1. Chapter 3: The Structure of Crystalline Solids
ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation?

2. The “unit cell” is the basic repeating unit of the arrangement of atoms, ions or molecules in a crystalline solid.
The “lattice” refers to the 3-D array of particles in a crystalline solid. One type of atom occupies a “lattice point” in the array.

3. Examples of Unit Cells

4.   SPACE LATTICE AND UNIT CELLS   Space Lattice atoms arranged in a pattern that repeats itself in three dimensions.   Unit cell smallest grouping which can be translated in three dimensions to recreate the space lattice.    

5. Crystalline Versus Amorphous
Quartz
Obsidian

6. Crystals

7. Crystal Structures
Atoms (and later ions) will be viewed as hard spheres. In the case of pure metals, the packing pattern often provides the greatest spatial efficiency (closest packing).
Ionic crystals can often be viewed as a close-packed arrangement of the larger ion, with the smaller ion placed in the “holes” of the structure.

8. Unit Cells
Crystals consist of repeating asymmetric units which may be atoms, ions or molecules. The space lattice is the pattern formed by the points that represent these repeating structural units.

9. Unit Cells
A unit cell of the crystal is an imaginary parallel-sided region from which the entire crystal can be built up.
Usually the smallest unit cell which exhibits the greatest symmetry is chosen. If repeated (translated) in 3 dimensions, the entire crystal is recreated.

10. Crystal Structures Types of crystal structures
Face centered cubic (FCC)
Body centered cubic (BCC)
Hexagonal close packed (HCP)

11. 3 2 1 1 4 2 1

12. BCC
FCC
HCP

13. Face Centered Cubic (FCC)
Atoms are arranged at the corners and center of each cube face of the cell.
Atoms are assumed to touch along face diagonals

14. Face Centered Cubic (FCC)
The lattice parameter, a, is related to the radius of the atom in the cell through:
Coordination number: the number of nearest neighbors to any atom. For FCC systems, the coordination number is 12.

15. Face Centered Cubic (FCC)
Atomic Packing Factor: the ratio of atomic sphere volume to unit cell volume, assuming a hard sphere model.
FCC systems have an APF of 0.74, the maximum packing for a system in which all spheres have equal diameter.

16. Body Centered Cubic
Atoms are arranged at the corners of the cube with another atom at the cube center.

17. ATOMIC PACKING FACTOR: BCC
Adapted from
Fig. 3.2,
Callister 6e.
• APF for a body-centered cubic structure = p3/8 = 0.68

18. Body Centered Cubic
Since atoms are assumed to touch along the cube diagonal in BCC, the lattice parameter is related to atomic radius through:

19. Principal Metallic Crystal Structures   We will concentrate on three of the more densely packed crystal structures, BCC - body centered cubic, FCC - face centered cubic, and HCP - hexagonal close packed.   BCC - 2 atoms per unit cell CN = 8 

20. FCC - 4 atoms per unit cell CN = 12 
HCP - 6 atoms per unit cell CN = 12

21. Hexagonal Close Packed
Cell of an HCP lattice is visualized as a top and bottom plane of 7 atoms, forming a regular hexagon around a central atom. In between these planes is a half-hexagon of 3 atoms.

22. Close Packing
Since metal atoms and ions lack directional bonding, they will often pack with greatest efficiency. In close or closest packing, each metal atom has 12 nearest neighbors.
The number of nearest neighbors is called the coordination number. Six atoms surround an atom in the same plane, and the central atom is then “capped” by 3 atoms on top, and 3 atoms below it.

23. Close Packing

24. # of Atoms/Unit Cell Atoms in corners are ⅛ within the cell
For atoms in a cubic unit cell:
Atoms in corners are ⅛ within the cell

25. # of Atoms/Unit Cell Atoms on faces are ½ within the cell
For atoms in a cubic unit cell:
Atoms on faces are ½ within the cell

26. # of Atoms/Unit Cell
A face-centered cubic unit cell contains a total of 4 atoms: 1 from the corners, and 3 from the faces.

27. # of Atoms/Unit Cell For atoms in a cubic unit cell:
Atoms in corners are ⅛ within the cell
Atoms on faces are ½ within the cell
Atoms on edges are ¼ within the cell

28. Other Metallic Crystal Structures
Body-centered cubic unit cells have an atom in the center of the cube as well as one in each corner. The packing efficiency is 68%, and the coordination number = 8.

