CRYSTAL STRUCTURE- Chapter 3
(atomic arrangement)
Why study this?
Ductile-brittle transition in metals.
Crystalline # amorphous - transparency
In gases, atoms have no order.
If atoms bonded to each other but there is no repeating
pattern (short range order) . e.g. water, glasses
(AMORPHOUS - non-crystalline)
If atoms bonded together in a regular 3-D pattern they
form a CRYSTAL
- long range order - like wall paper pattern or brick wall.
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ENERGY AND PACKING
• Non dense, random packing
• Dense, regular packing
Dense, regular-packed structures tend to have
lower energy.
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MATERIALS AND PACKING
Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers
crystalline SiO2
Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline
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noncrystalline SiO2
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FOR SOLID MATERIALS:
Most METALS (>99%) are CRYSTALLINE.
CERAMICS are CRYSTALLINE
except for GLASSES which are AMORPHOUS.
POLYMERS (plastics) tend to be:
either AMORPHOUS or
a mixture of CRYSTALLINE + AMORPHOUS
(known as Semi-crystalline)
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CRYSTALS
Different ways of arranging atoms in crystals.
Assume atoms are hard spheres and pack like
pool/snooker balls (touching).
Each type of atom has a preferred
arrangement depending on Temp. and
Pressure (most stable configuration).
These patterns known as SPACE LATTICES
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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.
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7 types of CRYSTAL SYSTEM
14 standard UNIT CELLS
METALLIC CRYSTAL STRUCTURES
Most metals crystallize into one of three densely
packed structures:
BODY CENTERED CUBIC
- BCC
FACE CENTERED CUBIC
- FCC
HEXAGONAL (CLOSE PACKED)
- HCP
Actual size of UNIT CELLS is VERY VERY SMALL!!
Iron unit cell length (0.287 x 10-9 m) (0.287 nm)
1 mm length of iron crystal has  3.5 million unit cells
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SIMPLE CUBIC STRUCTURE (SC)
• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
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ATOMIC PACKING FACTOR
• APF for a simple cubic structure = 0.52
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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.
• Coordination # = 8
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BCC STRUCTURE
Atoms at cube corners and one in cube centre.
Lattice Constant for BCC:
e.g.
4R
a
3
Fe (BCC) a = 0.287 nm
Two atoms in Unit Cell.
(1 x 1 (centre)) + (8 x 1/8 (corners)) = 2
Each atom in BCC is surrounded by 8 others.
COORDINATION number of 8.
Packing is not as good as FCC; APF = 0.68
BCC metals include:
Iron (RT), Chromium, Tungsten, Vanadium
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ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
R
Unit cell contains:
1 + 8 x 1/8
= 2 atoms/unit cell
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.
• Coordination # = 12
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FACE CENTERED CUBIC (FCC)
e.g. copper, aluminium, gold, silver, lead, nickel
Lattice constant (length of cube side in FCC) “a” for
FCC structure:
4R
a  8R 
 2R 2
2
where R = atomic radius
Each type of metal crystal structure has its own
lattice constant.
(1/8 at each corner x 8) + (½ at each face x 6 ) = 4
So 4 atoms per Unit Cell.
Each atom touches 12 others. Co-ordination
number = 12.
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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
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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
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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
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HEXAGONAL CLOSE PACKED
Note: not simple hexagonal but HCP
Simple Hex. very inefficient; HCP has extra
plane of atoms in middle.
1/6 of atom at each corner.
So (1/6) x 12 corners =
and (½) x (top + bottom) =
and (3) internal =
Total =
2 atoms
1 atom
3 atoms
6 atoms/cell
Because of Hexagonal arrangement (not
cubic), have 2 lattice parameters
“a” , and “c”
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a = basal side = 2R
c = cell height
By geometry, for IDEAL HCP:
c
 1.633
a
but this varies slightly for some HCP Metals.
HCP metals include: Magnesium, Zinc, Titanium,
Zirconium, Cobalt.
atomic packing factor for HCP = 0.74
(same
as FCC)
Atoms are packed as tightly as possible.
Each atom surrounded by 12 other atoms so coordination number = 12.
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CRYSTAL DENSITY
The true density, , of material (free from defects)
can be calculated knowing its crystal structure.
 nA



