1. Chapter 3: The Structure of Crystalline Solids
ISSUES TO ADDRESS...
• How do atoms assemble (arrange) into solid structures?
(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?
Structure Levels: Subatomic  now: Atomic

2. 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 have
lower energies.

3. 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

4. Crystal Systems
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
Fig. 3.4, Callister 7e.
There are 6 lattice parameters
Their different combinations lead to:
7 crystal systems (shapes)
14 crystal lattices
In metals: Cubic and Hexagonal
a, b, and c are the lattice constants

5. Table 3.2 Lattice Parameter Relationships and Figures Showing Unit Cell Geometries for the Seven Crystal Systems
See other shapes

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

7. 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...

8. Parameters in Studying Crystal Structures
Number and Position of Atoms in unit cell
Coordination No.
No. of touching-neighbor atoms
Cube edge (length)
Atomic Packing Factor (APF):
Fraction of volume occupied by atoms
Fraction of empty space = 1 – APF
APF = Total volume of atoms in the unit cell
Volume of the unit cell

9. 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)
* Po: Polonium

10. 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

11. Body Centered Cubic Structure (BCC)
Atoms touch each other along cube diagonals.
examples: Cr, Fe(), W, Ta, Mo*
Coordination # = 8
No of Atoms per unit cell = 1 center + 8 corners x 1/8 =2 atoms
--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
* Cr: Chromium, W: Tungsten, Ta: Tantalum, Mo: Molybdenum

12. 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
APF = 4 3 p (
a/4 ) 2
No. atoms
unit cell
atom
volume a

13. Face Centered Cubic Structure (FCC)
Atoms touch each other along face diagonals.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
Coordination # = 12
No of atoms per unit cell = 6 face x 1/2 + 8 corners x 1/8 = 4 atoms
--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

14. 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
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a

15. FCC Stacking Sequence • ABCABC... Stacking Sequence • 2D Projection
A sites B C
sites A B
sites C A C A A B C
• FCC Unit Cell

16. 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
No. of atoms/unit cell = 6
• APF = 0.74
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
Cd: Cadmium, Mg: Magnesium, Ti: Titanium, Zn: Zinc

17. Polymorphism Polymorphous Material: BCC FCC 1538ºC 1394ºC 912ºC -Fe
Same Material with two or more distinct crystal structures
Crystal Structure is a Property:
It may vary with temperature
Allotropy /polymorphism
Examples
Titanium: , -Ti
Carbon: diamond, graphite
BCC
FCC
1538ºC
1394ºC
912ºC
-Fe
-Fe
-Fe
liquid
iron system
Heating

18. 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

19. Theoretical Density, r  = a R a 3 52.00 2 atoms unit cell mol g
Example: For Cr
Known: A = g/mol and R = nm
Crystal Structure: BCC
For BCC: n = 2 and a = 4R/ 3 = nm
Calculate:
 = a 3
52.00 2
atoms
unit cell
mol g
volume
6.023 x 1023
theoretical
ractual
= 7.18 g/cm3
= 7.19 g/cm3
Compare

20. Densities of Material Classes
In general
Graphite/ r
metals r
ceramics r
polymers
Metals/
Composites/
>
>
Ceramics/
Polymers
Alloys
fibers
Semicond 30
Why? B
ased on data in Table B1, Callister
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).
Ceramics have...
• less dense packing
• often lighter elements G
raphite
Silicon
Glass -
soda
Concrete
Si nitride
Diamond
Al oxide
Zirconia 5 3 4
(g/cm ) 3
Wood
AFRE *
CFRE
GFRE*
Glass fibers
Carbon
fibers A
ramid fibers H
DPE, PS
PP, LDPE PC
PTFE
PET
PVC
Silicone
Polymers have...
• low packing density
(often amorphous)
• lighter elements (C,H,O) r 2
Water 1
Composites have...
• intermediate values
0.5
0.4
0.3

21. Structure-Properties
Materials-Structures
Amorphous: Random atomic orientation
Crystalline:
Single Crystal Structure:
One Crystal
Perfectly repeated unit cells.
Polycrystalline
Many crystals
Properties
Anisotropy: depend on direction
Isotropy: does not depend on direction

22. Single Crystals
• 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.

23. Properties for Single and 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
- Properties may/may not vary with direction.
If grains are randomly oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured, anisotropic.
200 mm

24. Polycrystals • Most engineering materials are polycrystals.
Anisotropic
1 mm
Isotropic
• Each "grain" is a single crystal (composed of many unit cells).
• 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).

