Body-centered Cubic Packing of 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... 

Germanium forms the same kinds of modifications as silicon at similar conditions (Fig. 12.4). Tin, however, does not exhibit this diversity )3-tin transforms to a body-centered cubic packing of spheres at 45 GPa. Lead already adopts a cubic closest-packing of spheres at ambient pressure. 

The space filling in the body-centered cubic packing of spheres is less than in the closest packings, but the difference is moderate. The fraction of space filled amounts to ns/3 = 0.6802 or 68.02 %. The reduction of the coordination number from 12 to 8 seems to be more serious however, the difference is actually not so serious because in addition to the 8 directly adjacent spheres every sphere has 6 further neighbors that are only 15.5 % more distant (Fig. 14.3). The coordination number can be designated by 8 + 6. 

Corresponding to its inferior space filling, the body-centered cubic packing of spheres is less frequent among the element structures. None the less, 15 elements crystallize with this structure. As tungsten is one of them, the term tungsten type is sometimes used for this kind of packing. 

Unit cell of the body-centered cubic packing of spheres and the coordination around one sphere... 

Figure 10.20b). Called body-centered cubic packing, each sphere has a coordination number of 8—four neighbors above and four below—and space is used quite efficiently 68% of the available volume is occupied. Iron, sodium, and 14 other metals crystallize in this way. 

The CsCl type offers the simplest way to combine atoms of two different elements in the same arrangement as in body-centered cubic packing the atom in the center of the unit cell is surrounded by eight atoms of the other element in the vertices of the unit cell. In this way each atom only has adjacent atoms of the other element. This is a condition that cannot be fulfilled in a closest-packing of spheres (cf. preceding section). 

A body-centered cubic unit cell has eight corner atoms plus an additional atom in the center of the cube (Figure 10.22b). This body-centered cubic unit cell, with two repeating offset layers and with the spheres in a given layer slightly separated, is the repeat unit found in body-centered cubic packing. 

When we determined the crystalline structure of solids in Chapter 4, we noted that most transitional metals form crystals with atoms in a close-packed hexagonal structure, face-centered cubic structure, or body-centered cubic arrangement. In the body-centered cubic structure, the spheres take up almost as much space as in the close-packed hexagonal structure. Many of the metals used to make alloys used for jewelry, such as nickel, copper, zinc, silver, gold, platinum, and lead, have face-centered cubic crystalline structures. Perhaps their similar crystalline structures promote an ease in forming alloys. In sterling silver, an atom of copper can fit nicely beside an atom of silver in the crystalline structure. 

So in the body-centered cubic arrangement 68.0% of the space is actually occupied by spheres. This is somewhat less than the space occupied in the closest packed structures (74.0%). That is, in contrast to cubic closest packing and hexagonal closest packing, the body-centered cubic method of packing spheres does not represent a closest packed structure. 

Close-packing arrangements of spheres of eqnal sizes. Shown are (a) cubic close-packing, (b) hexagonal close-packing, and (c) body-centered cubic packing. 

Figure 1 Common morphologies of microphase-separated block copolymers body-centered cubic packed spheres (BCC), hexagonally ordered cylinders (HEX), g)Toid (Ia3d), hexagonaUy perforated layers (HPLs), modulated lamellae (MLAM), lamellae (LAM), cylindrical micelles (CYL), and spherical micelles (MIC). (Reproduced from Ref. 2. Wiley-VCH, 1998.)...

It can be observed that Chong s equation approximates to Einstein s equation at very low solid loading where < )c is 0.605 (for monosize spheres under body centered cubic packing). Stedman et al (16) measure the rheology of silicon nitride particulates/silicon carbide whiskers composite mixed with a polypropylene based binder. The relative viscosity data was found to fit Chong s equation well. Zhang and Evans [17] investigated on... 

The morphology of the ABA-type linear block copolymers is strongly influenced by the volume fraction of the two components. For example, in PS-EB-PS-type block copolymer as the volume fraction of PS is increased, the shape of the dispersed PS phase changes from spherical (comprising body-centered cubic spheres of PS dispersed in continuous soft phase) to cylindrical form (hexagonal packed cylinders of PS) [10,133,134]. When the volume fraction of the two phases... 

Although the space filling of the body-centered cubic sphere packing is somewhat inferior to that of a closest-packing, the CsCl type thus turns out to be excellently suited for compounds with a 1 1 composition. Due to the occupation of the positions 0,0,0 and with different kinds of atoms, the structure is not... 

Body-centered cubic sphere packing => CsCl type => superstructures of the CsCl type... 

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