1. Why do we care about crystal structures, directions, planes ?
Physical properties of materials depend on the geometry of crystals
ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
(for now, focus on metals)
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation? 1

2. Energy and Packing • Non dense, random packing COOLING
typical neighbor
bond length
bond energy
COOLING
• Dense, ordered packing
Energy r
typical neighbor
bond length
bond energy
Dense, ordered packed structures tend to have
lower energies.

3. MATERIALS AND PACKING Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of:
-metals
-many ceramics
-some polymers
crystalline SiO2
Adapted from Fig. 3.18(a),
Callister 6e.
LONG RANGE ORDER
Noncrystalline materials...
• atoms have no periodic packing
• occurs for:
-complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2
Adapted from Fig. 3.18(b),
Callister 6e.
SHORT RANGE ORDER 3

4. Unit Cell Concept
The unit cell is the smallest structural unit or building block that uniquely can describe the crystal structure. Repetition of the unit cell generates the entire crystal. By simple translation, it defines a lattice .
Lattice Parameter : Repeat distance in the unit cell, one for in each dimension a b

5. Crystal Systems Units cells and lattices in 3-D:
When translated in each lattice parameter direction, MUST fill 3-D space such that no gaps, empty spaces left. c b
Lattice Parameter : Repeat distance in the unit cell, one for in each dimension a

6. Section 3.3 – Crystal Systems
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
Fig. 3.4, Callister 7e.
7 crystal systems
14 crystal lattices
a, b, and c are the lattice constants

7. Section 3.4 – Metallic Crystal Structures
How can we stack metal atoms to minimize empty space?
2-dimensions
vs.
Now stack these 2-D layers to make 3-D structures

8. METALLIC CRYSTALS • tend to be densely packed.
• have several 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.
• have the simplest crystal structures.
We will look at three such structures...
Remember metallic bond => non-directional, does not restrict number of nearest-neighbors ( covalent ; 8-N’ rule), allows for dense atomic packing 4

9. WEB SITE

10. SIMPLE CUBIC STRUCTURE (SC)
• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
Closed packed direction is where the atoms touch each other
• Coordination # = 6
(# nearest neighbors)
(Courtesy P.M. Anderson) 5

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

12. 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.
ex: Cr, W, Fe (), Tantalum, Molybdenum
• Coordination # = 8
2 atoms/unit cell: 1 center + 8 corners x 1/8
(Courtesy P.M. Anderson) 7

13. ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68 a R a 3 a a 2 8

14. 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.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
• Coordination # = 12
Adapted from Fig. 3.1, Callister 7e.
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
(Courtesy P.M. Anderson) 9

15. ATOMIC PACKING FACTOR: FCC
• APF for a body-centered cubic structure = 0.74 10

16. FCC STACKING SEQUENCE • ABCABC... Stacking Sequence • 2D Projection
• FCC Unit Cell 11

17. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection
Adapted from Fig. 3.3,
Callister 6e.
• Coordination # = 12
6 atoms/unit cell
• APF = 0.74
ex: Cd, Mg, Ti, Zn
• c/a = 1.633 12

18. STRUCTURE OF COMPOUNDS: NaCl
• Compounds: Often have similar close-packed structures.
• Structure of NaCl
• Close-packed directions
--along cube edges.
(Courtesy P.M. Anderson)
(Courtesy P.M. Anderson) 13

19. Quotes from People in the know
Arthur C. Clarke's Three Laws:
1)When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.
2)The only way to discover the limits of the possible is to go beyond them into the impossible.
3)Any sufficiently advanced technology is indistinguishable from magic. (from Profiles of the Future, 1961)

20. THEORETICAL DENSITY, r Example: Copper
Data from Table inside front cover of Callister (see next slide):
• crystal structure = FCC: 4 atoms/unit cell
• atomic weight = g/mol (1 amu = 1 g/mol)
• atomic radius R = nm (1 nm = 10 cm) -7 14

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

22. Characteristics of Selected Elements at 20C
Adapted from
Table, "Charac-
teristics of
Selected
Elements",
inside front
cover,
Callister 6e. 15

23. DENSITIES OF MATERIAL CLASSES
Why?
Metals have...
• close-packing
(metallic bonding)
• large atomic mass
Ceramics have...
• less dense packing
(covalent bonding)
• often lighter elements
Polymers have...
• poor packing
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
Data from Table B1, Callister 6e. 16

24. POLYMORPHISM & ALLOTROPY
Some materials may exist in more than one crystal structure, this is called polymorphism.
If the material is an elemental solid, it is called allotropy. An example of allotropy is carbon, which can exist as diamond, graphite, and amorphous carbon.

