MSE420/514: Session 1 Crystallography & Crystal Structure (Review)

Crystal Classes & Lattice Types 2

SIMPLE CUBIC STRUCTURE (SC) • Rare due to poor packing (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson) 5

ATOMIC PACKING FACTOR • APF for a simple cubic structure = 0.52 Adapted from Fig. 3.19, Callister 6e. 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

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 Adapted from Fig. 3.1(a), Callister 6e. (Courtesy P.M. Anderson) 9

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

Summary: Coordination Number & Atoms/Unit Cell Coordination Number (CN) Number of nearest neighboring atoms, e.g., 8 for inside atom of a BCC 6 for corner atoms 12 for each FCC & HCP atoms Number of Atoms Per Unit Cell Determine Total Number of Atom Fraction Shared by Unit Cell, e.g., SC: 8 (corner atoms)/8 (shared by 8 unit cells) = 1 BCC: [8/8] + [1 (atom inside unit cell) /1 (shared by 1 unit cell)] = 2 FCC: [8/8] + [6 (atom on unit cell faces) /2 (shared by 1 unit cell)] = 4 HCP: [12/12] + [2 /2] + [3/1] = 6

Crystal Structure & Unit Cell Densely Packed Atoms are in Lower & More Stable Energy arrangement SC: Densely packed along cube axis BCC: Densely packed along cube body diagonal FCC: Densely packed along face diagonal Inter-atomic Spacing, r Energy + Equilibrium r BCC SC FCC

Summary: Atomic Packing Factor (APF) Unit Cell Space Occupied (by atoms) Volume of atoms in each cell Total volume of unit cell Example: SC: [1. (4pr3/3)]/[a3]= p/6 = 0.53 ; (a=2r) BCC: [2 . (4pr3/3)]/[(4/3 r)3] = 0.68 ; (a= 4/3 r) FCC: [4 . (4pr3/3)]/[(22 r)3]= 0.74 ; (a=22 r) HCP: 0.74 APF = SC BCC FCC a 2r 4/3 r 22 r FD 2 a 2 a 2 a BD 3 a 3 a 3 a at/UC 1 2 4 CN 6 8 12 APF 0.53 0.68 0.74

Interstitial Sites

Interstitial Sites

Crystal Notations https://www.doitpoms.ac.uk/tlplib/crystallography3/index.php https://www.doitpoms.ac.uk/tlplib/miller_indices/printall.php

Crystallographic Notations: Coordinates Atom Coordinates Locating Atom Position in Unit Cell Point in space, coordinates in ref. to origin (1,1,1) (1,0,0) O

Crystallographic Notations: Direction Indices [uvw] & <uvw> Id. coordinate w.r.t. origin Transform to integers * Lattice vector in a,b,c direction All parallel direction vectors have the same direction indices “Crystallographically equivalent” directions (same atom spacing along each direction) are designated with <uvw> direction family 0,1,½ [021] 0,0,0 -1,-1,0 [110] - - 1,0,0 [100]

Crystal directions Crystal directions are defined in the following way, relative to the unit cell. 1) Choose a beginning point (X1, Y1, Z1) and an ending point (X2, Y2, Z2), with the position defined in terms of the unit cell dimensions. Beginning point: (X1, Y1, Z1): (1, 1, 0) Ending point: (X2, Y2, Z2): (1/2, 0, 1) 2) Calculate the differences in each direction, ΔX, ΔY, ΔZ. ΔX, ΔY, ΔZ : (-1/2, -1, 1) 3) Multiply the differences by a common constant to convert them to the smallest possible integers u, v, w (u, v, w) : (-1, -2, 2)

Crystallographic Notations: Plane Indices

Crystallographic Notations: Plane Indices Miller Indices, (hkl) & {hkl} family Reciprocal of Intercepts Select an appropriate origin Id. Intercept with axis (INTERCEPT) Determine Reciprocal of intercept (INVERT) Clear fractions to smallest set of whole numbers (INTEGER) Equivalent lattice planes related by symmetry of the crystal system are designated by {hkl} family of planes In cubic system: - [abc] direction  (abc) plane ,-1,½ 0,-1,2 (012) - ,-1,  (or ,1, ) 0 -1 0 (or 0 1 0) (010) (or (010) )

Miller Plane Indices 1 x y z æ è ç ö ø ÷ Plane: (hkl), or plane family {hkl} Methodology for determining Miller Indices Identify an Origin Id. Plane Intercept with 3-Axis Invert the Intercepts Clear Fractions to Lowest Integers Overbars Indicate < 0, e.g. (214) x y z 1 x y z ercept int æ è ç ö ø ÷

Hexagonal Closed Packed (HCP)

HCP Indices previous chart

Hexagonal Closed Packed (HCP) _ _ [2120]

Plane Indices in Hexagonal System

Summary Many materials form crystalline structure Material properties are influenced by Atomic Packing & Crystalline structure Crystal notations (direction indices & plane/Miller indices), a communication tool for material scientists & engineers Directions: [uvw] ; except [uvtw] for HCP Planes: (hkl) ; except [hkil] for HCP Families of direction (<>) and planes ({}) are associated with groups of direction & planes which are crystallographically equivalent (similar atomic arrangement