کریستالوگرافی بخش تقارن آینه

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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 

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تناع ([-2 Symmetry Elements 1. Rotation a. Two-fold rotation 6 A Symmetrical Pattern < 2 rotation to reproduce a motif ina 9 symmetrical pattern 

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2-D Symmetry Symmetry Elements 1. Rotation 1 0 ‏ا‎ ‎a. Two-fold / Motif rotation 6 yperation سمس .قم 3609/2 = rotation to reproduce ۱ a motif ina 9 symmetrical pattern 

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2-D Symmetry Symmetry Elements 1. Rotation ‏ی سم ام و‎ a. Two-fold 6 first rotation 5 \ operati Cee re 3 \on step \ = 360°/2 ۱ 0 | rotation \ / to reproduce secon¢ J a motif ina ‏لتساك‎ 9 ۳ ۱ aap ‏سستد‎ symmetrical pattern 

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تناع ([-2 Symmetry Elements 1. Rotation ‏تعسو‎ We ioe i Se ‏سک ها‎ a. Two-fold rotation Some familiar objects have an intrinsic symmetry 

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تناع ([-2 Symmetry Elements 1. Rotation ‏سب هو عم هو موز‎ Se a. Two-fold rotation Some familiar objects have an intrinsic symmetry 

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تناع ([-2 Symmetry Elements 1. Rotation ‏سب هو عم هو موز‎ Se a. Two-fold rotation Some familiar objects have an intrinsic symmetry 

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تناع ([-2 Symmetry Elements 1. Rotation ‏سب هو عم هو موز‎ Se a. Two-fold rotation Some familiar objects have an intrinsic symmetry 

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تناع ([-2 Symmetry Elements 1. Rotation ‏سب هو عم هو موز‎ Se a. Two-fold rotation Some familiar objects have an intrinsic symmetry 

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تناع ([-2 Symmetry Elements 1. Rotation ‏سب هو عم هو موز‎ Se a. Two-fold rotation Some familiar objects have an intrinsic symmetry 

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2-D Symmetry Symmetry Elements 1. Rotation a. Two-fold rotation Some familiar objects have an intrinsic Second 180° hr 8 bject baci: to its ortyinal ation makes it ‏و وم وز‎ motif here?? 

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تناع ([-2 Symmetry Elements 1. Rotation b. Three-fold rotation © = 360°/3 rotation to reproduce a motif in a symmetrical pattern 9 

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تناع ([-2 Symmetry Elements 1. Rotation b. Three-fold —————————— rotation = 360°/3 rotation to reproduce a motif in a symmetrical pattern 

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لتنا ([-2 Symmetry Elements 1 Ratatian 6 6 6 © © © ها 2 ‎A 5 a‏ و 9 9 و 3-fold 4-fold 6-fold | لك ۲ 2-0 symmeti y 1 1-fold Objects wit! u identity 5-fold and > 6-fold rotations will not work in combination with translations in crystals (as we shall see later). Thus we will exclude them now. 

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4-fold, 2-fold, and 3-fold rotations in a cube Click on image to run animation 

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تناع ([-2 Symmetry Elements 2. Inversion (i) ‘ inversion through a 6 center to reproduce a motif in a symmetrical ٩ pattern = symbol for an inversion center inversion is identical to 2-fold rotation in 2-D, 

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تناع ([-2 Symmetry Elements 3. Reflection (m) a “mirror plane” reproduces a motif Reflection across 6 = symbol for a mirror ل لكك ‎plane‏ 

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2-[( ‏تناع‎ ‎We now have 6 unique 2-D symmetry operations: 12203 54% 6M Rotations are congruent operations reproductions are identical Inversion and reflection are enantiomorphic operations reproductions are “opposite-handed” 

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2-[( ‏تناع‎ ‎Combinations of symmetry elements are also possible To create a complete analysis of symmetry about a point in space, we must try all possible combinations of these symmetry elements In the interest of clarity and ease of illustration, we continue to consider only 2-D examples 

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2-[( ‏لتنا‎ ‎Try combining a 2-fold rotation axis with a mirror 

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2-[( ‏تناع‎ ‎Try combining a 2-fold rotation axis with a mirror Step 1: reflect ۱ 6/90 (could do either step first) 

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2-[( ‏تناع‎ ‎Try combining a 2-fold rotation axis with a mirror Step 1: reflect ۱ 6/90 Step 2: rotate (everything) 

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2-[( ‏تناع‎ ‎Try combining a 2-fold rotation axis with a mirror Step 1: reflect ۱ 6/90 Step 2: rotate (everything) Is that all?? 

