GEOL 303A Mineralogy and Introduction to Petrology

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GEOL 303A Mineralogy and Introduction to
Petrology Lecture 4. Crystallography
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• Remember the definition of a mineral
• Naturally occurring
• Homogeneous
• Solid
• Definite (but generally not fixed) chemical
  composition
• Formed from an inorganic process
• Highly ordered atomic arrangement

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• What does Highly ordered atomic arrangement
  mean?
• Atoms (or ions) are present in exactly the same
  structural site throughout an essentially
  infinite atomic array
• This defines a motif, or an identical arrangement
  of neighboring atoms
• Motifs must exhibit periodic translations along a
  set of chosen coordinate axes
• This distinguishes a mineral from liquids, gases,
  or glasses

Si atoms in quartz crystal at very high
magnification using Scanning Tunneling
Microscopy
www.aip.org/history/ einstein/atoms.htm
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• Crystal Ordered Arrangements
• A crystal structure can be thought of as a
  repetition of a unit cell, or group of identical
  neighboring atoms on a lattice. This is
  referred to a motif
• The ordered patterns that characterize crystals
  represent a lower energy state (more stable)
  compared to random patterns (less stable)

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• Crystal Ordered Arrangements
• A crystal structure can be thought of as a
  repetition of a unit cell, or group of identical
  neighboring atoms on a lattice. This is
  referred to a motif (The Chemical
  Units of a Crystal Structure)
• The ordered patterns that characterize crystals
  represent a lower energy state (more stable)
  compared to random patterns (less stable)
• See how the motif repeats in regular sequences of
  new locations?

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• Crystal Ordered Arrangements
• Any motion that brings the original motif into
  coincidence with the same motif elsewhere in the
  pattern is referred to as an operation. Wallpaper
  patterns are excellent examples of this.

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• Crystal Ordered Arrangements
• Any motion that brings the original motif into
  coincidence with the same motif elsewhere in the
  pattern is referred to as an operation. Wallpaper
  patterns are excellent examples of this.

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• Crystal Ordered Arrangements
• Any motion that brings the original motif into
  coincidence with the same motif elsewhere in the
  pattern is referred to as an operation. Wallpaper
  patterns are excellent examples of this.

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• Crystal Ordered Arrangements
• Any motion that brings the original motif into
  coincidence with the same motif elsewhere in the
  pattern is referred to as an operation. Wallpaper
  patterns are excellent examples of this.

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• Translation Directions and Distances
• Remember that crystals must be homogenous and
  possess long-range, three-dimensional internal
  order
• This results from repeating motif units by
  regular translations in three dimensions
• The 3-D pattern is homogeneous if the angles and
  distances from one motif to surrounding motifs in
  one location of the pattern are the same in all
  parts of the pattern

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• Translation Directions and Distances
• Remember that crystals must be homogenous and
  possess long-range, three-dimensional internal
  order
• This results from repeating motif units by
  regular translations in three dimensions
• The 3-D pattern is homogeneous if the angles and
  distances from one motif to surrounding motifs in
  one location of the pattern are the same in all
  parts of the pattern

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• Translation Directions and Distances
• Remember that crystals must be homogenous and
  possess long-range, three-dimensional internal
  order
• This results from repeating motif units by
  regular translations in three dimensions
• The 3-D pattern is homogeneous if the angles and
  distances from one motif to surrounding motifs in
  one location of the pattern are the same in all
  parts of the pattern

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• Translation Directions and Distances
• If we concentrate only on the repetitions in
  space and replace the motifs by points,
• A lattice is an imaginary pattern of points where
  every point has an environment identical to that
  of any other point in the pattern. Lattices have
  no origin, and can be infinitely shifted in any
  direction parallel to itself

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• LATTICES one dimensional order (rows)
• A sequence of equally spaced equivalent points
  (or motifs) along a line represents order in one
  dimension, or a row
• The magnitude of the unit translation determines
  the spacing
• The pattern of the unit translation is determined
  by the motif

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• LATTICES two dimensional order (plane lattices)
• Translation of equally spaced equivalent points
  (or motifs) in two directions, designated x and y
  
• There are only 5 possible and distinct plane
  lattices (nets) in two dimensions based on
  repeating a row with
• 1. A translation distance b along direction y
• 2. A translation distance a along direction x
• 3. At some angle (?) between x and y

Net 1
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LATTICES two dimensional order (plane lattices)
Net 2
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LATTICES two dimensional order (plane lattices)
Net 3
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LATTICES two dimensional order (plane lattices)
Net 4
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LATTICES two dimensional order (plane lattices)
Net 5
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• Symmetry of Planar Motifs (2-D)
• 2-D motifs may contain a number of symmetry
  elements (perpendicular to the paper)
• Mirror Lines (m) and Rotation axes (1, 2, 3, 4,
  and 6)
• There are only 10 kinds of symmetry elements for
  2-D motifs
• 10 Planar Point Groups
• 1
• 2
• m
• 2mm
• 4
• 4mm
• 3
• 3m
• 6
• 6mm

Numerals refer to rotations about a stationary
point ms refer to mirror lines
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• Symmetry of Planar Motifs (2-D)
• 10 Planar Point Groups
• Numerals refer to rotations about a stationary
  pointms refer to mirror lines

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Symmetry of Plane Lattices (2-D) Lattice Point
Group Oblique 1, 2 Rectangular m,
2mm Square 4, 4mm Hexagonal 3, 3mm 6,
6mm
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• 3-Dimensional Order
• Same concepts as 2-D, but with a 3rd translation
  direction (vector)
• Changes
• Rotational symmetry about a point ? about a line
  (or axis)
• Reflection across a mirror line ? about a mirror
  plane
• New concepts
• Combining rotation and translation ? Screws,
  (Screw Axis)
• As a result, we will have 14 different 3-D
  lattice types (compared to only 5 types of 2-D
  lattices)

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• 3-D Lattice Types
• Ground Rules
• Vector space is defined with respect to x, y, and
  z axes
• Unit cell space is defined with respect to a, b,
  and c
• Resulting space lattices are referred to as
  either Primitive, where lattice points occur only
  at the corners, or Non-Primitive (A-, B-,
  C-centered or F-centered face centered, or
  I-centered body centered)

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