CHAPTER 3: Structures of Metals
and Ceramics
ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
• How do the structures of ceramic
materials differ from those of metals?
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation?
1
Structures of Metals & Ceramics
• Just got done talking about atoms
– Next questions
• How do atoms pack/arrange in the solid state?
• How does this influence the properties we will spend all semester
talking about?
• Very important concept for what follows – crystalline
versus non-crystalline (amorphous)
– A crystalline material is one in which the atoms are situated in a
periodic (repeating) array over the distance of many (i.e. several
hundred) atoms in three-dimensional space
• Long-range structural ordering!
• This matters a great deal – why?
ENERGY AND PACKING
• Non dense, random packing
• Dense, regular packing
Dense, regular-packed structures tend to have
lower energy.
2
Structures of Metals & Ceramics
• Crystal structure
– The manner in which atoms, ions, or molecules are spatially
arranged
– Why do you think the crystal structure matters?
• There are many, many different types of crystal structures
possible – we will focus on just a few of them
– These happen to be the most common structures for the
materials we will discuss
• Face-centered cubic (FCC), body-centered cubic (BCC), hexagonal
close-packed (hcp)
METALLIC CRYSTALS
• tend to be densely packed.
• have several 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.
• have the simplest crystal structures.
We will look at three such structures...
3
SIMPLE CUBIC STRUCTURE (SC)
• Rare due to poor packing (only Polonium has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
(Courtesy P.M. Anderson)
4
Structures of Metals & Ceramics
• Unit cell
– So in crystalline solids the atoms form a repetitive pattern
– A useful way to describe such structures is by unit cells
– The unit cell is a three dimensional construct, from which the
structure of the crystalline solid can be generated by replication in
all directions
– In other words, if you know the unit cell contents and size, you’ve
specified the crystal structure!
ATOMIC PACKING FACTOR
• APF for a simple cubic structure = 0.52
Adapted from Fig. 3.19,
Callister 6e.
5
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.
• Coordination # = 12
Adapted from Fig. 3.1(a),
Callister 6e.
(Courtesy P.M. Anderson)
6
ATOMIC PACKING FACTOR: FCC
• APF for a body-centered cubic structure = 0.74
a
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
Adapted from
Fig. 3.1(a),
Callister 6e.
7
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.
• Coordination # = 8
Adapted from Fig. 3.2,
Callister 6e.
(Courtesy P.M. Anderson)
8
ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
R
Adapted from
Fig. 3.2,
Unit cell contains:
1 + 8 x 1/8
= 2 atoms/unit cell
a
Callister 6e.
9
HEXAGONAL CLOSE-PACKED
STRUCTURE (HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection
A sites
B sites
A sites
Adapted from Fig. 3.3,
Callister 6e.
• Coordination # = 12
• APF = 0.74
10
FCC STACKING SEQUENCE
• ABCABC... Stacking Sequence
• 2D Projection
A
B
B
C
A
B
B
B
A sites
C
C
B sites
B
B
C sites
• FCC Unit Cell
20
THEORETICAL DENSITY, 
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 = 10-7 cm)
Result: theoretical Cu = 8.89 g/cm3
Compare to actual: Cu = 8.94 g/cm3
11
Characteristics of Selected Elements at 20C
At. Weight
Element
Symbol (amu)
Aluminum Al
26.98
Argon
Ar
39.95
Barium
Ba
137.33
Beryllium
Be
9.012
Boron
B
10.81
Bromine
Br
79.90
Cadmium
Cd
112.41
Calcium
Ca
40.08
Carbon
C
12.011
Cesium
Cs
132.91
Chlorine
Cl
35.45
Chromium Cr
52.00
Cobalt
Co
58.93
Copper
Cu
63.55
Flourine
F
19.00
Gallium
Ga
69.72
Germanium Ge
72.59
Gold
Au
196.97
Helium
He
4.003
Hydrogen
H
1.008
Density
(g/cm3)
2.71
-----3.5
1.85
2.34
-----8.65
1.55
2.25
1.87
-----7.19
8.9
8.94
-----5.90
5.32
19.32
-----------
Atomic radius
(nm)
0.143
-----0.217
0.114
Adapted from
-----Table, "Charac-----teristics of
0.149 Selected
0.197 Elements",
inside front
0.071 cover,
0.265 Callister 6e.
-----0.125
0.125
0.128
-----0.122
0.122
0.144
----------12
DENSITIES OF MATERIAL CLASSES
metals • ceramics• polymers
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.
