INTRODUCTION TO CERAMIC MINERALS

INTRODUCTION TO CERAMIC MINERALS ATOMIC STRUCTURE AND PACKING Lattice Constant, a & Atomic Packing Factor, APF

INTRODUCTION TO CERAMICMINERALS Learning Outcomes • Calculate lattice constant • Calculate the atomic packing factor • Calculate the ionic packing factor • Definition of coordination number • Determine the total atom in unit cell

Lattice Constant, a : BCC √3 . a r a 2r r √2 . a Given, ( Fe )radius, r = 0.124nm 4r = √3. a a = 4r / √3 = 4(0.124) / √3 = 0.2864 nm

INTRODUCTION TO CERAMIC MINERALS COORDINATION NUMBER, CN = the number of closest neighbors to which an atom is bonded = number of touching atoms,

INTRODUCTION TO CERAMIC MINERALS ATOMIC PACKING FACTOR (APF) = fraction of volume occupied by hard spheres = Sum of atomic volumes / Volume of unit cell

SIMPLE CUBIC STRUCTURE (SC) • Rare due to poor packing (only Po has this structure)  Does NOT occur in common engineering metals • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson)

ATOMIC PACKING FACTOR: SC • APF for a simple cubic structure = 0.52 Coordination #:6

BODY CENTERED CUBIC STRUCTURE (BCC) • Close packed directions are cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. • Coordination # = 8 Adapted from Fig. 3.2, Callister 6e. (Courtesy P.M. Anderson) 7

BODY CENTERD CUBIC STRUCTURED, BCC BCC • The hard spheres touch one another along cube diagonal . • The cube edge length, a= 4R/√3 • The coordination number, CN = 8 • Number of atoms per unit cell, n = 2 • Center atom (1) shared by no other cells: 1 x 1 = 1 • 8 corner atoms shared by eight cells: 8 x 1/8 = 1 • Atomic packing factor, APF = 0.68 • Corner and center atoms are equivalent

ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68 Adapted from Fig. 3.2,Callister 6e. 8

FACE CENTERED CUBIC STRUCTURE (FCC) • Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. • Coordination # = 12 9

Figure 25: FCC atomic packing FACE CENTERED CUBIC STRUCTURE, FCC Share with 2 cells Share with 8 cells a= 2R√2 CN= 12 • The cube edge length, a= 2R√2 • The coordination number, CN = 12 • Number of atoms per unit cell, n = 4. (For an atom that is shared with m adjacent unit cells, we only count a fraction of the atom, 1/m). • 6 face atoms shared by two cells: 6 x 1/2 = 3 • 8 corner atoms shared by eight cells: 8 x 1/8 = 1 • Atomic packing factor (APF) = (Sum of atomic volumes)/(Volume of cell) = 0.74 (maximum possible)

ATOMIC PACKING FACTOR: FCC • APF for a body-centered cubic structure = 0.74 Adapted from Fig. 3.1(a), Callister 6e. 10

Figure 26: HCP closed-pack HEXAGONAL CLOSE-PACKED STRUCTURE, HCP HCP is one more common structure of metallic crystals • Number of atoms per unit cell, n = 6. • The coordination number, CN = 12 (same as in FCC) • 3 mid-plane atoms shared by no other cells: 3 x 1 = 3 • 12 hexagonal corner atoms shared by 6 cells: • 12 x 1/6 = 2 • 2 top/bottom plane center atoms shared by 2 cells: • 2 x 1/2 = 1 • Unit cell has two lattice parameters a and c. Ideal ratio c/a = 1.633 • Atomic packing factor, APF = 0.74 (same as in FCC) • Cd, Mg, Zn, Ti have this crystal struct

HEXAGONAL CLOSE-PACKED STRUCTURE (HCP) • ABAB... Stacking Sequence • 3D Projection • 2D Projection Adapted from Fig. 3.3, Callister 6e. • APF = 0.74 • Coordination # = 12 12

Calculate the atomic factor (APF) for the BCC unit cell, assuming the atoms to be hard spheres. APF = vol of atoms in BCC unit cell vol of BCC unit cell V atoms= 2 x 4/3πR3 V unit cell = a3 √3a = 4R or a = 4R √3 APF = 2 x 4/3πR3 / a3 = (8/3 πR3 ) / ( 4R3/ √3) = 0.74

INTRODUCTION TO CERAMIC MINERALS