2-D D Symmetry. Symmetry (Part II) Point Group m 3 4. The set of symmetry operations that leave the appearance of the crystal structure
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1 Symmetry (Part II) Lecture 6 Point Group The set of symmetry operations that leave the appearance of the crystal structure unchanged. There are 32 possible point groups (i.e., unique combinations of symmetry operations). 3-D D Symmetry We now have 8 unique 3D symmetry operations: 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 Try combining a 2-fold 2 The result is Point Group 2mm 2mm 2mm indicates 2 mirrors The mirrors are different 1
2 Step 1: reflect Step 2: rotate 1 Step 1: reflect Step 1: reflect Step 2: rotate 2 2
3 Step 1: reflect Step 2: rotate 3 Any other elements? Yes, two more mirrors Any other elements? Any other elements? Yes, two more mirrors Point group name?? 3
4 Any other elements? Yes, two more mirrors Point group name?? 4mm Why not 4mmmm? 6-fold creates point group 6mm 3-fold creates point group 3m Why not 3mmm? The original 6 elements plus the 4 combinations creates 10 possible 2-D D Point Groups: m 2mm 3m 4mm 6mm Any 2-D D pattern of objects surrounding a point must conform to one of these groups 4
5 3-D D Symmetry As in 2-D, 2 the number of possible combinations is limited only by incompatibility and redundancy There are only 22 possible unique 3-D 3 combinations, when combined with the 10 original 3-D 3 D elements yields the 32 3-D 3 D Point Groups Crystal Systems A grouping point groups that require a similar arrangement of axes to describe the crystal lattice. There are seven unique crystal systems. 3-D D Symmetry The 32 3-D 3 D Point Groups Every 3-D 3 D pattern must conform to one of them. This includes every crystal, and every point within a crystal Increasing Rotational Symmetry Rotation axis only Rotoinversion axis only 1 (= i ) 2 (= m) (= 3/m) Combination of rotation axes One rotation axis mirror 2/m 3/m (= 6) 4/m 6/m One rotation axis mirror 2mm 3m 4mm 6mm Rotoinversion with rotation and mirror 3 2/m 4 2/m 6 2/m Three rotation axes and mirrors 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/m Additional Isometric patterns /m 3 2/m 2/m 3 43m Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 3-D D Symmetry The 32 3-D 3 D Point Groups Regrouped by Crystal System Crystal System No Center Center Triclinic 1 1 Monoclinic 2, 2 (= m) 2/m Orthorhombic 222, 2mm 2/m 2/m 2/m Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m Hexagonal 3, 32, 3m 3, 3 2/m 6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 5
6 Triclinic Three axes of unequal length Angles between axes are not equal Point group: 1 Orthorhombic Three axes of unequal length Angle between all axes is 90 Point groups: 222 2/m2/m/2/m, 2mm Monoclinic Three axes of unequal length Angle between two axes is 90 Point groups: 2, m, 2/m Tetragonal Two axes of equal length Angle between all axes is 90 Point groups: 4, 4, 4/m, 4mm, 422, 42m, 4/m2/m2/m 6
7 Hexagonal Four axes, three equal axes within one plane Angle between the 3 co-planar axes is 60 Angle with remaining axis is 90 Point groups: 6, 6, 6/m, 6mm, 622, 62m, 6/m2/m2/m Cubic / Isometric All axes of equal length Angle between all axes is 90 Point groups: 23, 423, 2/m3, 43m, 4/m32/m Trigonal (Subset of Hexagonal) Four axes, three equal axes within one plane Angle between the 3 co-planar axes is 60 Angle with remaining axis is 90 Point groups: 3, 3, 3/m, 32, 32/m Crystal System Characteristics ALL AXES EQUAL AXES UNEQUAL 7
8 Birefringence ISOTROPIC ANISOTROPIC Interference Figure UNIAXIAL BIAXIAL Crystal System Characteristics ALL AXES EQUAL TWO AXES EQUAL ALL AXES UNEQUAL Crystal System Characteristics ALL AXES EQUAL AXES ORTHOGONAL AXES NON-ORTHOGONAL 8
9 Extinction PARALLEL INCLINED 9
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