1. Chapter 3: Structures of Metals & Ceramics
ISSUES TO ADDRESS...
• What is the difference in atomic arrangement between crystalline and noncrystalline solids?
• What features of a metal’s/ceramic’s atomic structure determine its density?
• How do the crystal structures of ceramic materials differ from those for metals?
• Under what circumstances does a material property vary with the measurement direction?

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
Adapted from Fig. 3.41(a),
Callister & Rethwisch 4e. Si
Oxygen
Noncrystalline materials...
• atoms have no periodic packing
• occurs for:
-complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2
Adapted from Fig. 3.41(b),
Callister & Rethwisch 4e.

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

5. 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
• Metals have the simplest crystal structures.
We will examine three such structures...

6. 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)
Click once on image to start animation
(Courtesy P.M. Anderson)

7. 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. 3.43,
Callister & Rethwisch 4e.
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

8. 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: Cr, W, Fe (), Tantalum, Molybdenum
• Coordination # = 8
Adapted from Fig. 3.2,
Callister & Rethwisch 4e.
Click once on image to start animation
(Courtesy P.M. Anderson)
2 atoms/unit cell: 1 center + 8 corners x 1/8

9. VMSE Screenshot – BCC Unit Cell

10. 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 & Rethwisch 4e.
APF = 4 3 p (
a/4 ) 2
atoms
unit cell
atom
volume a

11. 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 & Rethwisch 4e.
Click once on image to start animation
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
(Courtesy P.M. Anderson)

12. 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
Adapted from
Fig. 3.1(a),
Callister & Rethwisch 4e.
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a

13. 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

14. Hexagonal Close-Packed Structure (HCP)
• 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 & Rethwisch 4e.
• Coordination # = 12
6 atoms/unit cell
• APF = 0.74
ex: Cd, Mg, Ti, Zn
• c/a = 1.633

15. VMSE Screenshot – Stacking Sequence and Unit Cell for HCP

16. 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

17. Theoretical Density, r  = a R Ex: Cr (BCC) A = 52.00 g/mol
R = nm
n = 2 atoms/unit cell
a = 4R/ 3 = nm
Adapted from
Fig. 3.2(a), Callister & Rethwisch 4e.
 = a 3
52.00 2
atoms
unit cell
mol g
volume
6.022 x 1023
theoretical
= 7.18 g/cm3
ractual
= 7.19 g/cm3

18. Atomic Bonding in Ceramics
-- Can be ionic and/or covalent in character.
-- % ionic character increases with difference in
electronegativity of atoms.
• Degree of ionic character may be large or small:
CaF2: large
SiC: small
Adapted from Fig. 2.7, Callister & Rethwisch 4e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition. Copyright 1939 and 1940, 3rd edition copyright © 1960 by Cornell University.

19. Ceramic Crystal Structures
Oxide structures
oxygen anions larger than metal cations
close packed oxygen in a lattice (usually FCC)
cations fit into interstitial sites among oxygen ions

20. Factors that Determine Crystal Structure
1. Relative sizes of ions – Formation of stable structures:
--maximize the # of oppositely charged ion neighbors. - +
unstable - + - +
Adapted from Fig. 3.4, Callister & Rethwisch 4e.
stable
stable
2. Maintenance of
Charge Neutrality :
--Net charge in ceramic
should be zero.
--Reflected in chemical
formula:
CaF 2 : Ca 2+
cation F -
anions + A m X p
m, p values to achieve charge neutrality

21. Coordination Number and Ionic Radii
cation
anion
• Coordination Number increases with
To form a stable structure, how many anions can
surround around a cation?
Adapted from Fig. 3.5, Callister & Rethwisch 4e.
Adapted from Fig. 3.6, Callister & Rethwisch 4e.
Adapted from Fig. 3.7, Callister & Rethwisch 4e.
ZnS
(zinc blende)
NaCl
(sodium
chloride)
CsCl
(cesium r
cation
anion
Coord.
Number
< 0.155 2
linear 3
triangular 4
tetrahedral 6
octahedral 8
cubic
Adapted from Table 3.3, Callister & Rethwisch 4e.

22. Computation of Minimum Cation-Anion Radius Ratio
Determine minimum rcation/ranion for an octahedral site (C.N. = 6)
a = 2ranion

23. Bond Hybridization
Bond Hybridization is possible when there is significant covalent bonding
hybrid electron orbitals form
For example for SiC
XSi = 1.8 and XC = 2.5
~ 89% covalent bonding
Both Si and C prefer sp3 hybridization
Therefore, for SiC, Si atoms occupy tetrahedral sites

24. Example Problem: Predicting the Crystal Structure of FeO
• On the basis of ionic radii, what crystal structure
would you predict for FeO?
Cation
Anion Al 3+ Fe 2 + Ca 2+ O 2- Cl - F
Ionic radius (nm)
0.053
0.077
0.069
0.100
0.140
0.181
0.133
• Answer:
based on this ratio,
-- coord # = 6 because
0.414 < < 0.732
-- crystal structure is NaCl
Data from Table 3.4, Callister & Rethwisch 4e.

