Body-centered cubic structure figure

Allotropy in the solid state can also arise because of differences in crystal structure. For example, solid iron has a body-centered cubic structure (recall Figure 9.16, page 246) at room temperature. This changes to a face-centered structure upon heating to 910°C. 

Figure 9.2 is schematic diagram of the crystal structure of most of the alkali halides, letting the black circles represent the positive metal ions (Li, Na, K, Rb, and Cs), and the gray circles represent the negative halide ions (F, Cl, Br, and I).The ions lie on two interpenetrating face-centered-cubic lattices. Of the 20 alkali halides, 17 have the NaCl crystal structure of Figure 9.1. The other three (CsCl, CsBr, and Csl) have the cesium chloride structure where the ions lie on two interpenetrating body-centered-cubic lattices (Figure 9.3). The plastic deformation on the primary glide planes for the two structures is quite different.

Figure 5.8 Interstitial diffusion (a) interstitial diffusion involving the direct migration of an interstitial atom to an adjacent site in the crystal (b, c) some of the octahedral and tetrahedral interstitial sites in the body-centered cubic structure of metals such as iron and tungsten and (d) the total number of octahedral and tetrahedral sites in a unit cell of the body-centered cubic structure. Diffusion paths parallel to the unit cell edges can occur by a series of alternating octahedral and tetrahedral site jumps, dashed line.

Figure 9.26. The body-centered cubic structure of M0AI12 shows Mo atoms at the corners and center. Only two M0AI12 molecules are shown.

Figure 9 shows a plot of Bobs - B against /B for metallic a-Fe particles in zeolite NaX. From the slope of the curve, a mean magnetic moment of /u. = 916 180/u-b has been obtained. With the saturation magnetization of a-Fe and assuming spherical particle sizes with body-centered cubic structure, one obtains a mean diameter of <7 = 2.1 0.1 nm. 

Plan Because an atom is spherical, we can find its radius from its volume. If we multiply the reciprocal of density (volume/mass) by the molar mass (mass/mole), we find the volume of 1 mol of Ba metal. The metal crystallizes in the body-centered cubic structure, so 68% of this volume is occupied by 1 mol of the atoms themselves (see Figure 12.26C). Dividing by Avogadro s number gives the volume of one Ba atom, from which we find the radius. 

Bismuth, core radius, 362 Bloch sum, 33, 72 of bond orbitals, 144 Body-centered cubic structure, 350 figure, 479... 

FIGURE 20-2 The body-centered cubic structure. The central atom sits at the center of a cube formed by k,2 its eight nearest neighbors, shaded to distinguish them, though every atom and its environment (in the extended crystal) is identical. The six second neighbors lie a distance 15 percent further away. We construct a Bloch sum with wave number in the z-direction, giving phase factors shown for atoms in each plane of constant z. 

The coordination number of first nearest neighbors in closed packed structures is 12. Nevertheless, several metals crystallize in the body centered cubic structure (W type, Pearson symbol cI2, Figure 2.5). The coordination number is reduced to 8 but the difference of the distance between the first and the second nearest neighbors (white spheres) is small (15.5%). This structure is known as the CsCl structure. 

Figure 11.1. (a) Small-angle X-ray scattering patterns and (b) nitrogen adsorption isotherms for SBA-16 silica with body-centered cubic structure (Im3m) calcined at different temperatures. The structures in the middle illustrate the pore structure symmetry and the pore connectivity. The figure is adapted from our earher pubheation. ... 

As a function of temperature, and up to 350 °C in some cases, magnesium and cadmium soaps exhibit only one or two eylindrical structures, whereas calcium soaps present both different successive struetures with disk and cylinders. These two classes of structure are found also in strontium and barium soaps, with, in addition, a body-centered cubic structure. These cubic phases have interesting features in that the polar groups are present on rods of finite length which belong to two, interwoven, infinite three-dimensional networks and the hydrocarbon chains constitute a eontinuous, paraffinic matrix (Figure 45). 

Figure 14.2 (a) and (b) Body-center cubic structure of BPl,(c) and (d) simple cubic stmcture of BPII filled with double-twist cylinders. The black lines in (b) and (d) represent the defect lines. Reproduced with permission from the Optical Society of America. 

