Download - M02_Element Symmetry - Symmetry Operation
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ELEMENT SYMMETRY & SYMMETRY OPERATION
M02_Element Symmetry & Symmetry Operation
LECTURE NOTE
DEPARTMENT OF GEOLOG
FACULTF GEOLOG! PAD"AD"ARAN UN#$ERS#T
ACADEM#C % 202'20(
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Mineralogy @ FTG - UNPAD 2014 2
*External Crystal Form (Morphology)- macroscopic observations/measurements of crystal faces
- 32 crystal classes (point groups) possible symmetry elements or
combinations of elements
*Internal Atomic Arrangement- determined from X-ray diffraction
- takes into account 3D translation
- 14 lattice types (ravais lattices)
- !ombination of 32 point groups and 14 lattice types yields 23"
space groups
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Mineralogy @ FTG - UNPAD 2014 3
Symmetry
) #otif (pola)#otif (pola)$ t%e fundamental part of asymmetric design t%at& '%en repeated&
creates t%e '%ole pattern
) peration (operasi)peration (operasi)$ some act t%atreproduces t%e motif to create t%e
pattern
) lement (unsur)lement (unsur)$ an operation located ata particular point in space
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*ymmetry
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*ymmetry +lane (idang *imetri) (mirror / m /
+)
,es (*umbu *imetri) (aes / ,)
!enter of *ymmetry (+usat *imetri) (center / !)
ELEMENT symmetry(Unsur Simetri)
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*a+ Rotation *,+ Re-le.tion *.+ Center o- Symmetry */+ Rotation it1
#ner3ion
*4lein & 56l,6rt! "R7! 88(+ 2/26/2014
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+.,* *#0.
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Diberi notasi p (plane) atau m (mirror) idang simetri adala%
bidang yang melalui pusat kristal dan membela% kristal menadi 2bagian yang sama& dimana bagian yang satu merupakanpencerminan bagian yang lainnya
Dibedakan $
erdasarkan kedudukannya menadi 3 macam $ a ertikal
b Diagonal
c ori5ontal
erdasarkan enisnya ada 2 macam $ a idang simetri utama
b idang simetri biasa/tamba%an
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*Bidang simetri utama (BSU) $ bidang simetri yang padanyaterdapat dua atau lebi% bidang simetri lain yang tegak lurusdan tegak lurus ter%adap sumbu simetri ber%arga palingtinggi
*Bidang simetri biasa $ bidang yang membagi kristalmenadi dua bagian yang simetris yang padanya terdapat
satu sumbu simetri
*perasi bidang simetri ini disebut operasi 6 pencerminan(Reflection)”
*4lein & 56l,6rt! "R7! 88(+ 2/26/2014
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9i/an: 3imetri erti;al! 1ori
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Adalah suatu garis lurus yang dibuat melalui pusat kristal dan bila kristaltersebut diputar 360° pada sumbu tersebut, 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 yangapabila diputar 360°, akan dijumpai konfigurasi yang sama atau halhal
yang sama yang muncul lebih dari satu kali.
Sumbu poler : sumbu khayal !seperti halnya sumbu bipoler", hanyakedua ujung sumbu menembus dua keadaan yang berbeda.
*#perasi dari sumbu lipat ini disebut sebagai operasi rotasi !perulanganperiodik dari motif asli yang dijumpai setelah terjadinya perputaranmotif tersebut dengan sudut sebesar 360° akibat beroperasinya sumburotasi atau sumbu lipat".