25. Rock Salt Structure
Same concepts can be applied to ionic solids in general.
Example: NaCl (rock salt) structure
rNa = nm
rCl = nm
rNa/rCl = 0.564
cations (Na+) prefer octahedral sites
Adapted from Fig. 3.5, Callister & Rethwisch 4e.

26. MgO and FeO MgO and FeO also have the NaCl structure
O2- rO = nm
Mg2+ rMg = nm
rMg/rO = 0.514
cations prefer octahedral sites
Adapted from Fig. 3.5, Callister & Rethwisch 4e.
So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms

27. AX Crystal Structures
AX–Type Crystal Structures include NaCl, CsCl, and zinc blende
Cesium Chloride structure:
 Since < < 1.0, cubic sites preferred
So each Cs+ has 8 neighbor Cl-
Adapted from Fig. 3.6, Callister & Rethwisch 4e.

28. AX2 Crystal Structures Fluorite structure Calcium Fluorite (CaF2)
Cations in cubic sites
UO2, ThO2, ZrO2, CeO2
Antifluorite structure –
positions of cations and anions reversed
Adapted from Fig. 3.8, Callister & Rethwisch 4e.

29. ABX3 Crystal Structures
Perovskite structure
Ex: complex oxide
BaTiO3
Adapted from Fig. 3.9, Callister & Rethwisch 4e.

30. VMSE Screenshot – Zinc Blende Unit Cell

31. Density Computations for Ceramics
Number of formula units/unit cell
Avogadro’s number
Volume of unit cell
= sum of atomic weights of all cations in formula unit
= sum of atomic weights of all anions in formula unit

32. 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). 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 H
DPE, PS
PP, LDPE PC
PTFE
PET
PVC
Silicone
Polymers have...
• low packing density
(often amorphous)
• lighter elements (C,H,O) r 2 1
Composites have...
• intermediate values
0.5
0.4
0.3
Data from Table B.1, Callister & Rethwisch, 4e.

33. Silicate Ceramics Most common elements on earth are Si & O
SiO2 (silica) polymorphic forms are quartz, crystobalite, & tridymite
The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material
Si4+
O2-
Adapted from Figs , Callister & Rethwisch 4e
crystobalite

34. Silicates
Bonding of adjacent SiO44- accomplished by the sharing of common corners, edges, or faces
Adapted from Fig. 3.12, Callister & Rethwisch 4e.
Mg2SiO4
Ca2MgSi2O7
Presence of cations such as Ca2+, Mg2+, & Al3+
1. maintain charge neutrality, and
2. ionically bond SiO44- to one another

35. Glass Structure • Basic Unit: Glass is noncrystalline (amorphous)
• Fused silica is SiO2 to which no impurities have been added
• Other common glasses contain impurity ions such as Na+, Ca2+, Al3+, and B3+
Si0 4
tetrahedron 4- Si 4+ O 2 -
• Quartz is crystalline
SiO2: Si 4+ Na + O 2 -
(soda glass)
Adapted from Fig. 3.42, Callister & Rethwisch 4e.

36. Layered Silicates Layered silicates (e.g., clays, mica, talc)
SiO4 tetrahedra connected together to form 2-D plane
A net negative charge is associated with each (Si2O5)2- unit
Negative charge balanced by adjacent plane rich in positively charged cations
Adapted from Fig. 3.13, Callister & Rethwisch 4e.

37. Layered Silicates (cont)
Kaolinite clay alternates (Si2O5)2- layer with Al2(OH)42+ layer
Adapted from Fig. 3.14, Callister & Rethwisch 4e.
Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces.

38. Polymorphic Forms of Carbon
Diamond
tetrahedral bonding of carbon
hardest material known
very high thermal conductivity
large single crystals – gem stones
small crystals – used to grind/cut other materials
diamond thin films
hard surface coatings – used for cutting tools, medical devices, etc.
Adapted from Fig. 3.16, Callister & Rethwisch 4e.

39. Polymorphic Forms of Carbon (cont)
Graphite
layered structure – parallel hexagonal arrays of carbon atoms
weak van der Waal’s forces between layers
planes slide easily over one another -- good lubricant
Adapted from Fig. 3.17, Callister & Rethwisch 4e.

40. Polymorphic Forms of Carbon (cont) Fullerenes and Nanotubes
Fullerenes – spherical cluster of 60 carbon atoms, C60
Like a soccer ball
Carbon nanotubes – sheet of graphite rolled into a tube
Ends capped with fullerene hemispheres
Adapted from Figs & 3.19, Callister & Rethwisch 4e.

