Krista Log Rafi

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Bentuk dan susunan mineral dengan komposisiBentuk dan susunan mineral dengan komposisiYang sama memiliki keteraturan susunanYang sama memiliki keteraturan susunanKristal yang sama pulaKristal yang sama pula

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SymmetrySymmetry

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

22--D SymmetryD Symmetry

Symmetry Elements1. Rotation

a. Two-fold rotation

= 360o/2 rotation to reproduce a motif in a symmetrical pattern

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A Symmetrical PatternA Symmetrical Pattern

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= the symbol for a two-fold rotation

Motif

Element

OperationOperation

6

6


6

6

first operation step

second operation step





= the symbol for a two-fold rotation

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Some familiar objects have an intrinsic symmetry






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6









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180o rotation makes it coincident

What’s the motif here??

Second 180o brings the object back to its original position



b. Three-fold rotation


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6


8

6

6

6

step 1

step 2

step 3



b. Three-fold rotation



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6 6

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1-fold 2-fold 3-fold 4-fold 6-fold

Z5-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.

aidentity

Objects with symmetry:


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44--fold, 2fold, 2--fold, and 3fold, and 3--fold fold rotations in a cuberotations in a cube

Click on image to run animation

Symmetry Elements2. Inversion (i)

inversion through a center to reproduce a motif in a symmetrical pattern= symbol for an inversion centerinversion is identical to 2-fold rotation in 2-D, but is unique in 3-D (try it with your hands)

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6


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Symmetry Elements3. Reflection (m)

Reflection across a “mirror plane”reproduces a motif

= symbol for a mirrorplane


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 operationsreproductions are “opposite-handed”


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•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


Try combining a 2-fold rotation axis with a mirror22--D SymmetryD Symmetry

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Try combining a 2-fold rotation axis with a mirror

Step 1: reflect

(could do either step first)



Step 1: reflect

Step 2: rotate (everything)


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Step 1: reflect


Is that all??



Step 1: reflect


No! A second mirror is required


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The result is Point Group 2mm

“2mm” indicates 2 mirrors

The mirrors are different(not equivalent by symmetry)


Now try combining a 4-fold rotation axis with a mirror22--D SymmetryD Symmetry

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Now try combining a 4-fold rotation axis with a mirror

Step 1: reflect



Step 1: reflect

Step 2: rotate 1


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Step 1: reflect

Step 2: rotate 2



Step 1: reflect

Step 2: rotate 3


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Any other elements?



Yes, two more mirrors

Any other elements?


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Point group name??


Any other elements?



4mm

Point group name??


Any other elements?


Why not 4mmmm?

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3-fold rotation axis with a mirror creates point group 3m

Why not 3mmm?


6-fold rotation axis with a mirror creates point group 6mm


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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-foldThis is the same as 2 + m → 2mm


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


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New 3-D Symmetry Elements4. Rotoinversion

a. 1-fold rotoinversion ( 1 )




Step 1: rotate 360/1(identity)

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Step 1: rotate 360/1(identity)

Step 2: invert

This is the same as i, so not a new operation

Sistem Kristal AsimetrikSistem Kristal Asimetrik

xx

xx

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New Symmetry Elements4. Rotoinversion

b. 2-fold rotoinversion ( 2 )

Step 1: rotate 360/2

Note: this is a temporary step, the intermediate motif element does not exist in the final pattern




Step 1: rotate 360/2

Step 2: invert

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The result:




This is the same as m, so not a new operation

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c. 3-fold rotoinversion ( 3 )




Step 1: rotate 360o/3 Again, this is a

temporary step, the intermediate motif element does not exist in the final pattern

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Step 2: invert through center




Completion of the first sequence

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2

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Rotate another 360/3




Invert through center

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Complete second step to create face 3

1

2

3




Third step creates face 4 (3 → (1) → 4)

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2

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4

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Fourth step creates face 5 (4 → (2) → 5)

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Fifth step creates face 6(5 → (3) → 6)

Sixth step returns to face 1

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This is unique1

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d. 4-fold rotoinversion ( 4 )

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1: Rotate 360/4

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1: Rotate 360/4

2: Invert




1: Rotate 360/4

2: Invert

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3: Rotate 360/4




3: Rotate 360/4

4: Invert

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3: Rotate 360/4

4: Invert




5: Rotate 360/4

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5: Rotate 360/4

6: Invert




This is also a unique operation

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A more fundamental representative of the pattern



e. 6-fold rotoinversion ( 6 )

Begin with this framework:

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e. 6-fold rotoinversion ( 6 ) 1


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1

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1

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1

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1

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Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane

(combinations of elements follows)Top View

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A simpler pattern

Top View

33--D SymmetryD SymmetryWe 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 spacerequires that we try all possible combinations of these symmetry elements

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33--D SymmetryD Symmetry3-D symmetry element combinations

a. Rotation axis parallel to a mirrorSame as 2-D2 || m = 2mm3 || m = 3m, also 4mm, 6mm

b. Rotation axis ⊥ mirror ------ beberapa mineral2 ⊥ m = 2/m3 ⊥ m = 3/m, also 4/m, 6/m

c. Most other rotations + m are impossible2-fold axis at odd angle to mirror?Some cases at 45o or 30o are possible, as we shall see

33--D SymmetryD Symmetry3-D symmetry element combinations

d. Combinations of rotations2 + 2 at 90o → 222 (third 2 required from combination)4 + 2 at 90o → 422 ( “ “ “ )6 + 2 at 90o → 622 ( “ “ “ )

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33--D SymmetryD SymmetryAs 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


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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33--D SymmetryD SymmetryThe 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

Rotation axis only 1 2 3 4 6

Rotoinversion axis only 1 (= i ) 2 (= m) 3 4 6 (= 3/m)

Combination of rotation axes 222 32 422 622

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/mAdditional Isometric patterns 23 432 4/m 3 2/m

2/m 3 43m

Increasing Rotational Symmetry

Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons


Regrouped by Crystal System(more later when we consider translations)

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

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