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mBENTUK MINERAL

DENGAN SUSUNAN

KRISTAL YANG SAMA

MEMILIKIKETERATURAN SISTEM

KRISTAL YANG SAMA

PULA

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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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2-D Symmetry

Symmetry Elements

1. Rotation

a. Two-fold rotation

= 360o/2 rotation

to reproduce a

motif in asymmetrical

pattern

6

6

A Symmetrical Pattern

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Symmetry Elements

1. Rotation


= 360o/2 rotation

to reproduce a


pattern

= the symbol for a two-fold

rotation

Motif

Element

Operation

6

6

2-D Symmetry

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6

6

first

operation

step

second

operation

step

2-D Symmetry

Symmetry Elements

1. Rotation


= 360o/2 rotation

to reproduce a


pattern

= the symbol for a two-fold

rotation

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Symmetry Elements

1. Rotation


Some familiar

objects have an

intrinsic

symmetry

2-D Symmetry

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Symmetry Elements

1. Rotation


Some familiar

objects have an

intrinsic

symmetry

2-D Symmetry

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Symmetry Elements

1. Rotation


Some familiar

objects have an

intrinsic

symmetry

2-D Symmetry

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Symmetry Elements

1. Rotation


Some familiar

objects have an

intrinsic

symmetry

2-D Symmetry

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Symmetry Elements

1. Rotation


Some familiar

objects have an

intrinsic

symmetry

2-D Symmetry

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Symmetry Elements

1. Rotation


Some familiar

objects have an

intrinsic

symmetry

2-D Symmetry

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Symmetry Elements

1. Rotation


Some familiarobjects have anintrinsic

symmetry180o rotation makes it coincident

Whats the motif here??

Second 180obrings the object

back to its original position

2-D Symmetry

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Symmetry Elements

1. Rotation

b. Three-fold rotation

= 360o/3 rotation

to reproduce a

motif in asymmetrical pattern

2-D Symmetry

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step 1

step 2

step 3

2-D Symmetry

Symmetry Elements

1. Rotation

b. Three-fold rotation

= 360o/3 rotation

to reproduce a

motif in asymmetrical pattern

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Symmetry Elements

1. Rotation

6

6

666

6

6

6

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

9t dZ5-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:

2-D Symmetry

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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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Symmetry Elements

2. Inversion (i)

inversion through acenter to reproduce a

motif in a symmetrical

pattern

= symbol for aninversion center

inversion is identical to 2-fold

rotation in 2-D, but is unique

in 3-D (try it with your hands)

6

6

2-D Symmetry

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Symmetry Elements

3. Reflection (m)

Reflection across amirror plane

reproduces a motif

= symbol for a mirror

plane

2-D Symmetry

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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

2-D Symmetry

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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 allpossible combinations of these symmetry

elements

In the interest of clarity and ease of illustration, we continue to

consider only 2-D examples

2-D Symmetry

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Try combining a 2-fold rotation axis with a mirror

2-D Symmetry

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Step 1: reflect

(could do either step first)

2-D Symmetry

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Step 1: reflect

Step 2: rotate (everything)

2-D Symmetry

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Step 1: reflect


Is that all??

2-D Symmetry

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Step 1: reflect


No! A second mirror is required

2-D Symmetry

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The result is Point Group 2mm

2mm indicates 2 mirrors

The mirrors are different

(not equivalentby symmetry)

2-D Symmetry

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

2-D Symmetry

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Step 1: reflect

2-D Symmetry

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Step 1: reflect

Step 2: rotate 1

2-D Symmetry

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Step 1: reflect

Step 2: rotate 2

2-D Symmetry

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Step 1: reflect

Step 2: rotate 3

2-D Symmetry

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Any other elements?

2-D Symmetry

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Yes, two more mirrors

Any other elements?

2-D Symmetry

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Point group name??


Any other elements?

2-D Symmetry

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4mm

Point group name??


Any other elements?

2-D Symmetry

Why not 4mmmm?

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

Why not 3mmm?

2-D Symmetry

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

2-D Symmetry

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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-fold

This is the same as 2 + m 2mm

2-D Symmetry

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The original 6 elements plus the 4 combinationscreates 10possible 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

2-D Symmetry

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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


4. Rotoinversion


Step 1: rotate 360/1

(identity)

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3-D Symmetry


4. Rotoinversion



(identity)

Step 2: invert

This is the same as i, so not a newoperation

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Sistem Kristal Asimetrik

x

x

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3-D Symmetry

New Symmetry Elements

4. Rotoinversion

b. 2-fold rotoinversion ( 2 )


Note: this is a temporary

step, the intermediate

motif element does not

exist in the final pattern

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3-D Symmetry


4. Rotoinversion



Step 2: invert

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3-D Symmetry


4. Rotoinversion


The result:

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3-D Symmetry


4. Rotoinversion


This is the same as m, so not

a new operation

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3-D Symmetry


4. Rotoinversion

c. 3-fold rotoinversion ( 3 )

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3-D Symmetry


4. Rotoinversion


Step 1: rotate 360o/3

Again, this is a

temporary step, theintermediate motif

element does not exist

in the final pattern

1

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3-D Symmetry


4. Rotoinversion


Step 2: invert through

center

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3-D Symmetry


4. Rotoinversion


Completion of the first

sequence

1

2

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3-D Symmetry


4. Rotoinversion


Rotate another 360/3

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3-D Symmetry


4. Rotoinversion


Invert through center

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3-D Symmetry


4. Rotoinversion


Complete second step to

create face 3

1

2

3

S

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3-D Symmetry


4. Rotoinversion


Third step creates face 4

(3 (1) 4)

1

2

3

4

3 D S

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3-D Symmetry


4. Rotoinversion


Fourth step creates face

5 (4 (2) 5)

1

2

5

3 D S

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3-D Symmetry


4. Rotoinversion


Fifth step creates face 6

(5 (3) 6)

Sixth step returns to face 1

1

6

5

3 D S

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3-D Symmetry


4. Rotoinversion


This is unique1

6

5

2

3

4

3 D S

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3-D Symmetry


4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

3 D S

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3-D Symmetry


4. Rotoinversion


3 D S t

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3-D Symmetry


4. Rotoinversion


1: Rotate 360/4

3 D S t

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3-D Symmetry


4. Rotoinversion


1: Rotate 360/4

2: Invert

3 D S t

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3-D Symmetry


4. Rotoinversion


1: Rotate 360/4

2: Invert

3 D S t

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3-D Symmetry


4. Rotoinversion


3: Rotate 360/4

3 D S t

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3-D Symmetry


4. Rotoinversion


3: Rotate 360/4

4: Invert

3 D S t

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3-D Symmetry


4. Rotoinversion


3: Rotate 360/4

4: Invert

3 D S t

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3-D Symmetry


4. Rotoinversion


5: Rotate 360/4

3 D S t

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3-D Symmetry


4. Rotoinversion


5: Rotate 360/4

6: Invert

3 D S mmetr

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3-D Symmetry


4. Rotoinversion


This is also a unique operation

3 D Symmetry

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3-D Symmetry


4. Rotoinversion


A more fundamental

representative of the

pattern

3 D Symmetry

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3-D Symmetry


4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Begin with this framework:

3 D Symmetry

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3-D Symmetry


4. Rotoinversion

e. 6-fold rotoinversion ( 6 ) 1

3 D Symmetry

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3-D Symmetry

1


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

1

2


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

1

2


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

13

2


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

13

2


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

13

4

2


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

1

2

3

4


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

1

2

3

4

5


4. Rotoinversion


3 D Symmetry

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3-D Symmetry

1

2

3

4

5


4. Rotoinversion


3-D Symmetry

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3-D Symmetry

1

2

3

4

5

6


4. Rotoinversion


3-D Symmetry

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3-D Symmetry


4. Rotoinversion


Note: this is the same as a 3-fold

rotation axis perpendicular to a

mirror plane

(combinations of elements follows)

Top View

3-D Symmetry

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3-D Symmetry


4. Rotoinversion


A simpler pattern

Top View

3-D Symmetry

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y y

We now have 10 unique 3-D symmetry operations:

1 2 3 4 6 i 2 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

3-D Symmetry 3rd

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y y

3-D symmetry element combinations

a. Rotation axis parallel to a mirror

Same as 2-D

2 || m = 2mm

3 || m = 3m, also 4mm, 6mmb. Rotation axis mirror ------ beberapa mineral

2 m = 2/m

3 m = 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 45o or 30o are possible, as we shall see

3-D Symmetry

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y y

3-D symmetry element combinations

d. Combinations of rotations

2 + 2 at 90o 222 (third 2 required from

combination)

4 + 2 at 90o 422 ( )6 + 2 at 90o 622 ( )

3-D Symmetry

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y y

As in 2-D, the number of possible combinations is

limited only by incompatibility and redundancy

There are only 22possible unique 3-D combinations,

when combined with the 10 original 3-D elementsyields the 32 3-D Point Groups

3-D Symmetry

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y y

But it soon gets hard tovisualize (or at least

portray 3-D on paper)

Fig. 5.18 of Klein (2002) Manual of

Mineral Science, John Wiley and

Sons

3-D Symmetry

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y yThe 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/mThree rotation axes and mirrors 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/m

Additional 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

3-D Symmetry

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y yThe 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

3-D Symmetry

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3 D SymmetryThe 32 3-D Point Groups

After Bloss, Crystallography and

Crystal Chemistry. MSA

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Documents

4 Symm