The internal order of minerals: Lattices, Unit Cell

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The internal order of minerals Lattices, Unit
Cell Bravais Lattices

• Geol 3055
• Klein (22nd ed), pages 213-221
• 229-234

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Definition of a mineral

• Naturally occuring
• Homogenous solid
• Definite (but not fixed) chemical composition
• Defined physical properties
• Highly ordered atomic arrangement
• Usually formed by inorganic processes

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• Ordered atoms distinguished crystals (solids)
  from liquids, gases and glasses
• Orderedperiodic repetition of atoms of atom or
  ion througout an infinite atomic array.
• An atom is surrounded by an identical arrangement
  of neighboring atoms, which are n quantity of
  unit cells
• Unit cells dimensions 5-20 angstroms
• (1A10-8cm)

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Translation

• Example of translation (vectors)
• , , , ,
• , , , ,

Translation in y-axis
Translation in x-axis
Translation symbols are t1 for the y axis
translation and t2 for the x-axis translation
for 2-D figures. 3-D figures have a t3
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One-dimensional order (rows)

• Motifs, nodes or objects in a row
• In a row the magnitude of one translation
  determines spacing (distance)

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Two dimensional order (plane lattices)

• Regular translation in two different directions
• The connection of four nodes in the figure
  represent a unit cell (smallest building unit).
  Various unit cells produce a plane lattice.

y
?
x
Unit cell
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Lattices

• When motifs (commas) are substitute by points
  (nodes) the pattern is called a lattice. The
  nodes represent atoms or ions.
• Lattice is an imaginary pattern of points (or
  nodes) in which every point has an environment
  that is identical to that of any other point
  (node) in a pattern. A lattice has no specific
  origin, as it can be shifted parallel to itself

a
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Plane lattices

• The are ONLY 5 possible and distinct plane
  lattices or nets (see figure 5.50)
• Result by the repetition of a row (translation
  along y)
• Depend on the angle ? between x and y, and the
  size of the b translation along y
• See Fig 5.50

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Unit cells produce by arrays of nodes
Parallelogram a?b, ??90o Fig 5.50a
Rectangles a?b, ?90o Figs 5.50a b
Square, a1a2, ?90o fig 5.50 e
Diamond a1a2, ??90o,60o,120o fig 5.50c
Rhombus a1a2, ?60o or 120o Fig 5.50d
P primitive (only nodes that produce the unit
cell are _at_ corners of figure C centered (node
at center of unit cell, is called non primitive
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Three-dimensional order

• Three vectors (a, b, c) instead of two (a b)
• The stacking in the c-axis, of the five planar
  nets discussed in 2-dimensional figures (fig.
  5.50), will produce 14 different lattice types
  known as the Bravais Lattices (see figs. 5.62
  5.63)
• ONLY possible ways which points can be arranged
  periodically in 3 dimensions
• Coincide with the 32 crystal classes studied in
  class!
• (see CD-ROM Three dimensional order Generation
  of the Bravais Lattices)

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Three-dimensional order unit cells

• Since a lot of unit cells are possible in 3-d
  figures, crystallographer drawn some rules to
  minimize the number
• Edges of unit cells should coincide, if possible,
  with symmetry axes of the lattice
• Edges should be related to each other by the
  symmetry of the lattice
• The smallest possible cell should be chosen in
  accordance with first two rules.

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14 Bravais Lattices
P primitive C centered I body centered
node at center of figure F face centered
(node at the center of face(s)