Miller indices/crystal forms/space groups

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Miller indices/crystal forms/space groups
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Crystal Morphology

• How do we keep track of the faces of a crystal?
• Sylvite a 6.293 Å
• Fluorite a 5.463 Å
• Pyrite a 5.418 Å
• Galena a 5.936 Å

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Crystal Morphology

• How do we keep track of the faces of a crystal?
• Remember, face sizes may vary, but angles can't

Note interfacial angle the angle between the
faces measured like this
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Crystal Morphology

• How do we keep track of the faces of a crystal?
• Remember, face sizes may vary, but angles can't
• Thus it's the orientation angles that are the
  best source of our indexing

Miller Index is the accepted indexing method It
uses the relative intercepts of the face in
question with the crystal axes
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Crystal Morphology

• Given the following crystal

2-D view looking down c
b
a
b
a
c
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Crystal Morphology

• Given the following crystal

How reference faces? a face? b face? -a and -b
faces?
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Crystal Morphology

• Suppose we get another crystal of the same
  mineral with 2 other sets of faces
• How do we reference them?

b
w
x
y
b
a
a
z
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• Miller Index uses the relative intercepts of the
  faces with the axes

Pick a reference face that intersects both
axes Which one?
b
b
w
x
x
y
y
a
a
z
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• Which one?

Either x or y. The choice is arbitrary. Just pick
one. Suppose we pick x
b
x
y
a
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• MI process is very structured (cook book)

a b c
unknown face (y)
1
1
?
reference face (x)
2
1
1
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• What is the Miller Index of the reference face?

a b c
unknown face (x)
1
1
?
reference face (x)
1
1
1
(2 1 0)
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• What if we pick y as the reference. What is the
  MI of x?

a b c
unknown face (x)
2
1
?
reference face (y)
1
1
1
(1 1 0)
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3-D Miller Indices (an unusually complex example)
a b c
c
2
2
2
unknown face (XYZ)
reference face (ABC)
1
4
3
C
Z
O
A
Y
X
B
a
b
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Miller indices

• Always given with 3 numbers
• A, b, c axes
• Larger the Miller index , closer to the origin
• Plane parallel to an axis, intercept is 0

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What are the Miller Indices of face Z?
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The Miller Indices of face z using x as the
reference
a b c
1


unknown face (z)
1
1
reference face (x)
1
b
w
(1 1 0)
(2 1 0)
(1 0 0)
a
z
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What do you do with similar faces on opposite
sides of crystal?
b
(1 1 0)
(2 1 0)
(1 0 0)
a
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b
(0 1 0)
(1 1 0)
(1 1 0)
(2 1 0)
(2 1 0)
(1 0 0)
a
(1 0 0)
(2 1 0)
(2 1 0)
(1 1 0)
(1 1 0)
(0 1 0)
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Demonstrate MI on cardboard cube model
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• If you dont know exact intercept
• h, k, l are generic notation for undefined
  intercepts

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You can index any crystal face
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Crystal habit

• The external shape of a crystal
• Not unique to the mineral
• See Fig. 5.2, 5.3, and 5.4

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• Crystal Form a set of symmetrically equivalent
  faces
• braces indicate a form 210

b
(0 1)
(1 1)
(1 1)
(2 1)
(2 1)
(1 0)
a
(1 0)
(2 1)
(2 1)
(1 1)
(1 1)
(0 1)
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• Form a set of symmetrically equivalent faces
• braces indicate a form 210
• Multiplicity of a form depends on symmetry

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• Form a set of symmetrically equivalent faces
• braces indicate a form 210
• What is multiplicity?
• pinacoid prism pyramid dipryamid

related by a mirror or a 2-fold axis
related by n-fold axis or mirrors
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• Form a set of symmetrically equivalent faces
• braces indicate a form 210

Quartz 2 forms Hexagonal prism (m
6) Hexagonal dipyramid (m 12)
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• Isometric forms include
• Cube Octahedron
• Dodecahedron

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Crystal forms

• Forms can be open or closed
• Cube block demo
• Forms on stereonets
• Cube faces on stereonet

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• General form
• hkl not on, parallel, or perpendicular to any
  symmetry element
• Special form
• On, parallel, or perpendicular to any symmetry
  element
• Rectangle block
• Find symmetry, plot symmetry, plot special face,
  general face, determine multiplicity

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Space groups

• Point symmetry symmetry about a point
• 32 point groups, 6 crystal systems
• Combine point symmetry with translation, you have
  space groups
• 230 possible combinations

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Symmetry
Translations (Lattices) A property at the atomic
level, not of crystal shapes Symmetric
translations involve repeat distances The origin
is arbitrary 1-D translations a row
a
?
a is the repeat vector
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Symmetry
Translations (Lattices) 2-D translations a net
b
a
Pick any point Every point that is exactly n
repeats from that point is an equipoint to the
original
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Translations
There is a new 2-D symmetry operation when we
consider translations The Glide Plane A combined
reflection and translation
repeat
Step 2 translate
Step 1 reflect (a temporary position)
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• 32 point groups with point symmetry
• 230 space groups adding translation to the point
  groups

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3-D translation

• Screw axes rotation and translation combined