Lecture 1 Crystallography

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Lecture 1 Crystallography

• Part 1
• Internal Order and 2-D Symmetry
• Plane Lattices
• Planar Point Groups
• Plane Groups

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Internal Order and Symmetry

• Repeated and symmetrical arrangement (ordering)
  of atoms and molecules in minerals creates a
  3-dimensional lattice array
• Arrays are generated by translation of a unit
  cell smallest unit of lattice points that
  define the basic ordering
• Spacing of lattice points (atoms) are typically
  measured in Angstroms (10-8 cm 10-10 m) About
  the scale of atomic and ionic radii

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Two-Dimensional Plane Lattice
Generating an 2D Lattice Array (Plane Lattice)
involves translation of a motif in two
directions possible directions not unique
Translation in two directions x and y
axes Angle between axes ? Translation
distance a along x and b along y Replacing
motifs with points (or nodes) creates a plane
lattice
Unit Cell defined by a choice of lengths and
directions.
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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

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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

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Symmetry

• Translations (Lattices)
• 2-D translations a net

A 2-D Unit Cell
Unit Cell the basic repeat unit that, by
translation only, generates the entire pattern
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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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There are 5 Types of Plane Lattices
Memorize these names and rules
Preferred
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Translations

• The lattice and point group symmetry interrelate,
  because both are properties of the overall
  symmetry pattern

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Defining a Unit Cell
Rules that help us
Choose Smallest Most orthogonal Most in line
with symmetry 2 Nodes per Lattice Vector Most
Primitive (non-centered)
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Translations

• The lattice and point group symmetry interrelate,
  because both are properties of the overall
  symmetry pattern

Choose Smallest Most orthogonal Most in line
with symmetry 2 Nodes per Lattice Vector Most
Primitive (non-centered)
Good unit cell choice.
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Translations

• There is a new 2-D symmetry operation when we
  consider translations
• The Glide Line, g
• A combined reflection
• and translation

repeat
Step 2 translate
Step 1 reflect (a temporary position)
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Symmetry Elements of Planar Motifs Planar Point
Groups
10 Possible symmetry combinations called Planar
Point Groups Limitations of rotational
symmetries (1,2,3,4, 6)
dark lines added found mirrors
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total 17 point groups
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• There are 5 unique 2-D plane lattices.

There are also 17 2-D Plane Groups that combine
translations with compatible symmetry operations.
The bottom row are examples of Plane Groups that
correspond to each lattice type
Note p refers to a primitive cell, as apposed to
c, a 2-end (opposite ends) centered cell
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17 Plane Groups
10 H-M Point Groups and 5 Lattices combine to
form 17 Plane Groups.
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Next Lecture
3-D Internal Symmetry