13
CERAMIC BONDING
• Bonding:
--Mostly ionic, some covalent.
--% ionic character increases with difference in
electronegativity.
• Large vs small ionic bond character:
Adapted from Fig. 2.7, Callister 6e. (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.
14
IONIC BONDING & STRUCTURE
• Charge Neutrality:
--Net charge in the
structure should
be zero.
--General form:
• Stable structures:
--maximize the # of nearest oppositely charged neighbors.
Adapted from Fig. 12.1,
Callister 6e.
15
COORDINATION # AND IONIC RADII
• Coordination # increases with
Issue: How many anions can you
arrange around a cation?
Adapted from Fig. 12.4,
Callister 6e.
Adapted from Fig. 12.2,
Callister 6e.
Adapted from Table
12.2, Callister 6e.
Adapted from Fig. 12.3,
Callister 6e.
16
EX: PREDICTING STRUCTURE OF FeO
• On the basis of ionic radii, what crystal structure
would you predict for FeO?
Cation
Al3+
Fe 2+
Fe 3+
Ca2+
Anion
O2ClF-
• Answer:
r cation 0.077

r anion
0.140
 0.550
based on this ratio,
--coord # = 6
--structure = NaCl
Data from Table 12.3,
Callister 6e.
17
AmXp STRUCTURES
r cation 0.100

 0.8
• Consider CaF2 :
r anion 0.133
• Based on this ratio, coord # = 8 and structure = CsCl.
• Result: CsCl structure w/only half the cation sites
occupied.
• Only half the cation sites
are occupied since
#Ca2+ ions = 1/2 # F- ions.
Adapted from Fig. 12.5,
Callister 6e.
18
DEMO: HEATING AND
COOLING OF AN IRON WIRE
• Demonstrates "polymorphism"
The same atoms can
have more than one
crystal structure.
19
STRUCTURE OF COMPOUNDS: NaCl
• Compounds: Often have similar close-packed structures.
• Structure of NaCl
• Close-packed directions
--along cube edges.
(Courtesy P.M. Anderson)
(Courtesy P.M. Anderson)
21
CRYSTALS AS BUILDING
BLOCKS
• Some engineering applications require single crystals:
--diamond single
crystals for abrasives
(Courtesy Martin Deakins,
GE Superabrasives,
Worthington, OH. Used
with permission.)
--turbine blades
Fig. 8.30(c), Callister 6e.
(Fig. 8.30(c) courtesy
of Pratt and Whitney).
• Crystal properties reveal features
of atomic structure.
--Ex: Certain crystal planes in quartz
fracture more easily than others.
(Courtesy P.M. Anderson)
22
POLYCRYSTALS
• Most engineering materials are polycrystals.
1 mm
Adapted from Fig. K,
color inset pages of
Callister 6e.
(Fig. K is courtesy of
Paul E. Danielson,
Teledyne Wah Chang
Albany)
• 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).
23
SINGLE VS POLYCRYSTALS
• Single Crystals
Data from Table 3.3,
Callister 6e.
(Source of data is
R.W. Hertzberg,
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
Deformation and
Fracture Mechanics of
Engineering Materials,
3rd ed., John Wiley
and Sons, 1989.)
• 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.
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].)
24
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.
25
Structures of Metals & Ceramics
• One more time …
– A crystalline material is one in which the atoms are situated in a
periodic (repeating) array over the distance of many (i.e. several
hundred) atoms in three-dimensional space
• Long-range structural ordering!
• This matters a great deal – why?
– Non-crystalline or amorphous materials – do not have long-range
ordering of atoms
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.
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2
Adapted from Fig. 3.18(b),
Callister 6e.
26
GLASS STRUCTURE
• Basic Unit:
4Si04 tetrahedron
Si4+
O2-
• Glass is amorphous
• Amorphous structure
occurs by adding impurities
(Na+,Mg2+,Ca2+, Al3+)
• Impurities:
interfere with formation of
crystalline structure.
