Carbon nanostructures
(http://www.mf.mpg.de/de/abteilungen/schuetz/index.php?lang=en&content=researchtopics&type=specific&name=h2storage)
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Crystal Structures
• Crystalline Material: atoms arrange into a periodic array (repetitive
three dimensional pattern/ arrangement of atoms– lattice structure).
– repeat unit  called unit cell
– repeated pattern  called crystal lattice
• Non-crystalline / Amorphous Material: do not crystallize, i.e. there is no
long-range atomic order
• Crystal Structure: manner in which atoms, ions or molecules are
spatially arranged
• Many possible structures defined by
– shape of cell
– arrangement of atoms within cell
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Atomic Packing in Solids
• regular atomic packing  crystalline
• random atomic packing  amorphous, glassy
Crystalline
Amorphous
Mixed
metals
usually
(e.g. steel, brass)
rarely
(e.g. metallic glass)
never
ceramics
often
(e.g. alumina)
often
(e.g. soda glass)
often
(e.g. silicon nitride)
polymers
never
(“crystalline” polymers
always partly amorphous)
usually
(e.g. polyethylene)
sometimes
(e.g. nylon)
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Unit Cells
• Basic structural unit or building block of
crystal structure
• Represents the symmetry of the crystal
structure
• Defines geometry and atom positions
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http://www.tutorvista.com/content/chemistry/chemistry-iv/solid-state/space-lattice.php
Properties / Crystal Structure
Relationship
• Many material properties are influenced
by the crystal structure including:
e.g. Dielectric constant (capacitance)
Strength
Ductility
Electrical conductivity
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Metallic Crystals
• Metallic bonding
 non-directional
 tends to form cubic,
close-packed structures
• 3 main variants
– face-centered cubic (FCC)
– body-centered cubic (BCC)
– hexagonal close packed (HCP)
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Close Packing in Two Dimensions
Square packing:
Each circle occupies an 'equivalent
area' of 4r2, because no other sphere
can use this area.
J. Hiscocks, 2003
Area = (2r)2 = 4r2
Hexagonal packing:
Each circle occupies a smaller
'equivalent area', making this a more
efficient packing system.
This is close packing in 2 dimensions.
J. Hiscocks, 2003
Area = (√3/2)(2r)2 = 3.464 r2
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See http://www.drking.worldonline.co.uk/hexagons/misc/area.html
Demo: Close Packing in Two Dimensions
• The curve of the watchglass
pushes the spheres together.
– equivalent to a bonding force
• 2-D close packing occupies
the smallest area and lowers
the overall energy.
– any spheres that achieve
hexagonal packing will stay
that way
• Any other arrangement (e.g.
square array) is unstable.
J. Hiscocks,
2003
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Metallic Crystal
Structures
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Packing atoms in Three
Dimensions
Most metals have one of the following
unit cells:
face-centered cubic (FCC)
body-centered cubic (BCC)
hexagonal close packed (HCP)
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SC Crystals
(Simple Cubic)
• Not very common.
• Atoms sit at
– cell corners
• 1 atom/cell
• Atomic Hard sphere
model
2R  a
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SC Crystals
Simple Cubic Crystal
Structure
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BCC Crystals
(Body-Centered Cubic)
• α-Fe, Cr, W, Mo
(transition metals)
• atoms sit at
•
– cell corners
– cell center
Number of Atoms/ Unit Cell: 2
4
a
R
3
APF: 0.68 (a portion of the unit cell
that is occupied by atoms)
CN: 8
Courtesy P. M. Anderson
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BCC Crystals
Body-Centered Cubic Crystal
Structure
(Click to Play)
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FCC Crystals
• Cu, Al, Ag, Au, γ-Fe
(Face-Centered Cubic)
• Atoms sit at
– cell corners
– middle of cell face
• Co-ordination number (CN) = 12
• Counting up atoms
– How many neighbouring
cells share each atom?
– 4 atoms/cell
• Atomic packing factor
(APF) = 0.74
4R  2a a  2 2R
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Crystal Structure
Closed packed lattices
FCC: ABCABC layers
HCP: ABABAB layers
FCC Crystal Structure
This image is the property of IBM Corporation.
http://www.kings.edu/~chemlab/vrml/clospack.html
Courtesy P. M. Anderson
Scanning tunnelling
microscope image of a Ni
surface.
