Atoms at “1” contribute 1/6 of an atom each. Miller Indices (100) 1. Show that the atomic packing factor for the FCC crystal structure is 0.74 Solution The APF is defined as the fraction of solid sphere volume in a unit cell, or C S V V total unit cell volume total sphere volume APF Both the total sphere and unit cell volumes may be calculated in terms of the atomic radius R. APFBCC = 0.68 or 68% The remaining 32% is empty space. Atomic Packing Factor (APF) APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres Adapted from Fig. 8. Coordination number in simple cubic crystal structure (a) 1 (b) 2 (c) 3 (d) 4 10. Hence, where R is the atomic radius. Miller Indices (110) example a b c z x y a b c 4. 9 • Coordination # = 12 Adapted from Fig. Atomic packing factor is (a) Distance between two adjacent atoms (b) Projected area fraction of atoms on a plane (c) Volume fraction of atoms in cell (d) None 9. Atomic Packing Factor (APF) for FCC maximum achievable APF Close-packed directions: 4R = 2 a a 2 a Adapted from Fig. Compute the atomic packing factor for the rock salt crystal structure in which r C/r A = 0.414. APF = 4 3 4 p ( 2a/4 ) 3 a 3 # of atoms per unit cell Volume of each atom Volume of an unit cell The atomic diameter of an BCC crystal (if a is lattice parameter) is Atomic Packing Factor (APF) เป็นการค านวณหาแฟกเตอร์ของปริมาตรที่อะตอมเข้าไปบรรจุใน unit cell • Coordination # ส ำหรับ FCC = 12 . •Solution: Pb is fcc, hence •FCC = HCP = 74% (26% void space in unit cell) •BCC = 68% . Atomic Packing Factor of BCC Structure Atomic Packing Factor = Volume of atoms in unit cell Volume of unit cell Vatoms = = 8.373R3 3 3 4 R = 12.32 R3 Therefore APF = 8.723 R3 12.32 R3 = … close-packed directions a R=0.5a contains 8 … Unit cell c ontains: Atoms at “2” contribute Atomic Packing Factor •The ratio of atomic sphere volume to unit cell volume, assuming a hard sphere model. (Courtesy P.M. Anderson) • Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded 3.1(a), Callister 6e. Coordination number = 6 Simple Cubic (SC) Structure •Coordination number is the number of nearest neighbors •Linear density (LD) is the number of atoms per unit length along a specific crystallographic direction a1 a2 a3 . 3.23, Callister 7e. 3.1(a), Callister & Rethwisch 8e. Hexagonal Close Packed (HCP) The isolated unit cell is called the primitive cell as in the figure below. Solution This problem asks that we compute the atomic packing factor for the rock salt crystal structure when r C/r A = 0.414. Coordination Number & Atomic Packing Factor Coordination number : number of nearest neighbors Atomic Packing Factor (APF) APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres 16 Miller Indices for Crystal Planes (BCC, FCC) z x y a b c 4. Atomic Packing Factor: BCC APF = a3 4 3 2 π ( 3a/4 )3 atoms unit cell atom volume unit cell volume. Problem 1 •If the atomic radius for Pb= 0.175nm, find the volume of the unit cell. • Rare due to poor packing (only Po [84] has this structure) • Close-packed directions are cube edges. . LD .