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

2. Crystal Classes & Lattice Types2

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. 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

11. 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 = ; (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 CN
APF

12. Interstitial Sites

13. Interstitial Sites

14. Crystal Notations

15. 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

16. 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]

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. 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, )
(or 0 1 0)
(010) (or (010) )

20. 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 ÷

21. Hexagonal Closed Packed (HCP)

22. HCP Indices
previous chart

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