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2-[( ‏تناع‎ ‎Try combining a 2-fold rotation axis with a mirror Step 1: reflect ۱ 6/90 Step 2: rotate pal No! A second mirror is require 

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2-[( ‏تناع‎ ‎Try combining a 2-fold rotation axis with a mirror The result is Point Group 2mm 9 “2mm” indicates 2 ‏ب‎ The mirrors are differen (not equivalent by symmetry) 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Step 1: reflect 6/90 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Step 1: reflect ۱ 6/90 Step 2: rotate 1 a 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Step 1: reflect ۱ 6/90 Step 2: rotate 2 a 0 ۱9 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Step 1: reflect ۱ 6/90 Step 2: rotate 3 ‏م‎ Oo 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Any other elements? 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors e 9 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Any other elements? 9 Yes, two more mirrors 0 9 Point group name?? ‏سس‎ 0 ۱9 

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2-[( ‏تناع‎ ‎Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors 0 9 Point group name?? 4mm Why not 0۱ 9 4mmmm? 

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تناع ([-2 3-fold rotation axis with a mirror creates point group 3m Why not 3mmm? 

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تناع ([-2 6-fold rotation axis with a mirror creates point group 6mm 

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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 

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تناع ([-2 The original 6 elements plus the 4 combinations creates 10 possible 2-D Point Groups: 12.3 4 6 m 2mm 3m 4mm 6mm Any 2-D pattern of objects surrounding a point must conform to one of these groups 

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3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion (1 ) @& | 

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3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion | a. 1-fold rotoinversion (1 ) ۵ ۲ | Step 1: rotate 360/1 (identity) 

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3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion (1 ) Step 1: rotate 360/1 (identity) Step 2: invert This is the same as i, so nota اد دمم ی 

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3-D Symmetry New Symmetry Elements 2 45.565. 2. 2-5010 9 (2) | Step 1: rotate 360/; Note: this is a temporary step, the intermediate motif —— 

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3-D Symmetry New Symmetry Elements 2 45.565. 2. 2-5010 (2) Step 1: rotate 360/; Step 2: invert 

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3-D Symmetry New Symmetry Elements 2 45.565. b. 2-fold rotoinversion ( 2 ) The result: 

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3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion 2) This is the same as m, so not anew operation 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion c. 3-fold rotoinversion 3) Step 1: rotate 3609/3 Again, this is a temporary step, the intermediate — ‏لل‎ ۱5 cy woe, IN ma aes Ss 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion c. 3-fold rotoinversion 3) Step 2: invert through center 

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3-D Symmetry New Symmetry Elements 7 45.565. c. 3-fold rotoinversion (3) Completion of the first sequence ‘ 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion c. 3-fold rotoinversion 3) Rotate another 360/3 #4 

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3-D Symmetry New Symmetry Elements an 4. Rotoinversion c. 3-fold rotoinversion (3) Invert through center 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion _ c. 3-fold rotoinversion 3) Complete second step to create face 3 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion c. 3-fold rotoinversion 3) Third step creates face 4 (3> (1) > 4) 

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3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion 3) Fourth step creates face 5 (4 (2) > 5) 

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3-D Symmetry New Symmetry Elements ive 4. Rotoinversion_ c. 3-fold rotoinversion (3) Fifth step creates face 6 (5 > (3) > 6) Sixth step returns to — 

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3-D Symmetry New Symmetry 2 Elements a 4. Rotoinversion c. 3-fold rotoinversion This is unique 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion d. 4-fold rotoinversion ( 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion d. 4-fold rotoinversion 4 (4) / 

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3-D Symmetry V New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion (4) 1: Rotate 360/4 

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3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 1: Rotate 360/4 2: Invert 

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3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 1: Rotate 360/4 2: Invert 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion d. 4-fold rotoinversion ‏ثم‎ ‎C4) ‎3: Rotate 360/4 Vy 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion ۱ d. 4-fold rotoinversion ( 4 3: Rotate 4 4: Invert 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion ۱ 0. 4-1010 rotoinversion ( 4) 3: Rotate 360/4 4: Invert 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion d. 4-fold rotoinversion C4) 5: Rotate 360/4 

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3-D Symmetry New Symmetry Elements 0 4. Rotoinversion d. 4-fold rotoinversion C4) 5: Rotate 360/4 6: Invert 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion d. 4-fold rotoinversion C4) This is also a unique operation 

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3-D Symmetry New Symmetry Elements a 4. Rotoinversion d. 4-fold rotoinversion (4) A more fundamental representative of the pattern 

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3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion (6 ) Begin with this framework: 

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3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6) 

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3-D Symmetry New Symmetry Elements ‏تس‎ ‎4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements ‏تس‎ ‎4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements 5 4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements 5 4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements 5 4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements 5 4. Rotoinversion e. 6-fold rotoinversion ( 

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3-D Symmetry New Symmetry Elements 5-6 4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements 5-6 4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements 5 4. Rotoinversion e. 6-fold rotoinversion (6) 

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3-D Symmetry New Symmetry Elements o 4. Rotoinversion 3-fold rotation axis Top View perpendicular to a mirror plane (combinations of elements ‏سح‎ 

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3-D Symmetry Top, View A New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion (6) A simpler pattern 

