Kristalografi

Bentuk dan susunan mineral dengan komposisi Yang sama memiliki keteraturan susunan Kristal yang sama pulam 1 Symmetry Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern Operation: some act that reproduces the motif to create the pattern Element: an operation located at a particular point in space 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation = 360o/2 rotation to reproduce a motif in a symmetrical pattern A Symmetrical Pattern 6 6 2 2-D Symmetry Symmetry Elements Operation 1. Rotation a. Two-fold rotation = 360o/2 rotation to reproduce a motif in a symmetrical pattern = the symbol for a two-fold rotation 6 Motif Element 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation = rotation to reproduce a motif in a symmetrical pattern = the symbol for a two-fold rotation 360o/2 6 6 second operation step 6 first operation step 3 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 4 Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 5 . Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1.2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic symmetry 2-D Symmetry Symmetry Elements 1. Rotation a. Rotation a.2-D Symmetry Symmetry Elements 1. Two-fold rotation Some familiar objects have an intrinsic symmetry 6 . Two-fold rotation Some familiar objects have an intrinsic symmetry 180o rotation makes it coincident Second 180o brings the object back to its original position What’s the motif here?? 2-D Symmetry Symmetry Elements 1. Three-fold rotation = 360o/3 rotation to reproduce a motif in a symmetrical pattern 6 6 6 7 .2-D Symmetry Symmetry Elements 1. Rotation b. Rotation a. Rotation b.2-D Symmetry Symmetry Elements 1. Three-fold rotation = rotation to reproduce a motif in a symmetrical pattern 360o/3 6 step 3 step 1 6 2-D Symmetry Symmetry Elements 1. Thus we will exclude them now. 6 6 6 6 6 step 2 6 6 6 6 8 . Rotation 6 6 6 6 6 6 6 6 1-fold 2-fold 3-fold 4-fold 6-fold Objects with symmetry: a identity Z 5-fold and > 6-fold rotations will not work in combination with translations in crystals (as we shall see later). Inversion (i) inversion through a center to reproduce a motif in a symmetrical pattern = symbol for an inversion center inversion is identical to 2-fold rotation in 2-D. and 3-fold rotations in a cube Click on image to run animation 2-D Symmetry Symmetry Elements 2. 2-fold. but is unique in 3-D (try it with your hands) 6 6 9 .4-fold. 2-D Symmetry Symmetry Elements 3. Reflection (m) Reflection across a “mirror plane” reproduces a motif = symbol for a mirror plane 2-D Symmetry We now have 6 unique 2-D symmetry operations: 1 2 3 4 6 m Rotations are congruent operations reproductions are identical Inversion and reflection are enantiomorphic operations reproductions are “opposite-handed” 10 . we must try all possible combinations of these symmetry elements •In the interest of clarity and ease of illustration.2-D Symmetry •Combinations of symmetry elements are also possible •To create a complete analysis of symmetry about a point in space. we continue to consider only 2-D examples 2-D Symmetry Try combining a 2-fold rotation axis with a mirror 11 . 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect (could do either step first) 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) 12 . 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) Is that all?? 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) No! A second mirror is required 13 . 2-D Symmetry Try combining a 2-fold rotation axis with a mirror The result is Point Group 2mm “2mm” indicates 2 mirrors The mirrors are different (not equivalent by symmetry) 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror 14 . 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 15 . 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 16 . 