur parallel to the face diagonal of the unit cell, i.e. spheres 2,5,6 in Fig. 1.16b. The length of the face diagonal: 4r The length of the cell edge: 2 2 r The volume of the cell: 16 2 r ( 2 2γ ) + ( 2 2γ ) = 4γ 3 3 ( 2 2γ ) = 16 2γ total sphere volume unit cell volume = πγ γ = Similar results are obtained for hcp. In non-cp structures, densities lower than are obtained - Body centered cubic, bcc, is (the cp directions (Fig. 1.12) in bcc are parallel to the body diagonals of the cube). 9

10 1.14 Unit cell projections and atomic coordinates To give 3D perspective to crystal structures, they are often drawn as oblique projections (Fig, 1.16a). For accurate and unambiguous descrriptionm, it is necessary to project them down particular crystallographic directions and/or on to crystallographic faces. Ex. Fig. 1.18b (Unit Cell Projections), a face centered cube projected down the z-axis onto the ab unit cell face. - All sense of vertical perspective is lost - To restore vertical perspective, their vertical height in the cell is given as a fraction of c, beside each atom. - It is not necessary to specify the x, y coordinates if the structure is drawn to scale. - The origin is often taken as the top left hand corner. - Two atoms at each corners and two at top and bottom face centers, with z-coordinates of 0 and 1. - The four side face centers, each with z =1/2 Fig. 1.18c (Atomic Coordinates): It is important to use Fig. 1.18b to relate the fractional atomic coordinates. - A face centered cube contains, effectively, four positions: one corner and three face centers, i.e. 000, 0, 0, Each coordinates specifies the fractional distance of the atom from the origin in the directions a, b, and c. - The more complete structure shown in Fig. 1.18b is obtained by 10

11 addition of extra, equivalent positions. (to Fig. 1.18c) 11

12 1.15 Materials that can be described as close packed a) Metals Most metals crystallize in one of the three arrangements: ccp, hcp, and bcc (Table 1.3) - No clear-cut trends are observed (reason for preferred structure not well understood) - Lattice energies of hcp and ccp metal structures are comparable - The structure observed probably depends on fine details of the bonding requirements or band structure of the metal. - Appendix 4: structure details of elements. b) Alloys Alloys are intermetallic phases or solid solutions. Many of them have cp structures. Ex: Cu and Au - both have ccp structures as pure elements Cu Au alloys have ccp structures - At high temperatures, a complete range of solid solutions between Cu and Au forms; the Cu and Au atoms are distributed statistically over the lattice points of the fcc unit cell, i.e. a random mixture of Cu and Au. - On annealing the compositions AuCu and AuCu 3 at lower temperatures, the Au and Cu atoms order themselves; ccp layers still occur but the arrangement is no longer statistical. (Such order-disorder phenomena are outside the scope of this introduction to crystal structures but occur commonly in both metallic and ionic structures). 12

13 c) Ionic structures Ionic compounds: such as NaCl, Al 2 O 3, Na 2 O, ZnO, - The anion is larger than the cation; the structures are built of cp layers of anions with the cations placed in the interstitial sites. - Many structures are possible: structural variables: 1) anion stacking sequence (either hcp or ccp); 2) number and type of interstitial sites occupied by cations. - To accommodate large cations, expanding the anion array is necessary. The anions in cp may not be in contact (eutactic). In the following discussions, use of the terms hcp and ccp for anion arrays may include the structures being in contact or eutactic. (i) Tetrahedral and octahedral sites Two types of interstitial sites in cp: tetrahedral and octahedral, the nature of the interstitial space between any pair of adjacent cp layers. Tetrahedral sites, T -3 anions that form the base of the tetrahedron belong to one cp layer the apex of the tetrahedron in the layer above the base (Fig.1.19a) the apex of the tetrahedron in the layer below the base (Fig.1.19b) the apex is up, giving the T + tetrahedral site the apex is down, giving the T tetrahedral site -Because the center of gravity of a tetrahedron is nearer to the base than the apex cations in tetrahedral sites are not located midway between 13

14 adjacent ion layers but are nearer one layer than the other (the layers seen edge-on in Fig.19c,d). 14

15 Octahedral sites, O -these sites are coordinated to 3 anions in both adjacent anion layers. (Fig.1.19e) -these sites are midway between the anion layers (Fig.1.19f edge-on) -A more common way to regard octahedral coordination is as 4 coplanar atoms with one atom at each apex above and below the plane. In Fig.1.19e, atoms 1,2,4,6 are coplanar. Also, atoms 2,3,4,5 and 1,3,5,6 are coplanar. -Fig.1.19g,h are more conventional perspective. Fig.1.19i : The distribution of interstitial sites between any two adjacent layers of cp anions: -below every sphere in the upper layer (solid line) is a T + -above every sphere in the lower layer (dashed line) is a T -many O sites -counting up the number of each, for every anion there is one octahedral site and two tetrahedral sites, one T + and one T. site number distribution anion : octahedral : T + : T = 1 : 1 : 1 : 1 Table 1.4: -It is rare that all the interstitial sites in a cp structure are occupied; often one set is full or partly occupied and the remaining sets are empty. -cp ionic structures, classified according to the anion layer stacking 15

16 sequence and the occupancy of the interstitial sites (Table 1.4) -how a wide range of structures are grouped into one large family; this helps to bring out similarities and differences between them. For example: (a) rock salt and nickel arsenide (NiAs) structures both have octahedrally coordinated cations and differ only in the anion stacking sequence; similar relationships for olivine and spinel, wurtzite and blende, CdI 2 and CdCl 2. (b) rutile TiO 2 and CdI 2 both have hcp anions with half the octahedral sites occupied by cations; in rutile, half the octahedral sites between any pair of cp anion layers are occupied by Ti; in CdI 2, layers of fully occupied octahedral site alternate with layers in which all sites are empty (this gives CdI 2 a layered structure). (c) In a few structures, it is useful to regard cations as forming the cp layers with the anions occupying the interstitial sites. -The fluorite structure, CaF 2, may be regarded as ccp Ca with all T + and T sites occupied by F. -The antifluorite structure of K 2 O is the exact inverse of fluorite with ccp layers of O and K in T +, T sites. (d) The cp concept may be extended to structures in which a mixture of anions and large cations form the packing layers and smaller cations occupy interstitial sites. In perovskite, BaTiO 3, ccp layers of composition BaO 3 occur and one quarter of the octahedral sites are occupied by Ti (only 16

17 those sites in which all six corners are O 2- ions). (e) Some structures are anion-deficient cp structures in which the anions form an incomplete cp array. -ReO 3 : has ccp oxide layers with 1/4 of O sites empty. It is analogous to perovskite in which Ti is replaced by Re and Ba sites are left vacant. -β-alumina (NaAl 11 O 17 ): cp oxide layers in which every fifth layer has approximately three quarters of the O 2- ions missing. (NaAl 11 O 17 3 ) 17

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19 Figure 2 (a) The unit cell of monoclinic NaTaO 3 ; (b) the refined structure of monoclinic phase NaTaO 3 with a Ta O Ta bond angle less than 180 ; (c) the refined structure of K-doped monoclinic NaTaO 3 with a Ta O Ta bond angle of approximately

20 Figure 6 Time course of photocatalytic H 2 evolution from 1100 ml pure water suspended with 0.1 g of the Na 1-x K x TaO 3 catalysts under mercury lamp irradiation. The inset shows the dependence of the H 2 and O 2 evolution rates on the K content. Reference: Journal of Materials Chemistry, 2011, 21,

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