m_03 symmetry operation

7/30/2019 M_03 Symmetry Operation

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MINERALOGY03 : SYMMETRY OPERATIONM03_Symmetry Operation

LECTURE NOTE

DEPARTMENT OF GEOLOGY

FACULTY OF GEOLOGY, PADJ ADJ ARAN UNIVERSITY

ACADEMIC : 2012/2013

Mineralogy@Rosana 2013 1


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REcOGNITION Of MINERAL SYMMETRY External Crystal Form (Morphology)

- macroscopic observations/measurements of crystal faces- 32 crystal classes (point groups) possible symmetry elements orcombinations of elements

Internal Atomic Arrangement

- determined from X-ray diffraction- takes into account 3D translation- 14 lattice types (Bravais lattices)- Combination of 32 point groups and 14 lattice types yields 230space groups



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Mineralogy@Rosana 2013

Symmetry

4


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OPERASISIMETRI



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Symmetry Plane (Bidang simetri)(mirror / m / P)

Axes (Sumbu Simetri) (axes / A)

Center of Symmetry (Pusat Simetri)(center / C)


ExTERNAL SYMMETRY ELEMENTScRYSTAL MORPhOLOGY

(ELEMENT SYMMETRY)

6


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ExTERNAL SYMMETRY ELEMENTScRYSTAL MORPhOLOGY

(SYMMETRY OPERATION)

(a) Rotation (b) Reflection (c) Center of Symmetry (d) Rotation withInversion


(Klein & Hulburt, J R., 1993)

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ExTERNAL SYMMETRY ELEMENTSMOTIf PATTERNS

Reflection

Center of Symmetry

Rotation with Inversion Mineralogy@Rosana 2013


Rotation

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Bidang simetri utama (BSU) : bidang simetri yang padanyaterdapat dua atau lebih bidang simetri lain yang tegak lurusdan tegak lurus terhadap sumbu simetri berharga palingtinggi.

Bidang simetri biasa : bidang yang membagi kristalmenjadi dua bagian yang simetris yang padanya terdapatsatu sumbu simetri

Operasi bidang simetri ini disebut operasi pencerminan

(Reflection)


SYMMETRY PLANE (BIdANG SIMETRI)

(Klein & Hulburt, J R., 1993) 10


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Bidang simetri vertikal, horizontal dan diagonal, serta simetri utama dantambahan

(Klein & Hulburt, J R., 1993)Mineralogy@Rosana 2013

SYMMETRY PLANE (BIdANG SIMETRI)

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SYMMETRY AxESAdalah suatu garis lurus yang dibuat melalui pusat kristaldan bila kristal tersebut diputar 360 pada sumbutersebut, maka pada kedudukan tertentu akan dijumpai

kenampakan yang sama dengan semula.

Sumbu simetri ada dua macam : Sumbu simetri biasa (bipoler) : sumbu khayal yang

melalui kristal yang apabila diputar 360, akan dijumpai

konfigurasi yang sama atau hal-hal yang sama yangmuncul lebih dari satu kali.

Sumbu poler : sumbu khayal (seperti halnya sumbubipoler), hanya kedua ujung sumbu menembus duakeadaan yang berbeda.

Operasi dari sumbu lipat ini disebut sebagai operasi rotasi(perulangan periodik dari motif asli yang dijumpai setelahterjadinya perputaran motif tersebut dengan sudutsebesar 360 akibat beroperasinya sumbu rotasi atau

sumbu lipat). Mineralogy@Rosana 2013 12


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Operasi Rotasi (Klein & Hulburt, J R., 1993)



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Dibedakan lagi menjadi dua berdasarkan atas macamoperasinya :a. Gyre

b. Gyroida

Gyre : sumbu simetri biasa yang besarnya sudut putaradalah 360/n ; n adalah bilangan yang utuh sehingga akandiperoleh nilai n 180, 120, 90,60n = 2 (()) digyre/diadn = 3 () trigyre / triadn = 4 () Tetragyre / tetradn = 6 ( . ) Hexagyre / hexad