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 17 . two more mirrors Point group name?? 4mm Why not 4mmmm? 18 .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. 2-D Symmetry 3-fold rotation axis with a mirror creates point group 3m Why not 3mmm? 2-D Symmetry 6-fold rotation axis with a mirror creates point group 6mm 19 . 2-D Symmetry All other combinations are either: Incompatible (2 + 2 cannot be done in 2-D) Redundant with others already tried m + m → 2mm because creates 2-fold This is the same as 2 + m → 2mm 2-D Symmetry The original 6 elements plus the 4 combinations creates 10 possible 2-D Point Groups: 1 2 3 4 6 m 2mm 3m 4mm 6mm Any 2-D pattern of objects surrounding a point must conform to one of these groups 20 . 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 ) 3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity) 21 . Rotoinversion a. so not a new operation Sistem Kristal Asimetrik x x 22 . 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity) Step 2: invert This is the same as i.3-D Symmetry New 3-D Symmetry Elements 4. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Step 2: invert 23 . the intermediate motif element does not exist in the final pattern 3-D Symmetry New Symmetry Elements 4. Rotoinversion b. Rotoinversion b.3-D Symmetry New Symmetry Elements 4. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Note: this is a temporary step. Rotoinversion b. 2-fold rotoinversion ( 2 ) The result: 3-D Symmetry New Symmetry Elements 4. so not a new operation 24 .3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) This is the same as m. the intermediate motif element does not exist in the final pattern 25 . Rotoinversion c. Rotoinversion c. this is a temporary step. 3-fold rotoinversion ( 3 ) 3-D Symmetry New Symmetry Elements 4. 3-fold rotoinversion ( 3 ) 1 Step 1: rotate 360o/3 Again.3-D Symmetry New Symmetry Elements 4. 3-D Symmetry New Symmetry Elements 4. 3-fold rotoinversion ( 3 ) 1 Completion of the first sequence 2 26 . Rotoinversion c. 3-fold rotoinversion ( 3 ) Step 2: invert through center 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. Rotoinversion c.3-D Symmetry New Symmetry Elements 4. 3-fold rotoinversion ( 3 ) Rotate another 360/3 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Invert through center 27 . 3-fold rotoinversion ( 3 ) Third step creates face 4 (3 → (1) → 4) 3 1 4 2 28 . Rotoinversion c. 3-fold rotoinversion ( 3 ) 3 Complete second step to create face 3 2 1 3-D Symmetry New Symmetry Elements 4. Rotoinversion c.3-D Symmetry New Symmetry Elements 4. 3-D Symmetry New Symmetry Elements 4. 3-fold rotoinversion ( 3 ) Fourth step creates face 5 (4 → (2) → 5) 5 1 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Fifth step creates face 6 (5 → (3) → 6) Sixth step returns to face 1 5 1 6 29 . Rotoinversion c. 3-D Symmetry New Symmetry Elements 4. 4-fold rotoinversion ( 4 ) 30 . Rotoinversion c. Rotoinversion d. 3-fold rotoinversion ( 3 ) 3 This is unique 5 4 6 1 2 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3-D Symmetry New Symmetry Elements 4. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 31 . Rotoinversion d.3-D Symmetry New Symmetry Elements 4. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert 32 . Rotoinversion d. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert 3-D Symmetry New Symmetry Elements 4.3-D Symmetry New Symmetry Elements 4. Rotoinversion d. Rotoinversion d.3-D Symmetry New Symmetry Elements 4. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert 33 . 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 34 . Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert 3-D Symmetry New Symmetry Elements 4.3-D Symmetry New Symmetry Elements 4. