Crystal lattices body-centered cubic

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

A similar effect occurs in highly chiral nematic Hquid crystals. In a narrow temperature range (seldom wider than 1°C) between the chiral nematic phase and the isotropic Hquid phase, up to three phases are stable in which a cubic lattice of defects (where the director is not defined) exist in a compHcated, orientationaHy ordered twisted stmcture (11). Again, the introduction of these defects allows the bulk of the Hquid crystal to adopt a chiral stmcture which is energetically more favorable than both the chiral nematic and isotropic phases. The distance between defects is hundreds of nanometers, so these phases reflect light just as crystals reflect x-rays. They are called the blue phases because the first phases of this type observed reflected light in the blue part of the spectmm. The arrangement of defects possesses body-centered cubic symmetry for one blue phase, simple cubic symmetry for another blue phase, and seems to be amorphous for a third blue phase. 

Only body-centered cubic crystals, lattice constant 428.2 pm at 20°C, are reported for sodium (4). The atomic radius is 185 pm, the ionic radius 97 pm, and electronic configuration is lE2E2 3T (5). Physical properties of sodium are given ia Table 2. Greater detail and other properties are also available... 

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.

Explain why the density of vanadium (6.1 L g-cm 3) is significantly less than that of chromium (7.19 g-cm-3). Both vanadium and chromium crystallize in a body-centered cubic lattice. 

For simple monovalent metals, the pseudopotential interaction between ion cores and electrons is weak, leading to a uniform density for the conduction electrons in the interior, as would obtain if there were no point ions, but rather a uniform positive background. The arrangement of ions is determined by the ion-electron and interionic forces, but the former have no effect if the electrons are uniformly distributed. As the interionic forces are mainly coulombic, it is not surprising that the alkali metals crystallize in a body-centered cubic lattice, which is the lattice with the smallest Madelung energy for a given density.46 Diffraction measurements... 

Figure 9.2 is schematic diagram of the crystal structure of most of the alkali halides, letting the black circles represent the positive metal ions (Li, Na, K, Rb, and Cs), and the gray circles represent the negative halide ions (F, Cl, Br, and I).The ions lie on two interpenetrating face-centered-cubic lattices. Of the 20 alkali halides, 17 have the NaCl crystal structure of Figure 9.1. The other three (CsCl, CsBr, and Csl) have the cesium chloride structure where the ions lie on two interpenetrating body-centered-cubic lattices (Figure 9.3). The plastic deformation on the primary glide planes for the two structures is quite different.

The most important metals for catalysis are those of Groups VIII and I-B of the periodic system. Three crystal structures are important, face-centered cubic (fee Ni, Cu, Rh, Pd, Ag, Ir, Pt, Au), hexagonal close-packed (hep Co, Ru, Os) and body-centered cubic (bcc Fe) [9, 10]. Before discussing the surfaces that these lattices expose, we mention a few general properties. 

Body-centered cubic (bcc) is the lattice symmetry of Fe, for instance (Fig. 16.2c). Bcc here refers to a crystal arrangement of atoms at the corners of a cube and one atom in the center of the cube equidistant from each face. 

Soft silvery metal body-centered cubic crystal lattice density 5.24 g/cm melts at 822°C vaporizes at 1,596°C electrical resistivity 81 microhm-cm reacts with water soluble in liquid ammonia. 

Crystal Systems. The cubic crystal system is composed of three space lattices, or unit cells, one of which we have already studied simple cubic (SC), body-centered cubic (BCC), anA face-centered cubic (FCC). The conditions for a crystal to be considered part of the cubic system are that the lattice parameters be the same (so there is really only one lattice parameter, a) and that the interaxial angles all be 90°. 

Fig. 3-3.—The atomic arrangement in the cubic crystal SiF4. The atoms form tetrahedral molecules, with four fluorine atoms surrounding a silicon atom. The molecules are arranged at tlie points of a body-centered cubic lattice.

The density of beryllium is 1.847 g/cm3 based upon average values of lattice parameters at 255C (a — 22.856 nm and c — 35.832 nm). Beryllium products generally have a density around 1.850 g/cm3 or higher because of impurities, such as aluminum and other metals, and beryllium oxide. The crystal structure is close-packed hexagonal. The alpha-form of beryllium transforms to a body-centered cubic structure at a temperature very close to the melung point. 

Figure 16.2. Conventional (non-primitive) unit cells of (a) the face-centered cubic and (b) the body-centered cubic lattices, showing the fundamental vectors a1 a2, and a3 of the primitive unit cells. (A conventional unit cell is one that displays the macroscopic symmetry of the crystal.)...

Crystals can be further divided into 14 space lattices, which describe the positions of lattice points. For example, there are three cubic space lattices. The simple cubic has lattice points only at the comers of the cubic cell the body-centered cubic (bcc) has lattice points at the comers and the body-centering position, and the face-centered cubic (fee) has lattice points at the comers and the centers of the faces. Table 2.2 and Figure 2.1 illustrate these. 

There are very many unit cell structures if we consider all the atoms or ions in the crystal. However, if we focus on just one atom or ion, we can reduce the number to just 14 primitive cells. Three of these, the simple cubic, face-centered cubic (fee), and body-centered cubic (bcc) unit cells, are shown in Figure 9-2. The lattice points, represented by small spheres in the drawings, correspond to the centers of the atoms, ions, or molecules occupying the lattice. 

Potassium crystallizes in a body-centered cubic lattice with a = 520 pm. (a) What is the distance between nearest neighbors (b) How many nearest neighbors does each K atom have (c) Compute the density of crystalline K. 

Potassium crystallizes in a body-centered cubic lattice (unit cell length a = 520 pm). 

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