Cubic atom structures

Ferritic stainless steels are magnetic, have body-centered cubic atomic structures and possess mechanical properties similar to those of carbon steel, though they are less ductile. 

Extensive computer simulations have been caiTied out on the near-surface and surface behaviour of materials having a simple cubic lattice structure. The interaction potential between pairs of atoms which has frequently been used for inert gas solids, such as solid argon, takes die Lennard-Jones form where d is the inter-nuclear distance, is the potential interaction energy at the minimum conesponding to the point of... 

Lewis s interest in chemical bonding and structure dated from 1902. In attempting to explain "valence" to a class at Harvard, he devised an atomic model to rationalize the octet rule. His model was deficient in many respects for one thing, Lewis visualized cubic atoms with electrons located at the corners. Perhaps this explains why his ideas of atomic structure were not published until 1916. In that year, Lewis conceived of the... 

This compound has the cubic fluorite structure with one octahedral interstice per Ce atom. Therefore, a =1, and S = 2 for CeH2. We can therefore write ... 

Two modifications are known for polonium. At room temperature a-polonium is stable it has a cubic-primitive structure, every atom having an exact octahedral coordination (Fig. 2.4, p. 7). This is a rather unusual structure, but it also occurs for phosphorus and antimony at high pressures. At 54 °C a-Po is converted to /3-Po. The phase transition involves a compression in the direction of one of the body diagonals of the cubic-primitive unit cell, and the result is a rhombohedral lattice. The bond angles are 98.2°. 

The transition-metal monopnictides MPn with the MnP-type structure discussed above contain strong M-M and weak Pn-Pn bonds. Compounds richer in Pn can also be examined by XPS, such as the binary skutterudites MPn , (M = Co, Rh, Ir Pn = P, As, Sb), which contain strong Pn-Pn bonds but no M-M bonds [79,80], The cubic crystal structure consists of a network of comer-sharing M-centred octa-hedra, which are tilted to form nearly square Pnn rings creating large dodecahedral voids [81]. These voids can be filled with rare-earth atoms to form ternary variants REM Pnn (RE = rare earth M = Fe, Ru, Os Pn = P, As, Sb) (Fig. 26) [81,82], the antimonides being of interest as thermoelectric materials [83]. 

Chemists have synthesized a spectacular array of submicron- and nano-particles with well-defined size and atomic structure and very special properties. Examples include CdSe quantum dots and novel spheres and rods. Transport enters the picture via fundamental studies of the physical processes that affect the synthesis, which must be understood for even modest scale-up from the milligram level. Likewise, processes for assembling fascinating face-centered-cubic crystals or ordered multilayers must concentrate on organizing the particles via flow, diffusion, or action of external fields. Near-perfection is possible but requires careful understanding and control of the forces and the rates. 

The atomic structure of the nuclei of metal deposits, which have the simplest form since they involve only one atomic species, appear to be quite different from those of the bulk metals. The structures of metals fall mainly into three classes. In the face-centred cubic and the hexagonal structures each atom has 12 co-ordination with six neighbours in the plane. The repeat patterns obtained by laying one plane over another in the closest fit have two alternative arrangements. In the hexagonal structure the repeat pattern is A-B-A-B etc., whereas in the face-centred cubic structure the repeat pattern is A-B-C-A-B-C. In the body-centred cubic structure in which each atom is eight co-ordinated, the repeat pattern is A-B-A-B. (See Figure 1.4.)... 

To illustrate this, take the situation in a very common and relatively simple metal structure, that of copper. A crystal of copper adopts the face-centered cubic (fee) structure (Fig. 2.8). In all crystals with this structure slip takes place on one of the equivalent 111 planes, in one of the compatible <110> directions. The shortest vector describing this runs from an atom at the comer of the unit cell to one at a face center (Fig. 3.10). A dislocation having Burgers vector equal to this displacement, i <110>, is thus a unit dislocation in the structure. 

The simplest of structures is the rock salt structure, depicted in Figure 2.2a. Magnesium oxide is considered to be the simplest oxide for a number of reasons. It is an ionic oxide with a 6 6 octahedral coordination and it has a very simple structure — the cubic NaCl structure. The structure is generally described as a cubic close packing (ABC-type packing) of oxygen atoms in the (111) direction forming octahedral cavities. This structure is exhibited by other alkaline earth metal oxides such as BaO, CaO, and monoxides of 3d transition metals as well as lanthanides and actinides such as TiO, NiO, EuO, and NpO. 

Fig. 6-1. TVo-dimensional atomic structure on the (100) plane of platinum crystals (1x1) = cubic close-packed surface plane identical with the (100) plane (5 x 20) = hexagonal dose-packed surface plane reconstructed finm the original (100) plane. [From Kolb, 1993.]...

You now know how to define a supercell for a DFT calculation for a material with the simple cubic crystal structure. We also said at the outset that we assume for the purposes of this chapter that we have a DFT code that can give us the total energy of some collection of atoms. How can we use calculations of this type to determine the lattice constant of our simple cubic metal that would be observed in nature The sensible approach would be to calculate the total energy of our material as a function of the lattice constant, that is, tot(a). A typical result from doing this type of calculation is shown in Fig. 2.1. The details of how these calculations (and the other calculations described in the rest of the chapter) were done are listed in the Appendix at the end of the chapter. 

The simple cubic crystal structure we discussed above is the simplest crystal structure to visualize, but it is of limited practical interest at least for elements in their bulk form because other than polonium no elements exist with this structure. A much more common crystal stmcture in the periodic table is the face-centered-cubic (fee) structure. We can form this structure by filling space with cubes of side length a that have atoms at the corners of each cube and also atoms in the center of each face of each cube. We can define a supercell for an fee material using the same cube of side length a that we used for the simple cubic material and placing atoms at (0,0,0), (0,g/2,g/2), (g/2,0,g/2), and (g/2,g/2,0). You should be able to check this statement for yourself by sketching the structure. 

Figure 1. LMTO DOS of cubic AI12M11 (13 atoms / unit cell) and cubic a-Aln4Mn24 approximant (experimental atomic structure of a-AlMnSi [26] with Si = Al, 138 atoms/ unit cell) [22].

Energy calculations for small clusters of atoms indicate that a cluster of 55 atoms should be reasonably stable (Mackay 1962, Allpress and Sanders 1970, Hore and Pal 1972). In addition, calculations suggest that the 55-atom cluster will take up an icosahedral structure in preference to the cubic cuboctahedral structure (Hore and Pal 1972). 

Figure 1.18 The face-centered cubic (FCC) structure showing (a) atoms touching and (b) atoms as small spheres. Reprinted, by permission, from W. Callister, Materials Science and Engineering An Introduction, 5th ed., p. 32. Copyright 2000 by lohn Wiley Sons, Inc.

Take, for iustauce, one of the forms of carbon diamond. Diamond has a cubic crystal structure with au F-ceutred lattice (Figure 1.47) the positious of the atomic... 

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