29. Other Metallic Crystal Structures
Simple cubic (or primitive cubic) unit cells are relatively rare. The atoms occupy the corners of a cube. The coordination number is 6, and the packing efficiency is only 52.4%.

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

31. Polymorphism
Many metals exhibit different crystal structures with changes in pressure and temperature.
It is temperature /pressure dependent phenomenon.

32. Example: Iron. liquid above 1539 C
Example:   Iron liquid above 1539 C. δ-iron (BCC) between 1394 and 1539 C. γ-iron (FCC) between 912 and 1394 C. α-iron (BCC) between -273 and 912 C.    

33. Single crystal
All unit cells interlock in the same way and have the same orientation.
Single crystals exist in nature, but they may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled.

34. Polycrystalline Materials
Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline
The Various stages in the solidification of a polycrystalline specimen are:
Initially, small crystals or nuclei form at various positions.
The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each.
The extremities of adjacent grains impinge on one another as the solidification process approaches completion
exists some atomic mismatch within the region where two grains meet; this area, called a grain boundary

36. Crystals as Building Blocks
• Some engineering applications require single crystals:
--diamond single
crystals for abrasives
--turbine blades
• Properties of crystalline materials
often related to crystal structure.
--Ex: Quartz fractures more easily along some crystal planes than others.

37. Polycrystals Anisotropic
• Most engineering materials are polycrystals.
1 mm
Isotropic
• Each "grain" is a single crystal.
• If grains are randomly oriented,
overall component properties are not directional.
• Grain sizes typ. range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).

38. Single vs Polycrystals
E (diagonal) = 273 GPa
E (edge) = 125 GPa
• Single Crystals
-Properties vary with
direction: anisotropic.
-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.

40. Theoretical Density, r
A knowledge of crystal structure of a metallic solid permits computation density :
Cell
Unit of
Volume
Total in
Atoms
Mass
Density =  =
VC NA
n A
 =
where n = number of atoms/unit cell
A = atomic weight g/mole
VC = Volume of unit cell = a3 for cubic
NA = Avogadro’s number
= x 1023 atoms/mol

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

42. Densities of Material Classes
In general
Graphite/ r
metals r
ceramics r
polymers
Metals/
Composites/
>
>
Ceramics/
Polymers
Alloys
fibers
Semicond 30
Why?
Metals have...
• close-packing
(metallic bonding)
• often large atomic masses 2
Magnesium
Aluminum
Steels
Titanium
Cu,Ni
Tin, Zinc
Silver, Mo
Tantalum
Gold, W
Platinum
*GFRE, CFRE, & AFRE are Glass,
Carbon, & Aramid Fiber-Reinforced
Epoxy composites (values based on
60% volume fraction of aligned fibers 10
in an epoxy matrix). G
raphite
Silicon
Glass -
soda
Concrete
Si nitride
Diamond
Al oxide
Zirconia
Ceramics have...
• less dense packing
• often lighter elements 5 3 4
(g/cm ) 3
Wood
AFRE *
CFRE
GFRE*
Glass fibers
Carbon
fibers A
ramid fibers r H
DPE, PS
PP, LDPE PC
PTFE
PET
PVC
Silicone
Polymers have...
• low packing density
(often amorphous)
• lighter elements (C,H,O) 2 1
Composites have...
• intermediate values
0.5
0.4
0.3

43. Energy and Packing • Non dense, random packing
typical neighbor
bond length
bond energy
• Dense, ordered packing
Energy r
typical neighbor
bond length
bond energy
Dense, ordered packed structures tend to
1.have lower energies and more stable energy arrangements.
2. Atoms come closer together and bond more tightly

44. Attractive Energy: the energy released when the ions come close together
Repulsive energy: the energy absorbed as the ions come close together
Net Energy: the sum of energies associated with the attraction and repulsion of the ions
It is minimum when the ions are at their equilibrium separation distance r0.
At the minimum energy, the force between the ions are zero. The ions will remain at an equilibrium separation distance r0 ( more stable).

