1. Crystallography and Structure
Lec. (3)
Crystallography and Structure

2. Overview: Crystal Structure – matter assumes a periodic shape
Non-Crystalline or Amorphous “structures” exhibit no long range periodic shapes
Xtal Systems – not structures but potentials
FCC, BCC and HCP – common Xtal Structures for metals
Point, Direction and Planer ID’ing in Xtals
X-Ray Diffraction and Xtal Structure

3. Energy and Packing • Non dense, random packing
typical neighbor
bond length
bond energy
Energy r
typical neighbor
bond length
bond energy
• Dense, ordered packing
Dense, ordered packed structures tend to have
lower energies & thus are more stable.

4. CRYSTAL STRUCTURES Means: periodic arrangement of atoms/ions
over large atomic distances
Leads to structure displaying long-range order that is
Measurable and Quantifiable
All metals, many ceramics, and some polymers exhibit this “High Bond Energy” and a More Closely Packed Structure

5. Are materials lacking long range order
Amorphous Materials
Are materials lacking long range order
These less densely packed lower bond energy “structures” can be found in Metals are observed in Ceramic GLASS and many “plastics”

6. Crystal Systems – Some Definitional information
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
Fig. 3.4, Callister 7e.
7 crystal systems of varying symmetry
are known
These systems are built by changing the lattice parameters: a, b, and c are the edge lengths
, , and  are interaxial angles

7. General Unit Cell Discussion
For any lattice, the unit cell &, thus, the entire lattice, is UNIQUELY determined by 6 constants (figure):
a, b, c, α, β and γ
which depend on lattice geometry.
As we’ll see, we sometimes want to calculate the number of atoms in a unit cell. To do this, imagine stacking hard spheres centered at each lattice point & just touching each neighboring sphere. Then, for the cubic lattices, only 1/8 of each lattice point in a unit cell assigned to that cell. In the cubic lattice in the figure, each unit cell is associated with (8)  (1/8) = 1 lattice point.

8. Primitive Unit Cells & Primitive Lattice Vectors
In general, a Primitive Unit Cell is determined by the parallelepiped formed by the Primitive Vectors a1 ,a2, & a3 such that there is no cell of smaller volume that can be used as a building block for the crystal structure.
As we’ve discussed, a Primitive Unit Cell can be repeated to fill space by periodic repetition of it through the translation vectors
T = n1a1 + n2a2 + n3a3.
The Primitive Unit Cell volume can be found by vector manipulation:
V = a1(a2  a3)
For the cubic unit cell in the figure, V = a3

9. Primitive Unit Cells A 2 Dimensional Example!
Note that, by definition, the Primitive Unit Cell must contain ONLY ONE lattice point.
There can be different choices for the Primitive Lattice Vectors, but the Primitive Cell volume must be independent of that choice.
A 2 Dimensional Example!
P = Primitive Unit Cell
NP = Non-Primitive Unit Cell

10. 2-Dimensional Unit Cells
Artificial Example: “NaCl”
Lattice points are points with identical environments.

11. 2-Dimensional Unit Cells: “NaCl”
The choice of origin is arbitrary - lattice points need not be atoms - but the unit cell size must always be the same.

12. 2-Dimensional Unit Cells: “NaCl”
These are also unit cells it doesn’t matter if the origin is at Na or Cl !

13. 2-Dimensional Unit Cells: “NaCl”
These are also unit cells the origin does not have to be on an atom!

14. 2-Dimensional Unit Cells: “NaCl”
These are NOT unit cells - empty space is not allowed!

15. 2-Dimensional Unit Cells: “NaCl”
In 2 dimensions, these are unit cells – in 3 dimensions, they would not be.

16. Crystal Systems
Crystal structures are divided into groups according to unit cell geometry (symmetry).

17. Metallic Crystal Structures
• Tend to be densely packed.
• Reasons for dense packing:
- Typically, only one element is present, so all atomic
radii are the same.
- Metallic bonding is not directional.
- Nearest neighbor distances tend to be small in
order to lower bond energy.
- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures (those of engineering importance) called: FCC, BCC and HCP – with a nod to Simple Cubic

18. Crystal Structure of Metals – of engineering interest

19. Simple Cubic Structure (SC)
• Rare due to low packing density (only Po – Polonium -- has this structure)
• Close-packed directions are cube edges.
• Coordination No. = 6
(# nearest neighbors) for each atom as seen
(Courtesy P.M. Anderson)

20. Atomic Packing Factor (APF)
Volume of atoms in unit cell*
APF =
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
Adapted from Fig. 3.23,
Callister 7e.
close-packed directions a
R=0.5a
contains (8 x 1/8) = 1
atom/unit cell
atom
volume
atoms
unit cell 4 3 p
(0.5a) 1
APF = 3 a
unit cell
volume
Here: a = Rat×2
Where Rat is the ‘handbook’ atomic radius

21. Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals within a unit cell.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
ex: Cr, W, Fe (), Tantalum, Molybdenum
Adapted from Fig. 3.2,
Callister 7e.
• Coordination # = 8
2 atoms/unit cell: (1 center) + (8 corners x 1/8)
(Courtesy P.M. Anderson)

22. Atomic Packing Factor: BCC3 a a 2
length = 4R =
Close-packed directions:
3 a
APF = 4 3 p (
a/4 ) 2
atoms
unit cell
atom
volume a
Adapted from
Fig. 3.2(a), Callister 7e.
• APF for a body-centered cubic structure = 0.68