• Quartz is crystalline
SiO2:
(soda glass)
Adapted from Fig. 12.11,
Callister, 6e.
28
SUMMARY
• 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.
27
Structures of Metals & Ceramics
• More on unit cells
– Can’t just choose any old unit cell!
– The unit cell is a result of the symmetry of the crystal structure
(e.g cubic), wherein all atomic positions can be generated by
translations of the unit cell integral differences along each of its
edges
– The unit cell is the “building block” of the crystal structure!
Structures of Metals & Ceramics
• Metallic Crystal Structures
– Atomic bonding in these materials is metallic – hence nondirectional
– No restrictions as to number of nearest neighbor atoms – can
have very high coordination numbers/very dense atomic packing
– “Hard sphere model” of atoms – each sphere represents an ion
– Three most common metal structures – face-centered cubic,
body-centered cubic, hexagonal close-packed
Structures of Metals & Ceramics
• Metallic Crystal Structures – face
centered cubic (FCC)
– Picture: imagine a cube
• Atoms sit at each corner of the cube (8)
• Atoms also sit in each of the faces of the
cube (6)
• Some metals that possess this structure –
copper, aluminum, gold
• How big is the cell – depends on the size
of the atoms!
– The cube edge length (i.e. unit cell
parameter a) is related to the atom/ion
radius by simple geometry
aFCC  2R 2
Structures of Metals & Ceramics
• Metallic Crystal Structures – face centered cubic (FCC)
– How many atoms are in a FCC unit cell?
• Careful!!
• There are 8 atoms on the cube vertices; but these are shared by
eight unit cells – so 8 x 1/8 = 1
• There are 6 atoms on the cube faces; but these are shared by two
unit cells – so 6 x ½ = 3
• 1 + 3 = 4 atoms / unit cell!
– Corners and face atoms are equivalent! If you translate the cube
corner from the original position to a face atom, you have the
same structure (good HW problem!)
Structures of Metals & Ceramics
• Metallic Crystal Structures – face centered cubic (FCC)
– Two last concepts … coordination number, and atomic packing
factor (APF)
• Coordination number – shouldn’t be new. The number of nearest
neighbor atoms – in the FCC structure this is 12
• Atomic packing factor – fraction of solid sphere volume in a unit cell,
assuming the atomic hard sphere model
volume of atoms in a unit cell
APF 
total unit cell volume
For a FCC structure
4 3
4 3
4 R  4 R 
3
3
  0.74
APF   3   
3
a
2R 2


*Note: this does not
depend on R!
Structures of Metals & Ceramics
• Metallic Crystal Structures – body-centered cubic (BCC)
– Picture: imagine a cube
•
•
•
•
Atoms sit at each corner of the cube (8)
Now, only other atom – in the middle of the cell
Some metals that possess this structure – chromium, iron, tungsten
Again, the unit cell size depends on the size of the atoms!
– The cube edge length (i.e. unit cell parameter a) is related to the
atom/ion radius by simple geometry
aBCC
4R

3
Structures of Metals & Ceramics
• Metallic Crystal Structures – body-centered cubic (BCC)
– In contrast to FCC, each BCC unit cell has 2 atoms associated
with it
– What does this mean?
– Lower packing densities
• Coordination number is 8
• APF is now:
4 3
2 R 
3

  0.68
APF 
3
 4R 


 3
Structures of Metals & Ceramics
• Metallic Crystal Structures – hexagonal close-packed
(hcp)
– So everything has cubic symmetry – NO!
– Many metals do, but not even all metals
– Another frequently observed structure – hexagonal close-packed
Structures of Metals & Ceramics
• Metallic Crystal Structures – hexagonal close-packed
(hcp)
– Students don’t like this as much – more complex
– Key point – there is a 6-fold symmetry axis (i.e. if you take the
structure below, and rotate it about G-C by 60o, you get the same
structure!