N
mass of unit cell
nA
A





volume of unit cell
Vc
Vc N A
n = number of atoms in unit cell
A = Atomic Weight of element (g/mol)
Vc = volume of unit cell
Nav = Avogadro’s number (6.023 x 1023 atoms/mol)
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e.g., copper, FCC  4 atoms/cell
n=4
Cu atoms have mass 63.5 g/mol
Vol. of cell = a3 , for FCC a =2R2
Atomic radius of copper = 0.128 nm

nACu
4 x63.5

 8.89 x106 gm3
Vc N A 
-9 3 
23
16
2
0.128x10

 x6.023 x 10




= 8.89 Mgm-3 (or 8.89 gcm-3 or 8890 kgm-3)
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POLYMORPHISM / ALLOTROPY
Some elements/compounds can exist in more than one
crystal form. Usually requires change in temperature or
pressure.
Carbon: Diamond (high pressure) or Graphite (low).
Can be IMPORTANT as some crystal structures more
dense (better packing, higher APF) than others, so a
change in crystal structure can often result in volume
change of material.
e.g.
Iron
APF
913oC FCC
911oC BCC
0.74
0.68
i.e. expands on cooling!
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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.
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CRYSTAL SYSTEMS
Group crystals depending on shape of Unit
Cell.
x, y and z are three axes of lattice separated
by angles ,  and .
A unit cell will have sides of length a, b and c.
(Note: for the cubic system all sides equal so
a = b = c)
SEVEN possible crystal systems (Table 3.2)
Cubic
…
…
Triclinic
most symmetry
least symmetry
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Positions in lattice
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CRYSTALLOGRAPHIC DIRECTIONS
Line between two points or vector.
Using 3 coordinate axes, x, y, and z.
•
•
•
•
Position vector so that it passes through origin
(parallel vectors can be translated).
Length of vector projected onto the three axes
(x, y and z) is determined in terms of unit cell
dimensions (a, b and c).
Multiply or divide by common factor to reduce
to lowest common integers.
Enclose in SQUARE brackets with no commas
[uvw], and minus numbers given by bar over
number; e.g.
[112], [111], [212]
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Parallel vectors have same indices.
Changing sign of all indices gives opposite
direction.
If directions are similar, (i.e., same atomic
arrangements - for example, the edges of a BCC
cube) they belong to a FAMILY of directions:
[100], [100], [010], [010], [001], [001]   100 
i.e. with < > brackets can change order and sign
of integers.
e.g.
cube internal diagonals
<111>
cube face diagonals
<110>
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HEXAGONAL CRYSTALS
Use a 4-axis system (Miller-Bravais).
a1, a2 and a3 axes in basal plane at 120
to each other and z axis in vertical
direction.
Directions given by [uvtw] or [a1 a2 a3 c]
Can convert from three-index to four index
system.
t=-(u+v)
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CRYSTAL PLANES
Planes specified by Miller Indices (hkl)
(Reciprocal Lattice).
Used to describe a plane (or surface) in a
crystal e.g., plane of maximum packing.
Any two planes parallel to each other are
equivalent and have identical Miller indices
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To find Miller Indices of a plane:
• If the plane passes through the selected origin, construct
a parallel plan in the unit cell or select an origin in another
unit cell.
• Determine where plane intercepts axes. (if no intercept
i.e.., plane is parallel to axis, then )
e.g., axis
x
y
z
intercept
a
b
c
• Take reciprocals of intercepts (assume reciprocal of  is
0):
1/a
1/b
1/c
• Multiply or divide to clear fractions: (hkl)
of plane
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Miller indices
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FAMILY of planes, use {hkl}
These planes are crystallographically
similar (same atomic arrangements).
e.g., for cube faces: {100}
(100), (100), (010), (010), (001), (001)  {100}
NOTE: In CUBIC system only, directions are
perpendicular to planes with same indices.
e.g., [111] direction is perpendicular to the
(111) plane.
HEXAGONAL CRYSTALS
Four-index system similar to directions; (hkil)
i = - (h+K)