45. Zincblende/Diamond Lattices
The Cubic Unit Cell
Zincblende Lattice
The Cubic Unit Cell
Other views of the cubic unit cell

46. Diamond Lattice
Diamond Lattice
The Cubic Unit Cell

47. Zincblende (ZnS) Lattice
Zincblende Lattice
The Cubic Unit Cell.

48. Wurtzite Structure Wurtzite Structure
We’ve also seen: Many semiconductors have the
Wurtzite Structure
Tetrahedral coordination: Each atom has 4 nearest-neighbors (nn). Basis set: 2 atoms. Primitive lattice  hexagonal close packed (hcp).
2 atoms per hcp lattice point
A Unit Cell looks like

49. Materials and Packing Si Oxygen Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of:
-metals
-many ceramics
-some polymers
crystalline SiO2 Si
Oxygen
Noncrystalline materials...
• atoms have no periodic packing
• occurs for:
-complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2

50. Crystal: is a solids in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. Long range order exist
Noncrystalline(amorphous): materials that don’t crystallize, this long range atomic order is absent
Crystal structure: it is the manner in which atoms, ions, or molecules are spatially arranged. Or
A regular 3 dimensional pattern of atoms or ions in space
There are an extremely large number of different crystal structures all having long range atomic order.
Crystal system: is described in terms of the unit cell geometry.
A crystal structure is described by both the geometry of, and atomic arrangements within the unit cell, whereas a
Crystal system is described only in terms of the unit cell geometry. For example, face-centered cubic and body-centered cubic are crystal structures that belong to the cubic crystal system.

51. Crystal Systems
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
7 crystal systems
14 crystal lattices
a, b, and c are the lattice constants
a, b, c, α, β, γ are lattice parameters

52. 3 2 1 1 4 2 1

53. 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

54. 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...

55. Simple Cubic Structure (SC)
• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)

56. 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
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

57. Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
ex: Fe (), Tantalum, Molybdenum
• Coordination # = 8
2 atoms/unit cell: 1 center + 8 corners x 1/8

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

59. 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, Pb, Ni, Ag
• Coordination # = 12
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

60. Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74 a
2 a
maximum achievable APF
Close-packed directions:
length = 4R =
2 a
Unit cell contains:
6 x 1/2 + 8 x 1/8 =
4 atoms/unit cell
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a

61. CLOSE-PACKED CRYSTAL STRUCTURES FOR METALS
Both face-centered cubic and hexagonal close-packed crystal structures have atomic packing factors of 0.74, which is the most efficient packing of equal sized spheres or atoms.
In addition to unit cell representations, these two crystal
structures may be described in terms of close-packed planes of atoms (i.e., planes having a maximum atom or sphere-packing density); a portion of one such plane is illustrated in Figures bellow.
Both crystal structures may be generated by the stacking of these close-packed planes on top of one another; the difference between the two structures lies in the stacking sequence.

62. Hexagonal Close-Packed Structure (HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection c a
A sites B
sites
Bottom layer
Middle layer
Top
layer
• Coordination # = 12
6 atoms/unit cell
• APF = 0.74
ex: Cd, Mg, Ti, Zn
• c/a = 1.633

63. CRYSTALLOGRAPHIC POINTS, DIRECT IONS AND PLANES
Crystallographic planes and directions are specified in terms of an indexing scheme.
Crystallographic directional and planar equivalencies are related to atomic linear and planar densities, respectively.
The atomic packing (i.e., planar density) of spheres in a crystallographic plane depends on the indices of the plane as well as the crystal structure.

64. Point Coordinates
To specify the position of any point located within unit cell. z 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
Lattice constants: a, b, c z 2c y b b

65. Linear Density 3.5 nm a 2 LD = Linear Density  LD = [110]j a
[110]
ex: linear density of Al in [110] direction  a = nm
½+1+ ½=2 atoms
# atoms
length 1
3.5 nm a 2 LD - =
LD is important relative to the process of slip-the mechanism by which metals plastically deform

66. 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

67. 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

68. SUMMARY • Atoms may assemble into crystalline or amorphous structures.
• Common metallic crystal structures are FCC, BCC, and
HCP. Coordination number and atomic packing factor
are the same for both FCC and HCP crystal 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).
• Crystallographic points, directions and planes are
specified in terms of indexing schemes.
Crystallographic directions and planes are related
to atomic linear densities and planar densities.

69. SUMMARY • Materials can be single crystals or polycrystalline.
Material properties generally vary with single crystal
orientation (i.e., they are anisotropic), but are generally
non-directional (i.e., they are isotropic) in polycrystals
with randomly oriented grains.
• Some materials can have more than one crystal
structure. This is referred to as polymorphism (or
allotropy).
• X-ray diffraction is used for crystal structure and
interplanar spacing determinations.

70. Alloys
Alloys are solid solutions of metals. They are usually prepared by mixing molten components. They may be homogeneous, with a uniform distribution, or occur in a fixed ratio, as in a compound with a specific internal structure.