Structures of Metals & Ceramics
• Metallic Crystal Structures – hexagonal close-packed
(hcp)
– More details
• The top and bottom faces of the cell have six atoms that form regular
hexagons, surrounding a single atom in the center of the plane
• The layers between the top and bottom faces contains three atoms
• ALSO NOTE: this structure has two unit cell constants, a and c
– In an ideal hcp structure c/a = 1.633; this is not always observed
• Coordination number, atomic packing factor for hcp structure is the
same as fcc – 12 and 0.74
• HCP materials – cadmium, magnesium, titanium, zinc
Structures of Metals & Ceramics
• What can I do when I know the structure
– Coordination #s, APF values – those are important!
– Estimate density
nA

Vc N A
n – number of atoms associated with one unit cell
A – atomic weight
VC – volume of the unit cell
NA – Avagadro’s number (6.023 x 1023 atoms/mol)
Structures of Metals & Ceramics
• Ceramic crystal structures
– These are more complex!
– Why?
• Typically ceramics are composed of at least two elements (e.g. SiO2)
• The bonding character of ceramics is between “pure ionic” and “pure
covalent”
• If the bonding has an appreciable ionic characteristic (how would you
estimate this?), can think of the structure as being comprised of
charged ions (anions and cations) versus atoms
– If you can think of the ceramic as somewhat ionic, then there are
two key points
• Magnitude of the electric charge – i.e. you must satisfy charge
neutrality  CaF2
• Relative size of the anions and cations
Structures of Metals & Ceramics
• Ceramic crystal structures
– Charge matching is not surprising
– Relative size of the anions and cations is critical in determining
the crystal structure formed
• Does this make sense? Why?
• Determines how the atoms can pack together!
– Before going into the details, a few points
• Since cations (metallic element) give up electrons to the anion,
cations are typically smaller than anions (rC < rA)
• Want to maximize nearest-neighbor contacts (coordination number)
• Form stable ceramics when the anions surrounding a cation are all in
contact with that cation (see Figure 3.4)
• rC/rA determines coordination number – hence strongly influences the
crystal structure!
Structures of Metals & Ceramics
• Ceramic crystal structures
– Relationships between rC/rA and the
coordination number
– Do the results in the table make
sense physically?
– Basis – as cation gets bigger it can
have more direct contacts (why?)
Structures of Metals & Ceramics
• Ceramic crystal structures
– So does it work?
– Take SiO2 – rC/rA ~ 0.286
• Coordination # is 4 – that’s good –
silicon is tetrahedral in SiO2
Structures of Metals & Ceramics
• Ceramic crystal structures
– These are more complicated – why? Multiple types of atoms
– Before getting into details – a few observations
– I will start with the simplest structures
• Binary mixture of atoms, 1:1
• Binary, 2:1
• Ternary, etc …
Structures of Metals & Ceramics
• AX-Type Crystal Structure (often referred as the sodium
chloride, or rock salt structure)
– Coordination number (CN) of anions & cations – 6
– What is the ratio of the radii of the cations and anions?
– Structure – the anions are FCC, cation at the cube center, also at
the center of each cube edge (2 FCC lattices, one of cations,
one of anions)
Structures of Metals & Ceramics
• Cesium Chloride
–
–
–
–
Coordination number – 8
Anions on cube (unit cell) edges
Cation in the middle of the cell
THIS IS NOT BCC – why?
Structures of Metals & Ceramics
• Zinc Blende Structure – these are getting complicated!
– AX structure
– Coordination number of atoms is 4
– Why called zinc blende? Because Zinc Sulfide (ZnS) is one of
the best known compounds with this structure. The bonding
here is highly covalent
• Can you explain/contrast that to the other two previous structures?
Structures of Metals & Ceramics
• AmXp – Type Crystal Structure
– General class when the charges on the cations/anions are not
the same (i.e. m, p ≠ 1)
– Prototypical example is fluorite (CaF2)
– rc/rA ~ 0.8  CN# 8
– Structurally somewhat similar to CsCl structure, but due to
stoichiometry only half the cube centers are full
Structures of Metals & Ceramics
• AmBnXp – Type Crystal Structure (Perovskite structure)
– Basic idea here as opposed to earlier structures – can now have
two types of cations (A and B)
– Example: Barium Titanate (BaTiO3)
– Barium – eight corners of the unit cell, Titanium – center of the
cell, oxygens – unit cell faces
– For this case the titanium has a CN of 6, what about the barium?
