Sphere body-centered cubic

Figure 2.5 Equilibrium phase morphologies of AB diblock copolymers as a function of component A volume fraction (S) Spheres (body-centered cubic lattice), (C) Cylinders (hexagonal lattice), (G) Gyroid, (L) Lamellar. Reproduced with permission from Ref. [7].

O (molecular) F molecular Ne (atomic). 1Z42 Covalent Si, C. Molecular Scg, HBr, CO2, P4O6, SiH4.12.44 Simple cubic one sphere body-centered cubic two spheres lace-centered cubic four spheres. 12.46 6.20 X 10 atom/mol. 12.48 458 pm. 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

It follows that 3a2 = (4r)1 and therefore that a = 4r/3112. Each unit cell contains one sphere at each of the eight corners and one sphere at the center 8 X 1/8 +1=2 spheres so the total mass of a body-centered cubic unit cell is 2M/NA. Therefore,... 

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... 

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... 

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). 

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... 

Figure 5.10 Self-organization of di-block copolymers. Block copolymers can form spherical and cylindrical micelles, vesicles, spheres with face-centered cubic (fee) and body-centered cubic (bcc) packing, hexagonally packed cylinders (hex) minimal surfaces (gyroid, F surface, and P surface), simple lamellae and modulated and perforated lamellae. (Adapted from Bucknall and Anderson, 2003.)...

The type of crystalline structure that is formed depends on the concentration of the particles as well as the magnitude of the Debye-Hiickel thickness. For large Debye-Hiickel thicknesses a body-centered cubic crystal is formed, whereas for smaller values a face-centered cubic crystal is preferred. An example of the latter observed experimentally in a dispersion of latex spheres is shown in Figure 13.3. Note that this crystallization phenomenon is analogous to crystallization of simple atomic fluids, as is evident from Figure 13.3a, which shows the coexistence of a crystal with a liquidlike structure. 

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