Unit cells simple cubic

Calculate the number of spheres in these unit cells simple cubic, body-centered cubic, and face-centered cubic cells. Assume that the spheres are of equal size and that they are only at the lattice points. 

Make a drawing of each unit cell simple cubic, body-centered... 

Some high-temperature superconductors adopt a crystal structure similar to that of perovskite (CaTiOs). The unit cell is cubic with a Ti" " ion in each comer, a Ca " ion in the body center, and 0 ions at the midpoint of each edge, (a) Is this unit cell simple, body-centered, or face-centered (b) If the unit cell edge length is 3.84 A, what is the density of perovskite (in g/cm ) ... 

Section 11.7 In a crystalline soUd, particles are arranged in a regularly repeating pattern. An amorphous solid is one whose particles show no such order. The essential structural features of a crystalline solid can be represented by its unit cell, toe smallest part of toe crystal that can, by simple displacement, reproduce the three-dimensional structure. The three-dimensional structures of a crystal can also be represented by its crystal lattice. The points in a crystal lattice represent positions in toe structure where there are identical environments. The simplest unit cells are cubic. There are three kinds of cubic unit cells primitive cubic, body-centered cubic, and face-centered cubic. 

The cubic crystal system has three possible cubic unit cells simple (or primitive) cubic, body-centered cubic, and face-centered cubic. These cells are illustrated in Figure 11.32. A simple cubic unit cell is a cubic unit cell in which lattice points are situated only at the corners. A body-centered cubic unit cell is a cubic unit cell in which there is a lattice point at the center of the cubic cell in addition to... 

Simple cubic unit cell a cubic unit cell in which lattice points are situated only at the corners of the unit cell. (11.7)... 

Practice Problem B The density of sodium metal is 0.971 g/cm, and the unit cell edge length is 4.285 A. Determine the unit cell (simple, body-centered, or face-centered cubic) of sodium metal. 

The hole left in the center of a simple cubic unit cell is referred to as, naturally enough, a cubic hole. Figure 7.16 shows a space-filling model of such a unit cell with one of the corner spheres removed and the hole shaded for clarity. The radius of this hole is about three-quarters that of the spheres forming it (see Problem 7.29). We decided earlier that there is one sphere per simple cubic unit cell. The cubic hole is completely within the unit cell so that there is also one hole per unit cell. It follows that there is one cubic hole per sphere in this lattice. 

The pattern from one layer to the next is ABCABC, with every fourth layer aligning with the first. Although not simple to visuahze, the unit cell for cubic closest packing is the face-centered cubic unit cell, as shown in Figure 11.49 . The cubic closest-packed structure is identical to the face-centered cubic unit cell structure. 

Metals A and B form an alloy or solid solution. To take a hypothetical case, suppose that the structure is simple cubic, so that each interior atom has six nearest neighbors and each surface atom has five. A particular alloy has a bulk mole fraction XA = 0.50, the side of the unit cell is 4.0 A, and the energies of vaporization Ea and Eb are 30 and 35 kcal/mol for the respective pure metals. The A—A bond energy is aa and the B—B bond energy is bb assume that ab = j( aa + bb)- Calculate the surface energy as a function of surface composition. What should the surface composition be at 0 K In what direction should it change on heaf)pg, and why ... 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

The summation is over the different types of ion in the unit cell. The summation ca written as an analytical expression, depending upon the lattice structure (the orij Mott-Littleton paper considered the alkali halides, which form simple cubic lattices) evaluated in a manner similar to the Ewald summation this typically involves a summc over the complete lattice from which the explicit sum for the inner region is subtractec... 

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. 

Below a temperature of Toi 260 K, the Ceo molecules completely lose two of their three degrees of rotational freedom, and the residual degree of freedom is a ratcheting rotational motion for each of the four molecules within the unit cell about a different (111) axis [43, 45, 46, 47]. The structure of solid Ceo below Tqi becomes simple cubic (space group Tji or PaS) with a lattice constant ao = 14.17A and four Ceo molecules per unit cell, as the four oriented molecules within the fee structure become inequivalent [see Fig. 2(a)] [43, 45]. Supporting evidence for the phase transition at Tqi 260 K is... 

The initial configuration is set up by building the field 0(r) for a unit cell first on a small cubic lattice, A = 3 or 5, analogously to a two-component, AB, molecular crystal. The value of the field 0(r) = at the point r = (f, 7, k)h on the lattice is set to 1 if, in the molecular crystal, an atom A is in this place if there is an atom B, 0, is set to —1 if there is an empty place, j is set to 0. Fig. 2 shows the initial configuration used to build the field 0(r) for the simple cubic-phase unit cell. Filled black circles represent atoms of type A and hollow circles represent atoms of type B. In this case all sites are occupied by atoms A or B. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

Significant figure A meaningful digit in a measured quantity, 9,20-2 lq ambiguity in, 10 in inverse logarithms, 645-647 in logarithms, 645-647 Silicate lattices, 243 Silicon, 242-243 Silver, 540-541 Silver chloride, 433,443-444 Simple cubic cell (SC) A unit cell in which there are atoms at each comer of a cube, 246... 

The easiest ciystal lattice to visualize is the simple cubic stracture. In a simple cubic crystal, layers of atoms stack one directly above another, so that all atoms lie along straight lines at right angles, as Figure 11-26 shows. Each atom in this structure touches six other atoms four within the same plane, one above the plane, and one below the plane. Within one layer of the crystal, any set of four atoms forms a square. Adding four atoms directly above or below the first four forms a cube, for which the lattice is named. The unit cell of the simple cubic lattice, shown in... 

The cesium chloride lattice consists of a simple cubic array of chloride anions with a cesium cation at the center of each unit cell. 

Thus, the planes of the lattice are found to be important and can be defined by moving along one or more of the lattice directions of the unitcell to define them. Also important are the symmetry operations that can be performed within the unit-cell, as we have illustrated in the preceding diagram. These give rise to a total of 14 different lattices as we will show below. But first, let us confine our discussion to just the simple cubic lattice. 

The corresponding unit cells are shown in Figure 1.1 and an examination of simple ball-and-stick models (which the reader is strongly urged to carry out) shows that the face-centred cubic (fee) and hexagonal close-packed (hep) structures correspond to the only two possible ways of close-packing spheres, in which each sphere has twelve nearest neighbours. 

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