symmetry, groups and crystal structures
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Symmetry, Groups and Crystal Structures. The Seven Crystal Systems. Minerals structures are described in terms of the unit cell. The Unit Cell. The unit cell of a mineral is the smallest divisible unit of mineral that possesses the symmetry and properties of the mineral. - PowerPoint PPT PresentationTRANSCRIPT
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Symmetry, Groups and Crystal Structures
The 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 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 interaxial angles (, , and )
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The Unit Cell
• 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 I
Identity• 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} is 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.