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FCC Crystals
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FCC Crystals
Face-Centered Cubic Crystal
Structure
(Click to Play)
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HCP Crystals
• FCC and HCP crystals are both
based on close-packed planes.
• FCC  ABCABC… sequence
• HCP  ABABAB… sequence
• For both  CN = 12
 APF = 0.74
• Ideally, the HCP c/a = 1.633 but
it often deviates from this.
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Crystal Structure
Closed packed lattices
FCC: ABCABC layers
HCP: ABABAB layers
Close Packed Structure
• FCC and HCP are close
packed structures.
– BCC is not.
• consider the FCC <111>
plane:
– 6-fold coordination of the
A-layer.
– two sets of positions for the
next layer, B or C.
• FCC uses ABCABC…
stacking.
• Whereas, HCP used
ABABAB…
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HCP Crystals
(Click to Play)
(Click to Play)
Hexagonal Close-Packed Crystal
Structure
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Other Structures
3-types: SC, BCC, FCC
1-type: HCP
2-types:
1-type:
4-types:
2-types:
1-type:
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Crystal Structures in the Periodic Table
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Crystal
Structure
Details
Structure
(Hard Sphere
Model)
Reduced
Sphere Unit
Cell
Number
of Atoms/
Unit Cell
Atomic
Packing
Factor
Coordination
Number
1
0.52
6
2
0.68
8
4
0.74
12
0.74
12
a=f(R)
Simple Cubic
Body Centered
Cubic
Face Centered
Cubic
Hexagonal Close
Packed
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a=2R
C=1.63a
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Density
• Density is a function of:
– atomic weight, A (g/mol)
– crystal structure
• cell volume, Vc (m3)
• No. of atoms/cell, n
n A
   
Vc  N A 
NA=Avogadro’s
number
Atoms/volume
Mass/atom
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Polymorphism
• Fe is polymorphic (has more than one
crystal structure)
– α - BCC at T < 910 oC
– γ - FCC at 910 oC < T < 1394 oC
– δ - BCC at 1394 oC < T <1538 oC
• Does anyone know what happened to βFe?
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ALS: Density Calculation
• Fe has two forms α (BCC) and  (FCC)
• Which has the higher density?
Hint: Keep in mind that A / NA is the same for both structures
BCC
~
~
2
a3
a
4
R
3
23 3 1
64 R3
~
0.16
R3
FCC
~
4
a3
a  2 2R
~
4 1
16 2 R3
~
0.177
R3
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Crystallographic
Directions
and Planes
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Miller Indices
• Need a nomenclature to describe crystal structures
in detail.
• In particular:
– directions
– planes within crystals
• The method should be independent of cell type.
 Can’t use Cartesian co-ordinates.
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Miller Direction Indices
1.
Start at any cell corner.
2.
Find coordinates of vector in units of a, b, c.
3.
Multiply or divide all the coordinates by a
common factor.
– To reduce all the coordinates to the smallest
possible integer values.
4.
Represent as [ u v w ]
– no commas
5.
Represent negative directions as ū.
This is called the Miller Index for direction.
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Miller Direction Indices
1.
Head point coordinate, a b c.
2.
Tail point coordinate, k l m.
3.
a – k= u,
4.
Represent as [ u v w ]
– no commas
5.
Multiply or divide all the coordinates by a
common factor.
- To reduce all the coordinates to the smallest
possible integer values.
b – l=v,
c-m=w
6. Represent negative directions as ū.
This is called the Miller Index for direction.
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ALS: Miller Index for Direction
• What is the Miller index for A?
a) [ 2 0 1 ]
b) [ 0 2 1 ]
c) [ 1 1 2 ]
d) other
• What is the Miller index
of B?
a) [ 2 2 1 ]
x
b) [ 1 1 2 ]
c) [ 1 2 1 ]
d) other
z
B
A
y
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Miller Indices for Planes
• The Miller index of a plane is the same as the Miller
index of the direction normal to the plane.
– Choose a starting point (origin) so that the plane does not pass
through the origin.
– Find the intercepts in units of x, y, z, (planes parallel to an axis
have an intercept at ).
– Find the reciprocals of the intercepts: 1/x, 1/y, 1/z.
– Multiply or divide by common factor to get the smallest
possible integer values.