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Operasi Rotasi
*4lein & 56l,6rt! "R7! 88(+
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3(22888=87e,3ite1ome7.o76777
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*$ibedakan lagi menjadi dua berdasarkan atas macamoperasinya :a. %yreb. %yroida
*%yre : sumbu simetri biasa yang besarnya sudut putaradalah 360°&n ' n adalah bilangan yang utuh sehingga akandiperoleh nilai n ()0°, (*0°, +0°,60°n * !!"" digyre&diadn 3 !-" trigyre & triad
n !◊" /etragyre & tetradn 6 ! . " e1agyre & he1ad
*%yroida ' sumbu simetri disini merupakan campuran daripemutaran melalui sumbu dan pencerminan pada bidang yang
tegak lurus pada bidang tadi()0° digyroida ' 2 pusat simetri(*0° /rigyroida+0° /etragyroida60° e1agyroida
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* ,s an eternal symmetry element& rotation
increments (n) range from 1 to 7
*8%en considering t%e
limits of translation&
rotation increments are
limited to 1& 2& 3& 4&
and 9-fold
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=
=
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Symmetry Element3
7 Rotation
!"a#$le % &y##e'ri(al Pa''ern
a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'i in
y##e'ri(al $a''ern
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Mo'i
!le#en'
=
=
> t1e 3ym,ol -or a to?-ol/rotation
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'i iny##e'ri(al $a''ern
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=
=
-ir3t
operation
3tep
: t%e symbol for a t'o-fold rotation
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'i
in y##e'ri(al
$a''ern
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=
=
-ir3t
operation
3tep
3e.on/
operation
3tep
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: t%e symbol for a t'o-fold rotation
a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'i
in y##e'ri(al
$a''ern
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a 0'o-fold rotation
*ome familiar obects%ave an intrinsic
symmetry
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'iin y##e'ri(al
$a''ern
: t%e symbol for a t'o-
fold rotation
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'iin y##e'ri(al
$a''ern
: t%e symbol for a t'o-
fold rotation
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'iin y##e'ri(al
$a''ern
: t%e symbol for a t'o-
fold rotation
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'iin y##e'ri(al
$a''ern
: t%e symbol for a t'o-
fold rotation
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'iin y##e'ri(al
$a''ern
: t%e symbol for a t'o-
fold rotation
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*econd 1;"o brings t%e obect
back to its original position
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a) T*o Fol+ ,o'a'ion
3600 /2 ro'a'ion 'o
re$ro+.(e a #o'iin y##e'ri(al
$a''ern
: t%e symbol for a t'o-
fold rotation
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3 D *
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3-D *
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=
=
=
'e$ 1
2-D *
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=
=
=
'e$ 1
'e$ 2
2-D *
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=
=
=
'e$ 1
'e$ 2
'e$ 3
2-D *
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2-D *
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* ,es at ="> (ecept 3-fold aes in cubicsymmetry at ?4>44@)
* ,es intersect atpoint
*+ossible symmetrycombinations$
422& 922& 222& 32& 23&432
(ie' 422 *ymmetryai)
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? moti- proe.tion3/o not re6ire a
3e.on/ 3et o- 2?-ol/ ae3
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* 432 +oint Aroup
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Adalah suatu titik yang apabila ditarik garis melalui titiktersebut dari sembarang titik pada permukaan kristal akanmembagi garis tsb sama panjang
*#perasi pusat simetri disebut juga operasi inversi !i"
*n4ersi : suatu operasi simetri yang dihasilkan dengan jalanmenghubungkan titiktitik dari salah satu bidang kristal melaluititik pusatnya, sehingga dihasilkan titik turunannya yang berjaraksama dari pusat simetri, tetapi berseberangan dan terbalik
*asil in4ersi suatu bidang kristal adalah bidang yang sejajar,sama dan sebangun, tetapi terbalik, dengan letak yangberseberangan terhadap pusat simetrinya dan berjarak samaterhadap titik in4ersi tersebut.
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Inversi = Pusat Simetri (C)Inversi = Pusat Simetri (C)
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5usat Simetri ! 2 " disebut juga titik simetri adalah suatu titik apabila
ditarik garis melalui titik tsb dari sembarang titikpada permukaan kristal akan membagi garis tsb samapanjang. #perasi pusat simetri disebut juga operasiin4ersi !i".
