Symmetry Elements II 3 D Symmetry We now

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Symmetry Elements II

Symmetry Elements II

3 -D Symmetry We now have 8 unique 3 D symmetry operations: 1 2

3 -D Symmetry We now have 8 unique 3 D symmetry operations: 1 2 3 4 6 m 3 4 Combinations of these elements are also possible A complete analysis of symmetry about a point in space requires that we try all possible combinations of these symmetry elements

Point Group l The set of symmetry operations that leave the appearance of the

Point Group l The set of symmetry operations that leave the appearance of the crystal structure unchanged. l There are 32 possible point groups (i. e. , unique combinations of symmetry operations).

2 -D Symmetry Try combining a 2 -fold rotation axis with a mirror The

2 -D Symmetry Try combining a 2 -fold rotation axis with a mirror The result is Point Group 2 mm “ 2 mm” indicates 2 mirrors The mirrors are different

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Step 1: reflect

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Step 1: reflect Step 2: rotate 1

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Step 1: reflect Step 2: rotate 2

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Step 1: reflect Step 2: rotate 3

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Any other elements?

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Any other elements? Yes, two more mirrors

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name? ?

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror

2 -D Symmetry Now try combining a 4 -fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name? ? 4 mm Why not 4 mmmm?

2 -D Symmetry 3 -fold rotation axis with a mirror creates point group 3

2 -D Symmetry 3 -fold rotation axis with a mirror creates point group 3 m Why not 3 mmm?

2 -D Symmetry 6 -fold rotation axis with a mirror creates point group 6

2 -D Symmetry 6 -fold rotation axis with a mirror creates point group 6 mm

2 -D Symmetry The original 6 elements plus the 4 combinations creates 10 possible

2 -D Symmetry The original 6 elements plus the 4 combinations creates 10 possible 2 -D Point Groups: 1 2 3 4 6 m 2 mm 3 m 4 mm 6 mm Any 2 -D pattern of objects surrounding a point must conform to one of these groups

3 -D Symmetry As in 2 -D, the number of possible combinations is limited

3 -D Symmetry As in 2 -D, the number of possible combinations is limited only by incompatibility and redundancy There are only 22 possible unique 3 -D combinations, when combined with the 10 original 3 -D elements yields the 32 3 -D Point Groups

3 -D Symmetry The 32 3 -D Point Groups Every 3 -D pattern must

3 -D Symmetry The 32 3 -D Point Groups Every 3 -D pattern must conform to one of them. This includes every crystal, and every point within a crystal Table 5. 1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Crystal Systems l A grouping point groups that require a similar arrangement of axes

Crystal Systems l A grouping point groups that require a similar arrangement of axes to describe the crystal lattice. | l There are seven unique crystal systems.

3 -D Symmetry The 32 3 -D Point Groups Regrouped by Crystal System Table

3 -D Symmetry The 32 3 -D Point Groups Regrouped by Crystal System Table 5. 3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Triclinic l Three axes of unequal length l Angles between axes are not equal

Triclinic l Three axes of unequal length l Angles between axes are not equal l Point group: 1

Monoclinic l Three axes of unequal length l Angle between two axes is 90°

Monoclinic l Three axes of unequal length l Angle between two axes is 90° l Point groups: 2, m, 2/m

Orthorhombic l Three axes of unequal length l Angle between all axes is 90°

Orthorhombic l Three axes of unequal length l Angle between all axes is 90° l Point groups: 222 2/m/2/m, 2 mm

Tetragonal l Two axes of equal length l Angle between all axes is 90°

Tetragonal l Two axes of equal length l Angle between all axes is 90° l Point groups: 4, 4, 4/m, 4 mm, 422, 42 m, 4/m 2/m

Hexagonal l Four axes, three equal axes within one plane l Angle between the

Hexagonal l Four axes, three equal axes within one plane l Angle between the 3 co-planar axes is 60° l Angle with remaining axis is 90° l Point groups: 6, 6, 6/m, 6 mm, 622, 62 m, 6/m 2/m

Trigonal (Subset of Hexagonal) l Four axes, three equal axes within one plane l

Trigonal (Subset of Hexagonal) l Four axes, three equal axes within one plane l Angle between the 3 co-planar axes is 60° l Angle with remaining axis is 90° l Point groups: 3, 3, 3/m, 32/m

Cubic / Isometric l All axes of equal length l Angle between all axes

Cubic / Isometric l All axes of equal length l Angle between all axes is 90° l Point groups: 23, 423, 2/m 3, 43 m, 4/m 32/m

Crystal System Characteristics l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l

Crystal System Characteristics l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic ALL AXES EQUAL AXES UNEQUAL

Birefringence l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic ISOTROPIC

Birefringence l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic ISOTROPIC ANISOTROPIC

Crystal System Characteristics l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l

Crystal System Characteristics l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic ALL AXES EQUAL TWO AXES EQUAL ALL AXES UNEQUAL

Interference Figure l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic

Interference Figure l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic UNIAXIAL BIAXIAL

Crystal System Characteristics l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l

Crystal System Characteristics l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic ALL AXES EQUAL AXES ORTHOGONAL AXES NON-ORTHOGONAL

Extinction l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic PARALLEL

Extinction l Isometric/Cubic l Hexagonal l Tetragonal l Orthorhombic l Monoclinic l Triclinic PARALLEL INCLINED

Symmetry Elements II 3 D Symmetry We nowSymmetry Elements II 3 D Symmetry We now
Symmetry Elements II 3 D Symmetry We nowSymmetry Elements II 3 D Symmetry We now
Symmetry Elements II 3 D Symmetry We nowSymmetry Elements II 3 D Symmetry We now
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