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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
mm
2
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
6
6
A Symmetrical PatternA Symmetrical Pattern
3
Symmetry Elements1. Rotation
a. Two-fold rotation
= 360o/2 rotation to reproduce a motif in a symmetrical pattern
= the symbol for a two-fold rotation
Motif
Element
OperationOperation
6
6
22--D SymmetryD Symmetry
6
6
first operation step
second operation step
22--D SymmetryD Symmetry
Symmetry Elements1. Rotation
a. Two-fold rotation
= 360o/2 rotation to reproduce a motif in a symmetrical pattern
= the symbol for a two-fold rotation
4
Symmetry Elements1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry Elements1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
5
Symmetry Elements1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry Elements1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
6
Symmetry Elements1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry Elements1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
7
Symmetry Elements1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
180o rotation makes it coincident
What’s the motif here??
Second 180o brings the object back to its original position
22--D SymmetryD Symmetry
Symmetry Elements1. Rotation
b. Three-fold rotation
= 360o/3 rotation to reproduce a motif in a symmetrical pattern
66
6
22--D SymmetryD Symmetry
8
6
6
6
step 1
step 2
step 3
22--D SymmetryD Symmetry
Symmetry Elements1. Rotation
b. Three-fold rotation
= 360o/3 rotation to reproduce a motif in a symmetrical pattern
Symmetry Elements1. Rotation
6
6
6
6
6
6 6
6
6
6
6
6
6
66
6
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:
22--D SymmetryD Symmetry
9
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)
6
6
22--D SymmetryD Symmetry
10
Symmetry Elements3. Reflection (m)
Reflection across a “mirror plane”reproduces a motif
= symbol for a mirrorplane
22--D SymmetryD 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 operationsreproductions are “opposite-handed”
22--D SymmetryD Symmetry
11
•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
22--D SymmetryD Symmetry
Try combining a 2-fold rotation axis with a mirror22--D SymmetryD Symmetry
12
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
(could do either step first)
22--D SymmetryD Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
22--D SymmetryD Symmetry
13
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
Is that all??
22--D SymmetryD Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
No! A second mirror is required
22--D SymmetryD Symmetry
14
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)
22--D SymmetryD Symmetry
Now try combining a 4-fold rotation axis with a mirror22--D SymmetryD Symmetry
15
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
22--D SymmetryD Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 1
22--D SymmetryD Symmetry
16
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 2
22--D SymmetryD Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 3
22--D SymmetryD Symmetry
17
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
22--D SymmetryD Symmetry
Now try combining a 4-fold rotation axis with a mirror
Yes, two more mirrors
Any other elements?
22--D SymmetryD Symmetry
18
Now try combining a 4-fold rotation axis with a mirror
Point group name??
Yes, two more mirrors
Any other elements?
22--D SymmetryD Symmetry
Now try combining a 4-fold rotation axis with a mirror
4mm
Point group name??
Yes, two more mirrors
Any other elements?
22--D SymmetryD Symmetry
Why not 4mmmm?
19
3-fold rotation axis with a mirror creates point group 3m
Why not 3mmm?
22--D SymmetryD Symmetry
6-fold rotation axis with a mirror creates point group 6mm
22--D SymmetryD Symmetry
20
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
22--D SymmetryD 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
22--D SymmetryD Symmetry
21
33--D SymmetryD Symmetry
New 3-D Symmetry Elements4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
33--D SymmetryD Symmetry
New 3-D Symmetry Elements4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1(identity)
22
33--D SymmetryD Symmetry
New 3-D Symmetry Elements4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
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
23
33--D SymmetryD Symmetry
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
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 2: invert
24
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
The result:
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
This is the same as m, so not a new operation
25
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
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
1
26
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Step 2: invert through center
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Completion of the first sequence
1
2
27
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Rotate another 360/3
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Invert through center
28
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Complete second step to create face 3
1
2
3
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Third step creates face 4 (3 → (1) → 4)
1
2
3
4
29
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fourth step creates face 5 (4 → (2) → 5)
1
2
5
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fifth step creates face 6(5 → (3) → 6)
Sixth step returns to face 1
1
6
5
30
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
This is unique1
6
5
2
3
4
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
31
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
32
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
33
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
34
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
35
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
6: Invert
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
This is also a unique operation
36
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
A more fundamental representative of the pattern
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
Begin with this framework:
37
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1
33--D SymmetryD Symmetry
1
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
38
33--D SymmetryD Symmetry
1
2
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
33--D SymmetryD Symmetry
1
2
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
39
33--D SymmetryD Symmetry
13
2
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
33--D SymmetryD Symmetry
13
2
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
40
33--D SymmetryD Symmetry
13
4
2
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
33--D SymmetryD Symmetry
1
2
3
4
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
41
33--D SymmetryD Symmetry
1
2
3
4
5
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
33--D SymmetryD Symmetry
1
2
3
4
5
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
42
33--D SymmetryD Symmetry
1
2
3
4
5
6
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane
(combinations of elements follows)Top View
43
33--D SymmetryD Symmetry
New Symmetry Elements4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
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
44
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 ( “ “ “ )
45
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
33--D SymmetryD Symmetry
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
46
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
33--D SymmetryD SymmetryThe 32 3-D Point Groups
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
47
33--D SymmetryD SymmetryThe 32 3-D Point Groups
After Bloss, Crystallography and Crystal Chemistry. © MSA