Unit cells hexagonal close packed

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Sohd hydrogen usually exists in the hexagonal close-packed form. The unit cell dimensions are = 378 pm and Cg = 616 pm. SoHd deuterium also exists in the hexagonal close-packed configuration, and = 354 pm, Cg = 591 pm (57—59). 

The differing malleabilities of metals can be traced to their crystal structures. The crystal structure of a metal typically has slip planes, which are planes of atoms that under stress may slip or slide relative to one another. The slip planes of a ccp structure are the close-packed planes, and careful inspection of a unit cell shows that there are eight sets of slip planes in different directions. As a result, metals with cubic close-packed structures, such as copper, are malleable they can be easily bent, flattened, or pounded into shape. In contrast, a hexagonal close-packed structure has only one set of slip planes, and metals with hexagonal close packing, such as zinc or cadmium, tend to be relatively brittle. 

Phase Type in Struk-turbericht Character Atoms per unit cell (mini mum) 0 B2 and Z,20 body- centered cubic 2 /3-Mn A13 complex cubic 20 7 D81, 82, 83 84 complex cubic 52 e hexagonal close packed, c/a <(8/3) 2 ... 

Figure 5.1. Unit cells of the face-centered cubic (fee), body-centered cubic (bcc), and hexagonally closed packed (hep) lattices.

The corresponding unit cells are shown in Figure 1.1 and an examination of simple ball-and-stick models (which the reader is strongly urged to carry out) shows that the face-centred cubic (fee) and hexagonal close-packed (hep) structures correspond to the only two possible ways of close-packing spheres, in which each sphere has twelve nearest neighbours. 

As a first approximation, let us consider the X atoms as rigid spheres arranged in an ideal hexagonal close-packed (hep) sublattice. On transforming the usual hexagonal unit cell (a, 6, c ) to an ortho-hexagonal cell, defined by a0 = c, bo = b —a, and Co = + b, and introducing... 

Answer (c) is correct atoms at the comers, along the edges, and on the faces of a unit cell are shared with adjacent unit cells, (a) is incorrect often there are several formula units within a unit cell, as in the case for NaCl. (b) is incorrect the unit cell need not be cubic that of hexagonal close packing is not. (d) is incorrect a unit cell will not contain the same number of cations as anions if their numbers are not equal in the formula of the compound. 

The unit cell of corundum is rhombohedral, but the structure is usually described with respect to hexagonal axes. Each of the cations is surrounded by six oxygen ions in a slightly distorted octahedral coordination. The anions are close to a hexagonal close-packed array and the cations occupy two-thirds of the available octahedral positions in this array in an ordered fashion. The structure is also adopted by a-Fe203 and Q2O3. 

Figure 3.20. A lateral view of different stacking sequences of triangular nets. They correspond to some typical close-packed structures. The first layer sequence shown corresponds to a superimposition according to the scheme ABABAB... (equivalent to BCBCBC... or CACACA... descriptions) characteristic of the hexagonal close-packed, Mg-type, structure. With reference to the usual description of its unit cell, the full stacking symbol indicating the element, the relative position of the superimposed layers and their distance is Mg Mg. The other sequences correspond to the schemes ABC.ABC. (Cu, cubic), ABAC.ABAC. (La, hexagonal), ACACBCBAB. (Sm, hexagonal). For Cu the constant ch of the (equivalent, non-conventional) hexagonal cell is shown which may be obtained by a convenient re-description of the standard cubic cell (see 3.6.1.3). ch = cV 3, body diagonal of the cubic cell.

Figure 16.2. Unit cells of the three most important lattice types (a) face-centered cubic (b) hexagonal close-packed (c) body-centered cubic. (From Ref 1, with permission from Noyes.)...

Fig. 2.4 Structure of goethite. a) Hexagonal close packed anion arrangement and distribution of cations in the octahedral interstices projection on (010) with the cation pairs indicated and the unit cell outlined, b) Projection on (001) with the unit cell and the octahedral arrangementindi-cated. c) Projection down [001]. Dashed circles represent Fe in the next lower layer, d) Arrangement of octahedral double chains. H atoms also shown, e) Ball-and-stick model with unit cell out-...

