Mineralogy

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Mineralogy

• Carleton College
• Winter 2003

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Lattice and its properties

• Lattice An imaginary 3-D framework, that can be
  referenced to a network of regularly spaced
  points each of which represents the position of a
  motif.

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Lattice and its properties

• line lattice
• plane lattice
• space lattice
• unit cell
• primitive and non-primitive cells

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Lattice and its properties

• I can generate a lattice line from a lattice
  point by translating my lattice point with a
  vector (a)

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Lattice and its properties

• I can generate a lattice line from a lattice
  point by translating my lattice point with a
  vector (a)

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Lattice and its properties

• I can generate a lattice line from a lattice
  point by translating my lattice point with a
  vector (a)

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Lattice and its properties

• I can generate a lattice line from a lattice
  point by translating my lattice point with a
  vector (a)

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Lattice and its properties

• Plane lattice by introducing another vector b,
  that is not in the same direction as a, I can
  produce a plane lattice

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Lattice and its properties

• Space lattice, by introducing another vector c,
  which is not in the same plane as a and b, I can
  generate a space lattice

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Unit Cell

• The smallest representative unit of structure
  which when repeated in 3-D gives the whole
  crystal.

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Structure

• Nearly all minerals are crystalline solids
  composed of atoms or ions held in an orderly, 3-D
  array by inter atomic forces. Such array of
  atoms are called crystal structure and are
  characterized by periodic duplication of any
  grouping of atoms along any line through the
  structure.
• In other wards the ordered arrangement of atoms
  or group of atoms within crystalline substance.

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Unit Cell

• How to choose a Unit cell from plane lattice?

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Choice of a Unit Cell
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Choice of a Unit Cell

• Look at this pattern, it is produced by simple
  translations.
• There are several possible choices for the Unit
  Cell.

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Choice of a Unit Cell
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Choice of a Unit Cell
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Choice of a Unit Cell
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Choice of a Unit Cell

• A lattice point occurs where the corners of four
  cells meet, and therefore, 1/4 point per corner
  lies in a give cell (1/4 41)

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Choice of a Unit Cell

• Unit Cells that include one lattice point, such
  as A, and B are called primitive Cells.
• Unit Cell C is Non-primitive.

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Choice of a Unit Cell

• Many different cells containing a single lattice
  point may be chosen.

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Choice of a Unit Cell

• How do you chose the Unit Cell?
• To keep the translations short
• To provide as highly specialized a lattice
  geometry as possible
• To have the cell shape comparable with the shape
  of the crystal

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Symmetry of a Lattice

• Lets see what symmetry exist in a lattice for a
  moment and we will come back to Unit Cell

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Elements of symmetry operations

• Symmetry operations Movements performed on an
  object such that when completed, the object looks
  the same as when you started.
• These include

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Elements of symmetry

• Translation
• Reflection
• Rotation
• Inversion
• Roto-inversion
• Roto-reflection
• Glide
• screw axis

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Elements of symmetry

• What elements of repetition exist?
• Translation

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Elements of symmetry

• What elements of repetition exist?
• Reflection/Mirror
• Mirror plane plane passed through object such
  that the images on opposite sides of the plane
  are mirror images of one another

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Elements of symmetry

• What elements of repetition exist?
• Reflection

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Elements of symmetry

• What elements of repetition exist?
• Rotation
• Rotation Axis - An axis through the object,
  around which the object is rotated such that the
  original "motif" (or appearance) is repeated a
  specific number of times during 360 degrees

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Elements of symmetry

• What elements of repetition exist?
• Rotation

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Elements of symmetry

• What elements of repetition exist?
• Rotation of 90 degrees will give me..

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Elements of symmetry

• What elements of repetition exist?
• Rotation of 90 degrees will give me..

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Elements of symmetry

• What elements of repetition exist?
• Rotation of 90 degrees will give me..

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Elements of symmetry

• What elements of repetition exist?
• Rotation of 90 degrees will give me..

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Elements of symmetry

• What elements of repetition exist?
• Rotation of 90 degrees will give me..

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Elements of symmetry

• What elements of repetition exist?
• Here is a different unit cell

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Elements of symmetry

• What elements of repetition exist?
• Here is a different unit cell

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Elements of symmetry

• What elements of repetition exist?
• Rotation of 60 degrees gives me another motif

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Elements of symmetry

• What elements of repetition exist?
• Rotation
• 1 axis 360 degrees
• 2 axes 180 degrees
• 3 axes 120 degrees
• 4 axes 90 degrees
• 6 axes 60 degrees

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Elements of symmetry

• What elements of repetition exist?
• Inversion

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Elements of symmetry

• What elements of repetition exist?
• Roto-inversion
• first a rotation, then an inversion of 180 degrees

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Elements of symmetry

• What elements of repetition exist?
• Roto-reflection

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Elements of symmetry

• What elements of repetition exist?
• Glide

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Elements of symmetry

• What elements of repetition exist?
• Glide

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Elements of symmetry

• What elements of repetition exist?
• Glide

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Elements of symmetry

• What elements of repetition exist?
• Glide

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Elements of symmetry

• What elements of repetition exist?
• Glide

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Elements of symmetry

• What elements of repetition exist?
• Glide

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Elements of symmetry

• What elements of repetition exist?
• Glide

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Elements of symmetry

• What elements of repetition exist?
• screw axis
• This include translation and rotation together

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Screw Axis
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Unit Cell

• Unit Cell parameters
• a, b, c (sides)
• a, b, g angles

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Unit Cell

• Unit Cell parameters
• a, b, c sides
• a, b, g angles

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Translation Symmetry

• A translation is simply moving an object in some
  direction (a, b, c) without a rotation. Hence a
  point (x, y, z) is translated to the point (xa,
  yb, zc).

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Translation Symmetry

• Crystalline materials have structures with
  translational symmetry. The unit cell of the
  crystal contains the smallest atomic group that
  is needed to define the structure under
  repetition.

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Translational Nets in 2-D

• There are five different ways to translate a
  point in two-dimensions. Here is the first
  simple net.

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Translational Nets in 2-D

• There are five different ways to translate a
  point in two-dimensions. Here is the second
  simple net.

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Translational Nets in 2-D

• There are five different ways to translate a
  point in two-dimensions. Here is the third
  simple net.

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Translational Nets in 2-D

• There are five different ways to translate a
  point in two-dimensions. Here is the fourth
  simple net.

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Translational Nets in 2-D

• There are five different ways to translate a
  point in two-dimensions. Here is the fifth
  simple net.

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Translational Nets in 2-D (cont.)

• The diamond net can also be defined in terms of a
  centered rectangular net with a1 a2 and g
  90degrees.