CRYSTAL STRUCTURE- Chapter 3

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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
noncrystalline SiO2
"Amorphous" Noncrystalline
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FOR SOLID MATERIALS Most METALS (gt99) 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. 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
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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
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
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FCC STACKING SEQUENCE
ABCABC... Stacking Sequence 2D Projection
FCC Unit Cell
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HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
ABAB... Stacking Sequence
3D Projection
2D Projection
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 2
atoms and (½) x (top bottom) 1 atom and (3)
internal 3 atoms Total 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
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 co-ordination
number 12.
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CRYSTAL DENSITY The true density, ?, of material
(free from defects) can be calculated knowing its
crystal structure.
n number of atoms in unit cell A Atomic
Weight of element (g/mol) Vc volume of unit
cell Nav Avogadros 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
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. APF e.g. Iron 913oC
FCC 0.74 911oC BCC 0.68 i.e. expands on
cooling!
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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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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 most symmetry Triclinic 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.

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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
i.e. with lt gt brackets can change order and sign
of integers. e.g. cube internal diagonals
lt111gt cube face diagonals lt110gt
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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-(uv)
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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)
  Miller indices of plane

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FAMILY of planes, use hkl These planes are
crystallographically similar (same atomic
arrangements). e.g., for cube faces 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 - (hK)
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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.
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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.
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
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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. (Epoly iron 210 GPa) -If grains
are textured, anisotropic.
200 mm
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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, qc, 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
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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.