– Represent the index as ( h k l ) – no commas.
– Represent negative values using the bar:
h
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ALS: Miller Index of a Plane
What is the Miller index of this
plane?
Intercepts:
a=
b = -1
c = 1/2
Reciprocals:
0
–1
2
Reduction:
not needed
Index:
(012)
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Common Miller Indices
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Examples
z
z
y
x
y
x
(110)
11) planes
Identify the Miller indices of( 2these
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Examples
Eg. 1
Eg. 2
Draw the line [321]!
1/3
Find the Miller indices of this line!
[1 1 1]
2/3
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Families of Directions
•
_
[ 1 0 0 ] direction has 5 cousins:
[001]
_
_
[010], [001], [1 00], [01 0], [001]
[100]
call this the
[010]
< 1 0 0 > family
[010]
[100]
[001]
The three most important families of directions are:
<100>, <110>, <111>
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Family of Planes
• The (111) plane also has many cousins:
e.g.
(111)
(111)
(111)
Call this the {111} family.
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How are Lines and Planes
Related?
• How can we tell if the [ u v w ] direction lies in
the ( h k l ) plane?
– recall: The ( h k l ) represents the vector normal to
the plane.
– recall: The dot product between normal vectors is
zero.
• You can treat Miller indices like ordinary
vectors:
[ u v w ] lies in ( h k l ) if
(hkl)·(uvw)=hu+kv+lw=0
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Additional Concepts
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Discovery of X-Rays
Wilhelm Conrad Röntgen
Rontgen's first x‐ray image.
The ring can be seen on his
wife's hand (1895).
Won 1st Nobel Prize in physics
(1901).
X-Ray
X-Ray Diffraction
• X-Rays help determine
atomic interplanar
distances and crystal
structures
• A form of electromagnetic
Radiation with high energy
and short wavelengths
• Diffraction occurs when a
wave encounters a series of
regularly spaced obstacles
that:
– Are capable of scattering
the wave
– Have spacings that are
comparable (in
magnitude) to the
wavelength
• Diffraction: Constructive
Interference of x-ray
beams that are scattered
by atoms of a crystal.
• When two scattered waves
are:
– In Phase
Constructive
Interference
– Out of Phase
Destructive
Interference
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How Do We Know It’s A Crystal?
• Crystals diffract X-rays
• Bragg’s law says constructive interference will occur if the extra
path is a multiple of the wavelength:
n=2dsin
Note: For practical reasons,
the “diffraction angle” is
2
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Miller Indices and Planar Spacing
• From Bragg’s Law:
n = 2 d sin 
d = spacing between the planes
•
We can show that for any ( h k l ) plane:
a
d 2
h  k2  l2
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Diffraction from a Crystalline Solid
• SiAlON is a Si3N4-Al2O3 alloy
– Used for cutting tools (very hard)
• 2 phases present (a-cubic, b-hexagonal)
θ
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Polycrystals
• Single Crystals:
– Some materials consist of one crystal.
– Rare in nature, difficult to grow.
• Examples:
– Gem Stone
– Si wafers, quartz oscillators.
• Most materials contain many crystals called grains 
polycrystal / polycrystalline.
• The region of atomic mismatch where grains meet is called a
grain boundary (atomic dimensions).
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Anisotropy
• Often, the physical properties of a
material differ depending on the
crystallographic direction in which the
measurement is taken
– anisotropy
– e.g. conductivity, elastic modulus, index of
refraction
Fuchsite Mica
• Isotropic: Properties which are
independent of the direction of
measurement are referred to as being
isotropic.
• As structural symmetry decreases,
anisotropy increases
• Highly anisotropic crystals include:
– graphite (hexagonal with a large c/a
value).
– mica (sheet silicate).
BCC Fe
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Non-Crystalline Solids
• Also called amorphous solids or glass.
• Caused by irregular arrangements of the molecular
units.
– eg. SiO44- tetrahedra
• Amorphous solids show short-range order, but not longrange order.
– no X-ray diffraction patterns
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Silica Glass
• Glass is a supercooled
liquid.
• Gglass > Gsolid
– Glass is unstable…
– … and therefore hard to
make.
Free energy
glass
G
solid
liquid
T
• Network modifiers help
(CaO, Na2O).
– They break up the SiO2
network.
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