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. 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. 3.23(a),
Callister & Rethwisch 8e. Si
Oxygen
Noncrystalline materials...
• atoms have no periodic packing
• occurs for:
-complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2
Adapted from Fig. 3.23(b),
Callister & Rethwisch 8e.

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
• 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*
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
Adapted from Fig. 3.24,
Callister & Rethwisch 8e.
close-packed directions a
R=0.5a
contains 8 x 1/8 = 1
atom/unit cell
atom
volume 1
atoms
unit cell 4 3 p
(0.5a)
APF = a 3
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 8e.
Click once on image to start animation
(Courtesy P.M. Anderson)
2 atoms/unit cell: 1 center + 8 corners x 1/8

9. Atomic Packing Factor: BCC
• APF for a body-centered cubic structure = 0.68 a R a 2 3
length = 4R =
Close-packed directions:
3 a
Adapted from
Fig. 3.2(a), Callister & Rethwisch 8e.
APF = 4 3 p (
a/4 ) 2
atoms
unit cell
atom
volume a

10. 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 8e.
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)

11. 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 8e.
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a

12. 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

13. HCP Stacking Sequence

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 8e.
• Coordination # = 12
6 atoms/unit cell
• APF = 0.74
ex: Cd, Mg, Ti, Zn
• c/a = 1.633

15. 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

16. 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 8e.
 = a3
52.00 2
atoms
unit cell
mol g
volume
6.022 x 1023
theoretical
ractual
= 7.18 g/cm3
= 7.19 g/cm3

17. 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, 8e.

18. f16_03_pg64 SINGLE CRYSTALS f16_03_pg64.jpg
Pyrite cubic crystals on marlstone, from Navajún, Rioja, Spain. Size: 95 mm x 78 mm. Main crystal: 31 mm on edge. Weight: 512 g.
Garnet single crystal from China
Ruby crystals from Tanzania
Pyrite (FeS2) (Fool’s Gold) crystals from Spain

19. Crystals as Building Blocks
• Some engineering applications require single crystals:
-- diamond single
crystals for abrasives
-- turbine blades
Fig. 8.33(c), Callister & Rethwisch 8e. (Fig. 8.33(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)

20. Polycrystals
Most crystalline solids are composed of many small crystals (also called grains).
Initially, small crystals (nuclei) form at various positions.
These have random orientations.
The small grains grow and begin to impinge on one another forming grain boundaries.

21. POLYCRYSATLLINE MATERIALS

22. 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 typically range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).

23. Single vs Polycrystals
Data from Table 3.3, Callister 7e.
(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)
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.
Adapted from Fig. 4.14(b), Callister 7e.
(Fig. 4.14(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].)

25. Crystal Systems
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
Fig. 3.4, Callister & Rethwisch 8e.
7 crystal systems
14 crystal lattices
a, b, and c are the lattice constants

27. (c) 2003 Brooks/Cole Publishing / Thomson Learning™
The fourteen (14) types of Bravais lattices grouped in seven (7) systems.

28. 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

29. 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>

30. 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 8e.

31. Crystallographic Planes
Adapted from Fig. 3.10, Callister & Rethwisch 8e.

32. 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)

33. 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 (200)

34. 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),

35. Planar Density of (100) Iron
Solution:  At T < 912C iron has the BCC structure.
(100)
Radius of iron R = nm R 3 4 a =
2D repeat unit
Adapted from Fig. 3.2(c), Callister 7e. a 2 1
atoms
2D repeat unit
area = 1 2 R 3 4 m2
atoms
= 1.2 x 1019 =
nm2
atoms
12.1
Planar Density =

36. FCC Unit Cell with (110) plane

37. BCC Unit Cell with (110) plane

38. 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.  

40. X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
X-ray
intensity
(from
detector) q c
d =
n l
2 sin
Measurement of critical angle, qc, allows computation of planar spacing, d.

41. X-RAY DIFFRACTOMETER

42. 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. 3.22, Callister 8e.

43. SAMPLE PROBLEM

44. Allotropy or Polymorphism
Two or more distinct crystal structures for the same material (allotropy/polymorphism).
Allotropy (Gr. allos, other, and tropos, manner) or allotropism is a behavior exhibited by certain chemical elements that can exist in two or more different forms, known as allotropes of that element.
In each allotrope, the element's atoms are bonded together in a different manner.
Note that allotropy refers only to different forms of an element within the same phase or state of matter (i.e. different solid, liquid or gas forms) - the changes of state between solid, liquid and gas in themselves are not considered allotropy.

45. 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

46. Polymorphic Forms of Carbon
Diamond and graphite are two allotropes of carbon: pure forms of the same element that differ in structure.
Diamond
An extremely hard, transparent crystal with
tetrahedral bonding of carbon
very high thermal conductivity
very low electric conductivity.
The large single crystals are typically used as gem stones
The small crystals are used to grind/cut other materials
diamond thin films
hard surface coatings – used for cutting tools, medical devices.
Diamond is a good conductor because of the strong covalent bonds between the atoms, it results in a very rigid lattice structure. Also the purer the diamond the better.
In quantum mechanics they think in terms of traveling phonons, the more rigid the lattice structure the faster the phonons travel. Also the lightness of the carbon atom results in a greater speed for the phonons. You can think of the photons as waves of vibration which travel at the "speed of sound" for the material. Also the more perfect the crystal the further the phonons travel before dispersing.
The thermal conductivity of diamond, which is four times that of copper, goes against the trend that most electrical conductors are better conductors of heat than insulators. This is usually the case because the free electrons of metals end up transporting the energy.

47. Polymorphic Forms of Carbon
Graphite
a soft, black, flaky solid, with a layered structure – parallel hexagonal arrays of carbon atoms
weak van der Waal’s forces between layers
planes slide easily over one another

48. Polymorphic Forms of Carbon 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

49. Tin, Allotropic Transformation
Volume increase of 27%
Density decrease from 7.3 to 5.77 g/cm3
Tin Disease:
Another common metal that experiences an allotropic change is tin. White tin (BCT) transforms into gray tin (diamond cubic) at 13.2 ˚C. The transformation rate is slow, though, the lower the temp, the faster the rate.
Due to the volume expansion, the metal disintegrates into the coarse gray powder.
In 1850 Russia, a particularly cold winter crumbled the tin buttons on some soldier uniforms. There were also problems with tin church organ pipes.

50. 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.

51. 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.