ch3 part 2(1)
TRANSCRIPT
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Chapter 3: Atomic & Ionic Arrangements
Week 2 Lecture Notes
R. Clemmer
ME 215 Winter 2011
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References
D.R. Askeland, P.P. Fulay, W.J. Wendelin, “The Science and
Engineering of Materials,” 6th ed., Cengage Learning, 2011.
W.D. Callister, “Material Science and Engineering: An Introduction”,
5th ed.
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Notes & Slides Outline
Short Range Order vs. Long Range Order
Amorphous Materials
Lattice, Basis, Unit Cells, & Crystal Structures
Lattice & Basis
Unit Cells
Lattice Parameters & Interaxial Angles
Number of Atoms per Unit Cell
Atomic Radius vs. Lattice Parameter
Coordination Number
Packing Factor
Hexagonal Close-Packed Structure
Allotropic or Polymorphic Transformations
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Notes & Slides Outline
Points, Directions, & Planes in Unit Cells
Directions
Planes
Miller Indices for Hexagonal Unit Cells
Close-Packed Planes and Directions
Isotropic & Anisotropic behaviour
Interstitial Sites
Crystal Structures of Ionic Materials
Covalent Structures
Diffraction Techniques for Crystal Structure Analysis
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Points, Directions, &
Planes in Unit Cells
38
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Coordinates of Points
Some properties of a crystal are dependent on
directions and planes of atoms
To define directions and planes, a special
coordinate system is used – Miller Indices
Rather than using a Cartesian coordinate
system using length units, coordinates of atoms
are given as the number of lattice parameters
moved in the x, y, and z directions
For a unit cell, coordinates will range from 0,0,0
to 1,1,1 since a unit cell has dimensions of
1x1x139
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Coordinates of Points
40(Askeland)
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Miller Indices for Directions
Steps for Determining Miller Indices for Directions
1. Find the coordinates of 2 points that lie in the
direction
2. Subtract the “tail” coordinates from the “head”
coordinates (gives number of lattice points travelled)
3. Clear fractions and reduce to lowest terms
4. Enclose numbers in square brackets, e.g. [hkl]
5. Negatives are represented by a bar over the
number
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Miller Indices of Directions
42(Askeland)
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Points regarding Miller Indices (Directions)
Directions are vectors – a direction and its
negative are not equal i.e. [100] ≠ [-100]
Directions and their multiples are equal, i.e.
[100] = [200]
Families of directions are groups of
equivalent directions and indicated by < >
brackets
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Equivalency of Directions
44(Askeland)
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Directions of the Form <110>
45(Askeland)
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Miller Indices for Planes
Steps for Determining Miller Indices for Planes
1. Identify the points where the plane intercepts
the x, y, z axes (may have to move origin if plane
passes through it)
2. Take reciprocals of the intercepts
3. Clear fractions but do NOT reduce to lowest
terms
4. Enclose numbers in parentheses, e.g. (hkl)
5. Negatives are represented by a bar over the
number
46
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Miller Indices of Planes
47(Askeland)
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Points regarding Miller Indices (Planes)
Planes and their negatives are equal i.e.
(100) = (-100)
Planes and their multiples are not equal, i.e.
[100] ≠ [200] (as demonstrated by planar
densities)
Families of planes are groups of equivalent
planes and indicated by { } brackets
48
face of area
atoms/face Density Planar
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Planar Density of Different Planes
49(Askeland)
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Planes of the Form {110}
50(Askeland)
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Miller Indices for Hexagonal Unit Cells
A special set of Miller-Bravais indices for
HCP structures was devised to account for
the symmetry in the crystal structure
3 – axis system: Similar to before (hkl) or [hkl]
4 – axis system: A third a-axis (a3) is added
where a1 = h, a2 = k, and a3 = i and
51
i- khor 0 i k h
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Miller Indices for HCP Unit Cells
52(Askeland)
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Converting from (h’k’l’) to (hkil)
Clear fractions and reduce to lowest terms as
required
53
'
''
''
''
l l
)kh(3
1- i
)h2k(3
1 k
)k2h(3
1 h
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Miller Indices for HCP Unit Cells
54(Askeland)
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Close-Packed Planes & Directions
Close-packed directions are directions in which
atoms are in contact
Close-packed planes are planes of atoms all in
contact
HCP has two parallel close-packed planes
(called basal planes):
A (0001) and B (0002)
Stacking these planes produces the HCP structure
FCC crystals have 4 non-parallel close-packed
planes of the form {111}
55
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Close-Packed Planes and
Directions
56(Askeland)
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Isotropic vs. Anisotropic Behaviour
A material may behave differently in one
crystallographic direction or another because
of differences in atomic arrangement
Isotropic: Properties are identical in all
directions
Anisotropic: Properties are dependent on the
crystallographic direction
A polycrystalline material may act isotropic
since many crystals are randomly oriented
within the structure.
57
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Interstitial Sites
In addition to the regular atom placement in a
crystal structure, other atoms (usually smaller)
may fit in the spaces between atoms
Interstitial sites are the locations between
regular atoms or ions in a crystal structure
SC – cubic interstitial sites
BCC – tetrahedral, octahedral sites
FCC – octahedral, tetrahedral sites
For an atom to fit in an interstitial site it cannot
be “too big or too small” (based on the radius ratio)
58
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Simple Cubic Unit Cell
59
The Simple Cubic Unit Cell has 1 interstitial site
(Askeland)
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BCC Unit Cell
60
The BCC Unit Cell has 24 Tetrahedral interstitial sites and
6 Octahedral interstitial sites
(Askeland)
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FCC Unit Cell
61
The FCC Unit Cell has 8 Tetrahedral and
12 Octahedral interstitial sites(Askeland)
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62
Octahedral Site in FCC
2r
2R
Ra2
40
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Crystal Structures of Ionic
and Covalent Materials
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Crystal Structures of Ionic Materials
Ionic crystals allow different sized ions to be
packed efficiently yet maintain electrical
neutrality
Typically the larger anions occupy the regular
lattice points with smaller cations occupying
interstitial sites
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Sodium Chloride Structure
65(Askeland)
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Zinc Blende
66(Askeland)
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Crystal Structures of Covalent Materials
The directional qualities of covalent bonds
results in complex crystal structures
67
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Diamond Cubic
68(Askeland)
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Crystalline Silica
69(Askeland)
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Crystalline Polymers
70(Askeland)
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Diffraction Techniques for
Crystal Structure Analysis
71
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How do we know the distance between
atoms and crystal structures?
X-ray diffraction (XRD-Max von Laue 1912 Nobel Prize winner)
Similar effects as diffraction grating in optics
X-rays have much smaller wavelength than visible light
Wavelengths have a similar order of magnitude to the spacing between atoms
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X-Ray Diffraction Patterns
73(Askeland)
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74
Bragg’s Law
n = 2d sin
= wavelength of x-rays
d = interplanar spacing
= angle of incidence and
diffraction of x-rays
n = an integer
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75
Interplanar Spacing, dhkl
Distance between two parallel planes of
atoms with the same Miller indices
For cubic crystals:
dhkl a0
h2 k2 l2
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76
X-ray
diffractometer
Diffraction
spectrum
for gold
X-Ray Detector
or photographic film strip
(Askeland)