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 6: Invert 3-D Symmetry New Symmetry Elements 4.3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) This is also a unique operation 35 . Rotoinversion d. Rotoinversion e. 6-fold rotoinversion ( 6 ) Begin with this framework: 36 . 4-fold rotoinversion ( 4 ) A more fundamental representative of the pattern 3-D Symmetry New Symmetry Elements 4. Rotoinversion d.3-D Symmetry New Symmetry Elements 4. 6-fold rotoinversion ( 6 ) 1 3-D Symmetry New Symmetry Elements 4. Rotoinversion e.3-D Symmetry New Symmetry Elements 4. 6-fold rotoinversion ( 6 ) 1 37 . Rotoinversion e. 6-fold rotoinversion ( 6 ) 2 1 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. Rotoinversion e. 6-fold rotoinversion ( 6 ) 2 1 38 .3-D Symmetry New Symmetry Elements 4. 3-D Symmetry New Symmetry Elements 4. 6-fold rotoinversion ( 6 ) 3 2 1 39 . 6-fold rotoinversion ( 6 ) 3 2 1 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. Rotoinversion e. Rotoinversion e.3-D Symmetry New Symmetry Elements 4. 6-fold rotoinversion ( 6 ) 3 2 1 4 40 . Rotoinversion e. 6-fold rotoinversion ( 6 ) 3 2 1 4 3-D Symmetry New Symmetry Elements 4. 6-fold rotoinversion ( 6 ) 3 5 2 1 4 41 . 6-fold rotoinversion ( 6 ) 3 5 2 1 4 3-D Symmetry New Symmetry Elements 4.3-D Symmetry New Symmetry Elements 4. Rotoinversion e. Rotoinversion e. 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 3 5 2 1 6 4 3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane Top View (combinations of elements follows) 42 . 3-D Symmetry New Symmetry Elements 4. 6-fold rotoinversion ( 6 ) A simpler pattern Top View 3-D Symmetry We now have 10 unique 3-D symmetry operations: 1 2 3 4 6 i m 3 4 6 •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 43 . Rotoinversion e. as we shall see 3-D Symmetry 3-D symmetry element combinations d. also 4/m. 6mm b. Rotation axis ⊥ mirror -----. 6/m c.beberapa mineral 2 ⊥ m = 2/m 3 ⊥ m = 3/m. Rotation axis parallel to a mirror Same as 2-D 2 || m = 2mm 3 || m = 3m. also 4mm. Combinations of rotations 2 + 2 at 90o → 222 (third 2 required from combination) 4 + 2 at 90o → 422 ( “ “ “ ) “ “ ) 6 + 2 at 90o → 622 ( “ 44 .3-D Symmetry 3-D symmetry element combinations a. Most other rotations + m are impossible 2-fold axis at odd angle to mirror? Some cases at 45o or 30o are possible. the number of possible combinations is limited only by incompatibility and redundancy There are only 22 possible unique 3-D combinations.3-D Symmetry As in 2-D. John Wiley and Sons 45 .18 of Klein (2002) Manual of Mineral Science. when combined with the 10 original 3-D elements yields the 32 3-D Point Groups 3-D Symmetry But it soon gets hard to visualize (or at least portray 3-D on paper) Fig. 5. 4/m 2/m 2/m 3. 2mm 4. John Wiley and Sons 3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System (more later when we consider translations) Crystal System Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal No Center 1 2. 6.3 of Klein (2002) Manual of Mineral Science. 43m Center 1 2/m 2/m 2/m 2/m 4/m. and every point within a crystal Increasing Rotational Symmetry Rotation axis only Rotoinversion axis only Combination of rotation axes One rotation axis ⊥ mirror One rotation axis || mirror Rotoinversion with rotation and mirror Three rotation axes and ⊥ mirrors Additional Isometric patterns 2/m 2/m 2/m 23 2/m 3 1 1 (= i ) 2 2 (= m) 222 2/m 2mm 3 3 32 3/m (= 6) 3m 3 2/m 4 4 422 4/m 4mm 4 2/m 4/m 2/m 2/m 432 43m 6 6 (= 3/m) 622 6/m 6mm 6 2/m 6/m 2/m 2/m 4/m 3 2/m Table 5. 32. 2 (= m) 222.1 of Klein (2002) Manual of Mineral Science. 4. 42m 3. 432. 3m 6.3-D Symmetry The 32 3-D Point Groups Every 3-D pattern must conform to one of them. 4mm. 422. This includes every crystal. John Wiley and Sons 46 . 3 2/m 6/m. 622. 6/m 2/m 2/m 2/m 3. 6mm. 4/m 3 2/m Table 5. 62m Isometric 23. © MSA 47 . Crystallography and Crystal Chemistry.3-D Symmetry The 32 3-D Point Groups After Bloss.