Symmetry Elements II Lecture 6 3 D Symmetry
- Slides: 33
Symmetry Elements II Lecture 6
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 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 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 Step 1: reflect
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 Step 1: reflect Step 2: rotate 2
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 Any other elements?
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 Any other elements? Yes, two more mirrors Point group name? ?
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 m Why not 3 mmm?
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 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 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 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 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 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 l Point group: 1
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° l Point groups: 222 2/m/2/m, 2 mm
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 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 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 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 Triclinic ALL AXES EQUAL AXES UNEQUAL
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 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 UNIAXIAL BIAXIAL
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 INCLINED
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