Structures of Metals & Ceramics
• How do you know if I give you a set of cations and
anions (i.e. NaBr) what structure it will adopt?
– Can estimate rc/rA – get CN from this
– Note that some of these structures have different CN values for
the anion and cation
– Good starting place
Structures of Metals & Ceramics
• Concept check 3.1
– Structure of K2O?
rK   0.138nm
rO 2  0.140nm
rK 
rO 2
0.138nm

 0.986
0.140nm
Coordination number is 8 (Note this is for the anion! More in a second)
The resulting crystal structure is the fluorite structure
However, by charge neutrality there is twice as many cations as
anions
• The K+ occupy the positions shown for F-, and the O2- anions
occupy the positions shown for Ca+2
Hence the name “anti-fluorite” structure (Fluorite=CaF2)
Structures of Metals & Ceramics
• Silicate Ceramics
– Why do we care about silicates (species containing silicon and
oxygen)?
– Two most abundant elements in the earth’s crust
– These materials are built of SiO4-4 units
– Not considered to be ionic materials (bonding has a significant
covalent character)
Structures of Metals & Ceramics
• Silica
– Most simple material – silicon dioxide (SiO2) or silica
– Three-dimensional materials
• Silicon atoms connected via bridging oxygen atoms
– Crystalline forms of silica
• Three polymorphs (materials with same composition, different
crystal structure)
– Quartz, cristobalite, tridymite
– Structures are comparable, and fairly open
» Density ~ 2.65 g/cm3
– Melting temperature ~ 2000 K
cristobalite
Structures of Metals & Ceramics
• Silicates
– Can have very complex structures – as you can imagine there
are many ways to connect the SiO4-4
– Some observations
• Solids are charge neutral – so you have cations around (Ca+2, Mg+2,
Al+3)
• Cations also hold the silicate anions together
– Simple silicates – those involving isolated silicate anions
• Forsterite (Mg2SiO4) – Mg+2 ions have six oxygen nearest neighbors
• Dimer – Akermanite (Ca2MgSi2O7)
Structures of Metals & Ceramics
• Silicates
Structures of Metals & Ceramics
• Layered Silicates
– Two dimensional silicate sheets – anionic structures
Side view
Structures of Metals & Ceramics
• Layered Silicates
–
–
–
–
–
Balance charges by having an additional “layer” rich in cations
Bonding within individual layers – covalent
Bonding between layers – due to van der Waals’ forces
Kaolinite – common clay
Very important commercial materials
• Catalyst supports
– Other layered silicates
• Talc (Mg3(Si2O5)2(OH)2)
Structures of Metals & Ceramics
• Carbon
– Numerous polymorphic forms as well as amorphous
– Does not cleanly fit into the polymer/metal/ceramic
categorization
– Very important form of matter
• Diamond – formed at high pressures
– Zinc blend structure where both atoms are C
• Also referred to as Diamond cubic structure
Structures of Metals & Ceramics
• Graphite
– More stable form of carbon at ambient T and P
– Layers of hexagonally arranged carbon atoms
– Weak van der waals type of bonding between layers
Structures of Metals & Ceramics
• Fullerenes/carbon nanotubes
– More “exotic” versions of carbon
– Fullerenes – groups of carbon atoms connected to form
hexagons/pentagons
– Carbon nanotubes
• Unusual mechanical, electrical properties
Structures of Metals & Ceramics
• Polymorphism/Allotropy
– Polymorphs are materials of identical composition with different
crystal structures
– In elemental solids this is referred to allotropy
– Which polymorph is most stable? Depends on
• P, T and the like
– Good example – carbon on last slide
• Graphite most stable at ambient T and P
• Diamond more stable at high P
Structures of Metals & Ceramics
• Crystal Systems
– As we have seen, there are many crystal structures
– Useful to divide them into groups
– This is typically done by considering the shape (or symmetry) of
the unit cell
– In general, unit cell is specified by 6 parameters
• Unit cell lengths a, b, c
• Unit cell angles a, b, g
Turns out there are 7 different
crystal systems!
Structures of Metals & Ceramics
• Most of the materials we
are interested in (metals,
ceramics) are usually
either cubic or
hexagonal
• Crystallographic Planes, Directions, and Points
– Need ways to describe crystal structures
• Directions within crystals – in many materials the structure “looks
different” when viewed in different directions
• How are the atoms related spatially to one another?
