Simple cubic structure

Figure Al.3.23. Phase diagram of silicon in various polymorphs from an ab initio pseudopotential calculation [34], The volume is nonnalized to the experimental volume. The binding energy is the total electronic energy of the valence electrons. The slope of the dashed curve gives the pressure to transfomi silicon in the diamond structure to the p-Sn structure. Otlier polymorphs listed include face-centred cubic (fee), body-centred cubic (bee), simple hexagonal (sh), simple cubic (sc) and hexagonal close-packed (licp) structures.

In compound materials - in the ceramic sodium chloride, for instance - there are two (sometimes more) species of atoms, packed together. The crystal structures of such compounds can still be simple. Figure 5.8(a) shows that the ceramics NaCl, KCl and MgO, for example, also form a cubic structure. Naturally, when two species of atoms are not in the ratio 1 1, as in compounds like the nuclear fuel UO2 (a ceramic too) the structure is more complicated (it is shown in Fig. 5.8(b)), although this, too, has a cubic unit cell. 

Other refractory oxides that can be deposited by CVD have excellent thermal stability and oxidation resistance. Some, like alumina and yttria, are also good barriers to oxygen diffusion providing that they are free of pores and cracks. Many however are not, such as zirconia, hafnia, thoria, and ceria. These oxides have a fluorite structure, which is a simple open cubic structure and is particularly susceptible to oxygen diffusion through ionic conductivity. The diffusion rate of oxygen in these materials can be considerable. 

Although the comer atoms must move apart to convert a simple cube into a body-centered cube, the extra atom in the center of the stracture makes the body-centered cubic lattice more compact than the simple cubic structure. All the alkali metals, as well as iron and the transition metals from Groups 5 and 6, form ciystals with body-centered cubic structures. 

Yet another common crystal lattice based on the simple cubic arrangement is known as the face-centered cubic structure. When four atoms form a square, there is open space at the center of the square. A fifth atom can fit into this space by moving the other four atoms away from one another. Stacking together two of these five-atom sets creates a cube. When we do this, additional atoms can be placed in the centers of the four faces along the sides of the cube, as Figure 11-28 shows. 

Now examine the symmetry elements for the cubic lattice. It is easy to seethat the number of rotation elements, plus horizontal and vertical symmetry elements is quite high. This is the reason why the Cubic Structure is placed at the top of 2.2.3. E)ven though the lattice points of 2.2.1. are deceptively simple for the cubic structure, the symmetry elements are not... 

Mono- or single-crystal materials are undoubtedly the most straightforward to handle conceptually, however, and we start our consideration of electrochemistry by examining some simple substances to show how the surface structure follows immediately from the bulk structure we will need this information in chapter 2, since modern single-crystal studies have shed considerable light on the mechanism of many prototypical electrochemical reactions. The great majority of electrode materials are either elemental metals or metal alloys, most of which have a face-centred or body-centred cubic structure, or one based on a hexagonal close-packed array of atoms. 

We can determine the amount of empty space in the simple cubic (a space-filling model is shown in Figure 7.15) structure by considering it to have an edge length l, which will be twice the radius of an atom. Therefore, the radius of the atom is 1/2, so the volume of one atom is (4/3)7r(l/2)3 = 0.52413, but the volume of the cube is P. From this we see that because the cube contains only one atom that occupies 52.4% of the volume of the cube, there is 47.6% empty space. Because of the low coordination number and the large amount of empty space, the simple cubic structure does not represent an efficient use of space and does not maximize the number of metal atoms bonded to each other. Consequently, the simple cubic structure is not a common one for metals. 

A unit, or perfect, dislocation is defined by a Burgers vector which regenerates the structure perfectly after passage along the slip plane. The dislocations defined above with respect to a simple cubic structure are perfect dislocations. Clearly, then, a unit dislocation is defined in terms of the crystal structure of the host crystal. Thus, there is no definition of a unit dislocation that applies across all structures, unlike the definitions of point defects, which generally can be given in terms of any structure. 

