Unit Cells read Klein 2002 p.229234

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Unit Cellsread Klein (2002) p.229-234
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Unit Cell

• After a basic look at symmetry of designs,
  motifs, commas etc we can consider
• The external morphology of the crystals have
  dis-similar symmetry, so we can divide the
  crystals into 6 crystal systems ( classes).
• Looking again at translations

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Unit Cell
2-dimensional
Plane lattice
Already looked at symmetry in 2-D
3-dimensional
Space lattice

• For Crystalline objects

Need 3-D Lattices Add an additional translation
direction (or vector) - in a different plane to
the 2-D net. Vector space referenced to 3 axes
known as x, y and z Unit cell vectors are a, b
and c (NOTE - x,y and z axes often called a, b
and c)
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Unit Cell

• REMINDER A lattice is an imaginary pattern of
  points in which every point has an environment
  that is identical to that of any other point in
  the lattice
• a lattice has no specific origin and can be
  shifted parallel to itself
• 2 types
• A primitive space lattice, p is a parallelipiped
  with lattice points only at its corners
• If the unit cell is non-primitive, c then the
  centering may occur on a pair of the opposite
  faces of the unit cell (A, B or C), on all faces
  of the unit cell (F) or in the center of the unit
  cell (I)

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Unit Cell

• Here we have 2 choices for a unit cell.
• 1st choice the diamond
• The 2nd choice has a point in the center
  non-primitive unit cell

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NOTE The shaded area is the smallest geometrical
unit

• Five plane lattices
• Parallelogram
• a b g 90o
• Rectangle
• a b g 90o
• Diamond
• a b g 90o,60o,120o
• Rhombus
• a b g 60o, 120o
• Square
• a b g 90o

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Unit Cell

• Notice that the various unit cells are compatible
  with symmetry elements or with combinations of
  these.

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Unit Cell

• In 3-dimensional crystalline patterns the choice
  of unit cell is generally made according to the
  following
• Rule 1 Edges of unit cells, if possible, should
  coincide with symmetry axes of lattice
• Rule 2 Edges should be related to each other by
  the symmetry of the lattice
• In all cases, the smallest possible cell should
  be chosen.

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Unit Cell

• The most general 3-dimensional structure is the
  parallelepiped (with lattice points only at its
  corners)
• 3 unequal angles that are not 90º, 60º or 120o
• 3 unequal edges a b c

Note Space lattice has three directions - a,
b, c
g

• Convention
• is angle between b and c
• b is angle between a and c
• g is angle between a and b

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Unit Cell

• Any regular 3-dimensional array can be outlined
  by a primitive unit cell (where points are on the
  lattice corners)
• Often desirable to choose a non-primitive unit
  cell (according to rule 1 and 2) coincident with
  symmetry axes and related by symmetry elements
• The space lattices compatible with the 32
  point-groups (lattice arrangements) are known as
  the 14 Bravais Lattices (variations of the 5
  plane lattice nets along a 3rd direction).
• The 14 different lattice types shown on p 230 -
  231 of Klein textbook
• These represent the only possible ways arranged
  in 3 dimensions
• The 14 Bravais lattices can be split into 6
  crystal systems

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Triclinic
Monoclinic

• Possible combinations

Primitive unit cell
Orthorhombic
Hexagonal
Non-primitive unit cell
Tetragonal
Auguste Bravais 1811-1863
Isometric cubic system