25. Boron Nitride Allotropy
Hexagonal (h-BN): Soft
Properties of BN change with crystal structure
sp2
Cubic (c-BN) : Exteremely Hard
First reports on BN is around 1850’s by Balmain.
BN is an synthetic compound.
Bn has 4 allomorphs, of which two are analogous to graphite-diamond pair.
sp3

26. Fullerenes
B36N36
B12N12 B N C
4.3 Å
C60
6.8 Å
7.2 Å

27. Crystallographic Points, Directions, and Planes
It is necessary to specify a particular point/location/atom/direction/plane in a unit cell
We need some labeling convention. Simplest way is to use a 3-D system, where every location can be expressed using three numbers or indices.
a, b, c and α, β, γ z α β y γ x

28. Crystallographic Points, Directions, and Planes
Crystallographic direction is a vector [uvw]
Always passes thru origin 000
Measured in terms of unit cell dimensions a, b, and c
Smallest integer values
Planes with Miller Indices (hkl)
If plane passes thru origin, translate
Length of each planar intercept in terms of the lattice parameters a, b, and c.
Reciprocals are taken
If needed multiply by a common factor for integer representation

29. Section 3.8 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 corner are 111
Translation: integer multiple of lattice constants  identical position in another unit cell z 2c y b b

30. Crystallographic Directionsz
Algorithm
1. Vector repositioned (if necessary) to pass
through origin. 2. Read off projections 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>

31. Linear Density 3.5 nm a 2 LD = Linear Density of Atoms  LD = [110]
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 - =

32. HCP Crystallographic Directions- a3 a1 a2 z
Algorithm
1. Vector repositioned (if necessary) to pass
through origin. 2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas
[uvtw]
dashed red lines indicate
projections onto a1 and a2 axes a1 a2 a3
-a3 2 a 1
Adapted from Fig. 3.8(a), Callister 7e.
[ 1120 ]
ex: ½, ½, -1, =>

33. HCP Crystallographic Directions
Hexagonal Crystals
4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.
Fig. 3.8(a), Callister 7e. - a3 a1 a2 z = ' w t v u ) ( + - 2 3 1 ]
uvtw [

34. Crystallographic Planes
Adapted from Fig. 3.9, Callister 7e.

35. 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 
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)

36. Crystallographic Planesz x y a b c
example
a b c
Intercepts 
Reciprocals
1/ / /
Reduction
Miller Indices (110)
example
a b c z x y a b c
Intercepts
1/  
Reciprocals
1/½ 1/ 1/
Reduction
Miller Indices (100)

37. Crystallographic Planesz x y a b c
example
a b c
Intercepts
1/ /4
Reciprocals
1/½ 1/ /¾ /3
Reduction
Miller Indices (634)
(001)
(010),
Family of Planes {hkl}
(100),
(001),
Ex: {100} = (100),

38. Crystallographic Planes (HCP)
In hexagonal unit cells the same idea is used a2 a3 a1 z
example
a a a c
Intercepts  -1 1
Reciprocals / -1 1
Reduction -1 1
Miller-Bravais Indices
(1011)
Adapted from Fig. 3.8(a), Callister 7e.

39. Crystallographic Planes
We want to examine the atomic packing of crystallographic planes
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.

40. 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
Adapted from Fig. 3.2(c), Callister 7e. =
Planar Density = a 2 1
atoms
2D repeat unit
nm2
12.1 m2
= 1.2 x 1019 R 3 4
area

41. Planar Density of (111) Iron
Solution (cont):  (111) plane
1 atom in plane/ unit surface cell 2 a
atoms in plane
atoms above plane
atoms below plane
2D repeat unit 3 h = a 2 3 2 R 16 4 a ah
area = ÷ ø ö ç è æ 1 =
nm2
atoms
7.0 m2
0.70 x 1019 3 2 R 16
Planar Density =
2D repeat unit
area

42. Single Crystals and Polycrystalline Materials
In a single crystal material the periodic and repeated arrangement of atoms is PERFECT This extends throughout the entirety of the specimen without interruption.
Polycrystalline material, on the other hand, is comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.

