Crystal structure packing spheres

The atomic bonding in this group of materials is metallic and thns nondirectional in natnre. Conseqnently, there are minimal restrictions as to the nnmber and position of nearest-neighbor atoms this leads to relatively large nnmbers of nearest neighbors and dense atomic packings for most metallic crystal structnres. Also, for metals, when we nse the hard-sphere model for the crystal structure, each sphere represents an ion core. Table 3.1 presents the atomic radii for a number of metals. Three relatively simple crystal structures are found for most of the common metals face-centered cubic, body-centered cnbic, and hexagonal close-packed. 

It is also possible to have a unit cell that consists of atoms situated only at the corners of a cube. This is called the simple cubic (SC) crystal structure-, hard-sphere and reduced-sphere models are shown, respectively, in Figures 33a and 33b. None of the metallic elements have this crystal structure because of its relatively low atomic packing factor (see Concept Check 3.1). The only simple-cubic element is polonium, which is considered to be a metalloid (or semi-metal). 

Even in a close-packed structure, hard spheres do not fill all the space in a crystal. The gaps the interstices—between the atoms are called holes. To determine just how much space is occupied, we need to calculate the fraction of the total volume occupied by the spheres. 

This bismuth-III structure is also observed for antimony from 10 to 28 GPa and for bismuth from 2.8 to 8 GPa. At even higher pressures antimony and bismuth adopt the body-centered cubic packing of spheres which is typical for metals. Bi-III has a peculiar incommensurate composite crystal structure. It can be described by two intergrown partial structures that are not compatible metrically with one another (Fig. 11.11). The partial structure 1 consists of square antiprisms which share faces along c and which are connected by tetrahedral building blocks. The partial structure 2 forms linear chains of atoms that run along c in the midst of the square antiprisms. In addition, to compensate for the... 

Figure 5.2 (a) Electron density contour map of the CI2 molecule (see Chapter 6) showing that the chlorine atoms in a CI2 molecule are not portions of spheres rather, the atoms are slightly flattened at the ends of the molecule. So the molecule has two van der Waals radii a smaller van der Waals radius, r2 = 190 pm, in the direction of the bond axis and a larger radius, r =215 pm, in the perpendicular direction, (b) Portion of the crystal structure of solid chlorine showing the packing of CI2 molecules in the (100) plane. In the solid the two contact distances ry + ry and ry + r2 have the values 342 pm and 328 pm, so the two radii are r 1 = 171 pm and r2 = 157, pm which are appreciably smaller than the radii for the free CI2 molecule showing that the molecule is compressed by the intermolecular forces in the solid state. 

Figure 11 Packing of the Tel62 anions in the crystal structure of [C6H5NH (CH3)2]2 2 io- Black spheres Te white spheres I dashed lines / T contacts...

Figure 3.8. Crystal structure of CsCl. The positions of the centres of the atoms in the unit cell are shown in (a). In (b) the same cell is described by means of its characteristic sections taken at the height 0, A, and 1 of the third axis. In (c) a projection of the cell on its square basis is presented the values of the third (fractional) coordinate are indicated. In (d) the shortest interatomic distances are shown dCs-ci = a)3/2 = 411.3 X 0.866025. = 356.2. In (e) the subsequent group of interatomic distances (d = a = 411.3) involving six atoms in the adjacent cells is presented. A group of eight cells is represented in (f) to suggest that the actual structure of CsCl corresponds to a three-dimensional infinite repetition of unit cells and to show that the coordination around the white atoms is similar to that around the black ones shown in (d). The unit cell of the CsCl structure is shown as a packed spheres model in (g).

It is not hard to understand why many metals favor an fee crystal structure there is no packing of hard spheres in space that creates a higher density than the fee structure. (A mathematical proof of this fact, known as the Kepler conjecture, has only been discovered in the past few years.) There is, however, one other packing that has exactly the same density as the fee packing, namely the hexagonal close-packed (hep) structure. As our third example of applying DFT to a periodic crystal structure, we will now consider the hep metals. 

The structures of many inorganic crystal structures can be discussed in terms of the simple packing of spheres, so we will consider this first, before moving on to the more formal classification of crystals. 

We know from quantum mechanics that atoms and ions do not have precisely defined radii. However, from the foregoing discussion of ionic crystal structures we have seen that ions pack together in an extremely regular fashion in crystals, and their atomic positions, and thus their interatomic distances, can be measured very accurately. It is a very useful concept, therefore, particularly for those structures based on close-packing, to think of ions as hard spheres, each with a particular radius. 

FIGURE 1.50 (a) The crystal structure of CO2, (b) packing diagram of the unit cell of CO2 projected on to the xy plane. The heights of the atoms are expressed as fractional coordinates of c. C, blue spheres 0, grey spheres. 

This opening chapter has introduced many of the principles and ideas that lie behind a discussion of the crystalline solid state. We have discussed in detail the structure of a number of important ionic crystal structures and shown how they can be linked to a simple view of ions as hard spheres that pack together as closely as possible, but can also be viewed as the linking of octahedra or tetrahedra in various ways. Taking these ideas further, we have investigated the size of these ions in terms of their radii, and... 

In order to be able to describe the ideal crystal structure , it is important to bear in mind that there are two tetrahedral cavities and one octahedral one present for every sphere in a close-packed structure. With the help of Table 4.1 and the rules for octahedral and tetrahedral coordination the description of the following crystal structures are now easily understood when we bear in mind that in an ionic crystal lattice the larger negative ions form the close-packed structure and that the octahedral and tetrahedral cavities are filled with positive ions. 

The English physicist William Barlow began as a London business man later he became interested in crystal structures and devoted his life to that study. In 1894, he published his findings of the 230 space groups. It is amazing that from consideration of symmetry three scientists in different countries arrived at the 230 space groups of crystals at about this time. Barlow then worked with ideas of close packing. He pictured the atoms in a crystal as spheres, which, under the influence... 

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