Ch 04 Unsur simetri-1
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Transcript of Ch 04 Unsur simetri-1
Crystal SymmetryCrystal SymmetryMotif: adalah bagian yang paling mendasar dari
t b t k i t i bil dsuatu bentuk simetri yang bila padanya dilakukan perulangan akan membentuk pola
l hyang menyeluruhOperation: ialah suatu perbuatan yang
memperbanyak motif untuk membuat sebuah pola
Element: merupakan sebuah operasi yang dilakukan pada sebuah titik tertentu dalam ruang
R t i RefleksiRotasi Refleksi
R t i iInversi
Rotoinversi
Operasi simetriOperasi simetri
Unsur Simetri kristalografi
11 Rotasi = Rotation ~ Sumbu lipat / sumbu simetri (A)Rotasi = Rotation ~ Sumbu lipat / sumbu simetri (A)1.1. Rotasi = Rotation ~ Sumbu lipat / sumbu simetri (A)Rotasi = Rotation ~ Sumbu lipat / sumbu simetri (A)2.2. Inversi = Inversion ~ Pusat simetri / titik simetri (C)Inversi = Inversion ~ Pusat simetri / titik simetri (C)3.3. Refleksi = Reflection ~ Bidang cermin / bidang simetri Refleksi = Reflection ~ Bidang cermin / bidang simetri g gg g
(m ~ P)(m ~ P)
Rotasi = operasi simetri (putar) yang dilakukanRotasi operasi simetri (putar) yang dilakukan pada sumbu simetri sebagai sumbu putar (axis = A)(axis A)
Sumbu simetri :Sumbu simetri : adalah suatu garis lurus yang dibuat melalui pusat kristal, dimanakristal tsb diputar 360° dengan garis tersebut sebagai sumbu
k d k d d k k i l b kperputaran, maka pada kedudukan tertentu kristal tersebut akan menunjukkan kenampakan yang sama dengan semula.
Sumbu simetri biasa atau sumbu bipoler:sumbu khayal yg melalui kristal dapat diputar 3600 akan sum u haya yg m a u r sta apat putar 6 a an dijumpai konfigurasi yg sama lebih dari satu kali.
Sumbu simetri dapat juga dibagi atas macam operasinya, yaitu:- Gyre: operasi sumbu simetri yang besarnya sudut putar 360/n;dimana n = 1, 2 (digyre), 3 (triad), 4 (tetrad), 6 (hexad) (Ingat: kristal mempunyai bentuk polihedral yg tertutup)
- Gyroida: operasi sumbu simetri, yg merupakan campuran dariGyroida: operasi sumbu simetri, yg merupakan campuran daripemutaran melalui sumbu dan pencerminan pada bidang ygtegak lurus pada bidang tadi.
Operasi Rotasi
Operasi Rotoinversi
22--D SymmetryD Symmetry
Symmetry Elements
A Symmetrical PatternA Symmetrical Pattern
y y1. Rotation
a. Two-fold rotation
6= 360o/2 rotation to reproduce a motif in a
t i l
6
symmetrical pattern 6
22--D SymmetryD Symmetry
Symmetry ElementsOperationOperation
Symmetry Elements1. Rotation
a Two-fold rotationMotif6
a. Two-fold rotation
= 360o/2 rotation
Element
360 /2 rotation to reproduce a motif in a
6
symmetrical pattern 6
= the symbol for a two-fold rotation
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotation
6 first operation
a. Two-fold rotation
= 360o/2 rotation pstep
= 360 /2 rotation to reproduce a motif in a
6second operation
symmetrical pattern 6
operation step= the symbol for a two-fold
rotation
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotationa. Two-fold rotation
Some familiarSome familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotationa. Two-fold rotation
Some familiarSome familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotationa. Two-fold rotation
Some familiarSome familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotationa. Two-fold rotation
Some familiarSome familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotationa. Two-fold rotation
Some familiarSome familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotationa. Two-fold rotation
Some familiarSome familiar objects have an intrinsic symmetry
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
a Two fold rotationa. Two-fold rotation
Some familiarSome familiar objects have an intrinsic symmetry
180o rotation makes it coincidentSecond 180o brings the object
What’s the motif here??
back to its original position
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
b Three fold rotationb. Three-fold rotation
= 360o/3 rotation= 360 /3 rotation to reproduce a motif in a symmetrical pattern
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements1. Rotation
b Three fold rotationstep 1
b. Three-fold rotation
= 360o/3 rotation= 360 /3 rotation to reproduce a motif in a
step 3symmetrical pattern
step 2
22--D SymmetryD SymmetrySymmetry Elements
1. Rotation
66 66 6
6 6
6
6
1-fold 2-fold 3-fold 4-fold 6-foldObjects with symmetry:
Zaidentity
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.