43. Example of Polycrystalline Growth

44. CRYSTALS AS BUILDING BLOCKS
• Some engineering applications require single crystals:
--diamond single
crystals for abrasives
--turbine blades
Fig. 8.30(c), Callister 6e.
(Fig. 8.30(c) courtesy
of Pratt and Whitney).
(Courtesy Martin Deakins,
GE Superabrasives, Worthington, OH. Used with permission.)
• Crystal properties reveal features
of atomic structure.
--Ex: Certain crystal planes in quartz
fracture more easily than others.
(Courtesy P.M. Anderson) 17

45. POLYCRYSTALS • Most engineering materials are polycrystals. 1 mm
Adapted from Fig. K, color inset pages of Callister 6e.
(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)
1 mm
• Nb-Hf-W plate with an electron beam weld.
• Each "grain" is a single crystal.
• If crystals are randomly oriented,
overall component properties are not directional.
• Crystal sizes typ. range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers). 18

46. SINGLE VS POLYCRYSTALS
• Single Crystals
-Properties vary with
direction: anisotropic.
Data from Table 3.3, Callister 6e.
(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)
-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
200 mm
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
Adapted from Fig. 4.12(b), Callister 6e.
(Fig. 4.12(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].) 19

47. Anisotropy and Texture
Different directions in a crystal have a different APF.
For instance, atoms along the edge of FCC unit cell are more separated than along the face diagonal. This causes anisotropy in the properties of crystals.
For example, the deformation amount depends on the direction in which a stress is applied, other properties are thermal conductivity, optical properties, magnetic properties, hardness, etc.
In some polycrystalline materials, grain orientations are random, hence bulk material properties are isotropic, i.e. equivalent in each direction
Some polycrystalline materials have grains with preferred orientations (texture), so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties.

48. Sample Question a c Ultimate Strength Rupture Strain Hardening Necking
The Young’s modulus defines the amount of elastic strain induced on a material when stressed, slope of a stress-strain curve.
Question: Plot tensile Young’s Modulus E (θ) as a function of θ for graphite single crystal. θ is the angle between c- and a- axis of the hexagonal crystal system graphite assumes.
1/E(θ) = S11 cos4θ + S33sin4θ + (S44+2S13).(cos2θ sin2θ)
Ultimate Strength
Rupture
Yield Strength θ
Basal Planes of
graphite a c
Strain Hardening
Necking
Modulus of Elasticity
Young’s Modulus

49. X-RAYS TO CONFIRM CRYSTAL STRUCTURE
• Incoming X-rays diffract from crystal planes.
Adapted from Fig. 3.2W, Callister 6e.
• Measurement of:
Critical angles, qc,
for X-rays provide
atomic spacing, d. 20

50. SCANNING TUNNELING MICROSCOPY
• Atoms can be arranged and imaged
Photos produced from the work of C.P. Lutz, Zeppenfeld, and D.M. Eigler. Reprinted with permission from International Business Machines Corporation, copyright 1995.
Carbon monoxide molecules arranged on a platinum (111) surface.
Iron atoms arranged on a copper (111) surface. These Kanji characters represent the word “atom”. 21

51. DEMO: HEATING AND COOLING OF AN IRON WIRE
The same atoms can have more than one crystal structure.
• Demonstrates "polymorphism" 22

52. SUMMARY (I) • Atoms may assemble into crystalline or
amorphous structures.
• We can predict the density of a material,
provided we know the atomic weight, atomic
radius, and crystal geometry (e.g., FCC,
BCC, HCP).
• Material properties generally vary with single
crystal orientation (i.e., they are anisotropic),
but properties are generally non-directional
(i.e., they are isotropic) in polycrystals with
randomly oriented grains. 23

53. Summary (II) Allotropy Amorphous Anisotropy
Atomic packing factor (APF)
Body-centered cubic (BCC)
Coordination number
Crystal structure
Crystalline
Face-centered cubic (FCC)
Grain
Grain boundary
Hexagonal close-packed (HCP)
Isotropic
Lattice parameter
Non-crystalline
Polycrystalline
Polymorphism
Single crystal
Unit cell

54. Midterm Dates And Places
Midterm 1: Place and Time
Midterm 2: Place and Time.
Final: To be decided by the Faculty of Engineering.

55. ANNOUNCEMENTS
Reading: Read and work on the examples of Chapter 3 multiple times. Visit the Virtual Materials Science Engineering Website link listed in your book and go thru the different Crystal Structures !
Core Problems: 3.14, 3.15, 3.20, 3.31, 3.39, 3.40, 3.54
Bonus Problems: 3.26 (?), 3.32,3.41,3.42, 3.51,3.53
Due Date: 6 March 2008