Gyroida ; sumbu simetri disini merupakan campuran dari

pemutaran melalui sumbu dan pencerminan pada bidangyang tegak lurus pada bidang tadi180 = digyroida ; C pusat simetri120 = Trigyroida90 = Tetragyroida

60 = Hexagyroida Mineralogy@Rosana 2013 14


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ROTATIONAL AxES As an external symmetry element, rotation

increments (n) range from 1 to

When considering the

limits of translation,

rotation increments are

limited to 1, 2, 3, 4,

and 6-fold



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2-D SYMMETRY

6

6

A Symmetr ical Pattern

Symmetry Elements

1. Rotation

a. Two-fold rotation= 360o/2 rotation

to reproduce a motif in asymmetrical pattern



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Motif

Element

Operation

6

6

2-D SYMMETRY

Symmetry Elements

1. Rotation

a. Two-fold rotation

= 360o/2 rotation

to reproduce a

motif in asymmetricalpattern

= the symbol for a two-fold

rotation



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6

6

firstoperationstep

2-D SYMMETRY

Symmetry Elements

1. Rotationa. Two-fold rotation

= 360o/2 rotationto reproduce a


= the symbol for a two-foldrotation


2 D S


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6

6

firstoperationstep

secondoperationstep

2-D SYMMETRY

Symmetry Elements


= 360o/2 rotationto reproduce a


= the symbol for a two-foldrotation


2 D S


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2-D SYMMETRY

Symmetry

Elements

1. Rotation

a. Two-fold

rotation

Some familiarobjects have an

intrinsicsymmetry


2 D S


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2-D SYMMETRY

Symmetry

Elements1. Rotation

a. Two-foldrotation


intrinsicsymmetry


2 D S


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2-D SYMMETRY

SymmetryElements

1. Rotation

a. Two-fold

rotation

Some familiar

objects have anintrinsicsymmetry


2 D S


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2-D SYMMETRY

Symmetry

Elements1. Rotation

a. Two-fold

rotation


intrinsicsymmetry


2 D S


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2-D SYMMETRY

SymmetryElements

1. Rotation

a. Two-foldrotation

Some familiarobjects have anintrinsic

symmetry


2 D S


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2-D SYMMETRY

Symmetry

Elements1. Rotation

a. Two-foldrotation


intrinsicsymmetry


2 D SYMMETRY


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2-D SYMMETRY

Symmetry Elements


Some familiar

objects have anintrinsic symmetry

180o rotation makes itcoincident

Whats the motif here??

Second 180o brings the objectback to its original position


3 D SYMMETRY


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3-D SYMMETRY

Symmetry Elements

1. Rotation

b. Three-fold

rotation

= 360o/3 rotation

to reproduce a



2 D SYMMETRY


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step 1

2-D SYMMETRY

Symmetry Elements

1. Rotation

b. Three-fold

rotation

= 360o/3 rotation

to reproduce a



2 D SYMMETRY


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step 1

step 2

2-D SYMMETRY

Symmetry Elements

1. Rotation

b. Three-foldrotation

= 360o

/3 rotationto reproduce amotif in asymmetricalpattern


29

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



2 D SYMMETRY


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2-D SYMMETRYSymmetry 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 incrystals (as we shall see later). Thus we will exclude them now.

aidentity

Objects with symmetry:



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Combination of Symmetry Elements Multiple Rotational Axes

Axes at 90 (except 3-fold axes in cubicsymmetry at 5444)

Axes intersect atpoint

Possible symmetrycombinations:

422, 622, 222, 32, 23,432

(View 422 Symmetry.ai)



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Combination of Symmetry Elements

Multiple Rotational Axes with 32 Symmetry

- motif projectionsdo not require asecond set of 2-fold

axes



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Combination of Symmetry Elements Multiple Rotational Axes in the Cubic System