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3-D Symmetry We now have 10 unique 3-D symmetry operations: re ‏ل ا ل‎ ee 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 

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3-D Symmetry 3-D symmetry element combinations a. Rotation axis parallel to a mirror Same as 2-D 2 || m = 2mm 3||m=3m, also 4mm, 6mm b. Rotation axis | mirror 21m =2/m 31m=3/m, also 4/m, 6/m c. Most other rotations + m are impossible 2-fold axis at odd angle to mirror? Some cases at 45° or 30° are possible, as Yeas Ce 

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3-D Symmetry 3-D symmetry element combinations d. Combinations of rotations 2+ 2 at 90° — 222 (third 2 required from combination) 9 = ۴ ) 422 900-۰ اج 2 + 4 ) 64-2 at 90? > 622-(* te e ) 

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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 

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But it soon gets hard to visualize (or at least portray 3-D on paper) Fig. 5.18 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 

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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 ‏سوب‎ Rotatiowol Qrorwetry ‏سس‎ [Potatiod axe eal 1 8 9 3 [Roteiawersion ‏عفدت‎ vole TED) | SEH) Qombiwation Protalio ame an Our retatios oiz + weiner Ow مه ات | جات ملاس 1 ‎einers 2im2im2im‏ | ] (Bidiiowal Gocortte pollens 9 Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 

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3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System (more later when we consider translations) ‏سس اسب‎ Oe Orster Orater Meliaic 0 a Ovaveliaie Qo w) Onkorkowbic: 8B ‏ممه‎ ‎Tetrawwasl 6 8 0669 iow, Fr ‏مه 68 9 او مومس‎ 9 6 608 ‏مت رمست‎ |_ Geo, deo tw Geo ‘bowen a 48 Fa 9 5 An Aw Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons 

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a al 43 uy 3-D Symmetry The 32 3-D Point Groups After Bloss, Crystallography and Crystal Chemistry. © MSA en) ‎fy ۳‏ ام ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ 

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Flow Chart for Determining 3-D Point Group ‘Go, eet be oof the) اس سس ند ‎tree‏ ‏و ‏و و | ‏کت‎ ml wales: ‏تان‎ lm Cues, mame ‏متشه‎ ‎sates ‎&” ‏مين سم‎ ‏داه‎ aa 0 ‘normal to n-fokd axes? ۳ ‏بسي ليو سم‎ ‏موز‎ Yes ‏ولو‎ ves. ae ۳ ‏ات‎ ‎eth | ‏ید سم سوت‎ ‏م‎ ‎eee ‏خضت‎ wl on + ae we ‏كات‎ Gaal ‏دم‎ ای اه ‎wl‏ لكا دده ‎“a ‎oo fl ‎ ‎ ‎ 

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3-D Symmetn Stereonet of Monoclinic (+) Axes Crystal Axes +C Gye 3 8 a We y¥( pe 2 +4 000 Axial convention: “Tight-hand rule” Conventional 2-D Projection +a (front) 

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3-D Symmetry Crystal Axes Crystal Axes Triclinic Monoclinic 2 ۰ b b at bee at bee ‏وعم عه‎ 0, 0 ‘The axes are chosen as parallel to the principal face intersections. b is selected as | to the 2-fold axis or 1 to the mirror. ‘The most pronounced zone is oriented vertically and the zone axis is ¢. a slopes down and forward so that fis typically > 90. ‘The axes are chosen as parallel tothe principal face intersections. There are no symmetric restrictions to the choice of @, b, and e, but, by convention, the most pronounced zone is oriented vertically and the zone axis is 

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3-D Symmetry Crystal Axes Crystal Axes ‏د‎ Tetragonal b citer choice az bee ‏ومع یود بو‎ 18 OK foraaxes ‘The axes are mutually perpendicular and | to ‘The axes are mutually perpendicular and e is 2-fold axes (conventionally ¢<a<b). When chosen to the 4-fold axis. Due to the 4-fold crystals are elongated (as above left) eis chosen symmetry, the other two axes are equal. a is as the direction of elongation. When crystals are oriented toward the front and a, to the right. flattened (top insert), ¢ is chosen as normal to the predominant plane. 

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3-D Symmetry Trigonal {also considered as the rhombohedral division of the Hexagonal system) ¢ a, wa, Crystal Axes 22 7 ‏با‎ ‎0 ‎wh ‎Wa, Crystal Axes Hexagonal wa, wa, either choise —™ isOK foraaxes (4) a =a; #0 aZa=120; aZe=90 ‘The Hexagonal system (and Trigonal sub-system) typically has four axes, three are of equal length at 120 degrees to one another, and all_L to ¢, which is | to either the 3-fold or 6-fold rotation, ‘The conventional choice of the three a axes is shown in the inset. 

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3-D Symmetry Crystal Axes Crystal Axes Isometric dodecahedron All three axes are mutually perpendicular and of equal length. They ere set | tothe 4-fold axes (ifpresent), otherwise to te 2-fold axes. all three forms combined 

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