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ATOMIC PACKING
Arrangement of atoms on different planes
and in different directions.
LINEAR ATOMIC DENSITIES
Tells us how well packed atoms are in a
given direction. If LD = 1 then atoms are
touching each other.
Linear Den sity, LD 
length of line intersecti ng atom centres, Lc
selected length, Ll
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PLANAR DENSITIES
Tells us how well packed atoms are on a given
plane. Similar to linear densities but on a plane
rather than just a line.
Planar density, PD 
Area of atoms intersecte d by plane, Ac
selected area, A p
gives fraction of area covered by atoms.
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e.g., BCC unit cell, (110)
plane:
2 whole atoms on plane in
unit cell.
So
Ac = 2(R2)
AD = a, DE = a2
And so Ap = a22
2 (R 2 )
PD 
a2 2
4R
(where a =
for BCC)
3
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PACKING ON PLANES
FCC and HCP are both CLOSE-PACKED structures.
APF = 0.74
(This is the maximum if all atoms are same size).
Atoms are packed in CLOSE-PACKED planes
In FCC, {111} are close packed planes
In HCP, (0001) is close packed
Both made of close packed planes, but different
stacking sequence.
FCC planes stack as
ABCABCABC
HCP planes stack as
ABABABABAB
BCC is not close packed (APF = 0.68)
most densely packed plane is {110}
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CRYSTALS AS BUILDING BLOCKS
• Some engineering applications require single crystals:
diamond single
crystals for abrasives
--turbine blades
• Crystal properties reveal features
of atomic structure.
--Ex: Certain crystal planes in quartz
fracture more easily than others.
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SINGLE CRYSTALS
This is when a piece of
material is made up of one
crystal; all the unit cells are
aligned up in the same
orientation.
POLYCRYSTAL
Many small crystals (grains)
with
different orientations joined
together. Most materials/metals
are POLYCRYSTALLINE.
Grain boundary - Regions
where grains (crystals) meet.
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POLYCRYSTALS
• Most engineering materials are polycrystals.
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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ANISOTROPY
Many properties depend on direction in crystal in which
they are measured.
E.g. Stiffness (rigidity) electrical conductivity, refraction.
If property varies with direction - Anisotropic.
If no variation with direction - Isotropic
Single crystals show this variation.
Polycrystalline materials are usually randomly
oriented so effect is evened out to give average values
in all directions.
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SINGLE VS POLYCRYSTALS
• Single Crystals
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
200 mm
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
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FINDING OUT WHAT THE CRYSTAL
STRUCTURE IS.
Analysed using X-Rays. X-Ray Diffraction.
X-rays are diffracted off atoms and either
constructively interfere (peak) or destructively
interfere (low) from layers of atoms depending on
interplanar spacing (dhkl) and angle.
n = 2dhklsin 
(Bragg's Law)
n = 1,2, 3, 4, 5.......
 = wavelength of incident X-rays
 = incident angle
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X-RAYS TO CONFIRM CRYSTAL STRUCTURE
• Incoming X-rays diffract from crystal planes.
• Measurement of: Critical angles, c, for X-rays provide
atomic spacing, d.
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So can measure peak and determine dhkl and
then “a”.
Distance between similar planes in the cubic
systems, e.g., (110) planes in adjacent unit
cells:
d hkl 
a
h2  k 2  l 2
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NON-CRYSTALLINE SOLIDS
Non crystalline solids are amorphous
materials.i.e.. they are not crystalline.
They have no long range order.
Short range order only.
Structure is usually too complex to form
crystals when cooled from liquid at normal
rates.
E.g.. Glasses, some plastics,
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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.
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