Why do we care about crystal structures, directions, planes ?

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Why do we care about crystal structures,
directions, planes ?
Physical properties of materials depend on the
geometry of crystals
ISSUES TO ADDRESS...
How do atoms assemble into solid structures?
(for now, 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?
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The Big Picture
Electronic Structure
Bonding
State of aggregation

• Primary
• Ionic
• Covalent
• Metallic

Bohr atom Bohr-Sommerfeld Quantum numbers Aufbau
principle Multielectron atoms Periodic table
patterns Octet stability
Gas Liquid Solid

• Classification of Solids
• Bonding type
• Atomic arrangement

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Atomic Arrangement
SOLID Smth. which is dimensionally stable, i.e.,
has a volume of its own classifications of
solids by atomic arrangement ordered
disordered atomic arrangement regular
random order long-range
short-range name crystalline
amorphous crystal glass
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Energy and Packing
Non dense, random packing
COOLING
Dense, ordered packed structures tend to have
lower energies.
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MATERIALS AND PACKING
Crystalline materials...
atoms pack in periodic, 3D arrays typical
of
-metals -many ceramics -some polymers
crystalline SiO2
Adapted from Fig. 3.18(a), Callister 6e.
LONG RANGE ORDER
Noncrystalline materials...
atoms have no periodic packing occurs for
-complex structures -rapid cooling
noncrystalline SiO2
"Amorphous" Noncrystalline
Adapted from Fig. 3.18(b), Callister 6e.
SHORT RANGE ORDER
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 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
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Robert Hooke 1660 - Cannonballs
Crystal must owe its regular shape to the
packing of spherical particles
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Niels Steensen 1670
observed that quartz crystals had the same angles
between corresponding faces regardless of their
size.
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SIMPLE QUESTION
If I see something has a macroscopic shape very
regular and cubic, can I infer from that if I
divide, divide, divide, divide, divide, if I get
down to atomic dimensions, will there be some
cubic repeat unit?
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Christian Huygens - 1690
Studying calcite crystals made drawings of atomic
packing and bulk shape.
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BERYL Be3Al2(SiO3)6
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Early Crystallography

• René-Just Haüy (1781) cleavage of calcite
• Common shape to all shards rhombohedral
• How to model this mathematically?
• What is the maximum number of distinguishable
  shapes that will fill three space?
• Mathematically proved that there are only 7
  distinct space-filling volume elements

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The Seven Crystal Systems
BASIC UNIT
Specification of unit cell parameters
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Does it work with Pentagon?
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August Bravais

• How many different ways can I put atoms into
  these seven crystal systems, and get
  distinguishable point environments?

When I start putting atoms in the cube, I have
three distinguishable arrangements.
SC
BCC
FCC
And, he proved mathematically that there are 14
distinct ways to arrange points in space.
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Last Day Atomic Arrangement
SOLID Smth. which is dimensionally stable, i.e.,
has a volume of its own classifications of
solids by atomic arrangement ordered
disordered atomic arrangement regular
random order long-range
short-range name crystalline
amorphous crystal glass
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MATERIALS AND PACKING
Crystalline materials...
atoms pack in periodic, 3D arrays typical
of
-metals -many ceramics -some polymers
crystalline SiO2
Adapted from Fig. 3.18(a), Callister 6e.
LONG RANGE ORDER
Noncrystalline materials...
atoms have no periodic packing occurs for
-complex structures -rapid cooling
noncrystalline SiO2
"Amorphous" Noncrystalline
Adapted from Fig. 3.18(b), Callister 6e.
SHORT RANGE ORDER
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Energy and Packing
Non dense, random packing
COOLING
Dense, ordered packed structures tend to have
lower energies.
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Three Types of Solids according to atomic
arrangement
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Unit Cell Concept

• The unit cell is the smallest structural unit or
  building block that uniquely can describe the
  crystal structure. Repetition of the unit cell
  generates the entire crystal. By simple
  translation, it defines a lattice .
• Lattice The periodic arrangement of atoms in a
  Xtal.

Lattice Parameter Repeat distance in the unit
cell, one for in each dimension
a
b
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Crystal Systems

• Units cells and lattices in 3-D
• When translated in each lattice parameter
  direction, MUST fill 3-D space such that no gaps,
  empty spaces left.

c
b
Lattice Parameter Repeat distance in the unit
cell, one for in each dimension
a
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The Importance of the Unit Cell

• One can analyze the Xtal as a whole by
  investigating a representative volume.
• Ex from unit cell we can
• Find the distances between nearest atoms for
  calculations of the forces holding the lattice
  together
• Look at the fraction of the unit cell volume
  filled by atoms and relate the density of solid
  to the atomic arrangement
• The properties of the periodic Xtal lattice
  determine the allowed energies of electrons that
  participate in the conduction process.

