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. The Big Picture Classification of Solids: Bonding type
Electronic Structure
Bonding
State of aggregation
Bohr atom
Bohr-Sommerfeld
Quantum numbers
Aufbau principle
Multielectron atoms
Periodic table patterns
Octet stability
Primary:
Ionic
Covalent
Metallic
Gas
Liquid
Solid
Classification of Solids:
Bonding type
Atomic arrangement

3. classifications of solids by atomic arrangement
SOLID: Smth. which is dimensionally stable, i.e., has a volume of its own
classifications of solids by atomic arrangement
ordered disordered
atomic arrangement regular random*
order long-range short-range
name crystalline amorphous
“crystal” “glass”

4. 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.

5. 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

6. 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 6 6

7. Robert Hooke – 1660 - Cannonballs
“Crystal must owe its regular shape to the packing of spherical particles”

8. Niels Steensen ~ 1670
observed that quartz crystals had the same angles between corresponding faces regardless of their size.

9. SIMPLE QUESTION:
If I see something has a macroscopic shape very regular and cubic, can I infer from that if I divide, divide, divide, divide, divide, if I get down to atomic dimensions, will there be some cubic repeat unit?

10. Christian Huygens
Studying calcite crystals made drawings of atomic packing and bulk shape.

11. BERYL
Be3Al2(SiO3)6

12. Early Crystallography
René-Just Haüy (1781): cleavage of calcite
Common shape to all shards: rhombohedral
How to model this mathematically?
What is the maximum number of distinguishable shapes that will fill three space?
Mathematically proved that there are only 7 distinct space-filling volume elements

13. The Seven Crystal Systems
BASIC UNIT
Specification of unit cell parameters

14. Does it work with Pentagon?

15. August Bravais
How many different ways can I put atoms into these seven crystal systems, and get distinguishable point environments?
When I start putting atoms in the cube, I have three distinguishable arrangements. SC
BCC
FCC
And, he proved mathematically that there are 14 distinct ways to arrange points in space.

17. Last Day: Atomic Arrangement
SOLID: Smth. which is dimensionally stable, i.e., has a volume of its own
classifications of solids by atomic arrangement
ordered disordered
atomic arrangement regular random*
order long-range short-range
name crystalline amorphous
“crystal” “glass”

18. 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

19. 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.

20. Three Types of Solids according to atomic arrangement

21. 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: The periodic arrangement of atoms in a Xtal.
Lattice Parameter : Repeat distance in the unit cell, one for in each dimension a b

22. 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

23. The Importance of the Unit Cell
One can analyze the Xtal as a whole by investigating a representative volume.
Ex: from unit cell we can
Find the distances between nearest atoms for calculations of the forces holding the lattice together
Look at the fraction of the unit cell volume filled by atoms and relate the density of solid to the atomic arrangement
The properties of the periodic Xtal lattice determine the allowed energies of electrons that participate in the conduction process.

24. 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

25. 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

27. SIMPLE CUBIC STRUCTURE (SC)
• Rare due to poor packing
• 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

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

29. 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

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

31. 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

32. ATOMIC PACKING FACTOR: FCC? HW

33. HW Fill here! SC BCC FCC Unit Cell Volume a3 Lattice Points per cell 1
1. Finish reading Chapter 3.
2.On a paper solve example problems:3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9
3. Fill in the blanks in Table below SC
BCC
FCC
Unit Cell Volume a3
Lattice Points per cell 1
Nearest Neighbor Distance a
Number of Nearest Neighbors 6
Atomic Packing Factor
0.52
Fill here!

34. 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

35. 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

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

37. 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

38. 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.

39. Polymorphism
Two or more distinct crystal structures for the same material (allotropy/polymorphism)     titanium
  , -Ti
carbon
diamond, graphite
BCC
FCC
1538ºC
1394ºC
912ºC
-Fe
-Fe
-Fe
liquid
iron system

40. 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

41. 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

42. 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

43. 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
=> 2, 0, 1
=> [ 201 ]
Lecture 2 ended here
-1, 1, 1
where overbar represents a negative index
[ 111 ]
=>
families of directions <uvw>

44. 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 - =

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

46. 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 
If plane passes thru origin, translate
Read off intercepts of plane with axes in terms of a, b, c
Take reciprocals of intercepts
Reduce to smallest integer values
Enclose in parentheses, no commas i.e., (hkl)

47. 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)

48. 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),

49. 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.

50. 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

51. Planar Density of (111) Iron
? HW

52. 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.

53. Example of Polycrystalline Growth

54. 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

55. 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

56. 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

57. Anisotropy and Texture
Different directions in a crystal have a different APF.
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.

58. 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

59. 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

60. 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