Unit cell close-packed

TABLE 18.1. Unit cells, close-packed structures, and crystal lattices. A. Simple Cubic Unit Cell... 

The double-helical structures of native A- and B-amyloses are found in the fourth group. It is interesting that in both h as well as the d and dyg spacings, they are comparable with the structure of amylose triacetate I (ATAI). In part, this may arise because the packing of the bulky acetate substituents in ATAI is similar to the close-packing of two amylose chains into a double helix. In the latter, one chain may act as the "substituent" for the other chain. At any rate, all three structures contain similar, cylindrical-shaped helices. Somewhat unexpectedly, the distances cL and d-yo are very close for the two native polymorphs, even though their unit cells and packing are... 

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 a-rhombohedral form of boron has the simplest crystal stmcture with slightly deformed cubic close packing. At 1200°C a-rhombohedral boron degrades, and at 1500°C converts to P-rhombohedral boron, which is the most thermodynamically stable form. The unit cell has 104 boron atoms, a central B 2 icosahedron, and 12 pentagonal pyramids of boron atom directed outward. Twenty additional boron atoms complete a complex coordination (2). 

Let us now look at the c.p.h. unit cell as shown in Fig. 5.4. A view looking down the vertical axis reveals the ABA stacking of close-packed planes. We build up our c.p.h. crystal by adding hexagonal building blocks to one another hexagonal blocks also stack so that they fill space. Here, again, we can use the unit cell concept to open up views of the various types of planes. 

The ultimate covalent ceramic is diamond, widely used where wear resistance or very great strength are needed the diamond stylus of a pick-up, or the diamond anvils of an ultra-high pressure press. Its structure, shown in Fig. 16.3(a), shows the 4 coordinated arrangement of the atoms within the cubic unit cell each atom is at the centre of a tetrahedron with its four bonds directed to the four corners of the tetrahedron. It is not a close-packed structure (atoms in close-packed structures have 12, not four, neighbours) so its density is low. 

Fig. 20.25 Unit cells of (a) the face-centred cubic (f.c.c.), (b) the close-packed hexagonal (c.p.h.) and (c) the body-centred cubic (b.c.c.) crystal structures...

Figure 15 shows a stereoscopic view of the crystalline 1 1 complex where R7 = i-CsHn and R8 = (CH2)2Ph 9). The packing mode of the four molecules in the unit cell of this complex corresponds to the association scheme of tetramer 17 (Fig. 8). Of particular interest is that a pair of groups with similar geometrical shape, NMe2 and CHMe2 [part of C6H4NMe2 and (CH2)2CHMe2], are in close contact.

The best way to determine the type of unit cell adopted by a metal is x-ray diffraction, which gives a characteristic diffraction pattern for each type of unit cell (see Major Technique 3 following his chapter). However, a simpler procedure that can be used to distinguish between close-packed and other structures is to measure the density of the metal we then calculate the densities of the candidate unit cells and decide which structure accounts for the observed density. Density is an intensive property, which means that it does not depend on the size of the sample (Section A). Therefore, it is the same for a unit cell and a bulk sample. Hexagonal and cubic close packing cannot be distinguished in this way, because they have the same coordination numbers and therefore the same densities (for a given element). 

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. 

Many ionic compounds are considered to pack in such as way that the anions form a close-packed lattice in which the metal cations fill holes or interstitial sites left between the anions. These lattices, however, may not necessarily he as tightly packed as the label close-packed implies. The radius of an F ion is approximately 133 pm. The edge distances of the cubic unit cells of LiF, NaF, KF, RbF, and CsF, all of which... 

Consider a metallic element that crystallizes in a cubic close-packed lattice. The edge length of the unit cell is 408 pm. If close-packed layers are deposited on a flat surface to a depth (of metal) of 0.125 mm, how many close-packed layers are present ... 

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 ... 

Fig. 9. — Antiparallel packing arrangement of the 3-fold helices of (1— 4)-(3-D-xylan (7). (a) Stereo view of two unit cells roughly normal to the helix axis and along the short diagonal of the ab-plane. The two helices, distinguished by filled and open bonds, are connected via water (crossed circles) bridges. Cellulose type 3-0H-0-5 hydrogen bonds stabilize each helix, (b) A view of the unit cell projected along the r-axis highlights that the closeness of the water molecules to the helix axis enables them to link adjacent helices.

Two helices are packed antiparallel in the orthorhombic unit cell. Association of the helices occurs through a series of periodic carboxylate potassium water - carboxylate interactions. An axial projection of the unit-cell contents (Fig. 23b) shows that the helices and guest molecules are closely packed. This is the first crystal structure of a polysaccharide in which all the guest molecules in the unit cell, consistent with the measured fiber density, have been experimentally located from difference electron-density maps. The final / -value is 0.26 for 54 reflections, of which 43 are observed, and it is based on normal scattering factors.15... 

Fig. 28. (continued)—(b) An axial projection of the unit cell shows that the close packing of the chains is more prominent along the diagonals than along the edges. 

Here Hd is the number of atoms in a unit cell, the volume of which is V, and is the shortest interatomic distance in the arrangement. The definition contains a division by /2 so that the parameter D becomes unity for close-packing structures. Kepler s conjecture ensures that the parameter D is always less than or equal to unity. The fraction of space occupied (fi in the rigid-sphere model, which is often used in the discussion of metallic structures, is proportional to the parameter D and the relation is as follows. 

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