MSE420/514: Session 1 Crystallography & Crystal Structure

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

2. Crystal Classes & Lattice Types 4 Lattice Types 7 Crystal Classes

3. 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

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

5. 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

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

7. 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

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

9. 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

10. Energy Inter-atomic Spacing, r Equilibrium r + + 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 BCC SC FCC

11. 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 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

12. Interstitial Sites

13. Interstitial Sites

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

15. (1,1,1) O (1,0,0) Crystallographic Notations: Coordinates • Atom Coordinates • Locating Atom Position in Unit Cell • Point in space, coordinates in ref. to origin

16. -1,-1,0 [110] - - 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> directionfamily 0,1,½ [021] 0,0,0 1,0,0 [100]

17. 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)

18. Crystallographic Notations: Plane Indices

19. ,-1,½ 0,-1,2 (012) - ,-1,  (or ,1, ) 0 -1 0 (or 0 1 0) (010) (or (010) ) Crystallographic Notations: Plane Indices Reciprocal of Intercepts • Miller Indices, (hkl) & {hkl} family • 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 cubicsystem: - [abc] direction  (abc) plane

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

21. Hexagonal Closed Packed (HCP)

22. previous chart previous chart HCP Indices

23. Hexagonal Closed Packed (HCP) _ _ [2120]

24. Plane Indices in Hexagonal System

25. 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