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. 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).

4. 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

5. 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

6. 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.
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.

7. 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

9. 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

10. 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...

11. 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)

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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.

20. 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

21. 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

22. 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

23. Single crystal
For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption.
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.

24. 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

26. 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.

27. 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).

28. 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.

29. 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

30. 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.

31. 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

32. Crystallographic Directions
Is defined as a line between two points, or a vector. z
Algorithm
1. Vector repositioned (if necessary) to pass
through origin.( parallelism maintained) 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values( multiplied
or divided) 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>

33. For each of the three axes, there will exist both positive and negative coordinates. Thus negative indices are also possible, which are represented by a bar over the appropriate index.
For example, the direction would have a component
in the - y direction. Also, changing the signs of all indices produces an antiparallel direction; that is, is directly opposite to

34. Ex. The [100], [110], and [111]
directions are common ones;
they are drawn in the
unit cell shown in Figure

35. Example

36. Example Draw a direction within a cubic unit cell.
SOLUTION First construct an appropriate unit cell and coordinate axes system. In the accompanying figure the unit cell is cubic, and the origin of the coordinate system, point O, is located at one of the cube corners.
For this direction, the projections along the x, y, z axes are 1a, -1a, and 0a, respectively.
This direction is defined by a vector passing from the origin to point P, which is located by first moving along the x axis a units, and from this position, parallel to the y axis a units, as indicated in the figure. There is no z component to the vector, since the z projection is zero.

38. 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

39. Crystallographic Planes

40. 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)

41. 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)

42. Crystallographic Planesz x y a b c
example
a b c
Intercepts
1/ /4
Reciprocals
1/½ 1/ /¾ /3
Reduction
Miller Indices (634)

43. 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.

44. 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

45. 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

46. 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.  

47. X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
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.
n is the order of reflection
Which may be any integer(1,2,3,…)

48. 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)

49. 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.

50. 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.