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Crystal symmetryCrystal symmetry
Part 2Part 2
Rotoinversion
Rotoinversi : Kombinasi dari operasi rotasi dan inversi (simbol : i )
Motif awal diputar sebesar 360 sampai kembali ke posisi semula, selanjutnya motif tersebut diinversikan melalui titik inversi pada pusatnya.
Dibedakan : 360/n 1 = pusat simetri (i~inversi) 2 = operasi pencerminan pada bidang ekuator 3 = kombinasi operasi sumbu lipat 3 dgn inversi 4 = unique krn tdk sama dgn yg lain 6 = operasi sumbu lipat 3 dengan bidang cermin yang tegak
lurus sumbu putar
Ilustrasi operasi rotoinversiIlustrasi operasi rotoinversi
Rotasi dengan sudut : 360 yang kemudian di inversikan melaluiRotasi dengan sudut : 360 yang kemudian di inversikan melaluiPusat simetri = identik dengan operasi inversiPusat simetri = identik dengan operasi inversi
Ilustrasi operasi rotoinversiIlustrasi operasi rotoinversi
Rotasi dengan sudut : 180, 120, 90,60 Rotasi dengan sudut : 180, 120, 90,60 Yang kemudian di inversikan melaluiYang kemudian di inversikan melaluiPusat Pusat
(a)(a) :: 2 : identik dengan bidang cermin2 : identik dengan bidang cermin(b)(b) : 3 : identik dengan rotasi sumbu : 3 : identik dengan rotasi sumbu
lipat 3 dan pusat simetrilipat 3 dan pusat simetri(d) : 6 : identik dengan sumbu lipat 3(d) : 6 : identik dengan sumbu lipat 3 serta bidang cermin tegaklurusserta bidang cermin tegaklurus dengan sumbu axisdengan sumbu axis
3-D Symmetry3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
3-D Symmetry3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)
3-D Symmetry3-D Symmetry
New 3-D Symmetry Elements
4. 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
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. 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
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 2: invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
The result:
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
This is the same as m, so not a new operation
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. 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
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Step 2: invert through center
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Completion of the first sequence
1
2
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Rotate another 360/3
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Invert through center
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Complete second step to create face 3
1
2
3
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Third step creates face 4
(3 (1) 4)
1
2
3
4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fourth step creates face 5 (4 (2) 5)
1
2
5
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fifth step creates face 6
(5 (3) 6)
Sixth step returns to face 1
1
6
5
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
This is unique1
6
5
2
3
4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
6: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
This is also a unique operation
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
A more fundamental representative of the pattern
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
Begin with this framework:
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1
3-D Symmetry3-D Symmetry
1
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
3
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
3
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
3
4
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
5
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
5
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
5
6
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. 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
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
A simpler pattern
Top View
3-D Symmetry3-D SymmetryWe now have 10 unique 3-D symmetry operations:
1 2 3 4 6 i m 3 4 6
Kombinasi Rotasi
Sumbu lipat 2 tegaklurus dengan sumbu lipat 2 222 Sumbu lipat 4 tegaklurus dengan sumbu lipat 2 422 Sumbu lipat 6 tegaklurus dengan sumbu lipat 2 622 Sumbu lipat 3 tegaklurus dengan sumbu lipat 2 32 Sumbu lipat 2 tegaklurus dengan sumbu lipat 3 32 Sumbu lipat 4 tegaklurus dengan permukaan kubus, 3
terletak pada sudut kubus dan 2 pada bagian tengah dari tepi kubus 432
3-D Symmetry3-D Symmetry
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 ( “ “ “ )
Kombinasi rotasi sb 6 dgn sb 2; 4 dgn 2; 3 dgn 2; Kombinasi rotasi sb 6 dgn sb 2; 4 dgn 2; 3 dgn 2; 4 dgn 3 dgn 24 dgn 3 dgn 2
Kombinasi Rotasi dan Refleksi Gabungan dari operasi sumbu rotasi dan pencerminan Kemungkinan gabungan yang bisa :
Sumbu lipat 6 dengan bidang cermin yang tegaklurus padanya 6/mSumbu lipat 4 dengan bidang cermin yang tegaklurus padanya 4/mSumbu lipat 3 dengan bidang cermin yang tegaklurus padanya 3/mSumbu lipat 2 dengan bidang cermin yang tegaklurus padanya 2/mPenggabungan operasi rotasi (622, 422, 222) dengan bidang cermin yag tegaklurus pada masing-msing sumbu rotasi 6/m 2/m 2/m; 4/m 2/m 2/m;2/m 2/m 2/m, 4mm, 6mm, 3m, 2mm
3-D Symmetry3-D Symmetry
3-D symmetry element combinations
a. Rotation axis parallel to a mirrorSame as 2-D
2 || m = 2mm
3 || m = 3m, also 4mm, 6mm
b. Rotation axis mirror2 m = 2/m
3 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
(a)(a) Kombinasi rotasi sb 4 dgn 2 set rotasi sb 2Kombinasi rotasi sb 4 dgn 2 set rotasi sb 2(b)(b) Kombinasi rotasi sb 4 dgn 4 set rotasi sb 2 dgn bidang cerminKombinasi rotasi sb 4 dgn 4 set rotasi sb 2 dgn bidang cermin yang tegak lurus pada sumbu rotasiyang tegak lurus pada sumbu rotasi(c)(c) Kombinasi rotasi sb 4 dengan 2 bidang cermin yang paralelKombinasi rotasi sb 4 dengan 2 bidang cermin yang paralel terhadap sumbu 4terhadap sumbu 4
Kombinasi rotasi sumbu 4, 6 dengan bidang cerminKombinasi rotasi sumbu 4, 6 dengan bidang cermin
3-D Symmetry3-D Symmetry
As 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
3-D Symmetry3-D SymmetryBut 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
(a)(a) Kombinasi rotasi sb 2 dgn 2 Kombinasi rotasi sb 2 dgn 2 bidang cermin bidang cermin 2mm 2mm(b) Kombinasi rotasi sb 4 dgn 4 (b) Kombinasi rotasi sb 4 dgn 4 bidang cermin bidang cermin 4mm 4mm(c) (c) Kombinasi rotasi sb 2 dgn 3 Kombinasi rotasi sb 2 dgn 3 bidang cermin bidang cermin 2/m 2/m 2/m 2/m 2/m 2/m(d) Kombinasi rotasi sb 4 dgn 2 dan (d) Kombinasi rotasi sb 4 dgn 2 dan 5 bidang cermin 5 bidang cermin 4/m 2/m 2/m 4/m 2/m 2/m
3-D Symmetry3-D SymmetryJumlah kombinasi operasi simetri adalah tidak
terbatas, tetapi total jumlah kombinasi elemen simetri yang tidak identik hanya ada 32 kelas yang kemudian disebut “32 Groups notasi Hermann-Mauguin” = Simbol International, SI
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/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 Symmetry3-D SymmetryThe 32 kelas kristal
Dikelompokan pada 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 Symmetry3-D SymmetryThe 32 3-D Point Groups
After Bloss, Crystallography and Crystal Chemistry. © MSA
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
Distribusi motif pada 32 Distribusi motif pada 32 kelas kristal dari 6 sistim kristalkelas kristal dari 6 sistim kristal
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
Distribusi motif pada 32 Distribusi motif pada 32 kelas kristal dari 6 sistim kristalkelas kristal dari 6 sistim kristal
Next week Next week Pembagian 32 kelas kristal dan jenis simetri dariPembagian 32 kelas kristal dan jenis simetri dari
EEnam sistim kristalnam sistim kristal