n4ersi:suatu operasi simetri yang dihasilkan dengan jalanmenghubungkan titiktitik 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)Inversi = Pusat Simetri (C)
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2 D *
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2-D *
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Operasi Inversi *4lein & 56l,6rt! "R7! 88(+
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9i/an: 3imetri
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2-D *
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2-D *
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2-D *
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Com,ination3 o- 3ymmetry element3 are al3opo33i,le
To .reate a .omplete analy3i3 o- 3ymmetrya,o6t a point in 3pa.e! e m63t try all po33i,le
.om,ination3 o- t1e3e 3ymmetry element3
#n t1e intere3t o- .larity an/ ea3e o- ill63tration!e .ontin6e to .on3i/er only 2?D eample3
2-D *
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2 D *
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2 D *
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2 D *
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2 D *
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2 D *
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2 D *
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Mineralogy @ FTG - UNPAD 2014 48
,ll ot%er combinations are eit%er$
ncompatible
(2 2 cannot be done in 2-D)
.edundant 'it% ot%ers already tried
m m → 2mm because creates 2-
fold
0%is is t%e same as 2 m → 2mm
2-D *ymmetry2-D *ymmetry
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2-D *
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2 D *
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2 D *
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2 D *
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2 D *
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No try .om,inin: a @?-ol/ rotation ai3 it1 a
mirror
Step % re-le.t
Step 2% rotate (
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Eo' try combining a 4-fold rotation ais 'it% amirror
Any ot1er element3I
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Eo' try combining a 4-fold rotation ais 'it% a
mirror
e3! to more mirror3
Any ot1er element3I
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Eo' try combining a 4-fold rotation ais 'it% a
mirror
Point :ro6p nameII
e3! to more mirror3
Any ot1er element3I
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Eo' try combining a 4-fold rotation ais 'it% a mirror
@mm
Point :ro6p nameII
e3! to more mirror3
Any ot1er element3I
1y not
@mmmmI
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3-fold rotation ais 'it% a mirror creates point group 3m
8%y not 3mmmF
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9-fold rotation ais 'it% a mirror creates pointgroup 9mm
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0%e original 9 elements plus t%e 4 combinationscreates 1" possible 2-D +oint Aroups$
1 2 3 4 9 m 2mm 3m 4mm 9mm
Any 2-D pattern of obects surrounding a point
must conform to one of t%ese groups
Ga,6n:an Rota3i /an #ner3i
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*4lein & 56l,6rt! "R7! 88(+
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Ga,6n:an Rota3i /an #ner3i
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Ee' 3-D *ymmetry lements
4 .otoinversion
a 2-fold rotoinversion ( 2 )
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Ee' *ymmetry lements
4 .otoinversion
b 2-fold rotoinversion ( 2 )
*tep 1$ rotate 39"/2
Eote$ t%is is a temporary
step& t%e intermediatet%e intermediate
motif element does notmotif element does not
eist in t%e final patterneist in t%e final pattern
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Ee' *ymmetry lements
4 .otoinversion
b 2-fold rotoinversion ( 2 )
0%e result$
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Ee' *ymmetry
lements
4 .otoinversion
b 2-fold rotoinversion ( 2 )
0%is is t%e same as0%is is t%e same as mm& so not& so not
a ne' operationa ne' operation
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Ee' *ymmetry lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
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Ee' *ymmetrylements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
*tep 1$ rotate 39"o/3
,gain& t%is is a temporary
step& t%e intermediatemotif element does noteist in t%e final pattern
1
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Ee' *ymmetry
lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
*tep 2$ invert t%roug%
center
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Ee' *ymmetry
lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
!ompletion of t%e first
seBuence
1
2
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Ee' *ymmetry
lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
.otate anot%er 39"/3
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Ee' *ymmetry
lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
!omplete second step to
create face 3
1
2
3
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Ee' *ymmetry
lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
0%ird step creates face 4
(3→ (1)→ 4)
1
2
3
4
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Ee' *ymmetry
lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
Gourt% step creates face ?
(4→
(2)→
?)
1
2
5
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Ee' *ymmetry lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
Gift% step creates face 9
(? → (3) → 9)
*it% step returns to face 1
1
6
5
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Ee' *ymmetry
lements
4 .otoinversion
c 3-fold rotoinversion ( 3 )
0%is is uniBue
1
6
5
2
3
4
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Ee' *ymmetry
lements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
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Ee' *ymmetry
lements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
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Ee' *ymmetry
lements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
1$ .otate 39"/4
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
1$ .otate 39"/4
2$ nvert
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
1$ .otate 39"/4
2$ nvert
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
3$ .otate 39"/4
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
3$ .otate 39"/4
4$ nvert
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
3$ .otate 39"/4
4$ nvert
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
?$ .otate 39"/4
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
?$ .otate 39"/4
9$ nvert
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Ee' *ymmetrylements
4 .otoinversion
d 4-fold rotoinversion ( 4 )
0%is is also a uniBue operation
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8e no' %ave ;; uniBue 3D3D symmetry operations$
1 2 3 4 9 m1 2 3 4 9 m 3 43 4
Com,ination3 o- t1e3e element3 are al3o po33i,le
A .omplete analy3i3 o- 3ymmetry a,o6t a point in 3pa.e
re6ire3 t1at e try all po33i,le .om,ination3 o- t1e3e
3ymmetry element3
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0o be !ontinue