Considering Au in 0, 0, 0 as the reference atom, the next neighbours Au atoms are the six Au shown in Fig. 3.29(a), corresponding to the same Wyckoff position and having, in comparison with the reference atom, the coordinates 0, 0, 1 0, 0, 1 0, 1, 0 0, 1, 0 1, 0, 0 1, 0, 0, all at a distance d = a = 374.8 pm, that is at a reduced distance dr = d/dmin = 1.414. Notice that in the analysis of the structure it may be necessary to consider not only the positions of the atoms in the reference cell but also those in the adjacent cells. Notice also that, in a simple cubic structure without free positional parameters such as the AuCu3 type, the reduced distances are independent of the values of the lattice parameters and are the same for all the isostructural compounds. 

The W body-centred cubic structure can be compared with the simple cubic CsCl-type structure (which can be obtained from the W type by an ordered substitution of the atoms) and with the MnCu2Al-type structure ( ordered superstructure of the CsCl type) see Fig. 3.31 and notice the typical eight (cubic) coordination. 

According to Pearson (1972) the rhombohedral structure of these elements can be considered a distortion of a simple cubic structure in which the d2/d ratio would be 1. The decrease of the ratio on passing from As to Bi, and the corresponding relative increase of the strength of the X-X interlayer bond (passing from a coordination nearly 3, as for the 8 — eat rule, to a coordination closer to 6) can be related to an increasing metallic character. 

Any study of colloidal crystals requires the preparation of monodisperse colloidal particles that are uniform in size, shape, composition, and surface properties. Monodisperse spherical colloids of various sizes, composition, and surface properties have been prepared via numerous synthetic strategies [67]. However, the direct preparation of crystal phases from spherical particles usually leads to a rather limited set of close-packed structures (hexagonal close packed, face-centered cubic, or body-centered cubic structures). Relatively few studies exist on the preparation of monodisperse nonspherical colloids. In general, direct synthetic methods are restricted to particles with simple shapes such as rods, spheroids, or plates [68]. An alternative route for the preparation of uniform particles with a more complex structure might consist of the formation of discrete uniform aggregates of self-organized spherical particles. The use of colloidal clusters with a given number of particles, with controlled shape and dimension, could lead to colloidal crystals with unusual symmetries [69]. 

Random substitutional models are used for phases such as the gas phase or simple metallic liquid and solid solutions where components can mix on any atial position which is available to the phase. For example, in a simple body-centred cubic phase any of the components could occupy any of the atomic sites which define the cubic structure as shown below (Fig. 5.1). 

Figure 5.1. Simple body-centred cubic structure with random occupation of atoms on all sites.

We now need to define a collection of atoms that can be used in a DFT calculation to represent a simple cubic material. Said more precisely, we need to specify a set of atoms so that when this set is repeated in every direction, it creates the full three-dimensional crystal stmcture. Although it is not really necessary for our initial example, it is useful to split this task into two parts. First, we define a volume that fills space when repeated in all directions. For the simple cubic metal, the obvious choice for this volume is a cube of side length a with a corner at (0,0,0) and edges pointing along the x, y, and z coordinates in three-dimensional space. Second, we define the position(s) of the atom(s) that are included in this volume. With the cubic volume we just chose, the volume will contain just one atom and we could locate it at (0,0,0). Together, these two choices have completely defined the crystal structure of an element with the simple cubic structure. The vectors that define the cell volume and the atom positions within the cell are collectively referred to as the supercell, and the definition of a supercell is the most basic input into a DFT calculation. 

The choices we made above to define a simple cubic supercell are not the only possible choices. For example, we could have defined the supercell as a cube with side length 2a containing four atoms located at (0,0,0), (0,0,a), (0,a,0), and (a,0,0). Repeating this larger volume in space defines a simple cubic structure just as well as the smaller volume we looked at above. There is clearly something special about our first choice, however, since it contains the minimum number of atoms that can be used to fully define the structure (in this case, 1). The supercell with this conceptually appealing property is called the primitive cell. 

A large number of solids with stoichiometry AB form the CsCl structure. In this structure, atoms of A define a simple cubic structure and atoms of B reside in the center of each cube of A atoms. Define the cell vectors... 

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