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 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
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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
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SIMPLE CUBIC STRUCTURE (SC)
Rare due to poor packing Close-packed
directions are cube edges.
Closed packed direction is where the atoms touch
each other
Coordination 6 ( nearest neighbors)
(Courtesy P.M. Anderson)
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ATOMIC PACKING FACTOR
APF for a simple cubic structure 0.52
Adapted from Fig. 3.19, Callister 6e.
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BODY CENTERED CUBIC STRUCTURE (BCC)
Close packed directions are 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
2 atoms/unit cell 1 center 8 corners x 1/8
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(Courtesy P.M. Anderson)
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ATOMIC PACKING FACTOR BCC
APF for a body-centered cubic structure 0.68
a
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FACE CENTERED CUBIC STRUCTURE (FCC)
Close packed directions are 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 7e.
4 atoms/unit cell 6 face x 1/2 8 corners x 1/8
(Courtesy P.M. Anderson)
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ATOMIC PACKING FACTOR FCC
? HW
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HW

• 1. Finish reading Chapter 3.
• 2.On a paper solve example problems3.1, 3.2,
  3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9
• 3. Fill in the blanks in Table below

BCC
FCC
SC
Unit Cell Volume a3
Lattice Points per cell 1
Nearest Neighbor Distance a
Number of Nearest Neighbors 6
Atomic Packing Factor 0.52
Fill here!
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THEORETICAL DENSITY, r
Example Copper
Data from Table inside front cover of Callister
(see next slide)
crystal structure FCC 4 atoms/unit cell
atomic weight 63.55 g/mol (1 amu 1 g/mol)
atomic radius R 0.128 nm (1 nm 10 cm)
-7
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Theoretical Density, r

• Ex Cr (BCC)
• A 52.00 g/mol
• R 0.125 nm
• n 2

52.00
2
?theoretical
7.18 g/cm3
?
ractual
7.19 g/cm3
3
a
6.023 x 1023
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Characteristics of Selected Elements at 20C
Adapted from Table, "Charac- teristics
of Selected Elements", inside front cover, Callist
er 6e.
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DENSITIES OF MATERIAL CLASSES
Why? Metals have... close-packing
(metallic bonding) large atomic mass
Ceramics have... less dense packing
(covalent bonding) often lighter elements
Polymers have... poor packing
(often amorphous) lighter elements (C,H,O)
Composites have... intermediate values
Data from Table B1, Callister 6e.
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POLYMORPHISM ALLOTROPY

• Some materials may exist in more than one crystal
  structure, this is called polymorphism.
• If the material is an elemental solid, it is
  called allotropy. An example of allotropy is
  carbon, which can exist as diamond, graphite, and
  amorphous carbon.

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Polymorphism

• Two or more distinct crystal structures for the
  same material (allotropy/polymorphism)  
  titanium
•   ?, ?-Ti
• carbon
• diamond, graphite

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Crystallographic Points, Directions, and Planes

• It is necessary to specify a particular
  point/location/atom/direction/plane in a unit
  cell
• We need some labeling convention. Simplest way is
  to use a 3-D system, where every location can be
  expressed using three numbers or indices.
• a, b, c and a, ß, ?

z
a
ß
y
?
x
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Crystallographic Points, Directions, and Planes

• Crystallographic direction is a vector uvw
• Always passes thru origin 000
• Measured in terms of unit cell dimensions a, b,
  and c
• Smallest integer values
• Planes with Miller Indices (hkl)
• If plane passes thru origin, translate
• Length of each planar intercept in terms of the
  lattice parameters a, b, and c.
• Reciprocals are taken
• If needed multiply by a common factor for integer
  representation

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Section 3.8 Point Coordinates

• Point coordinates for unit cell center are
• a/2, b/2, c/2 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
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Crystallographic Directions
Algorithm
z
1. Vector repositioned (if necessary) to pass
through origin.2. Read off projections in
terms of unit cell dimensions a, b, and
c3. Adjust to smallest integer values4. Enclose
in square brackets, no commas uvw
y
x
ex 1, 0, ½
gt 2, 0, 1
gt 201
-1, 1, 1
families of directions ltuvwgt
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Linear Density

