Symmetry a. Two-fold rotation = 30 o /2 rotation a. Two-fold rotation = 30 o /2 rotation Operation Motif = the symbol for a two-fold rotation EESC 2100: Mineralogy 1
a. Two-fold rotation = 30 o /2 rotation first operation step a. Two-fold rotation = 30 o /2 rotation first operation step = the symbol for a two-fold rotation = the symbol for a two-fold rotation second operation step 3-D Symmetry a. Two-fold rotation = 30 o /2 rotation = the symbol for a two-fold rotation New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 30/2 Note: this is a temporary step, the intermediate motif element does not exist in the final pattern EESC 2100: Mineralogy 2
b. Three-fold rotation b. Three-fold rotation step 1 = 30 o /3 rotation = 30 o /3 rotation b. Three-fold rotation step 1 b. Three-fold rotation step 1 = 30 o /3 rotation = 30 o /3 rotation step 3 step 2 step 2 EESC 2100: Mineralogy 3
b. Three-fold rotation = 30 o /3 rotation 3 1 2 1-fold 2-fold 3-fold 4-fold -fold 3. Reflection (m) Reflection across a mirror plane 3. Reflection (m) Reflection across a mirror plane = symbol for a mirror plane = symbol for a mirror plane EESC 2100: Mineralogy 4
Symmetry elements can be combined Step 1: reflect Step 1: reflect Step 2: rotate EESC 2100: Mineralogy 5
Step 1: reflect Step 1: reflect Step 2: rotate Step 2: rotate Crystal Systems Step 1: reflect Step 2: rotate Similar arrangement of axes to describe crystal lattice Seven unique crystal systems Predictive of optical properties EESC 2100: Mineralogy
3-D Symmetry The 32 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,, 22, mm, 2m /m, /m 2/m 2/m Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m Triclinic Three axes of unequal length Angles between axes are not equal Point group: 1 Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons Monoclinic Three axes of unequal length Angle between two axes is 90 Point groups: 2, m, 2/m Orthorhombic Three axes of unequal length Angle between all axes is 90 Point groups: 222 2/m2/m/2/m, 2mm EESC 2100: Mineralogy 7
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 Hexagonal Four axes, three equal axes within one plane Angle between the 3 co-planar axes is 0 Angle with remaining axis is 90 Point groups:,, /m, mm, 22, 2m, /m2/m2/m Trigonal (Subset of Hexagonal) Four axes, three equal axes within one plane Angle between the 3 co-planar axes is 0 Angle with remaining axis is 90 Point groups: 3, 3, 3/m, 32, 32/m Cubic / Isometric All axes of equal length Angle between all axes is 90 Point groups: 23, 423, 2/m3, 43m, 4/m32/m EESC 2100: Mineralogy 8
Crystal System Characteristics Birefringence ALL AXES EQUAL ISOTROPIC AXES UNEQUAL ANISOTROPIC Crystal System Characteristics Interference Figure ALL AXES EQUAL TWO AXES EQUAL UNIAXIAL ALL AXES UNEQUAL BIAXIAL EESC 2100: Mineralogy 9
Crystal System Characteristics Extinction ALL AXES EQUAL AXES ORTHOGONAL PARALLEL AXES NON-ORTHOGONAL INCLINED EESC 2100: Mineralogy 10