41. Crystals as Building Blocks
• Some engineering applications require single crystals:
-- diamond single
crystals for abrasives
-- turbine blades
Fig. 9.40(c), Callister & Rethwisch 4e. (Fig. 9.40(c) courtesy of Pratt and Whitney).
(Courtesy Martin Deakins,
GE Superabrasives, Worthington, OH. Used with permission.)
• Properties of crystalline materials
often related to crystal structure.
-- Ex: Quartz fractures more easily along some crystal planes than others.
(Courtesy P.M. Anderson)

42. Polycrystals Anisotropic
• Most engineering materials are polycrystals.
Adapted from Fig. K, color inset pages of Callister 5e.
(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)
1 mm
Isotropic
• Nb-Hf-W plate with an electron beam weld.
• 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).

43. Single vs Polycrystals
E (diagonal) = 273 GPa
E (edge) = 125 GPa
• Single Crystals
-Properties vary with
direction: anisotropic.
Data from Table 3.7, Callister & Rethwisch 4e. (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
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
200 mm
Adapted from Fig. 5.19(b), Callister & Rethwisch 4e.
(Fig. 5.19(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].)

44. 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

45. Crystal Systems
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
Fig. 3.20, Callister & Rethwisch 4e.
7 crystal systems
14 crystal lattices
a, b, and c are the lattice constants

46. Point Coordinates z Point coordinates for unit cell center are cx 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

47. 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>

48. VMSE Screenshot – [101] Direction

49. 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 - =
Adapted from
Fig. 3.1(a),
Callister & Rethwisch 4e.

50. Drawing HCP Crystallographic Directions (i)
Algorithm (Miller-Bravais coordinates)
1. Remove brackets 2. Divide by largest integer so all values are ≤ 1
3. Multiply terms by appropriate unit cell dimension a (for a1, a2, and a3 axes) or c (for z-axis) to produce projections 4. Construct vector by stepping off these projections
Adapted from Figure 3.25, Callister & Rethwisch 4e.

51. Drawing HCP Crystallographic Directions (ii)
Draw the direction in a hexagonal unit cell.
Algorithm
a a a z
Adapted from p. 62, Callister & Rethwisch 8e.
1. Remove brackets
2. Divide by 3
[1213] s
3. Projections
4. Construct Vector
start at point o
proceed –a/3 units along a1 axis to point p
–2a/3 units parallel to a2 axis to point q p q
a/3 units parallel to a3 axis to point r r
c units parallel to z axis to point s
[1213] direction represented by vector from point o to point s

52. Determination of HCP Crystallographic Directions (ii)
Algorithm
1. Vector repositioned (if necessary) to pass
through origin. 2. Read off projections in terms of three axis (a1, a2, and z) unit cell dimensions a and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas, for three-axis coordinates
5. Convert to four-axis Miller-Bravais lattice coordinates using equations below:
6. Adjust to smallest integer values and enclose in brackets [uvtw]
Adapted from p. 74, Callister & Rethwisch 4e.

53. Determination of HCP Crystallographic Directions (ii)
Adapted from p. 74, Callister & Rethwisch 4e.
Determine indices for green vector
Example
a a z
Reposition not needed
Projections
a a c
Reduction
Brackets [110]
Convert to 4-axis parameters
Reduction & Brackets
1/3, 1/3, -2/3, => 1, 1, -2, 0 =>
[ 1120 ]

54. Crystallographic Planes
Adapted from Fig. 3.26, Callister & Rethwisch 4e.

55. 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)

56. 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)

57. 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),

58. VMSE Screenshot – Crystallographic Planes
Additional practice on indexing crystallographic planes

59. 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. 3.24(b), Callister & Rethwisch 4e.

60. 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.

61. 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 & Rethwisch 4e. =
Planar Density = a 2 1
atoms
2D repeat unit
nm2
12.1 m2
= 1.2 x 1019 R 3 4
area

62. 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

63. VMSE Screenshot – Atomic Packing – (111) Plane for BCC

64. X-Ray Diffraction
Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
Can’t resolve spacings  
Spacing is the distance between parallel planes of atoms.  

65. X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
Adapted from Fig. 3.38, Callister & Rethwisch 4e.
reflections must
be in phase for
a detectable signal
spacing
between
planes d
incoming
X-rays
outgoing X-rays
detector q l
extra
distance
travelled
by wave “2”
“1”
“2”
X-ray
intensity
(from
detector) q c
d =
n l
2 sin
Measurement of critical angle, qc, allows computation of planar spacing, d.

66. 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)
Adapted from Fig. 3.40, Callister 4e.

67. 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).
• Interatomic bonding in ceramics is ionic and/or covalent.
• Ceramic crystal structures are based on:
-- maintaining charge neutrality
-- cation-anion radii ratios.
• 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.

68. 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.

69. ANNOUNCEMENTS
Reading:
Core Problems:
Self-help Problems: