Crystal Structure

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Crystal Structure
Cinnabar
2
Hermann-Mauguin Symbols

• Hermann-Maugin Symbols are used as a shorthand
  notation in order to indicate the symmetry of a
  crystal
• Mirror symmetry is denoted by an m
• Rotation is denoted by 1,2,3,4,5,6 for 1-6 fold
  rotation
• RotoInversion (Rotation Inversion) is denoted
  by 1,m,3,4,6

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Determining Hermann-Maugin

• Determine the Crystallographic axes for the
  mineral.
• Determine all of the symmetry elements for the
  mineral
• Determine which, if any, of the elements are
  perpendicular
• List the symmetry elements. Perpendicular
  elements should appear next to each other and
  separated by a /

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H-M Notation Examples

• 4mm
• 4 fold rotation and 2 mirror planes
• 4/m
• 4 fold rotation axis - to a mirror plane
• 2/m2/m2/m
• 3 2-fold rotation axes all of which are - to a
  mirror plane

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Point Groups

• The 32 Point Groups or Crystal Classes comprise
  all of the possible 3-D Symmetry Operations

m
Garnet 4/m32/m
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Miller Indices

• Miller indices describe the orientation of a
  crystal feature (face, plane, etc.)
• Form (hkl)
• h a axis
• k b axis
• l c axis
• hkl are whole numbers

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Possible Forms

• Unit Face (111)
• Points (6 Possible)
• (100) (010) (001) (100) (010) (001)
• Lines (12 Possible)
• (110) (011) (101)
• Planes

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Determining Miller Indices

• Determine the intercepts of the face along the
  crystallographic axes, in terms of unit cell
  dimensions.
• Take the reciprocals
• Clear fractions
• Reduce to lowest terms

9
Hexagonal Crystal System

• For clarity, the hexagonal crystal system is
  sometimes assigned 4 dimensions
• c, a1, a2, a3
• The c axis is the same as in the 3-D system
• a1, a2, a3 axes lie within
• the same plane and at
• an angle of 120 to each other

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Miller-Bravais Indices

• Hexagonal Crystal System
• Form (hkil)
• hki correspond to
• a1,a2,a3 axis
• l corresponds to c axis

c
a3
a2
a1
Corundum
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Crystal Forms

• A crystal form is described by the set of faces
  that make of the form
• All Miller Indices within a simple crystal form
  will be variations of each other
• The Crystal Form is described by one of the
  Miller Indices and enclosed in
• The same crystal form designation may apply to
  different shapes

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Crystal Form Example

• Cube (001) (100) (010) . . .
• 001
• Rhombic Dodecahedron (101) (011) (110)
• 011
• Tetartoid (213) (312) (123)
• 123

Rhombic Dodecahedron
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Similar Crystal Forms

• Octahedron 111
• Tetrahedron 111
• So how are they different?
• Variable number of faces
• Octahedron 8
• Tetrahedron - 4
• Differing Symmetry

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Variable of Faces

• Octahedron
• 8
• Tetrahedron
• 4

2
1
3
4
6
5
7
8
1
2
3
4
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Varying Symmetry

• Octahedron
• 5 mirror planes
• 3 4-fold rotation axis
• 2 2-fold rotation axis
• 4/m4/m4/m2/m2/m
• Tetrahedron
• 4 mirror planes
• 4 3-fold rotation axis
• 3/m3/m3/m3/m

4m
2m
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Isometric Crystal Forms

• All planes are duplicated throughout the crystal
  i.e. all Miller Indices are variations of each
  other
• 15 Total Isometric Forms
• All are closed forms

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Closed Form

• Closed Form
• The crystals faces completely enclose a space

Fluorite
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Open Form

• Open Form
• The crystals faces do not enclose a space

Wulfenite