• Place to start – point coordinates
– Way to describe where atoms are in unit cell
– The position of any atom in the cell can be described in terms of
the unit cell edge lengths (a, b, c – these are also referred to as
the lattice constants)
Here: point P position is q r s
Distance qa from x = 0
Distance rb from y = 0
Distance sc from z = 0
Structures of Metals & Ceramics
• Place to start – point coordinates
– Notes: also referred to as “fractional coordinates” in that q r s
are some multiple of a b c and usually less than 1
– Typically these are dimensionless – to get actual position in say
angstroms, would multiply by appropriate unit cell edge length
Example 3.8
What are the point coordinates
for atoms at positions 1-8?
Structures of Metals & Ceramics
• Crystallographic directions
– Defined as a line between two points (vector)
– How to define the direction (directional indices) of the line:
• Vector (of convenient length) is positioned to pass through the origin
of the coordinate system
• Length of the vector projection on each of the axes is determined
and expressed in terms of a b c (this is the tricky part – more in a
minute)
• Multiple/divide these numbers by a common factor to convert them
all to integers (e.g. 0 ½ 0  x 2  0 1 0)
• The indices correspond to the reduced projections along x y z; the
indices are by convention written in brackets [0 1 0]
• Crystallographic directions
– Few more points:
• Negative indices are possible (e.g [1 1 1])
• Inverting signs of all indices results in the antiparallel direction
– [1 1 1]  [1 1 1]
• Key  be consistent with sign (positive/negative) convention
How did they come up with the indices
In this figure?
• Hexagonal Crystals
– Another approach; instead of three coordinate system, using
four-axis Miller-Bravais system
• Why? Some crystallographic equivalent directions will not have the
same indices
• Defining Miller-Bravais coordinate system
• The three axes a1 a2 a3 are all in one plane (basal
plane) and at 120° angles to one another
• The z axis is perpendicular to this basal plane
• Use same procedure as described above –
however you now have four indices [u v t w]
• The first three refer to projections along a1 a2 and
a3; the fourth index is the projection along z
Structures of Metals & Ceramics
• Hexagonal Crystals
– Conversion between three- and four-index system
[u ' v' w' ]  [uvtw]
1
u  2u 'v'
3
1
v  2v'u '
3
t  u  v 
w  w'
Structures of Metals & Ceramics
• Hexagonal Crystals
– Time for an example
Explain how they came up with these
Perhaps it is easier to go from [u’ v’ w’]  [u v t w]
• Hexagonal Crystals
– Time for an example
Structures of Metals & Ceramics
• Crystallographic planes – Miller indices
– Way to describe orientation of planes in crystal structures
– Will seem similar to defining directions as we did above
• These planes are specified by three Miller indices (hkl) (note use of
parentheses)
• Any two planes parallel to one another are equivalent  have identical
indices
– Procedure to define Miller Indices
• If the plane passes through the selected origin, either another parallel plane
must be constructed within the unit cell by appropriate translation, or a new
origin must be defined
• The plane either intercepts or is parallel to each of the axes; the length of
the intercept (i.e. distance from origin to intercept) is determined in terms of
a b and c
• The reciprocals of these numbers are taken. Planes parallel to an axis have
an infinite intercept  index is 0
• Convert these numbers to smallest set of integers via division/multiplication
– Write integers in parentheses (hkl)
Structures of Metals & Ceramics
• Examples
• Why is the one on the far left
the (001)?
• Is parallel to x, y –
intercepts are thus ∞, ∞
• Intercepts z axis c
• So ∞ ∞ 1  take reciprocals
• (001)
• *Note – since plane in xy plane
intercepts origin, define indices
based on top plane (does not
pass through origin)
Structures of Metals & Ceramics
• Atomic Arrangement
– Why do we bother to define Miller indices/planes in crystalline solids?
• We are often interested in the arrangement of atoms in these planes
– Simple example – atomic arrangement along (110) plane in FCC and
BCC structure (below)
– Different packing  different bonding  different properties!
Which is which?