• Linear Density of Atoms ? LD  

Number of atoms
Unit length of direction vector
ex linear density of Al in 110
direction  a 0.405 nm
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Crystallographic Planes
Adapted from Fig. 3.9, Callister 7e.
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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 
• If plane passes thru origin, translate
• Read off intercepts of plane with axes in terms
  of a, b, c
• Take reciprocals of intercepts
• Reduce to smallest integer values
• Enclose in parentheses, no commas i.e., (hkl)

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Crystallographic Planes
4. Miller Indices (110)
4. Miller Indices (100)
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Crystallographic Planes
example
a b c
4. Miller Indices (634)
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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.

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Planar Density of (100) Iron

• Solution  At T lt 912?C iron has the BCC
  structure.

2D repeat unit
(100)
Radius of iron R 0.1241 nm
Adapted from Fig. 3.2(c), Callister 7e.
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Planar Density of (111) Iron

• ?
• HW

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Single Crystals and Polycrystalline Materials

• In a single crystal material the periodic and
  repeated arrangement of atoms is PERFECT This
  extends throughout the entirety of the specimen
  without interruption.

• Polycrystalline material, on the other hand, is
  comprised of many
• small crystals or grains. The grains have
  different crystallographic orientation. There
  exist atomic mismatch within
• the regions where grains meet.
• These regions are called
• grain boundaries.

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Example of Polycrystalline Growth
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CRYSTALS AS BUILDING BLOCKS
Some engineering applications require single
crystals
--turbine blades
--diamond single crystals for abrasives
Fig. 8.30(c), Callister 6e. (Fig. 8.30(c)
courtesy of Pratt and Whitney).
(Courtesy Martin Deakins, GE Superabrasives,
Worthington, OH. Used with permission.)
Crystal properties reveal features of
atomic structure.
--Ex Certain crystal planes in quartz
fracture more easily than others.
(Courtesy P.M. Anderson)
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POLYCRYSTALS
Most engineering materials are polycrystals.
Adapted from Fig. K, color inset pages of
Callister 6e. (Fig. K is courtesy of Paul E.
Danielson, Teledyne Wah Chang Albany)
1 mm
Nb-Hf-W plate with an electron beam weld.
Each "grain" is a single crystal. If crystals
are randomly oriented, overall component
properties are not directional. Crystal sizes
typ. range from 1 nm to 2 cm (i.e., from a
few to millions of atomic layers).
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SINGLE VS POLYCRYSTALS
Single Crystals
Data from Table 3.3, Callister 6e. (Source of
data is R.W. Hertzberg, Deformation and Fracture
Mechanics of Engineering Materials, 3rd ed., John
Wiley and Sons, 1989.)
-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. 4.12(b), Callister 6e. (Fig.
4.12(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.)
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Anisotropy and Texture

• Different directions in a crystal have a
  different APF.
• For example, the deformation amount depends on
  the direction in which a stress is applied, other
  properties are thermal conductivity, optical
  properties, magnetic properties, hardness, etc.
• In some polycrystalline materials, grain
  orientations are random, hence bulk material
  properties are isotropic, i.e. equivalent in each
  direction
• Some polycrystalline materials have grains with
  preferred orientations (texture), so properties
  are dominated by those relevant to the texture
  orientation and the material exhibits anisotropic
  properties.

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X-RAYS TO CONFIRM CRYSTAL STRUCTURE
Incoming X-rays diffract from crystal planes.
Adapted from Fig. 3.2W, Callister 6e.
Measurement of Critical angles, qc,
for X-rays provide atomic spacing, d.
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SUMMARY (I)
Atoms may assemble into crystalline or
amorphous 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).
Material properties generally vary with
single crystal orientation (i.e., they are
anisotropic), but properties are generally
non-directional (i.e., they are isotropic)
in polycrystals with randomly oriented
grains.
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Summary (II)

• Allotropy
• Amorphous
• Anisotropy
• Atomic packing factor (APF)
• Body-centered cubic (BCC)
• Coordination number
• Crystal structure
• Crystalline
• Face-centered cubic (FCC)
• Grain
• Grain boundary
• Hexagonal close-packed (HCP)
• Isotropic
• Lattice parameter
• Non-crystalline
• Polycrystalline
• Polymorphism
• Single crystal
• Unit cell