The simplest crystal structures are cubic unit cells with only one atom centered at each lattice point. Most metals have such structures. Nickel, for example, has a face-centered cubic unit cell, whereas sodium has a body-centered cubic one. Figure 11.34 T shows how atoms fill the cubic unit cells. Notice that the atoms on the comers and faces do not lie wholly within the unit cell. Instead, these atoms are shared between unit cells. Table 11.6 summarizes the fraction of an atom that occupies a unit cell when atoms are shared between unit cells. 

In body centered cubic structure, see Figure 4.6, the spheres come into contact by axis (the space diagonal) of the cube. 

Figure 4.1 shows a model for a body-centered cubic crystal. Review the Properties tables for all of the metals in the Elements Handbook (Appendix A). What metals exist in body-centered cubic structures ... 

Figure 20. Two possible cubic defect structures for blue phases (a) a simple cubic structure, (b) a body-centered cubic structure.

What foam structure will minimize energy, which is just the total surface area of all of the films This is the Kelvin s problem. The solution of the problem in 2D was conjectured by him to be the honeybee s comb structure. This conjecture was proven recently by Thomas Hales for infinite structure or for finite structures with periodic boundary conditions. Besides this, only the N = 2 case (the double-bubble problem) has been solved in 2D and 3D. Cases for N larger or equal to 3 in 3D have been studied only partially. Concerning 3D infinite structures, Kelvin came up with the body-centered cubic structure, which he called tetrakaidecahedron. However, recently an alternative structure with a lower energy was computed by Weaire and Phalen. This has a more complicated structure with two different kinds of cells (see Figure 2.15). 

Some alloys are composed of metals that have different crystal structures. For example, nickel crystallizes in the face-centered cubic structure and chromium in the body-centered cubic structure. Because of their different structures, these two metals do not form a miscible solid solution at all compositions. At some intermediate composition, the structure has to change from that of one of the metals to that of the other. Figure 23.7 shows the nickel and chromium phase diagram from 700 °C to 1900 °C. Notice that the diagram has two different solid phases face-centered cubic and body-centered cubic. From pure nickel (0 mol % chromium) to about 40-50 mol % chromium, the structure is face-centered cubic. 

We can determine the composition and relative amounts of the two different phases that coexist in a two-phase region from a phase diagram. Point A on Figure 23.7 in the Cr-Ni phase diagram represents 50% composition at 700 °C, and both phases are present. Some of the Cr atoms have substituted into the nickel-rich face-centered cubic structure, but there is too much Cr to all lit into the aystal. The leftover Cr atoms form the chromium-rich body-centered cubic structure with a small number of Ni atoms in the crystal. 

Figure 14.1 Schematic representation with TEM images of PVPh-h-PS/P4VP blend system showing phase transition from lamellar, gyroid, and hexagonally packed cylinder to body-centered cubic structures with increase in P4VP volume fractions. Undulated and distorted lamellar structures observed in PVPh-h-PS/PMMA blends due to weak hydrogen bonding between PMMA and PVPh. Chen et al. [23]. Reproduced with permission of American Chemical Society.

Another class of intermetallic compounds that has received consid able attention are those with the crystal structure designated as B2. Figure 2(b) shows that this structure is analogous to the body-centered cubic structure of disordered alloys. These compounds have the stoichiometry AB, where one atom type is at the center of the unit odl and the other is at the oght comas. 

FIGURE 5.32 The body-centered cubic (bcc) structure. This structure is not packed as closely as the others that we have illustrated. It is less common among metals than the close-packed structures. Some ionic structures are based on this model. 

The most important metals for catalysis are those of the groups VIII and I-B of the periodic system. Three crystal structures are important, face-centered cubic (fee Ni, Cu, Rh, Pd, Ag, Ir, Pt, Au), hexagonally dose-packed (hep Co, Ru, Os) and body-centered cubic (bcc Fe). Figure 5.1 shows the unit cell for each of these structures. Note that the unit cells contain 4, 2, and 6 atoms for the fee, bcc, and hep structure, respectively. Many other structures, however, exist when considering more complex materials such as oxides, sulfides etc, which we shall not treat here. Before discussing the surfaces that the metals expose, we mention a few general properties. 

Figure 6.9 Structure of the high-temperature form of Agl (a-Agl) (a) the body-centered cubic arrangement of iodide (I-) ions the unit cell is outlined (b) two (of four) tetrahedral sites on a cube face, indicated by filled circles and (c) the four tetrahedral sites found on each cube face, indicted by filled circles. Ag+ ions continuously jump between all of the tetrahedral sites.

---