432 Point Group



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Pusat Simetri / titik simetri

Adalah suatu titik yang apabila ditarik garis melalui titiktersebut dari sembarang titik pada permukaan kristalakan membagi garis tsb sama panjang

Operasi pusat simetri disebut juga operasi inversi (i)

Inversi : suatu operasi simetri yang dihasilkan denganjalan menghubungkan titik-titik dari salah satu bidangkristal melalui titik pusatnya, sehingga dihasilkan titikturunannya yang berjarak sama dari pusat simetri, tetapiberseberangan dan terbalik

Hasil inversi suatu bidang kristal adalah bidang yangsejajar, sama dan sebangun, tetapi terbalik, dengan letakyang berseberangan terhadap pusat simetrinya danberjarak sama terhadap titik inversi tersebut.


Inversi = Pusat Simetri (C)


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Pusat Simetri ( C )

disebut juga titik simetri adalah suatu titik apabiladitarik garis melalui titik tsb dari sembarang titikpada permukaan kristal akan membagi garis tsb samapanjang. Operasi pusat simetri disebut juga operasiinversi (i).

Inversi:suatu operasi simetri yang dihasilkan dengan jalanmenghubungkan titik-titik dari salah satu bidang kristalmelalui titik pusatnya, sehingga dihasilkan titik turunan-nya yang berjarak sama dari pusat simetri, tetapiberseberangan dan terbalik.

Inversi = Pusat Simetri (C)


2 D SYMMETRY


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2-D SYMMETRY

Symmetry Elements

2. Inversion (i)

inversion through acenter to reproduce amotif in a symmetricalpattern

= symbol for an inversioncenter

inversion is identical to 2-fold

rotation in 2-D, but is uniquein 3-D (try it with yourhands)

6

6



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Operasi Inversi




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Bidang simetri


2-D SYMMETRY


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2-D SYMMETRY

Symmetry Elements

3. Reflection (m)

Reflection across a

mirror planereproduces a motif

= symbol for a mirror

plane


2-D SYMMETRY


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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 enantiomorphicoperations

reproductions are opposite-handed


2-D SYMMETRY


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Combinations of symmetry elements are alsopossible

To create a complete analysis of symmetryabout a point in space, we must try all possiblecombinations of these symmetry elements

In the interest of clarity and ease of illustration,we continue to consider only 2-D examples

2 D SYMMETRY


2-D SYMMETRY


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2 D SYMMETRY

Try combining a 2-fold rotation axis with a mirror


2-D SYMMETRY


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2 D SYMMETRY


Step 1: reflect

(could do either step first)


2-D SYMMETRY


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2 D SYMMETRY


Step 1: reflectStep 2: rotate (everything)


2-D SYMMETRY


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2 D SYMMETRY


Step 1: reflect

Step 2: rotate (everything)

Is that all??


2-D SYMMETRY


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2 D SYMMETRY


Step 1: reflect

Step 2: rotate (everything)

No! A second mirror is required


2-D SYMMETRY


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2 D SYMMETRY


The result is Point Group 2mm

2mm indicates 2 mirrors

The mirrors are different

(notequivalent by symmetry)



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All other combinations are either:

Incompatible

(2 + 2 cannot be done in 2-D)

Redundantwith others already tried

m + m 2mm because creates 2-fold

This is the same as 2 + m 2mm

2-D Symmetry


2-D SYMMETRY


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2 D SYMMETRY

Now try combining a 4-fold rotation axis with a

mirror


2-D SYMMETRY


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S


mirror

Step 1: reflect


2-D SYMMETRY


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mirror

Step 1: reflect

Step 2: rotate 1


2-D SYMMETRY


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mirror

Step 1: reflect

Step 2: rotate 2


2-D SYMMETRY


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mirror

Step 1: reflect

Step 2: rotate 3


2 D Symmetry


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2-D Symmetry

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

Any other elements?


2-D Symmetry


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2-D SymmetryNow try combining a 4-fold rotation axis with a mirror

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?

