1. 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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2. • Non dense, random packing
ENERGY AND PACKING
• Non dense, random packing
• Dense, regular packing
Dense, regular-packed structures tend to have
lower energy.
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3. 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
noncrystalline SiO2
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4. Most METALS (>99%) are CRYSTALLINE. CERAMICS are CRYSTALLINE
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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5. 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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6. 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. 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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8. 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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9. ATOMIC PACKING FACTOR • APF for a simple cubic structure = 0.52
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10. 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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11. Atoms at cube corners and one in cube centre.
BCC STRUCTURE
Atoms at cube corners and one in cube centre.
Lattice Constant for BCC:
e.g. Fe (BCC) a = 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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13. ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
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14. 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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15. 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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17. ATOMIC PACKING FACTOR: FCC
• APF for a body-centered cubic structure = 0.74
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18. FCC STACKING SEQUENCE • ABCABC... Stacking Sequence • 2D Projection
• FCC Unit Cell
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19. 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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20. 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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22. By geometry, for IDEAL HCP:
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 = (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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23. n = number of atoms in unit cell A = Atomic Weight of element (g/mol)
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 = Avogadro’s number (6.023 x 1023 atoms/mol)
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24. 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 = nm
= 8.89 Mgm-3 (or 8.89 gcm-3 or 8890 kgm-3)
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25. 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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26. 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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27. Group crystals depending on shape of Unit Cell.
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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30. Positions in lattice
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31. 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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33. 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 < > brackets can change order and sign of integers.
e.g. cube internal diagonals <111>
cube face diagonals <110>
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34. Use a 4-axis system (Miller-Bravais).
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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36. Planes specified by Miller Indices (hkl) (Reciprocal Lattice).
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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40. 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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41. 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 = - (h+K)
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42. Arrangement of atoms on different planes and in different directions.
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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43. gives fraction of area covered by atoms.
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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44. 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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45. FCC and HCP are both CLOSE-PACKED structures. APF = 0.74
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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49. 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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50. Many small crystals (grains) with
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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51. 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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53. E.g. Stiffness (rigidity) electrical conductivity, refraction.
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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54. 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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56. 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,
 = wavelength of incident X-rays
 = incident angle
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59. 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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61. 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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62. 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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64. 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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