Structures of Metals & Ceramics
• Equivalent “families” of planes
– These are planes that crystallographically equivalent
• What does this mean? They have the same atomic packing
– Example:
• Cubic system the (111), (111), (111), (111), (111), (111), (111), and (111) are
all equivalent  denoted as the {111} family
• Physics of equivalent families of planes
– Atomic arrangement along the plane is identical
– Tetragonal
• What planes are contained by the {100} family?
• (100), (100), (010), (010), but not the (001) and (001) planes
– Why? For tetragonal symmetry a = b, but a ≠ c!
Structures of Metals & Ceramics
• Hexagonal crystals
– Again, for reasons above want equivalent planes to have the same
indices
– Again, use Miller-Bravais system
• Now have 4-notation scheme (hkil)
• Some redundancy in that i is defined by the sum of h and k
i  h  k 
• Find h k l as described above
• Let’s work an example!
Structures of Metals & Ceramics
• Example 3.14
• What are the Miller-Bravais indices for
the plane below?
• Intercepts a1 a distance a from the
origin
• Intercepts a2 a distance –a from the
origin
• Intercepts z a distance c from the origin
 h = 1, k = -1, l = 1
i  h  k   1 1  0
(1101) plane
Structures of Metals & Ceramics
• Ok, that is all well and good…why are Miller Indices/Cell vector
useful?
– Simple example – determine linear and planar densities
• Linear densities – this is related to the idea of equivalent
crystallographic directions
– How many atoms per unit length whose centers lie along the direction
vector?
LD 
# of atoms centered on the direction vector
length of direction vector
LD110 
FCC cell
[110] direction shown
2
1

or
4R 2R
1
LD  , LD[]length 1
r
r is the repeat distance between adjacent
atoms
Structures of Metals & Ceramics
• Planar densities
– In analogous manner, the planar density can be defined as the number
of atoms per unit area centered on a particular crystal plane
PD 
# of atoms centered on a plane
area of plane
• Example – (110) plane of FCC structure
• How many atoms on centered on the (110) plane?
• Atoms A, C, D, F contribute – ¼
• Atoms B, E contribute – ½
2
2

Area length  width
2
1
PD110 

4R  2R 2 4R 2 2
PD110 
FCC cell
(110) plane shown
Structures of Metals & Ceramics
• Another view of crystal structures --- close packed crystal structures
– Another way to think about the most common crystal structure
(FCC, BCC, HCP)
• Instead of in terms of unit cells – closed packed structures
• Basic idea … denote the centers of all atoms in one close-packed
plane as A
– Note “dimples” or triangular depressions formed by three adjacent
atoms
• Dimples with triangles pointed up (B), pointed down (C)
Structures of Metals & Ceramics
• Basic idea … can keep stacking planes. How they are stacked with
respect to one another determines FCC v HCP
• For HCP, the third layer of atoms would be place directly above
the first A layer
• For FCC, third layer is situated over the C
positions in the first layer
2nd layer
3rd layer
Structures of Metals & Ceramics
• Idea is not just useful for metals …
• Can also consider ceramics using a closed-pack crystal structure
model. Here though a few differences…
– Typically the close-packed planes are comprised of anions
– There are small interstitial voids between these planes … this is where
the cations reside
• Different types of void spaces
– Tetrahedral versus octahedral voids – 4 versus 6 neighbors
Structures of Metals & Ceramics
• Again, how does this relate to materials?
• Layers can pack either ABABABA.. Or ABCABCABC..
• Structure you get depends on whether
– Cations go into tetrahedral or octahedral sites, layer stacking
• For sodium chloride, the coordination
number for Na+ is 6, the crystal structure
is cubic
– FCC packing of layers (ABCABC…)
these are the anions (Cl¯)
– Since the CN of Na+ is 6, the cations go
into the octahedral sites O
Structures of Metals & Ceramics
• Crystalline versus non-crystalline solids
– So far we have talked about crystal structures…how
does this translate to real solids?