Why not4mmmm?


2-D Symmetry


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2-D Symmetry

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

Why not 3mmm?



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2-D Symmetry

6-fold rotation axis with a mirror creates point group6mm



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2-D Symmetry

The original 6 elements plus the 4 combinationscreates 10 possible 2-D Point Groups:

1 2 3 4 6 m 2mm 3m 4mm 6mm

Any 2-D pattern of objects surrounding a pointmust conform to one of these groups


Gabungan Rotasi dan Inversi


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Gabungan Rotasi dan Inversi

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3-D Symmetry

New 3-D SymmetryElements

4. Rotoinversion

a. 2-fold rotoinversion ( 2 )



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3-D Symmetry

New SymmetryElements

4. Rotoinversion

b. 2-fold rotoinversion ( 2 )

Step 1: rotate 360/2

Note: this is a temporarystep, the intermediate

motif element does not

exist in the final patternMineralogy@Rosana 2013 64


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3-D Symmetry

New Symmetry Elements

4. Rotoinversion


Step 1: rotate 360/2

Step 2: invert


3 S


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3-D Symmetry


4. Rotoinversion


The result:


3 D S


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3-D Symmetry


4. Rotoinversion


This is the same as m, so not

a new operation


3 D S


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3-D Symmetry


4. Rotoinversion

c. 3-fold rotoinversion ( 3 )


3 D S t


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3-D Symmetry


4. Rotoinversion


Step 1: rotate 360o/3

Again, this is a temporary

step, the intermediatemotif element does notexist in the final pattern

1


3 D S t


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3-D Symmetry


4. Rotoinversion


Step 2: invert throughcenter


3 D S t


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3-D Symmetry


4. Rotoinversion


Completion of the firstsequence

1

2


3 D S t


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3-D Symmetry


4. Rotoinversion


Rotate another 360/3


3 D S t


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3-D Symmetry


4. Rotoinversion


Invert through center


3 D S t


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3-D Symmetry


4. Rotoinversion


Complete second step tocreate face 3

1

2

3


3 D S t


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3-D Symmetry


4. Rotoinversion


Third step creates face 4

(3 (1) 4)

1

2

3

4


3 D S t


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3-D Symmetry


4. Rotoinversion


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

1

2

5


3 D S t


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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 t


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3-D Symmetry


4. Rotoinversion


This is unique

1

6

5

2

3

4


3 D S t


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3-D Symmetry


4. Rotoinversion

d. 4-fold rotoinversion ( 4 )


3 D S mmetr


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3-D Symmetry


4. Rotoinversion



3 D Symmetry


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3-D Symmetry


4. Rotoinversion


1: Rotate 360/4


3 D Symmetry


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3-D Symmetry


4. Rotoinversion


1: Rotate 360/4

2: Invert


3 D Symmetry


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3-D Symmetry


4. Rotoinversion


1: Rotate 360/4

2: Invert


3 D Symmetry


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3-D Symmetry


4. Rotoinversion


3: Rotate 360/4


3 D Symmetry


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3-D Symmetry


4. Rotoinversion


3: Rotate 360/4

4: Invert


3 D Symmetry


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3-D Symmetry


4. Rotoinversion


3: Rotate 360/4

4: Invert


3 D Symmetry


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3-D Symmetry


4. Rotoinversion


5: Rotate 360/4


3 D Symmetry


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3-D Symmetry


4. Rotoinversion


5: Rotate 360/4

6: Invert


3 D Symmetry


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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 fundamentalrepresentative of the pattern


3-D Symmetry


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y yWe now have 8 unique 3D symmetry operations:

1 2 3 4 6 m 3 4

Combinations of these elements are also possible

A complete analysis ofsymmetry about a point in spacerequires that we try all possible combinations of thesesymmetry elements



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SYSTEM & CLASSCRYSTALOGRAPHY

To be Continue

m_03 symmetry operation

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