23. Face Centered Cubic Structure (FCC)
• Atoms touch each other along face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
• Coordination # = 12
Adapted from Fig. 3.1, Callister 7e.
4 atoms/unit cell: (6 face x ½) + (8 corners x 1/8)
(Courtesy P.M. Anderson)

24. Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74
The maximum achievable APF!
Close-packed directions:
length = 4R =
2 a
(a = 22*R)
Unit cell contains:
6 x 1/2 + 8 x 1/8 =
4 atoms/unit cell
Adapted from
Fig. 3.1(a),
Callister 7e.
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a

25. Comparison of the 3 Cubic Lattice Systems
Unit Cell Contents
Counting the number of atoms within the unit cell
Atom Position Shared Between: Each atom counts:
corner cells /8
face center cells /2
body center cell
edge center cells /2
Lattice Type Atoms per Cell
P (Primitive) [= 8  1/8]
I (Body Centered) [= (8  1/8) + (1  1)]
F (Face Centered) [= (8  1/8) + (6  1/2)]
C (Side Centered) [= (8  1/8) + (2  1/2)] 25

26. 2- HEXAGONAL CRYSTAL SYSTEMS
In a Hexagonal Crystal System, three equal coplanar axes intersect at an angle of 60°, and another axis is perpendicular to the others and of a different length.
The atoms are all the same.

27. Hexagonal Close-Packed Structure (HCP)
ex: Cd, Mg, Ti, Zn
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection c a
A sites B
sites
Bottom layer
Middle layer
Top
layer
Adapted from Fig. 3.3(a),
Callister 7e.
6 atoms/unit cell
• Coordination # = 12
• APF = 0.74
• c/a = (ideal)

28. Hexagonal Close Packed (HCP) Lattice
Bravais Lattice :
Hexagonal Lattice
He, Be, Mg, Hf, Re
(Group II elements)
ABABAB Type of Stacking 
a = b
Angle between a & b = 120° c = 1.633a, 
basis:
(0,0,0) (2/3a ,1/3a,1/2c)
Crystal Structure 28

29. We find that both FCC & HCP are highest density packing schemes (APF =0.74) – this illustration shows their differences as the closest packed planes are “built-up”

30. Comments on Close PackingB A B C
Sequence AAAA…
- simple cubic
Sequence ABABAB..
hexagonal close pack
Sequence ABAB…
- body centered cubic
Sequence ABCABCAB..
-face centered cubic close pack
Crystal Structure 30

31. Theoretical Density, r Density =  = n A  = VC NA
Cell
Unit of
Volume
Total in
Atoms
Mass
Density =  =
VC NA
n A
 =
where n = number of atoms/unit cell
A = atomic weight
VC = Volume of unit cell = a3 for cubic
NA = Avogadro’s number
= x 1023 atoms/mol

32. Theoretical Density, r  = a R Ex: Cr (BCC) A = 52.00 g/mol
R = nm
n = 2
 a = 4R/3 = nm
theoretical
ractual
= 7.18 g/cm3
= 7.19 g/cm3
 = a 3
52.00 2
atoms
unit cell
mol g
volume
6.023 x 1023

33. Locations in Lattices: Point Coordinatesz x y a b c
000
111
Point coordinates for unit cell center are
a/2, b/2, c/ ½ ½ ½
Point coordinates for unit cell (body diagonal) corner are 111
Translation: integer multiple of lattice constants  identical position in another unit cell z 2c y b b

34. Crystallographic Directionsz
Algorithm
1. Vector is repositioned (if necessary) to pass through the Unit Cell origin. 2. Read off line projections (to principal axes of U.C.) in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas
[uvw] y x
ex: 1, 0, ½
Lecture 2 ended here
=> 2, 0, 1
=> [ 201 ]
-1, 1, 1
where ‘overbar’ represents a negative index
[ 111 ]
=>
families of directions <uvw>

35. x y z What is this Direction ????? Projections:
Projections in terms of a,b and c:
Reduction:
Enclosure [brackets]
x y z 0c
a/2 b 1
1/2 1 2
[120]

36. Linear Density – considers equivalance and is important in Slip
Linear Density of Atoms  LD = 
Number of atoms
Unit length of direction vector a
[110]
ex: linear density of Al in [110] direction  a = nm
# atoms
length 1
3.5 nm a 2 LD - =
# atoms CENTERED on the direction of interest!
Length is of the direction of interest within the Unit Cell

37. Defining Crystallographic Planes
Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.
Algorithm (in cubic lattices this is direct)
1.  Read off intercepts of plane with axes in
terms of a, b, c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no
commas i.e., (hkl)  families {hkl}

38. Crystallographic Planes
We want to examine the atomic packing of crystallographic planes – those with the same packing are equivalent and part of families
Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important.
Draw (100) and (111) crystallographic planes
for Fe.
b) Calculate the planar density for each of these planes.

39. Planar Density of (100) Iron
Solution:  At T < 912C iron has the BCC structure.
2D repeat unit R 3 4 a =
(100)
Radius of iron R = nm =
Planar Density = a 2 1
atoms
2D repeat unit
nm2
12.1 m2
= 1.2 x 1019 R 3 4
area
Atoms: wholly contained and centered in/on plane within U.C., area of plane in U.C.

40. Planar Density of (111) Iron
Solution (cont):  (111) plane
1/2 atom centered on plane/ unit cell 2 a
atoms in plane
atoms above plane
atoms below plane
2D repeat unit 3 h = a 2
Area 2D Unit: ½ hb = ½*[(3/2)a][(2)a]=1/2(3)a2=8R2/(3)
3*1/6 =
nm2
atoms
7.0 m2
0.70 x 1019
Planar Density =
2D repeat unit
area