Inversi = Pusat Simetri (C)Inversi = Pusat Simetri (C)
Pusat Simetri ( C )disebut juga titik simetri adalah suatu titik apabila ditarik garis melalui titik tsb dari sembarang titik ditarik garis melalui titik tsb dari sembarang titik pada permukaan kristal akan membagi garis tsb sama panjang. Operasi pusat simetri disebut juga operasi
( )inversi (i).
Inversi: suatu operasi simetri yang dihasilkan dengan jalan menghubungkan titik-titik dari salah satu bidang kristal melalui titik pusatnya sehingga dihasilkan titik turunanmelalui titik pusatnya, sehingga dihasilkan titik turunan-nya yang berjarak sama dari pusat simetri, tetapi berseberangan dan terbalik.
22--D SymmetryD Symmetry
Symmetry Elements2. Inversion (i)
inversion through a center to reproduce a motif in a symmetrical 6ot a sy et capattern= symbol for an i iinversion centerinversion is identical to 2-fold rotation in 2-D, but is unique 6
in 3-D (try it with your hands)
6
Reflection = pencerminan (m)Reflection = pencerminan (m)
Bidang Simetri (m)id i i d l h bid l l i k i lBidang simetri adalah suatu bidang yang melalui pusat kristal
dan membelah kristal menjadi dua bagian yang sama, dimana bagian yang satu merupakan pencerminan bagian yang lainnya. g y g p p g y g y
Operasi bidang simetri disebut juga operasi pencerminan.
Berdasarkan kedudukannya dibedakan menjadi 3 macam yaitu: vertikal,di ldiagonal,horizontal.
Bidang simetri
22--D SymmetryD Symmetry
Symmetry ElementsSymmetry Elements3. Reflection (m)
Reflection across a “mirror plane” reproduces a motif
= symbol for a mirror= symbol for a mirrorplane
22--D SymmetryD SymmetryWe 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 SymmetryCombinations 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 symmetryspace, 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 SymmetryTry combining a 2-fold rotation axis with a mirror
22--D SymmetryD SymmetryTry combining a 2-fold rotation axis with a mirror
Step 1: reflectp
(could do either step first)
22--D SymmetryD SymmetryTry combining a 2-fold rotation axis with a mirror
Step 1: reflectp
Step 2: rotate (everything)Step 2: rotate (everything)
22--D SymmetryD SymmetryTry combining a 2-fold rotation axis with a mirror
Step 1: reflectp
Step 2: rotate (everything)Step 2: rotate (everything)
Is that all??Is that all??
22--D SymmetryD SymmetryTry combining a 2-fold rotation axis with a mirror
Step 1: reflectp
Step 2: rotate (everything)Step 2: rotate (everything)
N ! A d i i i dNo! A second mirror is required
22--D SymmetryD SymmetryTry 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 SymmetryNow try combining a 4-fold rotation axis with a mirror
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
Step 1: reflectp
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
Step 1: reflectp
Step 2: rotate 1Step 2: rotate 1
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
Step 1: reflectp
Step 2: rotate 2Step 2: rotate 2
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
Step 1: reflectp
Step 2: rotate 3Step 2: rotate 3
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
A th l t ?Any other elements?
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
A th l t ?Any other elements?
Yes, two more mirrors
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
A th l t ?Any other elements?
Point group name??
Yes, two more mirrors
g p
22--D SymmetryD SymmetryNow try combining a 4-fold rotation axis with a mirror
A th l t ?Any other elements?
Point group name??
Yes, two more mirrors
4mm
g p
Why not 4mmmm?
22--D SymmetryD Symmetry
3-fold rotation axis with a mirror creates point group 3m
Why not 3mmm?y
22--D SymmetryD Symmetry
6-fold rotation axis with a mirror creates point group 6mm
22 D S tD S t22--D SymmetryD Symmetry
All other combinations are either:IncompatibleIncompatible
(2 + 2 cannot be done in 2-D)R d d i h h l d i dRedundant with others already tried
m + m → 2mm because creates 2-foldThis is the same as 2 + m → 2mm
Rotasi : b, c, d
Refleksi : a Inversi : e
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
Tugas kelompokTugas kelompok1 Buat 10 kelompok : @ 10~13 orang1. Buat 10 kelompok : @ 10~13 orang2. Gambar setiap model kristal (2D~3D) skala 1:1
(sesuai ukuran kristal)3 T t k j l h i t i t k ti d l k i t l i3. Tentukan jumlah unsur simetri untuk tiap model kristal sesuai no
a. jumlah sumbu simetri/lipat (1, 2, 3, 4, dan 6)b. titik simetric. bidang simetri (vertikal, horizontal, diagonal)
4. Tugas di tulis/di ketik rapih perkelompok, dikumpulkan tgl 16 oktober 2004 (pada saat kuliah kristalografi)tgl 16 oktober 2004 (pada saat kuliah kristalografi)
Catatan : gambar bila di buat/di foto digital 3 D, h j hk d l b t k CD/di k t h il f tharus juga menyerahkan dalam bentuk CD/disket hasil fotonya