Body-centered cubic lattice unit cell

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

Construct a single body-centered cubic lattice unit cell. 

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. 

The dructure of CsCI diould not be referred to, incorrectly, as "body-centered cube". True body-centered cubic lattices have the seme specks on the comers and the center of the unit cell. as in the... 

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.)...

Manganese crystallizes in a structure that has a body-centered cubic lattice. What is the coordination number (number of nearest neighbors) for Mn in this structure, and how many atoms are there in each unit cell ... 

Potassium crystallizes in a body-centered cubic lattice, with a unit cell length a = 520pm. (a) What is the distance between nearest neighbors (h) What is the distance between next-nearest neighbors (c) How many nearest neighbors does each K atom have (d) How many next-nearest neighbors does each K have (e) What is the calculated density of crystalline K ... 

Barium metal crystallizes in a body-centered cubic lattice (the Ba atoms are at the lattice points only). The unit cell edge length is 502 pm, and the density of the metal is 3.50 g/cm. Using this information, calculate Avogadro s number. [Hint First calculate the volume (in cm ) occupied by 1 mole of Ba atoms in the unit cells. Next calculate the volume (in cm ) occupied by one Ba atom in the unit cell. Assume that 68% of the unit cell is occupied by Ba atoms.]... 

Vanadium crystallizes in a body-centered cubic lattice (the V atoms occupy only the lattice points). How many V atoms are present in a unit cell ... 

The calculation of a-factor involves two steps (1) calculation of which varies depending on crystal face and (2) calculation of thermodynamic properties [using AH and solubility with Eq. (3.32), or AH with Eq. (3.29)]. From the unit cell, the number of nearest neighbors of the HMT body centered cubic lattice is known to be 8. Since the number of nearest neighbors on the 110 face is 4, the value of for this face is = /tj/n = 0.5 (Davey 1986). 

Figure 1. The body-centered cubic lattice, drawn to emphasize the coordination of each atom by eight atoms at the corners of a cube (distance dbcc), and six next-nearest neighbors at the comers of an octahedron (distance a pm, a = unit cell length = 2dbcc/ 3).

The assignation of axes of reference in relation to the rotational symmetry of the crystal systems defines six lattices that, by definition, are primitive or P-lattices. To determine if new lattices can be formed from these P-lattices, one must determine if more points can be added so that the lattice condition is still maintained, and whether this addition of points alters the crystal system. For example, if one starts with a simple cubic primitive lattice and adds other lattice points in such a way that a lattice still exists, it must happen that the resulting new lattice still possesses cubic symmetry. Since the lattice condition must be maintained when new points are added, the points must be added to hightly symmetric positions of the P-lattice. These types of positions are (a) a single point at the body center of each unit cell, (b) a point at the center of each independent face of the unit cell, (c) a point at the center of one face of the unit cell, and (d) the special centering positions in the trigonal system that give a rhombohedral lattice. 

A cubic phase of space group Pmln is usually observed in type I systems [164]. Several structures have been suggested for the Pmln phase [168-171]. It is now agreed that it contains two types of micelles [172] two quasi-spherical micelles packed on a body-centered cubic lattice and six slightly asymmetrical micelles arranged in parallel rows on opposite faces of the unit cell. The asymmetrical micelles are assumed to be disklike [173] or rodlike with rotational disorder around one of the short axes [174,175]. In order to pack space completely, each asymmetrical micelle, together with the water that surrounds it, takes the shape of a... 

In two dimensions the unit cell is a parallelogram whose size and shape are defined by two lattice vectors (a and b). There are four primitive lattices, lattices where the lattice points are located only at the corners of the unit cell square, hexagonal, rectangular, and oblique. In three dimensions the unit cell is a parallelepiped whose size and shape are defined by three lattice vectors (a, b and c), and there are seven primitive lattices cubic, tetragonal, hexagonal, rhombohedral, orthorhombic, monoclinic, and triclinic. Placing an additional lattice point at the center of a cubic unit cell leads to a body-centered cubic lattice, while placing an additional point at the center of each face of the unit cell leads to a face-centered cubic iattice. 

What is the minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice ... 

An element crystallizes in a body-centered cubic lattice. The edge of the unit cell is 2.86 A, and the density of the crystal is 7.92 g/cm. Calculate the atomic weight of the element. Define the term alloy. Distinguish among solid solution alloys, heterogeneous alloys, and intermetallic compounds. Distinguish between substitutional and interstitial alloys. What conditions favor formation of substitutional alloys ... 

Figure 2.36 Unit cell and Wigner-Seitz cell (thick lines) of the body centered cubic lattice.

The spacing in a crystal lattice can be measured very accurately by X-ray diffraction, and this provides one way to determine Avogadro s number. One form of iron has a body-centered cubic lattice, and each side of the unit cell is 286.65 pm long. The density of this crystal at 25°C is 7.874 g/cm Use these data to determine Avogadro s number. 

Barium metal crystallizes in a body-centered cubic lattice (the Ba atoms are at the lattice points only). The unit cell edge length is 502 pm, and the density... 

Metals in their elemental state mostly crystallize in dose packed structures or in the body centered cubic lattice. For a face centered cubic metal the unit cell shown in Figure 5-1 is normally chosen as the conventional description of the structure. As an alternative, the close packed arrangement can be decomposed into a network of condensed regular octahedra whidi share corners and edges as indicated in Figure 5-1 for the case of cubic close packing, or of deformed octa-... 

There are three kinds of cubic unit cells, as illustrated in Figure 11.33 A. When lattice points are at the comers only, the unit cell is called primitive cubic. When a lattice point also occurs at the center of fhe unif cell, fhe cell is body-centered cubic. When the cell has lattice points at the center of each face, as well as at each comer, it is face-centered cubic. 

Fig. 3.2. Primitive unit cells (a, c) and Brillouin zones (b, d) for face-centered and body-centered cubic lattices...

Metallic iron has a body-centered cubic lattice with all atoms at lattice points and a unit cell whose edge length is 286.6 pm. The density of iron is 7.87 g/cm. What is the mass of an iron atom Compare this value with the value you obtain from the molar mass. 

Tungsten has a body-centered cubic lattice with all atoms at the lattice points. The edge length of the unit cell is 316.5 pm. The atomic weight of tungsten is 183.8 amu. Calculate its density. 

FIGURE 13.8 The structure of the face-centered cubic, hexagonal closest packing, and body-centered cubic lattices. To form each lattice, start with the first layer (green atoms), add the second layer as shown (blue atoms), and then the third layer (yellow atoms). The cycle repeats, with every other layer being identical in each lattice. One face of one unit cell is outlined on the first layer of each lattice. This representation illustrates the (100) surface of each lattice. 

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