The host crystal of chrysoberyl has a hexagonal-close-packed structure. The space group is orthorhombic Pnma with four molecules per unit cell. The AP ions are octahedrally coordinated by the oxygen ions and occur in two not equivalent crystal field sites in the lattice. The AP" sites lying in the mirror-... 

Figure 1.17 The extended unit cell of the hexagonal close-packed (HCP) structure.

The hexagonal ferrites have a structure related to the spinel structure but with hexagonal close-packed oxygen ions and a unit cell made up of two formulae of MN12O19, where M is divalent (Ba, Sr, or Pb) and N is trivalent (Al, Ga, Cr, or Fe),... 

Adsorption in many different adsorption sites simultaneously is expected for overlayers with an incommensurate lattice (cf. Sect. III). This has been confirmed by LEED intensity analyses for the case of an incommensurate overlayer of Xe on Ag(l 11), where both the substrate and the overlayer consist of hexagonally close-packed layers (with unrelated unit cells) parallel to the surface. 

The concept of close packing is particularly useful in describing the crystal structures of metals, most of which fall into one of three classes hexagonal close packed, cubic close packed (i.e., fee), and body-centered cubic (bcc). The bcc unit cell is shown in Fig. 4.8 its structure is not close packed. The stablest structures of metals under ambient conditions are summarized in Table 4.1. Notable omissions from Table 4.1, such as aluminum, tin, and manganese, reflect structures that are not so conveniently classified. The artificially produced radioactive element americium is interesting in that the close-packed sequence is ABAC..., while one form of polonium has... 

Wurtzite structure. Zinc sulfide can also crystallize in a hexagonal form called wurtzite that is formed slightly less exothermically than the cubic zinc blende (sphalerite) modification (Afff = —192.6 and —206.0 kJ mol-1, respectively) and hence is a high temperature polymorph of ZnS. The relationship between the two structures is best described in terms of close packing (Section 4.3) in zinc blende, the anions (or cations) form a cubic close-packed array, whereas in wurtzite they form hexagonal close-packed arrays. This relationship is illustrated in Fig. 4.13 note, however, that this does not represent the actual unit cell of either form. 

Fig. 35 Crystallization of non-spherical particles homogeneously coated with DNA. (a, b) Schematic of a hexagonal close-packed 2D layer in assemblies of gold nanorods and corresponding structure factor S(q) obtained from SAXS measurements (blue line) and simulations (red line), (c, d) Schematic of ID columnar assembly of gold triangular nanoprisms and corresponding SAXS patterns, (e, f) Schematic of the 3D fee assembly of gold rhombic dodecahedra (the lines indicate the fee unit cell) and corresponding SAXS patterns. Adapted with permission from [152]...

Figure 16.3. (a) Construction of the Wigner-Seitz cell in a 2-D hexagonal close-packed (hep) lattice, (b) Primitive unit cell of the hep lattice. 

Figure 4.2. The 2P structure (hexagonal close-packed, hep). The hexagonal unit cell is outlined by double lines.

Crystalline solids consist of periodically repeating arrays of atoms, ions or molecules. Many catalytic metals adopt cubic close-packed (also called face-centred cubic) (Co, Ni, Cu, Pd, Ag, Pt) or hexagonal close-packed (Ti, Co, Zn) structures. Others (e.g. Fe, W) adopt the slightly less efficiently packed body-centred cubic structure. The different crystal faces which are possible are conveniently described in terms of their Miller indices. It is customary to describe the geometry of a crystal in terms of its unit cell. This is a parallelepiped of characteristic shape which generates the crystal lattice when many of them are packed together. 

The rhombohedral unit cells for rhodium and iridium trifluorides (44) contain two formula units. The structure can be related to the first structure type by considering anion positions, which here correspond to a hexagonal, close-packed array. There are no vacant anion sites and the cations occupy one-third of the octahedral holes. This leads to M—F—M angles of 132°, characteristic for filling adjacent, octahedral holes in a hexagonal close-packed lattice. Alternatively, the structure can be described as a linking of octahedra through all corners, but the octahedra are now tilted with respect to each other. 

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