symmetry, groups and crystal structures the seven crystal systems
TRANSCRIPT
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Symmetry, Groups and Symmetry, Groups and Crystal StructuresCrystal Structures
The Seven Crystal SystemsThe Seven Crystal Systems
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Minerals structures are described in terms of the unit cell
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The Unit Cell• The unit cell of a mineral is the smallest
divisible unit of mineral that possesses all the symmetry and properties of the mineral.
• It is a small group of atoms arranged in a “box” with parallel sides that is repeated in three dimensions to fill space.
• It has three principal axes (a, b and c) and• Three inter-axial angles (, , and )
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The Unit Cell• Three unit cell vectors a, b, c• Three angles between
vectors: • is angle between b and c
• is angle between a and c
• is angle between a and b
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Seven Crystal Systems
• The presence of symmetry operators places constraints on the geometry of the unit cell.
• The different constraints generate the seven crystal systems.– Triclinic Monoclinic– Orthorhombic Tetragonal– Trigonal Hexagonal– Cubic (Isometric)
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Seven Crystal Systems
• Triclinic a b c; 90º 120º• Monoclinic a b c; = º 90º 120º• Orthorhombic a b c; = º• Tetragonala = b c; = º• Trigonal a = b c; = º; 120º• Hexagonal a = b c; = º; 120º• Cubic a = b = c; = º
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Symmetry Operations
• A symmetry operation is a transposition of an object that leaves the object invariant.– Rotations
• 360º, 180º, 120º, 90º, 60º
– Inversions (Roto-Inversions)• 360º, 180º, 120º, 90º, 60º
– Translations: • Unit cell axes and fraction thereof.
– Combinations of the above.
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Rotations
• 1-fold 360º IIdentity
• 2-fold 180º 2• 3-fold 120º 3• 4-fold 90º 4• 6-fold 60º 6
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Roto-Inversions(Improper Rotations)
• 1-fold 360º• 2-fold 180º• 3-fold 120º• 4-fold 90º• 6-fold 60º
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Translations
• Unit Cell Vectors• Fractions of unit cell vectors
– (1/2, 1/3, 1/4, 1/6)
• Vector Combinations
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Groups
• A set of elements form a group if the following properties hold:– Closure: Combining any two elements gives a
third element– Association: For any three elements: (ab)c =
a(bc).– Identity: There is an element, I, such that Ia = aI
= a– Inverses: For each element, a, there is another
element, b, such that ab = I = ba
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Groups
• The elements of our groups are symmetry operators.
• The rules limit the number of groups that are valid combinations of symmetry operators.
• The order of the group is the number of elements.
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Point Groups (Crystal Classes)
• We can do symmetry operations in two dimensions or three dimensions.
• We can include or exclude the translation operations.
• Combining proper and improper rotation gives the point groups (Crystal Classes)– 32 possible combinations in 3 dimensions– 32 Crystal Classes (Point Groups)– Each belongs to one of the (seven) Crystal
Systems
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Space Groups
• Including the translation operations gives the space groups.– 17 two-dimensional space groups– 230 three dimensional space groups
• Each space group belongs to one of the 32 Crystal Classes (remove translations)
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Crystal Morphology
• A face is designated by Miller indices in parentheses, e.g. (100) (111) etc.
• A form is a face plus its symmetric equivalents (in curly brackets) e.g {100}, {111}.
• A direction in crystal space is given in square brackets e.g. [100], [111].
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Halite Cube
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Miller Indices
• Plane cuts axes at intercepts (,3,2).
• To get Miller indices, invert and clear fractions.
• (1/, 1/3, 1/2) (x6)=• (0, 2, 3)• General face is (h,k,l)
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Miller Indices
• The cube face is (100)• The cube form {100}
comprises faces (100),(010),(001), (-100),(0-10),(00-1)
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Halite Cube (100)
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Stereographic Projections
• Used to display crystal morphology.
• X for upper hemisphere.
• O for lower.
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Stereographic Projections
• We will use stereographic projections to plot the perpendicular to a general face and its symmetry equivalents (general form hkl).
• Illustrated above are the stereographic projections for Triclinic point groups 1 and -1.
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Anatase TiO2
(tetragonal)
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Halite Cube
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Halite Cube