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Crystal symmetry Crystal symmetry Part 2 Part 2

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Page 1: Ch 05 Unsur simetri-2

Crystal symmetryCrystal symmetry

Part 2Part 2

Page 2: Ch 05 Unsur simetri-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

Page 3: Ch 05 Unsur simetri-2

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

Page 4: Ch 05 Unsur simetri-2

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

Page 5: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New 3-D Symmetry Elements

4. Rotoinversion

a. 1-fold rotoinversion ( 1 )

Page 6: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New 3-D Symmetry Elements

4. Rotoinversion

a. 1-fold rotoinversion ( 1 )

Step 1: rotate 360/1

(identity)

Page 7: Ch 05 Unsur simetri-2

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

Page 8: Ch 05 Unsur simetri-2

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

Page 9: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

b. 2-fold rotoinversion ( 2 )

Step 1: rotate 360/2

Step 2: invert

Page 10: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

b. 2-fold rotoinversion ( 2 )

The result:

Page 11: Ch 05 Unsur simetri-2

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

Page 12: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

c. 3-fold rotoinversion ( 3 )

Page 13: Ch 05 Unsur simetri-2

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

Page 14: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

c. 3-fold rotoinversion ( 3 )

Step 2: invert through center

Page 15: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

c. 3-fold rotoinversion ( 3 )

Completion of the first sequence

1

2

Page 16: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

c. 3-fold rotoinversion ( 3 )

Rotate another 360/3

Page 17: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

c. 3-fold rotoinversion ( 3 )

Invert through center

Page 18: Ch 05 Unsur simetri-2

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

Page 19: Ch 05 Unsur simetri-2

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

Page 20: Ch 05 Unsur simetri-2

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

Page 21: Ch 05 Unsur simetri-2

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

Page 22: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

c. 3-fold rotoinversion ( 3 )

This is unique1

6

5

2

3

4

Page 23: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

Page 24: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

Page 25: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

1: Rotate 360/4

Page 26: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

1: Rotate 360/4

2: Invert

Page 27: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

1: Rotate 360/4

2: Invert

Page 28: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

3: Rotate 360/4

Page 29: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

3: Rotate 360/4

4: Invert

Page 30: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

3: Rotate 360/4

4: Invert

Page 31: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

5: Rotate 360/4

Page 32: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

5: Rotate 360/4

6: Invert

Page 33: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

This is also a unique operation

Page 34: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

d. 4-fold rotoinversion ( 4 )

A more fundamental representative of the pattern

Page 35: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Begin with this framework:

Page 36: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 ) 1

Page 37: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 38: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

2

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 39: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

2

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 40: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

3

2

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 41: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

3

2

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 42: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

3

4

2

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 43: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

2

3

4

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 44: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

2

3

4

5

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 45: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

2

3

4

5

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 46: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

1

2

3

4

5

6

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

Page 47: Ch 05 Unsur simetri-2

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

Page 48: Ch 05 Unsur simetri-2

3-D Symmetry3-D Symmetry

New Symmetry Elements

4. Rotoinversion

e. 6-fold rotoinversion ( 6 )

A simpler pattern

Top View

Page 49: Ch 05 Unsur simetri-2

3-D Symmetry3-D SymmetryWe now have 10 unique 3-D symmetry operations:

1 2 3 4 6 i m 3 4 6

Page 50: Ch 05 Unsur simetri-2

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

Page 51: Ch 05 Unsur simetri-2

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 ( “ “ “ )

Page 52: Ch 05 Unsur simetri-2

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

Page 53: Ch 05 Unsur simetri-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

Page 54: Ch 05 Unsur simetri-2

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

Page 55: Ch 05 Unsur simetri-2

(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

Page 56: Ch 05 Unsur simetri-2

Kombinasi rotasi sumbu 4, 6 dengan bidang cerminKombinasi rotasi sumbu 4, 6 dengan bidang cermin

Page 57: Ch 05 Unsur simetri-2

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

Page 58: Ch 05 Unsur simetri-2

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

Page 59: Ch 05 Unsur simetri-2

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

Page 60: Ch 05 Unsur simetri-2

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

Page 61: Ch 05 Unsur simetri-2

3-D Symmetry3-D SymmetryThe 32 3-D Point Groups

After Bloss, Crystallography and Crystal Chemistry. © MSA

Page 62: Ch 05 Unsur simetri-2
Page 63: Ch 05 Unsur simetri-2

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

Page 64: Ch 05 Unsur simetri-2

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

Page 65: Ch 05 Unsur simetri-2
Page 66: Ch 05 Unsur simetri-2
Page 67: Ch 05 Unsur simetri-2

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