25. Point Coordinates z c y a b x z 2c y b b 111
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

26. Crystallographic Directions
Vectors or lines between two points
Described by indices
Algorithm (for obtaining indices)
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 (multiply by a factor)
4. Enclose in square brackets, no commas
[uvw]
(note: If the value is –ve put the – sign over the number)
Instead of 1 and 2: you can
1. Determine coordinates of head (H) and tail (T)
2. Subtract: head – tail
Then as before (steps 3 and 4) z x y
Ex 1: 1, 0, ½
2, 0, 1
[ 201 ]
Lecture 2 ended here
Ex 2: -1, 1, 1
(overbar represents a negative index)
[ 111 ]
=>
families of directions <uvw>

27. Linear Density 3.5 nm a 2 LD = Linear Density of Atoms  LD = [110]
Number of atoms
Unit length of direction vector
ex: linear density of Al in [110] direction  a = nm a
[110]
# atoms
length 1
3.5 nm a 2 LD - =

28. HCP Crystallographic Directions
Algorithm - a3 a1 a2 z
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
[ 1120 ]
ex: ½, ½, -1, =>

29. 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 [

30. Crystallographic Planes

31. 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 (for obtaining Miller Indices)
If the plane passes through the origin, choose another origin
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)

32. Crystallographic Planesz x y a b c
Example 1
a b c
Intercepts 
Reciprocals
1/ / /
Integers
Miller Indices (110)
Example 2
a b c z x y a b c
Intercepts
1/  
Reciprocals
1/½ 1/ 1/
Integers
Miller Indices (100)

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

34. 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
Integers -1 1
Miller-Bravais Indices
(1011)

35. No. of atoms per area of a plane
Planar Density
Concept
The atomic packing of crystallographic planes
No. of atoms per area of a plane
Importance
Iron foil can be used as a catalyst.
The atomic packing of the exposed planes is important.
Example
Draw (100) and (111) crystallographic planes
for Fe (BCC).
b) Calculate the planar density for each of these planes.

36. 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

37. 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

38. Determination of Crystal Structure
Wave: has an amplitude and wave length (l)
Diffraction: phase relationship between two (or more) waves that have been scattered.
Scattering:
In-phase Constructive
Out-of-phase Destructive
Detecting scattered waves after encountering an object:
Identify positions of atoms مثل الرؤية البصرية
Diffraction occurs when a wave encounters a series of regularly spaced atoms – IF:
The obstacles are capable of scattering the wave
Spacing between them nearly = wave length
Wave length depend on the source (i.e. material of the anode).

39. X-Ray Diffraction
Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
Can’t resolve spacings  

40. Constructive versus Destructive Diffraction

41. Constructive X-Ray Diffraction

42. X-Ray Experiment

43. X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
detector
“1”
incoming
X-rays
reflections must be in phase for
“2”
“1”
a detectable signal
outgoing X-rays l
“2” q q
extra distance traveled by wave 2 d
Interplaner spacing
spacing between planes
In-phase
Constructive diffraction occurs when
the second wave travels a distance = n l
From Geometry
extra traveled distance = 2 d sin q
Then
X-ray intensity q
Constructive
d =
n l
2 sin q c q c

44. Calculations d = n l 2 sin q c d = a for plane (h k l) (h2+k2+l2)0.5
Measurement of critical angle, qc, allows computation of planar spacing, d.
d =
n l
2 sin q c
Bragg’s Law, n: order of reflection
d = a for plane (h k l)
(h2+k2+l2)0.5
Atoms in other planes cause extra scattering
For validity of Bragg’s law
For BCC h+k+l = even
For FCC h, k, l all even or odd
Sample calculations
Given the plane indices and
Crystal structure,
get a (from C.S. and R) and d (from indices)
Then with wavelength get angle or wavelength

45. Solved Problem BCC For (310) Brag’s Law d = a for plane (h k l)
3.58 Determine the expected diffraction angle for the first-order reflection
from the (310) set of planes for BCC chromium
when monochromatic radiation of wavelength nm is used.
BCC
d = a for plane (h k l)
(h2+k2+l2)0.5
For (310)
Brag’s Law

46. X-Ray Diffraction Patternz x y a b c z x y a b c z x y a b c
(110)
(211)
Intensity (relative)
(200)
Diffraction angle 2q
Diffraction pattern for polycrystalline a-iron (BCC)

47. 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.

48. 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.