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Unformatted text preview: Chapter 8 The unit cell is the smallest repeat unit of the crystalline lattice that generates the entire lattice with translation. Crystalline solids are orderly, repeating, 3-D arrays of particles, which can be atoms, ions, or groups of atoms, such as polyatomic ions or molecules. The pattern of the array is called the crystal lattice, and the individual positions are called lattice sites . The simplest portion of the lattice that makes up the repeating unit is called the unit cell. When the unit cell is repeated in all three directions, it generates the entire crystalline lattice.-All unit cells can be uniquely characterized by their three edge lengths, (a, b, and c) and the angles There are three cubic unit cells that differ in how the particles fill the cube. In each cubic unit cell, the same atom-type occupies each of the eight corners of the cube. The unit cell type is then dictated by where else in the unit cell that atom-type is found. • simple cubic (sc) : The atom-type on the corners is found nowhere else in the unit cell • body-centered cubic (bcc) : The atom-type on the corners is also found in the center of the unit cell • face-centered cubic (fcc) : The atom-type on the corners is also found in the center of each of six faces of the unit cell The particles on the corners of a simple cubic (sc) unit cell are not present in any other locations within the cell.-Simple cubic unit cells have the greatest amount of void space, which means that the particles are not packed in the unit cell very efficiently.-The particles on the corners of a body-centered cubic (bcc) unit cell are also found in the center of the cell.-The identical particles on the corners of a bcc unit cell are also found in the body center. Note that there is less void space in the bcc unit cell than in the sc unit cell-The particles on the corners of a face-centered cubic (fcc) unit cell are also found in the centers of the six faces of the cell.-Note that the particles of the fcc unit cell are packed more efficiently than those of either sc or bcc unit cells. Indeed, this arrangement of spheres is called closest packed Figure 8.6 Atom contact in cubic unit cells sc 2r = a fcc fd = 4r fd 2 = a 2 + a 2 (4r) 2 = 2a 2 bcc bd = 4r bd 2 = fd 2 + a 2 (4r) 2 = 2a 2 + a 2 = 3a Setting the edge length equal to 2r in the sc unit cell, and applying the Pythagorean theorem to the triangles shown in Figure 8.6 , we obtain the relationships between the atomic radius (r) of the atom and the edge length of the unit cell (a) given in Equation 8.1 Relationship of atomic radii and unit cell edge lengths Equation 8.1 The radius obtained from the structure of a metallic solid is referred to as either the metallic radius or the the atomic radius....
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