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Crystallographic Concepts

GLY 4200 Fall, 2012

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Atomic Arrangement

• Minerals must have a highly ordered atomic arrangement

• The crystal structure of quartz is an example

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Quartz Crystals

• The external appearance of the crystal may reflect its internal symmetry

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Quartz Blob

• Or the external appearance may show little or nothing of the internal structure

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Building Blocks

• A cube may be used to build a number of forms

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Fluorite

• Fluorite may appear as octahedron (upper photo)

• Fluorite may appear as a cube (lower photo), in this case modified by dodecahedral crystal faces

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Crystal Growth

• Ways in which a crystal can grow: Dehydration of a solution Growth from the molten state (magma or lava) Direct growth from the vapor state

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Unit Cell

• Simplest (smallest) parallel piped outlined by a lattice

• Lattice: a two or three (space lattice) dimensional array of points

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Lattice Requirements

• Environment about all lattice points must be identical

• Unit cell must fill all space, with no “holes”

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Auguste Bravais

• Found fourteen unique lattices which satisfy the requirements

• Published Études Crystallographiques in 1849

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Isometric Lattices

• P = primitive• I = body-centered (I for German innenzentriate)• F = face centered• a = b = c, α = β = γ = 90 ̊

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Tetragonal Lattices

• a = b ≠c• α = β = γ = 90 ̊

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Tetragonal Axes

• The tetragonal unit cell vectors differ from the isometric by either stretching the vertical axis, so that c > a (upper image) or compressing the vertical axis, so that c < a (lower image)

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Orthorhombic Lattice

• a ≠ b ≠c• α = β = γ = 90 ̊• C - Centered: additional point in the center of each end of

two parallel faces

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Orthorhombic Axes

• The axes system is orthogonal

• Common practice is to assign the axes so the the magnitude of the vectors is c > a > b

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Monoclinic Lattice

• a ≠ b ≠c• α = γ = 90 ̊ (β ≠ 90 ̊)

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Monoclinic Axes

• The monoclinic axes system is not orthogonal

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Triclinic Lattice

• a ≠ b ≠c• α ≠ β ≠ γ ≠ 90 ̊

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Triclinic Axes

• None of the axes are at right angles to the others

• Relationship of angles and axes is as shown

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Hexagonal

• Some crystallographers call the hexagonal group a single crystal system, with two divisions Rhombohedral division Hexagonal division

• Others divide it into two systems, but this practice is discouraged

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Hexagonal Lattice

• a = b ≠ c• α = γ = 90 ̊ • β = 120 ̊

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Rhombohedral Lattice

• a = b = c• α = β = γ ≠ 90 ̊

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Hexagonal Axes

• The hexagonal system uses an ordered quadruplicate of numbers to designate the axes

• a1, a2, a3, c

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Arrangement of Ions

• Ions can be arranged around the lattice point only in certain ways

• These are known as point groups

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Crystal Systems

• The six different groups of Bravais lattices are used to define the Crystal Systems

• The thirty-two possible point groups define the crystal classes

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Point Group

• Point indicates that, at a minimum, one particular point in a pattern remains unmoved

• Group refers to a collection of mathematical operations which, taken together, define all possible, nonidentical, symmetry combinations