• Can imagine a few limiting cases
– The solid has perfect crystallinity
• This would mean that unit cells repeat perfectly throughout the
whole solid and are all in the same orientation
• These are called single crystals
– Usually this is not observed
• The crystal is not perfectly ordered
• There are crystalline domains of different orientations
Structures of Metals & Ceramics
• Polycrystalline materials
– Domains with different
sizes/orientations
– These different domains are often
referred to as grains
• Physical picture
– Start with solid that has no
crystallinity
– Crystalline phases nucleate and
grow
– Clusters start to fuse together
(typically not w/correct orientation)
– Areas between grains – grain
boundaries
Structures of Metals & Ceramics
• Polycrystalline materials
– Can have polycrystalline films – that is the picture on the last slide
– Powders are often polycrystalline
• Crystals are small (~ 10-6 m)
• Individual single crystals, but very small
Zeolite nanocrystals
~50 unit cells a side
100 nm
Structures of Metals & Ceramics
• Material anisotropy
– The properties of single crystals can depend on the
crystallographic direction in which the measurements are
performed
– Does this make sense?
– If properties are dependent on direction (i.e. they exhibit
directionality), they are said to be anisotropic
– Materials in which measured properties do not depend on direction
are said to be isotropic
– This directional dependency varies strongly based on material
composition, symmetry, etc.
– Polycrystalline materials will often appear isotropic
• Many grains – crystals appear to be randomly oriented
Structures of Metals & Ceramics
• X-Ray Diffraction – determination of crystal structures
– So you have a solid – how do you determine the arrangement of
atoms? (The solid does not announce its structure to you!)
– Diffraction methods are essential!
• The diffraction phenomenon
– Basic idea – diffraction occurs when a wave encounters a series of
regularly spaced obstacles (in our case atoms) that
• Are capable of scattering the waves
• Have spacings that are comparable in magnitude to the wavelength
– Diffraction is a consequence of specific phase relationships that
are established between two or more waves that have been
scattered by the obstacles
Structures of Metals & Ceramics
Constructive interference
Destructive interference
Structures of Metals & Ceramics
• Diffraction phenomenon – interference
– Consider two waves that are in phase. Now let’s suppose there is
a scattering event (e.g. the x-ray wave encounters an atom) and
that the two waves traverse different paths
• The phase relationship between the waves, depends on the difference
in path length, is essential
– Two limiting cases
• First, the path length difference is an integral number of wavelengths
– These are in phase – they mutually reinforce one another
(constructive interference) -- this is a manifestation of diffraction
• Other extreme – the path length differences are integrals of half
wavelengths – complete destructive interference
Structures of Metals & Ceramics
• X-Ray diffraction and Bragg’s Law
– X-rays: form of electromagnetic radiation, high energy, short
wavelength (~ 1-5 Å)
– This wavelength is comparable to the atomic spacing in solids
• What happens when an x-ray beam impinges on a solid
material?
– A portion of the beam will be scattered in all directions by the
electron cloud of the atoms
– What has to happen to get constructive interference of the x-rays
(i.e. diffraction)
Structures of Metals & Ceramics
• Constructive interference occurs for 1’, 2’ at an angle q to the planes A
and B. If the path length difference between 1-P-1’ and 2-Q-2’ is an
integral number (n) of wavelengths, then
n  SQ  QT
n  d hkl sin q  d hkl sin q
n  2d hkl sin q
 Bragg’s Law
Structures of Metals & Ceramics
• Bragg’s law is the “bridge” linking the diffraction
measurements to the atomic structure
• If the waves don’t constructively interfere (i.e. Bragg’s
law is not satisfied), no intensity is observed
• Relation between Bragg’s law and structure? dhkl
• There are equations for each Bravais lattice relating dhkl
to lattice constants
• For example, for a cubic structure
d hkl 
a
h2  k 2  l 2
Note appearance of Miller indices!
Structures of Metals & Ceramics
• Amorphous Solids (Non crystalline)
• Many important materials (most notably glass) are not
crystalline
• So, the atoms do not have a well-defined spatial
arrangement
• Very hard to characterize the structure of amorphous
materials
ANNOUNCEMENTS
Reading: Chapter 3
HW # 2. Due Friday, February 2
3.3; 3.4; 3.6; 3.9; 3.13; 3.21; 3.24; 3.27; 3.35;
3.43; 3.49; 3.51; 3.55; 3.58; 3.59; 3.70
Self-help Problems: Read all examples in
Chapter 3
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